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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Digit. Health</journal-id>
<journal-title>Frontiers in Digital Health</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Digit. Health</abbrev-journal-title>
<issn pub-type="epub">2673-253X</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.3389/fdgth.2025.1498072</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Digital Health</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Consent mechanisms and default effects in health information exchange in Japan</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes"><name><surname>Ito</surname><given-names>Atsushi</given-names></name>
<xref ref-type="aff" rid="aff1"><sup>1</sup></xref>
<xref ref-type="corresp" rid="cor1">&#x002A;</xref><uri xlink:href="https://loop.frontiersin.org/people/2842635/overview"/><role content-type="https://credit.niso.org/contributor-roles/conceptualization/"/><role content-type="https://credit.niso.org/contributor-roles/data-curation/"/><role content-type="https://credit.niso.org/contributor-roles/formal-analysis/"/><role content-type="https://credit.niso.org/contributor-roles/funding-acquisition/"/><role content-type="https://credit.niso.org/contributor-roles/investigation/"/><role content-type="https://credit.niso.org/contributor-roles/methodology/"/><role content-type="https://credit.niso.org/contributor-roles/project-administration/"/><role content-type="https://credit.niso.org/contributor-roles/resources/"/><role content-type="https://credit.niso.org/contributor-roles/supervision/"/><role content-type="https://credit.niso.org/contributor-roles/validation/"/><role content-type="https://credit.niso.org/contributor-roles/visualization/"/><role content-type="https://credit.niso.org/contributor-roles/writing-original-draft/"/><role content-type="https://credit.niso.org/contributor-roles/writing-review-editing/"/></contrib>
<contrib contrib-type="author"><name><surname>Nakamura</surname><given-names>Fumihiko</given-names></name>
<xref ref-type="aff" rid="aff2"><sup>2</sup></xref><uri xlink:href="https://loop.frontiersin.org/people/2894499/overview" /><role content-type="https://credit.niso.org/contributor-roles/conceptualization/"/><role content-type="https://credit.niso.org/contributor-roles/data-curation/"/><role content-type="https://credit.niso.org/contributor-roles/formal-analysis/"/><role content-type="https://credit.niso.org/contributor-roles/methodology/"/><role content-type="https://credit.niso.org/contributor-roles/software/"/><role content-type="https://credit.niso.org/contributor-roles/validation/"/><role content-type="https://credit.niso.org/contributor-roles/visualization/"/><role content-type="https://credit.niso.org/contributor-roles/writing-review-editing/"/></contrib>
</contrib-group>
<aff id="aff1"><label><sup>1</sup></label><institution>Faculty of Public Policy, Kyoto Prefectural University</institution>, <addr-line>Kyoto</addr-line>, <country>Japan</country></aff>
<aff id="aff2"><label><sup>2</sup></label><institution>Faculty of Engineering, Kitami Institute of Technology</institution>, <addr-line>Hokkaido</addr-line>, <country>Japan</country></aff>
<author-notes>
<fn fn-type="edited-by"><p><bold>Edited by:</bold> Stephen Fashoto, University of Eswatini, Eswatini</p></fn>
<fn fn-type="edited-by"><p><bold>Reviewed by:</bold> Richard Giordano, University of Southampton, United Kingdom</p>
<p>Yeong-Tae Song, Towson University, United States</p></fn>
<corresp id="cor1"><label>&#x002A;</label><bold>Correspondence:</bold> Atsushi Ito <email>iatsushi@kpu.ac.jp</email></corresp>
</author-notes>
<pub-date pub-type="epub"><day>24</day><month>02</month><year>2025</year></pub-date>
<pub-date pub-type="collection"><year>2025</year></pub-date>
<volume>7</volume><elocation-id>1498072</elocation-id>
<history>
<date date-type="received"><day>18</day><month>09</month><year>2024</year></date>
<date date-type="accepted"><day>03</day><month>02</month><year>2025</year></date>
</history>
<permissions>
<copyright-statement>&#x00A9; 2025 Ito and Nakamura.</copyright-statement>
<copyright-year>2025</copyright-year><copyright-holder>Ito and Nakamura</copyright-holder><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License (CC BY)</ext-link>. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p></license>
</permissions>
<abstract><sec><title>Background</title>
<p>Health information exchange (HIE) is an information system that efficiently shares patient information across medical institutions. However, traditional consent methods, represented by opt-in and opt-out, face a trade-off between efficiency and ethical, making it difficult to fundamentally improve consent rates. To address this issue, we focused on default settings and proposed an innovative approach called the &#x201C;two-step consent model,&#x201D; which leverages the advantages of existing models using utility theory. We evaluated the acceptability of this method.</p>
</sec><sec><title>Methods</title>
<p>An online survey was conducted with 2,000 participants registered with Japan&#x0027;s largest internet survey company. We compared and analyzed the consent rates of the opt-in, opt-out, and two-step consent models.</p>
</sec><sec><title>Results</title>
<p>The opt-in model had a 29.5&#x0025; consent rate, maximizing patient autonomy but increasing the burden and reducing efficiency. The opt-out model had a 95.0&#x0025; consent rate but raised concerns among half of the respondents. The two-step consent model had a 68.5&#x0025; consent rate, demonstrating its cost-effectiveness compared with traditional models.</p>
</sec><sec><title>Discussion</title>
<p>The two-step consent model, involving implicit and explicit consent when needed, ensures efficient consent acquisition while respecting patient autonomy. It is a cost-effective policy option that can overcome the ethical issues associated with the opt-out model. Introducing methods that leverage both opt-in and opt-out advantages is expected to address HIE stagnation.</p>
</sec><sec><title>Conclusion</title>
<p>The two-step consent model is expected to improve consent rates by balancing the efficiency and quality of consent acquisition. To achieve this, patient education is crucial for raising awareness and understanding of HIE and its consent methods.</p>
</sec>
</abstract>
<kwd-group>
<kwd>health information exchange</kwd>
<kwd>consent mechanism</kwd>
<kwd>utility-based consent rate model</kwd>
<kwd>default settings</kwd>
<kwd>online survey</kwd>
</kwd-group><contract-num rid="cn001">18K12831</contract-num><contract-num rid="cn002">619</contract-num><contract-sponsor id="cn001">JSPS KAKEN</contract-sponsor><contract-sponsor id="cn002">JST RISTEX</contract-sponsor><counts>
<fig-count count="6"/>
<table-count count="3"/><equation-count count="162"/><ref-count count="37"/><page-count count="13"/><word-count count="0"/></counts><custom-meta-wrap><custom-meta><meta-name>section-at-acceptance</meta-name><meta-value>Ethical Digital Health</meta-value></custom-meta></custom-meta-wrap>
</article-meta>
</front>
<body><sec id="s1" sec-type="intro"><label>1</label><title>Introduction</title>
<p>In recent years, interest in health information exchange (hereinafter HIE) has been growing due to population aging and increased healthcare expenditure worldwide (<xref ref-type="bibr" rid="B1">1</xref>). The HIE is an information network that allows medical facilities such as hospitals and clinics to share patient information, including medical records and test results, online. The widespread use of HIE as a medical infrastructure is expected to improve the overall quality and efficiency of medical care, especially by enabling small clinics to provide advanced medical care through enhanced collaboration with large hospitals. Additionally, information sharing among medical facilities could curb duplication of care and overmedication, ultimately conserving medical resources and lessening the economic burden on patients (<xref ref-type="bibr" rid="B2">2</xref>, <xref ref-type="bibr" rid="B3">3</xref>). In response to these expectations, the Japanese government has invested a significant amount of R&#x0026;D funds to establish more than 400 HIEs since the 2000s (<xref ref-type="bibr" rid="B4">4</xref>). However, despite the widespread use of these HIEs, the expected results have been slow to materialize. Specifically, the patient registration rate (hereinafter the &#x201C;consent rate&#x201D;) and the utilization of services by medical institutions in these HIEs have been stagnant, and the actual number of registered patients (hereinafter the &#x201C;number of consenters&#x201D;) is only 1&#x0025; of the national population (<xref ref-type="bibr" rid="B5">5</xref>). Why then do these problems occur? One possible cause is the consent mechanism exchanged between the patients and medical staff. The conventional method of obtaining consent is the opt-in method, in which the default is set to &#x201C;disagree.&#x201D; Since the medical staff directly approach patients and explicitly obtain consent through explanation, appeal, and persuasion, a significant burden is placed on both patients and medical staff (<xref ref-type="bibr" rid="B6">6</xref>). Thus, these burdens may hinder the improvement of consent rates.</p>
<p>This issue is not limited to Japan but represents a significant policy challenge for countries promoting HIE. In response, the Japanese government has adopted a policy to simplify the consent process through implied consent, such as in-hospital postings, even under the current Personal Information Protection Law. Additionally, it has recommended switching from an opt-in model to an opt-out model to lessen the burden on both patients and medical staff (<xref ref-type="bibr" rid="B7">7</xref>). However, in reality, 85.5&#x0025; of consent acquisition in domestic medical settings is still conducted through direct face-to-face consent forms, indicating that the transition to opt-out has not progressed sufficiently (<xref ref-type="bibr" rid="B8">8</xref>). Indeed, the opt-out model, which sets implied consent as the default, automatically considers consent unless the patient explicitly expresses refusal, thereby contributing to the efficiency of consent acquisition and reducing the burden on medical settings.</p>
<p>However, the protection of patient privacy and respect for self-determination (hereinafter the quality of consent acquisition) are not guaranteed. Differences from previous practices in medical settings can cause problems, making the opt-out mechanism not necessarily accepted as a reasonable means (<xref ref-type="bibr" rid="B9">9</xref>). This situation suggests a discrepancy between the goal of efficiently registering patient information and the emphasis on the quality of consent acquisition in medical settings, highlighting the necessity of finding a third solution to resolve this dilemma. Indeed, within the domain of medical ethics, there is an extensive body of literature on informed consent, acknowledging significant contributions of previous research in this field. However, prior research in this policy area has focused on the dichotomy between the opt-in and opt-out models, without exploring a third method that leverages the benefits of both (<xref ref-type="bibr" rid="B10">10</xref>&#x2013;<xref ref-type="bibr" rid="B12">12</xref>). Conversely, the role of defaults has been extensively examined in behavioral economics. For example, research on nudge theory has demonstrated that default settings significantly influence consumer decision-making and behavior (<xref ref-type="bibr" rid="B13">13</xref>). Specifically, studies on status quo bias by Kahneman and Tversky, as part of prospect theory, reveal that people&#x0027;s tendency to adhere to default options stems from a strong psychological inclination to maintain the status quo and resist change (<xref ref-type="bibr" rid="B14">14</xref>). However, decisions to join or register for new and unfamiliar services like HIE are not entirely governed by expectations and losses as framed by prospect theory. This perspective does not fully explain why patients express satisfaction with the default option. Therefore, addressing the inefficiencies and ethical challenges of consent acquisition requires reformulating the consent mechanism using utility theory.</p>
<p>Therefore, can the stagnant consent rate be overcome by setting defaults that balance efficiency and quality in obtaining consent? To test this hypothesis, this study mathematically formulates conventional methods of obtaining consent, with a focus on default settings, and introduces a new method of obtaining consent: a two-step consent model that combines opt-out and opt-in mechanisms (hereinafter the &#x201C;two-step consent model&#x201D;). Furthermore, we determine its acceptability. Although there are various possible policy options for improving consent rates, policies utilizing default settings are likely to be implemented in society because of their superior cost-effectiveness (<xref ref-type="bibr" rid="B15">15</xref>).</p>
<p>The remainder of this paper is structured as follows. <xref ref-type="sec" rid="s2">Section 2</xref> formulates a model to predict the consent mechanism of patients and the consent rate of the entire population by assuming that each individual acts rationally to maximize his or her utility. We define the model by applying utility theory and summarize the strengths and limitations of the traditional model in terms of balancing efficiency and quality in obtaining consent. In addition, we formulated a model to predict patient consent behavior and the overall consent rate, assuming that individuals act rationally to maximize their utility derived from providing consent for HIE. In <xref ref-type="sec" rid="s3">Section 3</xref>, an online survey is conducted to gauge the acceptability of the two-step consent model under various scenarios, including the traditional model. In this section, we estimate the consent rate for each model based on survey results. In <xref ref-type="sec" rid="s4">Section 4</xref>, we identify the strengths and limitations of the traditional model based on the results of these analyses and evaluate the acceptability of the two-step consent model. Furthermore, we discuss the limitations of this study and highlight areas for future research. Finally, <xref ref-type="sec" rid="s5">Section 5</xref> summarizes the main conclusions of the study.</p>
