Güth et al. first published the ultimatum game in 1982. In this game, the proposer can split a certain amount between themselves and a responder. After the responder receives the information about the payoffs that the proposer and responder will receive, they decide to reject or accept the proposal (Güth et al., 1982; Güth and Kocher, 2014). If the responder accepts the proposer's offer, both the proposer and the responder receive the amount proposed by the proposer. If the responder rejects the offer, both the proposer and responder receive nothing. The proposer's offer can be a decision influenced by fairness or by fear of an unfair offer being rejected; the proposers' average offer is between 40 and 50% (Güth and Kocher, 2014).

Many researchers have studied ultimatum game experiments in the gains domain. For example, time pressure (Sutter et al., 2003), gender differences (García-Gallego et al., 2012), and stake size effects (Andersen et al., 2011) have been subject to prior research in ultimatum game experiments (Güth and Kocher, 2014).

Previous research on the distribution of losses in the loss domain is scarce. For example, Buchan et al. (2005) showed that proposers' offers and responders' demands in ultimatum games are higher over losses than over gains. Further investigating those questions in three different countries (the United States, China, and Japan), they found consistent results. Meanwhile, Lusk and Hudson (2010) showed that proposers demand more for themselves and therefore make more aggressive offers when bargaining over losses than over gains. Baquero et al. (2013) found a significant generosity effect for bargaining in the loss domain compared to the gains domain. In a recent study, Neumann et al. (2017) found that proposers make higher offers in the loss domain than in the gains domain. However, one finding that comes to light when studying the literature is that the previous research is inconclusive, whereby various approaches in the experimental design have been implemented. For example, Neumann et al. (2017) used the strategic vector method (Rauhut and Winter, 2010). Furthermore, recent research has shown that the question of how to distribute losses compared to gains can also be investigated with various experimental games, such as the trust game with negative endowments (Füllbrunn and Vyrastekova, 2023). In Füllbrunn and Vyrastekova (2023), one main finding was that trust is higher in the loss context than in the gains context in the way that the loss context leads to an incentive toward trust. The results underline the difference between sharing losses and sharing gains. Nontheless, this research focuses on different aspects.

From the previous literature, we derive the first hypothesis as follows:

(H1) In ultimatum games over gains and losses, the demands in the loss domain differ from the demands in the gains domain.

Norm following is not a new concept, neither generally to describe human behavior in decision-making situations, nor especially in sociology. To structure the concept of norm following, Bicchieri (2006, 2017) defined social norms in her norm theory as the preference to follow and expect that others will also follow.

The literature distinguishes between descriptive and injunctive norms (Cialdini et al., 1991). Descriptive norms involve the perception of what a common pattern of behavior is in a given situation (Cialdini et al., 1991), whereas injunctive norms involve behavior that is normatively expected by others (Cialdini et al., 1991).

In ultimatum games, according to Güth and Kocher (2014), “proposers are either aware of the responder's willingness to reject unfair offers or guided by own fairness concerns when offering sizable or even fair shares for Y” (p. 398). The structure of the game induces the proposer's expectation that the responder demands a fair split. The proposer has either internalized a norm of fair giving (intrinsic fairness based on morality) or gives a fair split in fear of punishment by the responder (strategic fairness).

Empirical findings from ultimatum game experiments have shown that the equal split is a common behavioral type in these experiments (Güth and Kocher, 2014). The rejection rate increases when the proposers' offer is below 20% (Güth and Kocher, 2014).

The standard game theoretical prediction for the ultimatum game is the subgame perfect equilibrium (Selten, 1965, 1975), where the proposer offers the minimum possible amount and the responder accepts it. However, as Binmore (2010) argued, there is a continuum of Nash equilibria for the ultimatum game, and there is no reasonable argument that only the subgame perfect equilibrium can be used to empirically predict how subjects will play. Binmore (2010) further stated that subjects face an equilibrium selection problem regarding which of the many Nash equilibria they should coordinate. Social norms (e.g., dividing fairly) serve as one attempt to reach a solution. Often, empirically, it can be seen that subjects behave according to the egalitarian norm so that the equal split is chosen (Güth and Kocher, 2014).