</sec>
<sec id="s2" sec-type="methods"><label>2</label><title>Materials and methods</title>
<sec id="s2a"><label>2.1</label><title>Assumptions of the consent model</title>
<p>Since the consent rate <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM1"><mml:mi>p</mml:mi><mml:mspace width="thickmathspace" /></mml:math></inline-formula> is the ratio of the number of persons who have consented to the registration of patient information in the HIE (hereinafter &#x201C;the number of consenters&#x201D;) to the number of subjects, it can be obtained through dividing the number of consenters by the number of subjects. For clarity, &#x201C;subjects&#x201D; in this context refers to the entire population within the area where HIE is implemented. To improve the sluggish consent rate, it is necessary to understand the consent mechanism. First, the number of consenters on the numerator side is modeled by default, as it is expected to differ depending on the default settings (<xref ref-type="bibr" rid="B16">16</xref>). In this context, it is assumed that the decision to register patient information in this HIE is related to the utility of each individual and that the utility depends on the surrounding consent status and the subject&#x0027;s own need for medical care (<xref ref-type="bibr" rid="B17">17</xref>), in addition to the time and effort involved in exchanging consent acquisition. The HIE serves as a platform where medical institutions and patients collaborate. As the level of consent increases, so does the value of the subject&#x0027;s own use, leading to a greater willingness to consent and demonstrating a network effect (<xref ref-type="bibr" rid="B18">18</xref>). Unlike other platforms, the use of HIE is limited to medical care; therefore, the degree of need for medical care for each individual is an important consideration. However, even if the number of subjects in the denominator is constant, it is debatable whether the scope should be limited to patients or extended to include local residents. In fact, there are various medical information networks, ranging from those used as private goods among specific medical institutions, such as medical malls and healthcare groups, to those regarded as quasi-public goods, mainly in wide areas such as municipalities, medical regions, and entire prefectures (<xref ref-type="bibr" rid="B19">19</xref>, <xref ref-type="bibr" rid="B20">20</xref>). If we consider the latter case, it is more reasonable to expand the target audience to include local residents, including patients, because the system is publicly funded and established as a local medical infrastructure. Based on the above, we define the following symbols:</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM2"><mml:mi>n</mml:mi></mml:math></inline-formula>: The number of subjects, which is assumed to be constant,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM3"><mml:mi>S</mml:mi></mml:math></inline-formula>: A set of alternatives. The choices include &#x201C;agree&#x201D; and &#x201C;disagree,&#x201D;</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM4"><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>: The choice made by each subject <italic>i</italic> from the set of alternatives, <xref ref-type="fn" rid="FN0001"><sup>1</sup></xref></p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM5"><mml:mi>p</mml:mi></mml:math></inline-formula>: Consent rate <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM6"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>p</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:math></inline-formula></p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM7"><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>: Medical necessity of subject <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM8"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>q</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM9"><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>: For each model, a utility function that quantifies the utility determined by subject <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM10"><mml:mi>i</mml:mi><mml:mspace width="thickmathspace" /></mml:math></inline-formula>&#x0027;s choice <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM11"><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> when the consent rate is <italic>p</italic> and subject<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM12"><mml:mspace width="thickmathspace" /><mml:mi>i</mml:mi><mml:mspace width="thickmathspace" /></mml:math></inline-formula>&#x0027;s medical necessity is <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM13"><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM14"><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>),</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM15"><mml:msub><mml:mi>U</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>: For each model, the average utility across subjects when the consent rate is <italic>p</italic>.</p>
<p>Utilizing these variables, we construct a model with utility functions to understand the consent mechanisms for each of the previously discussed opt-in and opt-out options.</p>
</sec>
<sec id="s2b"><label>2.2</label><title>Definition of the opt-in model</title>
<p>First, we identify the mechanism of the opt-in model. By default, the choice is to &#x201C;disagree.&#x201D; Hence, there is no difference in utility for the subjects choosing to &#x201C;disagree.&#x201D; That is, for any <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM16"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>, we assume <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM17"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM18"><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>. For agreeing, we assume a monotonically increasing function with respect to <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM19"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>. Specifically,
<disp-formula id="disp-formula1"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM1"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mrow><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi mathvariant="normal">for</mml:mi></mml:mrow></mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">any</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi>q</mml:mi><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">if</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mi>p</mml:mi></mml:mrow><mml:mo>&#x2264;</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p>
<disp-formula id="disp-formula2"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM2"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mspace width="thinmathspace" /><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi mathvariant="normal">for</mml:mi></mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">any</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi>p</mml:mi><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">if</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mi>q</mml:mi></mml:mrow><mml:mo>&#x2264;</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>This condition implies that external factors, such as higher consent rates among peers, and internal factors, such as increased personal medical needs, influence the decision to agree. From these assumptions, when <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM20"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula> are small, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM21"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, indicating that the utility of disagreeing is greater, leading to the choice of &#x201C;disagree.&#x201D; As <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM22"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula> increase, there comes a point where <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM23"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, leading to the choice of &#x201C;agree&#x201D; (<xref ref-type="fig" rid="F1">Figure&#x00A0;1</xref>, left side).</p>
<fig id="F1" position="float"><label>Figure 1</label>
<caption><p>Examples of utility functions in the Opt-in model (left) and the Opt-out model (right). See <xref ref-type="sec" rid="s11">Supplementary Material</xref> for the functions used.</p></caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="fdgth-07-1498072-g001.tif"/>
</fig>
<p>This model aims to pre-register patient information in anticipation of future health issues, with medical staff directly approaching patients to explicitly explain the process and obtain consent. Consequently, the quality of consent acquisition is maximized for the patient; however, the process incurs high consent costs for both patients and medical staff due to the need for verbal explanations, appeals, and persuasion, making it inefficient (<xref ref-type="bibr" rid="B21">21</xref>). Therefore, the consent rate tends to remain low overall (<xref ref-type="bibr" rid="B22">22</xref>).</p>
</sec>
<sec id="s2c"><label>2.3</label><title>Definition of the opt-out model</title>
<p>Next, we identify the mechanism of the opt-out model. By default, the choice is to &#x201C;agree.&#x201D; Therefore, there is no difference in utility for the subjects choosing to &#x201C;agree.&#x201D; That is, for any <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM24"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>, we assume <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM25"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi>a</mml:mi><mml:mi>g</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>. For disagreeing, we assume a monotonically decreasing function with respect to <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM26"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>. Specifically,
<disp-formula id="disp-formula3"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM3"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagre</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mspace width="thinmathspace" /><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi mathvariant="normal">for</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">any</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">if</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>&#x2264;</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="disp-formula4"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM4"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mtd><mml:mtd><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thinmathspace" /><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi mathvariant="normal">for</mml:mi></mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">any</mml:mi></mml:mrow><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi>p</mml:mi></mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">if</mml:mi></mml:mrow><mml:mspace width="thickmathspace" /><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mi>q</mml:mi></mml:mrow><mml:mo>&#x2264;</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>.</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
This assumption implies that the utility of disagreeing decreases as the surrounding consent rates and personal medical needs increase.</p>
<p>From these assumptions, when <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM27"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula> are small, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM28"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, that is, the utility of disagreeing is greater, leading to the choice of &#x201C;disagreeing.&#x201D; As <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM29"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula> increase, there comes a point where <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM30"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, implying that the utility of agreeing becomes greater, leading to the choice of &#x201C;agreeing&#x201D; (<xref ref-type="fig" rid="F1">Figure&#x00A0;1</xref>, right side).</p>
<p>Unlike the opt-in model, this model sets the default to &#x201C;agree&#x201D; when registering patient information. Hence, unless the patient explicitly indicates &#x201C;disagreeing,&#x201D; consent is assumed automatically. This presumed consent reduces the consent costs compared with the opt-in model, making it more efficient (<xref ref-type="bibr" rid="B23">23</xref>). However, it should be noted that this may not always reflect the true intentions of patients and could potentially raise ethical concerns.</p>
</sec>
<sec id="s2d"><label>2.4</label><title>Measurement of average utility in the opt-in and opt-out models</title>
<p>Under the above assumptions, the average utility in both models is defined by the following equation:
<disp-formula id="disp-formula5"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM5"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:munderover><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:mo>&#x2061;</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:mstyle></mml:math></disp-formula>
This equation calculates the average utility for all subjects <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM31"><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM32"><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo>&#x2264;</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> in both models <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM33"><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:math></inline-formula>, given the surrounding consent rate <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM34"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula> and individual medical need <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM35"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> of subject <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM36"><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math></inline-formula> when they choose option <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM37"><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>. Here, if the utility coefficient is given by
<disp-formula id="disp-formula1111"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM6"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:mrow><mml:mn>1</mml:mn><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mn>0</mml:mn><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo fence="true" stretchy="true" symmetric="true"></mml:mo></mml:mrow></mml:math></disp-formula>
then the utility is determined independently of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM38"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM39"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>, indicating that <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM40"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> matches the consent rate. This represents a simple ratio measurement by counting. Furthermore, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM41"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> can be transformed as follows:
<disp-formula id="disp-formula6"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM7"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:mrow><mml:mn>1</mml:mn><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mn>0</mml:mn><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo fence="true" stretchy="true" symmetric="true"></mml:mo></mml:mrow></mml:math></disp-formula>
then the utility is determined independently of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM42"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM43"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>, indicating that <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM44"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> matches the consent rate. This represents a simple ratio measurement by counting. Furthermore, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM45"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> can be transformed as follows:
<disp-formula id="disp-formula7"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM8"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:msub><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2061;</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>+</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:msub><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2061;</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:msub><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">&#x0023;</mml:mi><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mstyle></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:msub><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
Here, we used <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM46"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM47"><mml:mi mathvariant="normal">&#x0023;</mml:mi><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>.</mml:mo></mml:math></inline-formula> Similarly,
<disp-formula id="disp-formula8"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM9"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:msub><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mstyle></mml:math></disp-formula></p>
</sec>
<sec id="s2e"><label>2.5</label><title>Differences in average utility between the opt-in and opt-out models</title>
<p>Next, we examine the extent of the differences in average utility between these two models. If the default changes, the surrounding consent situation changes; thus, the average utility is expected to show different values. Therefore, we assume the consent rate in the opt-in and opt-out models as <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM48"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM49"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, respectively, with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM50"><mml:mspace width="thickmathspace" /><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> due to the default difference. The difference in consent rates between these two models, <italic>&#x0394;U</italic>, is given by the following equation:
<disp-formula id="disp-formula9"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM10"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:mi>U</mml:mi></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mspace width="1em" /><mml:mo>+</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
Here, in the opt-out model, if &#x201C;disagreeing&#x201D; is chosen, it satisfies
<disp-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM11"><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>
While in the opt-in model, if &#x201C;agreeing&#x201D; is chosen, it satisfies
<disp-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM12"><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>
Using these, we get
<disp-formula id="disp-formula10"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM13"><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:mi>U</mml:mi><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:msubsup><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:mstyle></mml:math></disp-formula>
This is positive if the average utility when everyone chooses to &#x201C;agree&#x201D; in the opt-in model is below <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM51"><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>. Since <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM52"><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> until <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM53"><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> satisfies <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM54"><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x003C;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, and considering <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM55"><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> if <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM56"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is relatively low and there are few subjects with high <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM57"><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM58"><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:mi>U</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> is satisfied. Conversely, if <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM59"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is high and there are many subjects with high <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM60"><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM61"><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:mi>U</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> is satisfied. Assuming the former case, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM62"><mml:mi mathvariant="normal">&#x0394;</mml:mi><mml:mi>U</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, indicating that the average utility of the opt-out model is higher.</p>
<p>This mechanism, by which people follow the default choice, has been called the &#x201C;default effect (<xref ref-type="bibr" rid="B24">24</xref>).&#x201D; If this premise is correct, in the opt-out model, people automatically choose to &#x201C;agree,&#x201D; theoretically resulting in a 100&#x0025; consent rate unless they explicitly indicate &#x201C;disagreeing.&#x201D;</p>
<p>However, the opt-out model has a fundamental flaw in that it does not consider the ethical aspects of consent acquisition, namely quality. Typically, informed consent places the highest importance on respecting patient autonomy by ensuring that the patient has the freedom of choice (<xref ref-type="bibr" rid="B25">25</xref>). However, this model simplifies the process, which may lead to distrust and doubts from patients. Additionally, medical staff may be concerned about being held responsible for issues such as information leaks or privacy violations due to differences from previous practices (<xref ref-type="bibr" rid="B23">23</xref>). Therefore, in societies like Japan, where the opt-in model is already customary, transitioning to the opt-out model is challenging.</p>
<p>Based on the above considerations, <xref ref-type="table" rid="T1">Table&#x00A0;1</xref> reexamines the consent models from the perspectives of default settings, efficiency, and quality.</p>
<table-wrap id="T1" position="float"><label>Table 1</label>
<caption><p>Relationship between default, efficiency, quality, and consent rate (utility) in each model.</p></caption>
<table frame="hsides" rules="groups">
<colgroup>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
</colgroup>
<thead>
<tr>
<th valign="top" align="left">Consent model</th>
<th valign="top" align="left">Default</th>
<th valign="top" align="left">Efficiency</th>
<th valign="top" align="left">Quality</th>
<th valign="top" align="left">Consent rate (utility)</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">Opt-in model</td>
<td valign="top" align="left">Disagree</td>
<td valign="top" align="left">Low</td>
<td valign="top" align="left">High</td>
<td valign="top" align="left">Low</td>
</tr>
<tr>
<td valign="top" align="left">Opt-out model</td>
<td valign="top" align="left">Agree</td>
<td valign="top" align="left">High</td>
<td valign="top" align="left">Low</td>
<td valign="top" align="left">High</td>
</tr>
<tr>
<td valign="top" align="left">Two-step consent model</td>
<td valign="top" align="left">Step 1: agree Step 2: disagree</td>
<td valign="top" align="left">High</td>
<td valign="top" align="left">High</td>
<td valign="top" align="left">High to medium</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn id="table-fn1"><p>Source: Created by the authors.</p></fn>
</table-wrap-foot>
</table-wrap>
</sec>
<sec id="s2f"><label>2.6</label><title>Definition of the two-step consent model</title>
<p>As mentioned above, the opt-in and opt-out models face a trade-off between the efficiency and quality of consent acquisition. To overcome these limitations, this study proposes a new two-step consent model as a compromise between these two models (bottom of <xref ref-type="table" rid="T1">Table&#x00A0;1</xref>). Our model involves preregistering patient information through implied consent via in-hospital notices and obtaining explicit consent from the patient when the medical staff needs to access the patient information. The details of this model are as follows:</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM63"><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>: first step of consent acquisition.</p>
<p>In the first step, implied consent is obtained in advance through in-hospital notices and public relations to register patient information. The default setting at this stage is set to &#x201C;agree.&#x201D; Similar to the opt-out model, medical staff do not directly approach patients to explain and obtain consent, thus reducing consent costs (<xref ref-type="bibr" rid="B26">26</xref>).</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM64"><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>: second step of consent acquisition.</p>
<p>In the second step, explicit consent is obtained from patients during their visits when the medical staff needs to access patient information. The default setting at this stage is set to &#x201C;disagree.&#x201D; Notably, at this stage, the medical staff directly obtains consent from the patient using the opt-in model. Therefore, the quality of consent acquisition is ultimately ensured. This preparation is expected to be particularly effective in urgent situations (<xref ref-type="bibr" rid="B26">26</xref>).</p>
<p>Based on these assumptions, we define the following variables to formulate this model:</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM65"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>: Consent rate in the first step <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM66"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM67"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>: Consent rate in the second step <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM68"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM69"><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>: Number of subjects in the first step,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM70"><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>: Number of subjects in the second step <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM71"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM72"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>: Average utility in the first step when the consent rate is <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM73"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:math></inline-formula></p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM74"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>: Average utility in the second step when the consent rate is <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM75"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula><italic>,</italic></p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM76"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>: Utility of the two-step consent model.</p>
<p>Using these, we define the following equations:
<disp-formula id="disp-formula11"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM14"><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mo>&#x2061;</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mo>,</mml:mo></mml:mstyle></mml:math></disp-formula>
<disp-formula id="disp-formula12"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM15"><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mo>&#x2061;</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow></mml:mstyle></mml:math></disp-formula></p>