According to Chen et al. (2017) and Windrich et al. (2022), we intervened in this equilibrium selection problem by focusing participants on other possible solutions. This means that we asked subjects (i.e., proposers) to freely indicate their offer. However, in our first treatment, we focused the subjects on the average offer of other proposers (in prior experiments) by telling them what the average offer was. This focus should coordinate players on the Nash equilibrium nearest to the focal point of the average offer. In addition, this focus on the average offer induces a descriptive norm of what is usually done in the game. We know from the experimental literature that empirical expectations of what others do are a better predictor for behavior than normative expectations of what should be done when the social norm demands an egalitarian offer (cf. Bicchieri and Xiao, 2009). Hence, we expect proposers to orientate their offers around the given focus on the average offer. Our corresponding second hypothesis reads as follows:

(H2) Focusing on “Average behavior” leads to a concentration of demands around the focal point of the previous average proposal.¹

In a second treatment, we focused proposers on the subgame perfect equilibrium of offering the smallest possible amount. We knew from the empirical literature on ultimatum games (Güth and Kocher, 2014) that this Nash equilibrium seldom holds as a good prediction for behavior. By focusing participants on it, we made it more focal for the proposer and more acceptable for the responder. Therefore, our third hypothesis reads as follows:

(H3) Focusing on “Self-interested behavior” leads to a dispersion of the standard deviation of proposers' demands and a decrease in fairness.

Part of the ultimatum game analysis is the responders' behavior toward the proposed amount. In the ultimatum game, the responder has two options, namely rejecting the offer or accepting the offer. Previous literature on the analysis of responder behavior has shown that offer acceptance decreases with increasing proposer demand under an average offer of 40–50%, with the responders' acceptance approaching zero for offers below 20% (Güth and Kocher, 2014). As described in Section 2.2., a proposer's offer is driven by their own fairness considerations or by the awareness of the responder's willingness to reject the offer. The ultimatum game over losses leads to a costly rejection for the responder and the proposer. In the ultimatum game over gains, if the responder rejects the proposer's offer, both receive nothing. However, in the ultimatum game over losses, the responder and the proposer have to pay an amount (−10 EUR) if the responder rejects the offer. Neumann et al. (2017) found that responders are more frequently willing to accept higher proposer demands in the loss domain than in the gains domain (Neumann et al., 2017). Nevertheless, the ultimatum game design differs from our design as it uses the strategy vector method (Rauhut and Winter, 2010). In an early study, Camerer et al. (1993) found that the proposers' average offer does not differ between the gains and loss domains but that rejections are more frequent in the loss domain compared to the gains domain (Camerer et al., 1993). In a more recent study, the responders' behavior was analyzed by implementing a design that asymmetrically endowed the responder in the loss domain while the proposer is in the gains domain. With the strategic vector method (Rauhut and Winter, 2010), the modified ultimatum game, particularly the responders' behavior, was analyzed (Neumann et al., 2018b). Neumann et al. (2018b) found that the responder's demand is guided strongly by their desire to break even. The previous literature has implied that responders' behavior can differ between the gains and loss domains. Nevertheless, the fact that rejection by the responder is costly has two implications. First, the rejection frequency can differ between the gains and loss domains. Second, the proposer may anticipate and realize that the rejection is costly for the responder and that this could impact their offer. We herein analyze the responders' rejection frequency and compare it, in particular, between the gains and loss domains.

We decided to conduct our study using incentivized rather than hypothetical experiments. Previous literature has provided various approaches to implement losses in the laboratory in this regard. According to our reference treatment—reported in Neumann et al. (2018a), we used the prepaid mechanism with real monetary payoffs to induce losses.³ As a consequence, the sessions in the loss domain were divided into two subsessions. We invited the participants in the loss treatments to the laboratory 2 weeks before playing the ultimatum game experiments (Rosenboim and Shavit, 2012). In the first part of the sessions, the participants received 15 EUR to compensate for their potential losses in the ultimatum game experiments. In the second part, the participants played the ultimatum game and had to pay back their losses. The prepaid mechanism has two main advantages: (1) Due to mental accounting, participants feel that they are experiencing a real loss,⁴ and (2) this overcomes the “house-money-effect”⁵ (Thaler and Johnson, 1990) because participants feel they own the money they receive. It is important that enough time has passed between the two parts of the sessions. We kept it constant at 2 weeks in advance for all sessions, which was considered sufficient time. To our knowledge, this is the best way to implement monetary losses in the laboratory, corresponding to Rosenboim and Shavit (2012).