<p>Owing to the difference in default settings, the first and second steps are given by <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM77"><mml:mspace width="thickmathspace" /><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM78"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></inline-formula> respectively. By multiplying these two equations, the utility of the two-step consent model is defined as follows:
<disp-formula id="disp-formula13"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM16"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x00D7;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>
For example, if the utility function for consent is set to 1 at each step, and the consent rate in the first and second steps is 90&#x0025; and 80&#x0025;, respectively, then the consent rate of the two-step consent model can be easily calculated as
<disp-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM17"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x00D7;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.9</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mn>0.8</mml:mn><mml:mo>=</mml:mo><mml:mn>0.72.</mml:mn></mml:math></disp-formula>
Thus, as a result of setting a default that obtains consent from patients in two steps, the consent rate may be slightly lower than the opt-out model; however, it will ensure the quality of consent acquisition compared with the opt-in model, thereby resolving the trade-off issue.</p>
</sec>
<sec id="s2g"><label>2.7</label><title>Order of average utility among the three consent models</title>
<p>Furthermore, we observe that the following inequality holds for the average utility of these three consent models under general conditions:
<disp-formula id="disp-formula14"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM18"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>
In fact, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM79"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> always holds as follows:
<disp-formula id="disp-formula15"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM19"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd /><mml:mtd><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:mrow><mml:mspace width="thinmathspace" /><mml:mo fence="false" stretchy="false">{</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>&#x2212;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:mrow><mml:mspace width="thinmathspace" /><mml:mo fence="false" stretchy="false">{</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>&#x00D7;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mspace width="thinmathspace" /><mml:mo fence="false" stretchy="false">{</mml:mo></mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mspace width="thickmathspace" /><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mspace width="thinmathspace" /><mml:mo fence="false" stretchy="false">{</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>&#x2212;</mml:mo><mml:mspace width="thinmathspace" /><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>&#x00D7;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mspace width="thickmathspace" /></mml:mstyle></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mspace width="thinmathspace" /><mml:mo fence="false" stretchy="false">{</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="0.056em" /></mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2265;</mml:mo><mml:mn>0.</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
Here, the inequality <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM80"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> (provided that <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM81"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>) is used. Therefore, the average utility is always higher in the opt-out model. However, when the term inside the parentheses
<disp-formula id="disp-formula16"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM20"><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:msubsup><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mstyle></mml:math></disp-formula>
approaches zero, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM82"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> approaches <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM83"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>. This implies that the subjects who did not agree in the first stage derive utility from agreeing in the second stage. This is expected to be sufficiently satisfied as <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM84"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> becomes large enough in the second stage. Therefore, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM85"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> provides an average utility close to <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM86"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>. More specific comparisons of average utility are presented in <xref ref-type="sec" rid="s2h">Section 2.8</xref>.</p>
<p>Meanwhile, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM87"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> does not generally hold, but
<disp-formula id="disp-formula17"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM21"><mml:mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mml:mtr><mml:mtd><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mspace width="1em" /><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi mathvariant="normal">&#x0023;</mml:mi><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mspace width="1em" /><mml:mo>&#x00D7;</mml:mo><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mspace width="thinmathspace" /><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mspace width="thickmathspace" /></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p>
<p>Thus, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM88"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /></mml:mrow></mml:math></inline-formula> is calculated by having <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM89"><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> as the baseline and adding the utility of non-consenting subjects. Here, the term is summarized, as
<disp-formula id="disp-formula18"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM22"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi mathvariant="normal">&#x0023;</mml:mi><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mrow></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mstyle></mml:math></disp-formula>
is the average value of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM90"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">disagree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> for non-consenting subjects, satisfying <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM91"><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>. Applying similar transformations to <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM92"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>, we obtain
<disp-formula id="disp-formula19"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM23"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>
Thus, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM93"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> has <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM94"><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> as the baseline, with the utility of agreeing subjects added. However,
<disp-formula id="disp-formula20"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM24"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi mathvariant="normal">&#x0023;</mml:mi><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mrow></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi>a</mml:mi><mml:mi>g</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mstyle></mml:math></disp-formula>
satisfying <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM95"><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>. Using these, we obtain
<disp-formula id="disp-formula21"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM25"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>
Applying similar calculations to <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM96"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, we obtain
<disp-formula id="disp-formula22"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM26"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>
<disp-formula id="disp-formula23"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM27"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi mathvariant="normal">&#x0023;</mml:mi><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mrow></mml:mfrac></mml:mrow><mml:munder><mml:mrow><mml:mo movablelimits="false">&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow></mml:munder><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow><mml:mi mathvariant="normal">agree</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:mstyle></mml:math></disp-formula>
For <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM97"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> to hold, the following must be satisfied:
<disp-formula id="disp-formula24"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM28"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>
Here, since the relationships among <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM98"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> vary, this does not always hold. However, if
<disp-formula id="disp-formula25"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="UDM29"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>&#x22C6;</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>
is satisfied, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM99"><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> can be chosen from the interval satisfying
<disp-formula id="disp-formula26"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="DM1"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mstyle></mml:math></disp-formula>
regardless of the value of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM100"><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, ensuring <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM101"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mspace width="thickmathspace" /></mml:math></inline-formula>(see <xref ref-type="sec" rid="s11">Supplementary Material</xref> for details). Since <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM102"><mml:mspace width="thickmathspace" /><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> are almost the same value, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM103"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>&#x22C6;</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> is sufficient if <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM104"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mspace width="thickmathspace" /></mml:math></inline-formula> is satisfied. This means that the consent rate in the opt-in model is lower than that in the second stage, which is expected to be sufficiently satisfied. However, these are only sufficient conditions, and not satisfying them does not necessarily mean that <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM105"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>.</p>
</sec>
<sec id="s2h"><label>2.8</label><title>Numerical experimental examples</title>
<p>Finally, we conducted numerical experiments on these three models; the results are presented in this section. As real-world patient populations have different characteristics, the factors influencing the choice of consent may differ. Therefore, considering patient diversity, we assume representative distributions A-D to ascertain the average utility of each model. For details on the functions and parameters used in these numerical calculations and the shapes of the distributions, see <xref ref-type="sec" rid="s11">Supplementary Material</xref>. Based on the above, <xref ref-type="table" rid="T2">Table&#x00A0;2</xref> shows the results of calculating the average utility and average consent rate for each model for five cases, consisting of commonly assumed Cases 1, 2, and 3, and exceptional Cases 4 and 5.</p>
<table-wrap id="T2" position="float"><label>Table 2</label>
<caption><p>Numerical calculation examples.</p></caption>
<table frame="hsides" rules="groups">
<colgroup>
<col align="left"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
</colgroup>
<thead>
<tr>
<th valign="top" align="left" rowspan="2"/>
<th valign="top" align="center" colspan="2">Opt-in model</th>
<th valign="top" align="center" colspan="2">Two-step consent model</th>
<th valign="top" align="center" colspan="2">Opt-out model</th>
</tr>
<tr>
<th valign="top" align="center">Utility</th>
<th valign="top" align="center">Consent rate</th>
<th valign="top" align="center">Utility</th>
<th valign="top" align="center">Consent rate</th>
<th valign="top" align="center">Utility</th>
<th valign="top" align="center">Consent rate</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">Case 1</td>
<td valign="top" align="center">0.533</td>
<td valign="top" align="center">0.097</td>
<td valign="top" align="center">0.778</td>
<td valign="top" align="center">0.929</td>
<td valign="top" align="center">0.926</td>
<td valign="top" align="center">0.655</td>
</tr>
<tr>
<td valign="top" align="left">Case 2</td>
<td valign="top" align="center">0.536</td>
<td valign="top" align="center">0.102</td>
<td valign="top" align="center">0.769</td>
<td valign="top" align="center">0.993</td>
<td valign="top" align="center">0.903</td>
<td valign="top" align="center">0.954</td>
</tr>
<tr>
<td valign="top" align="left">Case 3</td>
<td valign="top" align="center">0.690</td>
<td valign="top" align="center">0.472</td>
<td valign="top" align="center">0.758</td>
<td valign="top" align="center">0.988</td>
<td valign="top" align="center">0.904</td>
<td valign="top" align="center">0.942</td>
</tr>
<tr>
<td valign="top" align="left">Case 4</td>
<td valign="top" align="center">0.532</td>
<td valign="top" align="center">0.096</td>
<td valign="top" align="center">0.493</td>
<td valign="top" align="center">0.649</td>
<td valign="top" align="center">0.926</td>
<td valign="top" align="center">0.649</td>
</tr>
<tr>
<td valign="top" align="left">Case 5</td>
<td valign="top" align="center">0.704</td>
<td valign="top" align="center">0.498</td>
<td valign="top" align="center">0.488</td>
<td valign="top" align="center">0.948</td>
<td valign="top" align="center">0.904</td>
<td valign="top" align="center">0.942</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn id="table-fn2"><p>Source: Authors&#x2019; creation.</p></fn>
</table-wrap-foot>
</table-wrap>
<p>Case 1: The distribution of patients in the first and second phases follows distributions A and D, respectively.</p>
<p>Case 2: The distribution of patients in the first and second phases follows distributions B and D, respectively.</p>
<p>Case 3: The distribution of patients in the first and second phases follows distributions C and D, respectively.</p>
<p>Case 4: The distribution of patients in the first and second phases follows distribution A.</p>
<p>Case 5: The distribution of patients in the first and second phases follows distributions C and A, respectively.</p>
<p>The results of the numerical experiments show that the relationship <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM106"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> is established in Cases 1, 2, and 3, with the two-step consent model, in particular, showing a high consent rate. However, in Cases 4 and 5, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM107"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, which is a specific uncommon situation wherein the respondents do not feel the need for medical care at the second step.</p>
<p>In all of the above situations, the opt-out model automatically obtains a large number of consents and has the highest utility, since it leads to the selection of &#x201C;I agree.&#x201D; Although the opt-out model is superior in terms of its low cost, it includes the portion of &#x201C;consent&#x201D; that is considered &#x201C;consent&#x201D; without the subject&#x0027;s knowledge. By contrast, the utility and consent rate are low in the opt-in model. It can be seen that this is inefficient because of the cost of consent. Meanwhile, the proposed two-step consent model is cost-effective since the utility and consent rate are comparable to those of the opt-out model, except for Case 4, even though the subject is aware of and gives their consent.</p>
</sec>
</sec>
<sec id="s3" sec-type="results"><label>3</label><title>Results</title>
<sec id="s3a"><label>3.1</label><title>Online survey</title>
<p>An online survey was conducted from late January to early February 2024, using a panel of 2.2 million monitors managed by one of the largest online survey companies in Japan. To collect a target sample size of 2,000, which aligns with Japan&#x0027;s population distribution, the questionnaire was distributed after prior allocation by gender, age group, and prefecture. Consequently, we obtained a sufficiently randomized sample with a valid collection of 2,000 samples (100&#x0025;). The questionnaire received prior ethical review from the &#x201C;Ethical Review Committee for Research Involving Human Subjects&#x201D; of the Kitami Institute of Technology (October 23, 2018, Approval No.: 1015, July 27, 2022, Approval: KIT-2022-01, Kitami Institute of Technology).</p>
<p>Based on the concept of the contingent valuation method (CVM), we presented the monitors with questions that assumed four situations related to obtaining consent and asked them to respond regarding the acceptability of these situations. The following four situations and symbols were assumed for each model:</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM108"><mml:msub><mml:mi>P</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:msub><mml:mo>&#x003A;</mml:mo><mml:mspace width="thickmathspace" /></mml:math></inline-formula>A scenario assuming administrative staff in the opt-in model,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM109"><mml:msub><mml:mi>P</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub><mml:mo>&#x003A;</mml:mo><mml:mspace width="thickmathspace" /></mml:math></inline-formula>A scenario assuming a physician in the opt-in model,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM110"><mml:msub><mml:mi>P</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x003A;</mml:mo><mml:mspace width="thickmathspace" /></mml:math></inline-formula>A scenario assuming the opt-out model,</p>
<p><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM111"><mml:msub><mml:mi>P</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x003A;</mml:mo><mml:mspace width="thickmathspace" /></mml:math></inline-formula>A scenario assuming the second stage of the two-step consent model.</p>
<p>Here, the response to the opt-in model is divided into two scenarios: administrative staff and physicians, considering that the trust relationship between the medical staff and patients may also influence the response (<xref ref-type="bibr" rid="B27">27</xref>). The details of the questions and response items are shown in <xref ref-type="sec" rid="s11">Supplementary Material</xref>. Additionally, a goodness-of-fit test was conducted to confirm whether the differences in opinions for each scenario were statistically significant. The expected ratios were set assuming a normal distribution.</p>
<p>The attribute information of the 2,000 samples was as follows. In terms of gender, 49.5&#x0025; were males and 50.6&#x0025; were females. The average age was 50.24 years, with 13.6&#x0025; in their 20s, 15.8&#x0025; in their 30s, 20.0&#x0025; in their 40s, 16.8&#x0025; in their 50s, 18.7&#x0025; in their 60s, and 15.3&#x0025; in their 70s. The residential locations of the monitors are described in <xref ref-type="sec" rid="s11">Supplementary Material</xref>.</p>
</sec>
<sec id="s3b"><label>3.2</label><title>Survey results</title>
<p>The survey results suggested that differences in the distribution of responses in all scenarios were statistically significant. First, we asked about the scenario in which administrative staff handled the opt-in model. The results were as follows: 29.5&#x0025; of the respondents chose &#x201C;I think I would sign,&#x201D; 17.9&#x0025; chose &#x201C;I think I would not sign,&#x201D; 52.4&#x0025; chose &#x201C;I don&#x0027;t know if I would sign or not,&#x201D; and 0.3&#x0025; chose &#x201C;Other&#x201D; (<xref ref-type="fig" rid="F2">Figure&#x00A0;2</xref>). This indicates that the most common response was the ambiguous &#x201C;I don&#x0027;t know if I would sign or not.&#x201D;</p>
<fig id="F2" position="float"><label>Figure 2</label>
<caption><p>Results for the scenario assuming administrative staff in the Opt-in model. Goodness-of-fit test, <italic>&#x03C7;</italic>&#x00B2; (1)&#x2009;&#x003D;&#x2009;1,831.1536, <italic>p</italic>&#x2009;&#x003C;&#x2009;0.001.</p></caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="fdgth-07-1498072-g002.tif"/>
</fig>
<p>Next, we asked about the scenario in which a physician handled the opt-in model. The results were as follows: 42.7&#x0025; of the participants chose &#x201C;I think I would sign,&#x201D; 15.7&#x0025; chose &#x201C;I think I would not sign,&#x201D; 41.4&#x0025; chose &#x201C;I don&#x0027;t know if I would sign or not,&#x201D; and 0.7&#x0025; chose &#x201C;Other&#x201D; (<xref ref-type="fig" rid="F3">Figure&#x00A0;3</xref>). This indicates that when a physician is involved, the response &#x201C;I think I would sign&#x201D; increases by about 10&#x0025;, while the ambiguous response &#x201C;I don&#x0027;t know if I would sign or not&#x201D; decreases by 10&#x0025;.</p>
<fig id="F3" position="float"><label>Figure 3</label>
<caption><p>Results for the scenario assuming a physician in the Opt-in model. Goodness-of-fit test, <italic>&#x03C7;</italic>&#x00B2; (1)&#x2009;&#x003D;&#x2009;1,657.8201, <italic>p</italic>&#x2009;&#x003C;&#x2009;0.001.</p></caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="fdgth-07-1498072-g003.tif"/>
</fig>
<p>Additionally, we asked about the scenario assuming the opt-out model. The results were as follows: 47.7&#x0025; of the respondents chose &#x201C;No worries, will continue visiting,&#x201D; 29.1&#x0025; chose &#x201C;Have worries, but will continue visiting,&#x201D; 17.4&#x0025; chose &#x201C;Have worries, will ask questions, may transfer,&#x201D; 5.0&#x0025; chose &#x201C;Have worries, will transfer (won&#x0027;t visit again),&#x201D; and 0.9&#x0025; chose &#x201C;Other&#x201D; (<xref ref-type="fig" rid="F4">Figure&#x00A0;4</xref>). In this model, unless explicit dissent is expressed, consent is assumed. Therefore, if the 5.0&#x0025; who answered &#x201C;Have worries, will transfer (won&#x0027;t visit again)&#x201D; are considered dissenters, 95&#x0025; are considered to have consented. However, the number of consenters might decrease depending on the response of the medical staff, as 17.4&#x0025; answered &#x201C;Have worries, will ask questions, may transfer.&#x201D;</p>
<fig id="F4" position="float"><label>Figure 4</label>
<caption><p>Results for the scenario assuming the Opt-out model. Goodness-of-fit test, <italic>&#x03C7;</italic>&#x00B2; (1)&#x2009;&#x003D;&#x2009;2,324.9391, <italic>p</italic>&#x2009;&#x003C;&#x2009;0.001.</p></caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="fdgth-07-1498072-g004.tif"/>
</fig>
<p>Finally, we asked about the scenario assuming the second stage of the two-step consent model. The results were as follows: 74.6&#x0025; chose &#x201C;I think I would sign,&#x201D; 9.4&#x0025; chose &#x201C;I think I would not sign,&#x201D; 15.8&#x0025; chose &#x201C;I don&#x0027;t know if I would sign or not,&#x201D; and 0.3&#x0025; chose &#x201C;Other&#x201D; (<xref ref-type="fig" rid="F5">Figure&#x00A0;5</xref>). This indicates that in the two-step consent model, the proportion of the ambiguous response &#x201C;I don&#x0027;t know if I would sign or not&#x201D; was lower than in the opt-out model, while the proportion of the &#x201C;I think I would sign&#x201D; response was higher than in the opt-in model.</p>
<fig id="F5" position="float"><label>Figure 5</label>
<caption><p>Results for the scenario assuming the second stage of the Two-step consent model. Goodness-of-fit test, <italic>&#x03C7;</italic>&#x00B2; (2)&#x2009;&#x003D;&#x2009;803.0122, <italic>p</italic>&#x2009;&#x003C;&#x2009;0.001.</p></caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="fdgth-07-1498072-g005.tif"/>
</fig>
<p>Among the four scenarios, it is clear that the opt-out model has the highest likelihood of obtaining signatures. However, approximately one-third of respondents answered, &#x201C;I have worries or concerns, but I will continue visiting,&#x201D; indicating that it is not unconditionally acceptable. By contrast, in the two-step consent model, about three-quarters of respondents answered, &#x201C;I think I would sign,&#x201D; suggesting that it is generally well-received.</p>
</sec>
<sec id="s3c"><label>3.3</label><title>Estimated consent rates</title>
<p>Based on these results, we estimated the consent rates for each scenario. <xref ref-type="fig" rid="F6">Figure&#x00A0;6</xref> shows the mean value graph, and <xref ref-type="table" rid="T3">Table&#x00A0;3</xref> shows the descriptive statistics. We converted the responses into dummy variables to estimate the consent rates: for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM112"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>,<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM113"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM114"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, &#x201C;I think I would sign&#x201D;&#x2009;&#x003D;&#x2009;1, and &#x201C;Other&#x201D;&#x2009;&#x003D;&#x2009;0; for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM115"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, &#x201C;I have worries or concerns, so I will consider transferring (never visit again)&#x201D;&#x2009;&#x003D;&#x2009;0, and &#x201C;Other&#x201D;&#x2009;&#x003D;&#x2009;1. Note that the first stage of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM116"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> uses the values from <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM117"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>.</p>
<fig id="F6" position="float"><label>Figure 6</label>
<caption><p>Mean value graph of consent rates by model. One-way ANOVA, F (3,7996)&#x2009;&#x003D;&#x2009;1,033.020, <italic>p</italic>&#x2009;&#x003C;&#x2009;0.000.</p></caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="fdgth-07-1498072-g006.tif"/>
</fig>
<table-wrap id="T3" position="float"><label>Table 3</label>
<caption><p>Descriptive statistics.</p></caption>
<table frame="hsides" rules="groups">
<colgroup>
<col align="left"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
<col align="center"/>
</colgroup>
<thead>
<tr>
<th valign="top" align="left">Consent model</th>
<th valign="top" align="center">Mean</th>
<th valign="top" align="center">SD</th>
<th valign="top" align="center">Min</th>
<th valign="top" align="center">Max</th>
<th valign="top" align="center">Median</th>
<th valign="top" align="center">CI (min-max)</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left"><italic>P<sub>inS</sub></italic></td>
<td valign="top" align="center">0.295</td>
<td valign="top" align="center">0.456</td>
<td valign="top" align="center">0</td>
<td valign="top" align="center">1</td>
<td valign="top" align="center">0</td>
<td valign="top" align="center">0.275&#x2013;0.315</td>
</tr>
<tr>
<td valign="top" align="left"><italic>P<sub>inD</sub></italic></td>
<td valign="top" align="center">0.427</td>
<td valign="top" align="center">0.495</td>
<td valign="top" align="center">0</td>