We are interested in (1) the potential differences between behavior in the gains and loss domains and (2) the robustness of the egalitarian norm. Regarding (1), we implemented all treatments in both domains (i.e., gains and losses, respectively). With regard to (2), we studied the behavior in three different treatments: the control treatment, the average behavior treatment, and the self-interested behavior treatment.

We conducted our incentivized ultimatum game experiments in two laboratories: 236 participants in the Magdeburg Experimental Laboratory of Economic Research (MaXLab) and 314 participants in the Leipzig Experimental Laboratory for Social Science (LEx). Experiments were implemented using z-Tree (Fischbacher, 2007), and participants were recruited using hroot (Bock et al., 2014) from the subject pools of MaXLab and LEx. Both subject pools have similar structures in terms of subjects' characteristics. The majority of subjects in both pools were university-enrolled students from different faculties.

In total, subjects (N = 550) participated in the ultimatum game experiments, with 312 female and 238 male participants. We recruited participants who were mostly students (87%) with a predominance toward human science and social science. The age of participants varies from 18 to 82 (mean= 26.9, SD = 8.1).

Based on the payoff structure in Neumann et al. (2018a), Table 1 shows the payoffs of the ultimatum game experiments in the gains and loss domains.

In Section 4, we describe and analyze the results of our experimental investigations.

In the “Average behavior” treatments, the proposers are less generous in the loss domain than in the gains domain. In T3, Average behavior—Gains, the proposers' average demand is 51.07 (SD = 6.13) and in T4, Average behavior—Loss, it is 55.35 (SD = 7.95) (see Table 2). At first glance, the difference is not surprising because, even in the “Control” treatments, proposers behave less generously in the loss domain than in the gains domain.

However, only in this norm focus treatment is the difference significant (Wilcoxon rank-sum test, z = −2.8103, p = 0.0050), which is a moderate effect with a Cohen's d = −0.59. As shown in the reference treatment in Neumann et al. (2018a), there is only a small but non-significant difference in the proposers' demand between the gains and loss domains in the “Control” treatments.

One explanation for this significant difference in the “Average behavior” treatments is the low variance, which is significantly lower according to the F-test [F_{(41, 49)} = 0.4910, p = 0.0211] in T3, “Average behavior—Gains”, compared to T1, “Control—Gains”, and according to the F-Test [F_{(42, 54)} = 0.5225, p = 0.0308] between T2,” Control—Losses”, and T4, “Average behavior—Losses”. We expected that the proposers' average demand would be centered around the previous proposers' average demand shown in the instruction, resulting in a lower variance in the “Average behavior” treatment compared to the “Control” treatment. Thus, (H2) Focusing on “Average behavior” leads to a concentration of demands around the focal point of the previous average proposal is confirmed by the significantly lower variance in both the gains and losses. The effect of norm focusing on the prior proposers' average demand has two significant effects: (1) The proposers' demand centers on the prior proposers' average demand with lower variance (2) and the difference in the proposers' demand between the gains and loss domains becomes significant, with proposers being less generous in the loss domain. The hypothesis, (H1) In ultimatum games over gains and losses, the demands in the loss domain differ from the demands in the gains domain, can also be confirmed for the “Average behavior treatments.

In addition, we compare the distribution of the proposers' demand in the “Control” treatments, shown in Figure 2, with the distribution of the proposers' demand in the “Average behavior” treatments, in Figure 3. In the loss domain, there is a shift from an equal split in the “Control” treatments toward the previous proposers' average demand indicated in the participant instructions. This suggests that the “Average behavior” focus appears to have a stronger influence on behavior in the loss domain than in the gains domain.

Comparing Figures 2, 4 shows a shift toward self-interested proposer demands as the proposers' average demand in the “Self-interested behavior” treatments is higher than in the “Control” treatments in both the gains and loss domains. Nevertheless, the difference within the “Self-interested” treatments between the prospers' demands in the gains and loss domains is not significant (for the gains domain: Wilcoxon rank-sum test, z = −0.6538, p = 0.5133; for the loss domain: Wilcoxon rank-sum test, z = −1.5608, p = 0.1186). Although the variance in the “Average behavior” treatments is lower than in the “Control” treatments, the variance in the “Self-interested behavior” treatments is significantly higher in the gains domain between T5 and T1, according to the F-test [F_{(49, 44)} = 0.5549, p = 0.0459], and in the loss domain between T6 and T2 [F_{(54, 39)} = 0.29931, p = 0.00005].