<td valign="top" align="center">1</td>
<td valign="top" align="center">0</td>
<td valign="top" align="center">0.405&#x2013;0.448</td>
</tr>
<tr>
<td valign="top" align="left"><italic>P<sub>out</sub></italic></td>
<td valign="top" align="center">0.950</td>
<td valign="top" align="center">0.218</td>
<td valign="top" align="center">0</td>
<td valign="top" align="center">1</td>
<td valign="top" align="center">1</td>
<td valign="top" align="center">0.940&#x2013;0.960</td>
</tr>
<tr>
<td valign="top" align="left"><italic>P<sub>2step</sub></italic></td>
<td valign="top" align="center">0.685</td>
<td valign="top" align="center">0.435</td>
<td valign="top" align="center">0</td>
<td valign="top" align="center">1</td>
<td valign="top" align="center">1</td>
<td valign="top" align="center">0.666&#x2013;0.704</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn id="table-fn3"><p>Source: Created by the authors.</p></fn>
</table-wrap-foot>
</table-wrap>
<p>The analysis revealed that the relationship <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM118"><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> derived from the models in Chapter 2 holds true (<xref ref-type="fig" rid="F6">Figure&#x00A0;6</xref>). To determine whether the differences in the means of these four values are statistically significant, a one-way analysis of variance (ANOVA) was conducted, with the results highlighting a significant difference [F (3, 7996)&#x2009;&#x003D;&#x2009;1,033.020, <italic>p</italic>&#x2009;&#x003C;&#x2009;0.000]. Descriptive statistics are shown in <xref ref-type="table" rid="T3">Table&#x00A0;3</xref> for further details.</p>
<p>First, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM119"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> had a low mean of 0.295 and a relatively large standard deviation of 0.456, indicating considerable variability in the consent rate. The median was 0, suggesting that more than half of the respondents tended not to consent. Next, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM120"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> had a higher mean of 0.427 and a slightly larger standard deviation of 0.495, compared to <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM121"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, indicating that despite the considerable variability in the consent rate, there was a higher tendency to consent. However, the median was 0, indicating that even when a physician was involved, more than half of the respondents did not consent.</p>
<p>Furthermore, regarding <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM122"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, the average was 0.950, the highest, and the standard deviation was 0.218, the smallest, indicating a consistently high agreement rate among the subjects. Additionally, the effectiveness ratios with the opt-in model, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM123"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>/<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM124"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> &#x003D; 3.220 and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM125"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>/<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM126"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>&#x2009;&#x003D;&#x2009;2.225 suggest that <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM127"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> is the most efficient. Finally, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM128"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> had an average of 0.685 and a standard deviation of 0.435, indicating a lower average and greater variability compared with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM129"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>. However, the effectiveness ratios with the opt-in model, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM130"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>/<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM131"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>&#x2009;&#x003D;&#x2009;2.322 and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM132"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>/<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM133"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>&#x2009;&#x003D;&#x2009;1.604, were both above 1, confirming its efficiency.</p>
<p>In summary, the most efficient model is <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM134"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, followed by the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM135"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> model. Conversely, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM136"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="IM137"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula> had low agreement rates, suggesting that even if the person obtaining consent was a physician instead of administrative staff, there is a limit to the improvement in agreement rates.</p>
</sec>
</sec>
<sec id="s4" sec-type="discussion"><label>4</label><title>Discussion</title>
<p>In this study, we raised the question: can the stagnant consent rate be overcome by setting defaults that balance both the efficiency and quality of consent acquisition? To answer this question, we formulated a new two-step consent model using utility theory and evaluated its acceptability along with that of the traditional model. The results show that the opt-in model has limitations in improving the consent rate, even if the physician responds, and that the opt-out model may cause the highest level of distrust on the part of the patient. Meanwhile, the proposed two-step consent model showed an intermediate consent rate between the opt-in and opt-out models, confirming that it can balance both efficiency and quality in obtaining consent. In this section, we will discuss the limitations of the conventional model and how the two-step consent model may overcome these limitations.</p>
<sec id="s4a"><label>4.1</label><title>Advantages and limitations of the Opt-in model</title>
<p>First, we examined the Opt-in model, which sets the default consent option to &#x201C;disagree.&#x201D; While the opt-in model is expected to improve consent rates when handled by physicians rather than administrative staff, it has limitations as a means of improving consent rates due to its impact on clinical efficiency. Indeed, the consent rate was slightly over 10&#x0025; higher when handled by physicians (42.7&#x0025;) than when handled by administrative staff (29.5&#x0025;). Therefore, if physicians attempt to persuade patients directly from a medical standpoint, the low consent rates can be improved while ensuring the quality of consent acquisition.</p>
<p>However, owing to the nature of this model, the cost of obtaining consent becomes significant, potentially negatively affecting the physicians&#x0027; primary clinical duties. In fact, Japan has a higher number of outpatient visits and a higher frequency of visits than other countries, making it difficult for physicians to spend sufficient time explaining medical situations and obtaining consent from patients (<xref ref-type="bibr" rid="B28">28</xref>). Therefore, it is not reasonable for physicians to spend a significant amount of their time to obtain consent from patients. Moreover, this could contradict the recent work style reforms for physicians, which the Japanese government is focusing on at present. Further, while increasing the number of administrative staff or enhancing public relations to promote consent acquisition are possible options, they require budget adjustments. Additionally, this model has challenges from a user experience perspective (<xref ref-type="bibr" rid="B29">29</xref>). If patients have no experience using HIE, even if medical staff attempt to obtain consent, they may not be able to comprehend its benefits, thereby making it less persuasive.</p>
</sec>
<sec id="s4b"><label>4.2</label><title>Advantages and limitations of the opt-out model</title>
<p>Next, we examined the opt-out model, where the default consent option is set to &#x201C;agree.&#x201D; In Japan&#x0027;s medical settings, where the opt-in method is path-dependent, fundamentally changing the default to an opt-out model may be difficult for both patients and medical institutions to accept. Indeed, the consent rate for the opt-out model demonstrated the highest level at 95.0&#x0025;, which was 2.2&#x2013;3.2 times higher than that of the opt-in model. According to a survey in England, 94&#x0025; of respondents were satisfied with the secondary use of medical data obtained through the opt-out method, which aligns with the results of this study (<xref ref-type="bibr" rid="B30">30</xref>). Therefore, this model may be considered the most efficient in terms of effectiveness. However, looking at the detailed survey results in <xref ref-type="fig" rid="F3">Figure&#x00A0;3</xref>, 29.1&#x0025; of respondents continued their visits despite having concerns or worries, and 17.4&#x0025; considered changing hospitals depending on the answers to their questions. Combined, nearly half of the respondents had concerns, which cannot be ignored. This suggests that respondents might change their consent status depending on the situation. The reasons for the distrust and doubts among respondents included serious issues where the qualitative aspects of consent acquisition were not guaranteed.</p>
<p>Recently, the Japanese government has been promoting implied consent to improve the consent rate for HIEs (<xref ref-type="bibr" rid="B31">31</xref>). However, the transition to the opt-out model in medical settings has not progressed, possibly due to concerns about registering personal information in HIEs amidst ongoing incidents of system downtimes and information leaks caused by cyberattacks (<xref ref-type="bibr" rid="B32">32</xref>, <xref ref-type="bibr" rid="B33">33</xref>). Additionally, some patients only use medical institutions to consult their primary care physician. If there is no need to use multiple medical institutions, the actual use may not increase even if the opt-out model superficially raises the consent rate. Moreover, in Japan&#x0027;s medical settings, obtaining consent through the opt-in method has already become the norm, making it difficult to overturn such practices fundamentally.</p>
<p>Therefore, while the opt-out model is superior in terms of the efficiency of consent acquisition, it has limitations due to the potential for superficial use caused by qualitative issues.</p>
</sec>
<sec id="s4c"><label>4.3</label><title>Acceptability of the two-step consent model</title>
<p>Furthermore, we evaluated the Two-Step Consent model. In this approach, the default for initial consent is set to &#x201C;agree,&#x201D; thereby obtaining implied consent. At a later stage, when a physician requires access to patient information, explicit consent is obtained by setting the default to &#x201C;disagree.&#x201D; This model effectively leverages the advantages of both the Opt-In and Opt-Out models. Based on the results, the proposed two-step consent model has the potential to improve consent rates through default settings that balance both the efficiency and quality of consent acquisition. Indeed, while the consent rate for this model is 68.5&#x0025;, which is 26.5&#x0025; lower than the opt-out model, only 9.4&#x0025; of respondents answered that they &#x201C;disagree.&#x201D; As previously mentioned, cyberattacks and data breaches targeting hospitals have been frequent in Japan (<xref ref-type="bibr" rid="B32">32</xref>, <xref ref-type="bibr" rid="B33">33</xref>). Strengthening measures to address these vulnerabilities may improve consent rates to levels comparable to those achieved by the opt-out model. Thus, this model is considered cost-effective as it can overcome the limitations of previous methods by leveraging the benefits of both opt-in and opt-out methods.</p>
<p>Furthermore, this model offers significant benefits for medical institutions. Although the HIE is a rational system that contributes to society as a whole, in reality, few medical institutions are connected to HIEs. One reason is that when hospitals are in a competitive relationship, the medical information and information systems accumulated by hospitals become important strategic management resources (<xref ref-type="bibr" rid="B34">34</xref>), which are not necessarily in line with economic rationality. In other words, hospitals may be concerned about the risk of losing patients to other hospitals or having valuable information extracted. However, in this model, patient information can only be viewed when the patient visits the hospital and the physician examines the patient. In other cases, access to other patient information is restricted, alleviating hospital concerns about opportunistic behavior, such as unauthorized viewing or extraction of information. Therefore, it is essential to implement measures that align with the economic rationality of hospitals in advance to gain cooperation from hospitals. In this regard, the initiatives of Inland Empire Health Plan (IEHP), the largest Medicaid plan in the United States, are instructive (<xref ref-type="bibr" rid="B35">35</xref>). This organization has introduced financial incentives for medical institutions that share patient data across institutional boundaries, improving utilization rates by compensating for management.</p>
<p>In conclusion, the two-step consent model can overcome the shortcomings of traditional models through the reconfiguration of default settings, making it a valuable policy option for future implementation. However, ensuring the robust adoption of this model necessitates a focus on patient education and comprehension.</p>
</sec>
<sec id="s4d"><label>4.4</label><title>Limitations and future research directions</title>
<p>Finally, we summarize the limitations of this study. First, although we proposed and verified the acceptability of the two-step consent model, we did not delve into the specific characteristics of the patients. Since the preferences and medical needs of the subjects vary (<xref ref-type="bibr" rid="B36">36</xref>), identifying utilities according to attributes and needs by future research may allow for more effective strategies. For instance, even with the two-step consent model, 15&#x0025; of respondents showed ambiguous reactions, indicating that they were unsure whether they would consent. Similar negative reactions have been reported in surveys from other fields (<xref ref-type="bibr" rid="B37">37</xref>). Therefore, while it may not be realistic to achieve a 100&#x0025; consent rate, examining the details of such ambiguous reactions and refusals in the future could help determine a reasonable level of consent rates.</p>
<p>Next, this study did not discuss information security or privacy protection measures aimed at improving consent rates. Further, it did not explore the effect of budgetary constraints on individual responses at the medical institution level. Therefore, future research should consider support measures at the policy level. Furthermore, this model assumed a constant number of subjects; however, the actual scale of information networks varies, and the number of subjects differs. Since the number of subjects affects the utility and consent rates of patients, future analyses should focus on the number of subjects and the scale of information networks.</p>
<p>Addressing the stagnation of HIE requires more than just improving consent rates. It is imperative to ensure operational cost-effectiveness, EHR standardization, interoperability, privacy, availability, and reliability to build trust among hospitals and encourage information sharing. These policy issues warrant further discussion.</p>
<p>Nevertheless, this study not only establishes the relationship between consent mechanisms and defaults using utility theory but also proposes a new consent model that overcomes the shortcomings of previous methods of obtaining consent. This model quantitatively demonstrates that it can guarantee both efficiency and quality in obtaining consent. Previous studies viewed the opt-in and opt-out approaches as mutually exclusive, which limited their ability to balance the efficiency and quality of consent acquisition. Conversely, this study introduces a new model that resolves these shortcomings with its novel results, thus making significant contributions to the extant literature as well as healthcare policymaking.</p>
</sec>
</sec>
<sec id="s5" sec-type="conclusions"><label>5</label><title>Conclusion</title>
<p>This study aimed to overcome the issue of low consent rates in HIEs by mathematically formulating a consent mechanism and proposing a new two-step consent model. We verified its acceptability through an online survey. The results showed that the acceptability of this model reached about 70&#x0025;, indicating its potential for improving consent rates while balancing the efficiency and quality of consent acquisition compared with traditional models. On the one hand, the traditional opt-in model, while maximizing patient autonomy by having medical staff directly explain medical situations and obtain explicit consent, faced issues of increased consent burden and decreased clinical efficiency, with acceptability falling below 50&#x0025;. On the other hand, the opt-out model, despite being the most efficient approach, was found to be fundamentally flawed, as approximately half of the respondents had some concerns. The proposed two-step consent model initially obtains implicit consent and then explicit consent when the medical staff needs to access patient information, thus efficiently obtaining consent while ensuring patient autonomy. By leveraging the default effect in this way, it can become a cost-effective policy option. Moving forward, rather than viewing opt-in and opt-out as opposing approaches, we hope that the introduction of new methods that leverage the advantages of both methods can help overcome stagnation issues in HIEs. Educating patients on the significance of HIE and the two-step consent model is critical to fostering understanding and acceptance.</p>
</sec>
</body>
<back>
<sec id="s6" sec-type="data-availability"><title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/<xref ref-type="sec" rid="s11">Supplementary Material</xref>, further inquiries can be directed to the corresponding author.</p>
</sec>
<sec id="s7" sec-type="ethics-statement"><title>Ethics statement</title>
<p>The studies involving humans were approved by Ethical Review Committee for Research Involving Human Subjects of the Kitami Institute of Technology. The studies were conducted in accordance with the local legislation and institutional requirements. The participants provided their written informed consent to participate in this study. Written informed consent was obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article.</p>
</sec>
<sec id="s8" sec-type="author-contributions"><title>Author contributions</title>
<p>AI: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Validation, Visualization, Writing &#x2013; original draft, Writing &#x2013; review &#x0026; editing. FN: Conceptualization, Data curation, Formal Analysis, Methodology, Software, Validation, Visualization, Writing &#x2013; review &#x0026; editing.</p>
</sec>
<sec id="s9" sec-type="funding-information"><title>Funding</title>
<p>The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work was supported by JSPS KAKEN Grant Number 18K12831 and JST RISTEX Grant 619.</p>
</sec>
<sec id="s10" sec-type="COI-statement"><title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec id="s12" sec-type="disclaimer"><title>Publisher&#x0027;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
<sec id="s11" sec-type="supplementary-material"><title>Supplementary material</title>
<p>The Supplementary Material for this article can be found online at: <ext-link ext-link-type="uri" xlink:href="https://www.frontiersin.org/articles/10.3389/fdgth.2025.1498072/full#supplementary-material">https://www.frontiersin.org/articles/10.3389/fdgth.2025.1498072/full&#x0023;supplementary-material</ext-link></p>
<supplementary-material id="SD1" content-type="local-data">
<media mimetype="application" mime-subtype="pdf" xlink:href="Datasheet1.pdf"/></supplementary-material>
</sec>
<fn-group>
<fn id="FN0001"><p><sup>1</sup>&#x201C;In reality, the set of options also includes choices such as &#x2018;undecided&#x2019; and &#x2018;unaware.&#x2019; For example, if the default is set to &#x2018;agree/disagree,&#x2019; selecting &#x2018;undecided&#x2019; or &#x2018;unaware&#x2019; would be evaluated for both &#x2018;agree&#x2019; and &#x2018;disagree,&#x2019; and the higher utility would be chosen.&#x201D;</p></fn>
</fn-group>
<ref-list><title>References</title>
<ref id="B1"><label>1.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Sarkar</surname><given-names>IN</given-names></name></person-group>. <article-title>Transforming health data to actionable information: recent progress and future opportunities in health information exchange</article-title>. <source>Yearbook Med Inform</source>. (<year>2022</year>) <volume>31</volume>:<fpage>203</fpage>&#x2013;<lpage>14</lpage>. <pub-id pub-id-type="doi">10.1055/s-0042-1742519</pub-id></citation></ref>
<ref id="B2"><label>2.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Menachemi</surname><given-names>N</given-names></name><name><surname>Rahurkar</surname><given-names>S</given-names></name><name><surname>Harle</surname><given-names>CA</given-names></name><name><surname>Vest</surname><given-names>JR</given-names></name></person-group>. <article-title>The benefits of health information exchange: an updated systematic review</article-title>. <source>J Am Med Inform Assoc</source>. (<year>2018</year>) <volume>25</volume>:<fpage>1259</fpage>&#x2013;<lpage>65</lpage>. <pub-id pub-id-type="doi">10.1093/jamia/ocy035</pub-id><pub-id pub-id-type="pmid">29718258</pub-id></citation></ref>
<ref id="B3"><label>3.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Payne</surname><given-names>TH</given-names></name><name><surname>Lovis</surname><given-names>C</given-names></name><name><surname>Gutteridge</surname><given-names>C</given-names></name><name><surname>Pagliari</surname><given-names>C</given-names></name><name><surname>Natarajan</surname><given-names>S</given-names></name><name><surname>Yong</surname><given-names>C</given-names></name><etal/></person-group> <article-title>Status of health information exchange: a comparison of six countries</article-title>. <source>J Glob Health</source>. (<year>2019</year>) <volume>9</volume>:<fpage>0204279</fpage>. <pub-id pub-id-type="doi">10.7189/jogh.09.020427</pub-id><pub-id pub-id-type="pmid">31673351</pub-id></citation></ref>
<ref id="B4"><label>4.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Ito</surname><given-names>A</given-names></name><name><surname>Okumura</surname><given-names>T</given-names></name></person-group>. <article-title>Factors for stagnation in the regional healthcare networks project: analysis of initial investment and management model</article-title>. <source>Kaikei-Kensa Kenkyu</source>. (<year>2021</year>) <volume>64</volume>:<fpage>63</fpage>&#x2013;<lpage>84</lpage>.</citation></ref>
<ref id="B5"><label>5.</label><citation citation-type="other"><person-group person-group-type="author"><name><surname>Shimbun</surname><given-names>NK</given-names></name></person-group>. <article-title>Sharing of medical data becomes hollow</article-title>. (<year>2019</year>). <comment>Available online at: <ext-link ext-link-type="uri" xlink:href="https://www.nikkei.com/article/DGXMZO42441870U9A310C1SHA000/">https://www.nikkei.com/article/DGXMZO42441870U9A310C1SHA000/</ext-link> (Accessed March 20, 2024)</comment>.</citation></ref>
<ref id="B6"><label>6.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Esmaeilzadeh</surname><given-names>P</given-names></name><name><surname>Sambasivan</surname><given-names>M</given-names></name></person-group>. <article-title>Patients&#x2019; support for health information exchange: a literature review and classification of key factors</article-title>. <source>BMC Med Inform Decis Mak</source>. (<year>2017</year>) <volume>17</volume>:<fpage>33</fpage>. <pub-id pub-id-type="doi">10.1186/s12911-017-0436-2</pub-id><pub-id pub-id-type="pmid">28376785</pub-id></citation></ref>
<ref id="B7"><label>7.</label><citation citation-type="other"><collab>Ministry of Health, Labour and Welfare</collab>. <article-title>Examples of consent acquisition methods in regional medical information collaboration networks</article-title> (<year>2020</year>).</citation></ref>
<ref id="B8"><label>8.</label><citation citation-type="other"><person-group person-group-type="author"><name><surname>Ai</surname><given-names>W</given-names></name></person-group>. <article-title>Overview of National Regional Medical Information Collaboration Network Using ICT (2022 Edition), Nichii soken Working Paper, No. 475</article-title> (<year>2022)</year>. <comment>Available online at:</comment> <ext-link ext-link-type="uri" xlink:href="https://www.jmari.med.or.jp/result/working/post-3866/">https://www.jmari.med.or.jp/result/working/post-3866/</ext-link> <comment>(Accessed March 20, 2024)</comment>.</citation></ref>
<ref id="B9"><label>9.</label><citation citation-type="book"><person-group person-group-type="author"><name><surname>Hill</surname><given-names>D</given-names></name><name><surname>Walker</surname><given-names>J</given-names></name><name><surname>Hale</surname><given-names>J</given-names></name></person-group>. <source>Privacy Considerations for Health Information Exchanges, Medical Data Privacy Handbook</source>. <publisher-loc>Cham</publisher-loc>: <publisher-name>Springer</publisher-name> (<year>2015</year>). p. <fpage>289</fpage>&#x2013;<lpage>311</lpage>.</citation></ref>
<ref id="B10"><label>10.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Suzumoto</surname><given-names>J</given-names></name><name><surname>Mori</surname><given-names>Y</given-names></name><name><surname>Kuroda</surname><given-names>T</given-names></name></person-group>. <article-title>Health information exchange use by healthcare workers in Japan: retrospective cohort study</article-title>. <source>JMIR Med Inform</source>. (<year>2024</year>) <volume>12</volume>(<issue>1</issue>):<fpage>e56263</fpage>. <pub-id pub-id-type="doi">10.2196/preprints.56263</pub-id><pub-id pub-id-type="pmid">39382566</pub-id></citation></ref>
<ref id="B11"><label>11.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Chimonas</surname><given-names>S</given-names></name><name><surname>Lipitz-Snyderman</surname><given-names>A</given-names></name><name><surname>Matsoukas</surname><given-names>K</given-names></name><name><surname>Kuperman</surname><given-names>G</given-names></name></person-group>. <article-title>Electronic consent in clinical care: an international scoping review</article-title>. <source>BMJ Health Care Inform</source>. (<year>2023</year>) <volume>30</volume>:<fpage>e100726</fpage>. <pub-id pub-id-type="doi">10.1136/bmjhci-2022-100726</pub-id><pub-id pub-id-type="pmid">37423643</pub-id></citation></ref>
<ref id="B12"><label>12.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Esmaeilzadeh</surname><given-names>P</given-names></name><name><surname>Mirzaei</surname><given-names>T</given-names></name></person-group>. <article-title>Comparison of consumers&#x2019; perspectives on different health information exchange (HIE) mechanisms: an experimental study</article-title>. <source>Int J Med Inform</source>. (<year>2018</year>) <volume>119</volume>:<fpage>1</fpage>&#x2013;<lpage>7</lpage>. <pub-id pub-id-type="doi">10.1016/j.ijmedinf.2018.08.007</pub-id><pub-id pub-id-type="pmid">30342677</pub-id></citation></ref>
<ref id="B13"><label>13.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Samuelson</surname><given-names>W</given-names></name><name><surname>Zeckhauser</surname><given-names>R</given-names></name></person-group>. <article-title>Status quo bias in decision making</article-title>. <source>J Risk Uncertain</source>. (<year>1988</year>) <volume>1</volume>(<issue>1</issue>):<fpage>7</fpage>&#x2013;<lpage>59</lpage>. <pub-id pub-id-type="doi">10.1007/BF00055564</pub-id></citation></ref>
<ref id="B14"><label>14.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Tversky</surname><given-names>A</given-names></name><name><surname>Kahneman</surname><given-names>D</given-names></name></person-group>. <article-title>Loss aversion in riskless choice: a reference-dependent model</article-title>. <source>Q J Econ</source>. (<year>1991</year>) <volume>106</volume>(<issue>4</issue>):<fpage>1039</fpage>&#x2013;<lpage>61</lpage>. <pub-id pub-id-type="doi">10.2307/2937956</pub-id></citation></ref>
<ref id="B15"><label>15.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Jachimowicz</surname><given-names>JM</given-names></name><name><surname>Duncan</surname><given-names>S</given-names></name><name><surname>Weber</surname><given-names>EU</given-names></name><name><surname>Johnson</surname><given-names>EJ</given-names></name></person-group>. <article-title>When and why defaults influence decisions: a meta-analysis of default effects</article-title>. <source>Behav Public Policy</source>. (<year>2019</year>) <volume>3</volume>:<fpage>159</fpage>&#x2013;<lpage>86</lpage>. <pub-id pub-id-type="doi">10.1017/bpp.2018.43</pub-id></citation></ref>
<ref id="B16"><label>16.</label><citation citation-type="book"><collab>OECD</collab>. <source>Behavioral Public Policy: New Policy Design Utilizing Insights from Behavioral Economics</source>. <publisher-loc>Tokyo</publisher-loc>: <publisher-name>Akashi Shoten</publisher-name> (<year>2016</year>). p. <fpage>61</fpage>&#x2013;<lpage>87</lpage>.</citation></ref>
<ref id="B17"><label>17.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Dhingra</surname><given-names>N</given-names></name><name><surname>Gorn</surname><given-names>Z</given-names></name><name><surname>Kener</surname><given-names>A</given-names></name><name><surname>Dana</surname><given-names>J</given-names></name></person-group>. <article-title>The default pull: an experimental demonstration of subtle default effects on preferences</article-title>. <source>Judg Decis Mak</source>. (<year>2012</year>) <volume>7</volume>:<fpage>69</fpage>&#x2013;<lpage>76</lpage>. <pub-id pub-id-type="doi">10.1017/S1930297500001844</pub-id></citation></ref>
<ref id="B18"><label>18.</label><citation citation-type="book"><person-group person-group-type="author"><name><surname>Cabral</surname><given-names>LMB</given-names></name></person-group>. <source>Introduction to Industrial Organization</source>. <edition>2nd ed</edition>. <publisher-loc>Cambridge, MA</publisher-loc>: <publisher-name>The MIT Press</publisher-name> (<year>2017</year>).</citation></ref>
<ref id="B19"><label>19.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Ito</surname><given-names>A</given-names></name><name><surname>Nakamura</surname><given-names>F</given-names></name></person-group>. <article-title>Alliance of medical malls: conditions for the establishment of network practice</article-title>. <source>J Jpn Manag</source>. (<year>2021</year>) <volume>5</volume>:<fpage>1</fpage>&#x2013;<lpage>16</lpage>.</citation></ref>
<ref id="B20"><label>20.</label><citation citation-type="web"><collab>Ministry of Health, Labour and Welfare</collab>. <article-title>Current status of regional medical information collaboration networks</article-title> (<year>2020</year>). <comment>Available online at:</comment> <ext-link ext-link-type="uri" xlink:href="https://www.mhlw.go.jp/content/10800000/000683765.pdf">https://www.mhlw.go.jp/content/10800000/000683765.pdf</ext-link> <comment>(Accessed March 20, 2024)</comment>.</citation></ref>
<ref id="B21"><label>21.</label><citation citation-type="other"><article-title>Yokohama City Government. <italic>Patient consent and information related to mutual collaboration, corresponding to Chapter 3 of the guidelines</italic>, <italic>Yokohama city ICT utilization regional medical collaboration network guidelines</italic></article-title> (<year>2018</year>). <comment>Available online at:</comment> <ext-link ext-link-type="uri" xlink:href="https://www.city.yokohama.lg.jp/kurashi/kenkoiryo/iryo/seisaku/ICT/guideline.files/0011_20180927.pdf">https://www.city.yokohama.lg.jp/kurashi/kenkoiryo/iryo/seisaku/ICT/guideline.files/0011_20180927.pdf</ext-link> <comment>(Accessed March 20, 2024)</comment>.</citation></ref>
<ref id="B22"><label>22.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>de Man</surname><given-names>Y</given-names></name><name><surname>Wieland-Jorna</surname><given-names>Y</given-names></name><name><surname>Torensma</surname><given-names>B</given-names></name><name><surname>de Wit</surname><given-names>K</given-names></name><name><surname>Francke</surname><given-names>AL</given-names></name><name><surname>Oosterveld-Vlug</surname><given-names>MG</given-names></name><etal/></person-group> <article-title>Opt-In and opt-out consent procedures for the reuse of routinely recorded health data in scientific research and their consequences for consent rate and consent bias: systematic review</article-title>. <source>J Med Internet Res</source>. (<year>2023</year>) <volume>25</volume>:<fpage>e42131</fpage>. <pub-id pub-id-type="doi">10.2196/42131</pub-id><pub-id pub-id-type="pmid">36853745</pub-id></citation></ref>
<ref id="B23"><label>23.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Apathy</surname><given-names>NC</given-names></name><name><surname>Holmgren</surname><given-names>AJ</given-names></name></person-group>. <article-title>Opt-in consent policies: potential barriers to hospital health information exchange</article-title>. <source>Am J Manag Care</source>. (<year>2020</year>) <volume>26</volume>:<fpage>e14</fpage>&#x2013;<lpage>20</lpage>. <pub-id pub-id-type="doi">10.37765/ajmc.2020.42148</pub-id><pub-id pub-id-type="pmid">31951362</pub-id></citation></ref>
<ref id="B24"><label>24.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Johnson</surname><given-names>EJ</given-names></name><name><surname>Goldstein</surname><given-names>D</given-names></name></person-group>. <article-title>Medicine &#x2013; do defaults save lives?</article-title> <source>Science</source>. (<year>2003</year>) <volume>302</volume>:<fpage>1338</fpage>&#x2013;<lpage>9</lpage>. <pub-id pub-id-type="doi">10.1126/science.1091721</pub-id><pub-id pub-id-type="pmid">14631022</pub-id></citation></ref>
<ref id="B25"><label>25.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Hall</surname><given-names>DE</given-names></name><name><surname>Prochazka</surname><given-names>AV</given-names></name><name><surname>Fink</surname><given-names>AS</given-names></name></person-group>. <article-title>Informed consent for clinical treatment</article-title>. <source>CMAJ</source>. (<year>2012</year>) <volume>184</volume>:<fpage>533</fpage>&#x2013;<lpage>40</lpage>. <pub-id pub-id-type="doi">10.1503/cmaj.112120</pub-id><pub-id pub-id-type="pmid">22392947</pub-id></citation></ref>
<ref id="B26"><label>26.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Medford-Davis</surname><given-names>LN</given-names></name><name><surname>Chang</surname><given-names>L</given-names></name><name><surname>Rhodes</surname><given-names>KV</given-names></name></person-group>. <article-title>Health information exchange: what do patients want?</article-title> <source>Health Inform J</source>. (<year>2017</year>) <volume>23</volume>:<fpage>268</fpage>&#x2013;<lpage>78</lpage>. <pub-id pub-id-type="doi">10.1177/1460458216647190</pub-id></citation></ref>
<ref id="B27"><label>27.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Kim</surname><given-names>KK</given-names></name><name><surname>Sankar</surname><given-names>P</given-names></name><name><surname>Wilson</surname><given-names>MD</given-names></name><name><surname>Haynes</surname><given-names>SC</given-names></name></person-group>. <article-title>Factors affecting willingness to share electronic health data among California consumers</article-title>. <source>BMC Med Ethics</source>. (<year>2017</year>) <volume>18</volume>:<fpage>25</fpage>. <pub-id pub-id-type="doi">10.1186/s12910-017-0185-x</pub-id><pub-id pub-id-type="pmid">28376801</pub-id></citation></ref>
<ref id="B28"><label>28.</label><citation citation-type="book"><collab>OECD</collab>. <source>Consultations with Doctors, Health at a Glance: Asia/Pacific 2022 Measuring Progress Towards Universal Health Coverage</source>. <publisher-loc>Paris</publisher-loc>: <publisher-name>OECD</publisher-name> (<year>2022</year>). p. <fpage>106</fpage>&#x2013;<lpage>7</lpage>.</citation></ref>
<ref id="B29"><label>29.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Bhaskaran</surname><given-names>V</given-names></name></person-group>. <article-title>Designing for trust: the crucial role in digital user experience</article-title>. <source>J User Experience</source>. (<year>2024</year>) <volume>19</volume>:<fpage>53</fpage>&#x2013;<lpage>9</lpage>.</citation></ref>
<ref id="B30"><label>30.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Atkin</surname><given-names>C</given-names></name><name><surname>Crosby</surname><given-names>B</given-names></name><name><surname>Dunn</surname><given-names>K</given-names></name><name><surname>Price</surname><given-names>G</given-names></name><name><surname>Marston</surname><given-names>E</given-names></name><name><surname>Crawford</surname><given-names>C</given-names></name><etal/></person-group> <article-title>Perceptions of anonymised data use and awareness of the NHS data opt-out amongst patients, carers and healthcare staff</article-title>. <source>Res Involvement Engagement</source>. (<year>2021</year>) <volume>7</volume>:<fpage>40</fpage>. <pub-id pub-id-type="doi">10.1186/s40900-021-00281-2</pub-id></citation></ref>
<ref id="B31"><label>31.</label><citation citation-type="other"><collab>Ministry of Health, Labour and Welfare</collab>. <article-title>Mechanisms for obtaining consent</article-title> (<year>2023</year>). <comment>Available online at:</comment> <ext-link ext-link-type="uri" xlink:href="https://www.mhlw.go.jp/content/10808000/001045573.pdf">https://www.mhlw.go.jp/content/10808000/001045573.pdf</ext-link> <comment>(Accessed March 20, 2024)</comment>.</citation></ref>
<ref id="B32"><label>32.</label><citation citation-type="other"><person-group person-group-type="author"><name><surname>Hata</surname><given-names>M</given-names></name></person-group>. <article-title>Japanese hospitals increasingly at risk of cyberattacks. The Japan News</article-title> (<year>2021</year>). <comment>Available online at:</comment> <ext-link ext-link-type="uri" xlink:href="https://japannews.yomiuri.co.jp/society/crime-courts/20210926-33177/">https://japannews.yomiuri.co.jp/society/crime-courts/20210926-33177/</ext-link> <comment>(Accessed July 20, 2024)</comment>.</citation></ref>
<ref id="B33"><label>33.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Zhu</surname><given-names>P</given-names></name><name><surname>Shen</surname><given-names>J</given-names></name><name><surname>Xu</surname><given-names>M</given-names></name></person-group>. <article-title>Patients&#x2019; willingness to share information in online patient communities: questionnaire study</article-title>. <source>J Med Internet Res</source>. (<year>2020</year>) <volume>22</volume>:<fpage>1</fpage>&#x2013;<lpage>13</lpage>. <pub-id pub-id-type="doi">10.2196/16546</pub-id></citation></ref>
<ref id="B34"><label>34.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Kuo</surname><given-names>KM</given-names></name><name><surname>Liu</surname><given-names>CF</given-names></name><name><surname>Talley</surname><given-names>PC</given-names></name><name><surname>Pan</surname><given-names>SY</given-names></name></person-group>. <article-title>Strategic improvement for quality and satisfaction of hospital information systems</article-title>. <source>J Healthc Eng</source>. (<year>2018</year>) <volume>2018</volume>:<fpage>1</fpage>&#x2013;<lpage>14</lpage>. <pub-id pub-id-type="doi">10.1155/2018/3689618</pub-id></citation></ref>
<ref id="B35"><label>35.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Juhn</surname><given-names>E</given-names></name><name><surname>Galvez</surname><given-names>E</given-names></name></person-group>. <article-title>Incentivizing data sharing among health plans, hospitals, and providers to improve quality</article-title>. <source>Am J Manag Care</source>. (<year>2022</year>) <volume>28</volume>:<fpage>e426</fpage>&#x2013;<lpage>7</lpage>. <pub-id pub-id-type="doi">10.37765/ajmc.2022.89277</pub-id><pub-id pub-id-type="pmid">36525661</pub-id></citation></ref>
<ref id="B36"><label>36.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Tong</surname><given-names>V</given-names></name><name><surname>Krass</surname><given-names>I</given-names></name><name><surname>Robson</surname><given-names>S</given-names></name><name><surname>Aslani</surname><given-names>P</given-names></name></person-group>. <article-title>Opt-in or opt-out health-care communication? A cross-sectional study</article-title>. <source>Health Expect</source>. (<year>2021</year>) <volume>24</volume>:<fpage>731</fpage>&#x2013;<lpage>43</lpage>. <pub-id pub-id-type="doi">10.1111/hex.13230</pub-id><pub-id pub-id-type="pmid">33729648</pub-id></citation></ref>
<ref id="B37"><label>37.</label><citation citation-type="journal"><person-group person-group-type="author"><name><surname>Lutomski</surname><given-names>JE</given-names></name><name><surname>Manders</surname><given-names>P</given-names></name></person-group>. <article-title>From opt-out to opt-in consent for secondary use of medical data and residual biomaterial: an evaluation using the RE-AIM framework</article-title>. <source>PLoS One</source>. (<year>2024</year>) <volume>19</volume>:<fpage>e0299430</fpage>. <pub-id pub-id-type="doi">10.1371/journal.pone.0299430</pub-id><pub-id pub-id-type="pmid">38547214</pub-id></citation></ref></ref-list>
</back>
</article>