Thus, we can confirm the hypothesis through the analysis of the data: (H3) Focusing on “Self-Interested behavior” leads to a dispersion of the standard deviations of proposers' demands and a decrease in fairness.

In particular, the “Self-interested behavior” treatment has a stronger effect in the loss domain than in the gains domain in terms of both the variance and the shift toward self-interested proposers' average demand. The initial situation is different; that is, in the “Control” treatments, the ultimatum game was predominantly solved by an equal split between the players, with the legitimation of the subgame perfect solution offering a large margin in which the proposer can at least somewhat adjust in the direction of selfish behavior.

As described above, the proposers' demands in the loss domain are higher than in the gains domain in all three treatments variants. Nevertheless, only in the “Average behavior” treatments is the difference between the proposers' demand significant with a moderate effect size due to a low variance in this treatment with norm focus. We find a difference in fairness behavior in the ultimatum games over losses in the norm focus treatment for “Average behavior”. Therefore, we can accept the following: (H1) In ultimatum games over gains and losses, the demands in the loss domain differ from the demands in the gains domain for the “Average behavior” treatments.

As shown in Table 3, selfish solutions exist, and we learned from the previous literature that selfish proposals are frequently rejected, mostly if the proposals are under 20% (Güth and Kocher, 2014). In the gains domain, we found only three responders rejecting proposer demands of 50, 60, and 70%, whereas we found 15 such rejections in the loss domain. Specifically, the responders' rejections in the loss domain are distributed as follows: 60% (4 times), 65% (1 time), 70% (2 times), 80% (1 time), 90% (3 times), 95% (3 times), and 100% (1 time) of proposers' demands. We thus observed a tendency toward a more frequent proposal rejection in the loss domain than in the gains domain. The distribution of responders' rejections between the different norm foci in the gains domain is as follows: no rejections in the “Control” treatment, two in the “Average behavior” treatment, and one in the “Self-interested behavior treatment. For the loss domain, there were four rejections in the control treatment, five in the “Average behavior” treatment, and six in the “Self-interested” treatment. Consequently, the tendency toward more frequent rejections by responders in the loss domain compared to the gains domain occurs in each norm foci treatment. In Table 4, we show the proposers' demands and responders' rejection per treatment and domain.

Social norms and losses are of great importance in distributive bargaining situations on both the individual and societal levels. The current energy crisis and shortage of gas is only one example that highlights the relevance of understanding the distribution of losses among various actors. While norm theory straightforward applies to the problem of distributing of losses in social situations, the consequential predictions had been hardly tested empirically. We tackled the research gap using ultimatum game experiments in the gains and loss domains, wherein we designed ultimatum game experiments with two different norm foci, namely (1) on the previous average behavior and (2) on the self-interested solution. We used the prepaid mechanism to implement losses in the laboratory.

In our experimental study, we found that the proposers demanded more in the loss domain than in the gains domain in both norm foci treatments. However, only in the “Average behavior” treatments was the difference in the proposers' demands significant, and it had a moderate effect. One explanation for this is the small variance created by the focus on the prior average proposer demand. In line with previous literature (e.g., Lusk and Hudson, 2010), we showed that proposers demanded more for themselves in the loss domain than in the gains domain. Our results confirm that there is a difference in the distribution of losses compared to gains (Buchan et al., 2005; Lusk and Hudson, 2010; Baquero et al., 2013; Neumann et al., 2017).

In line with norm theory (Bicchieri, 2006; Bicchieri and Xiao, 2009), we hypothesized that norm focusing leads to a different distribution of proposers' demands between the different norm foci. This was confirmed by our results from the ultimatum game experiments. We found that the standard deviation in proposers' demands in the ultimatum game experiments with the focus on “Average behavior” was smaller than in the “Control” experiments in both the gains and loss domains, as the proposers' demands concentrated around the shown average offer of the “Control” treatments. In contrast, we found significantly higher standard deviations in the norm focus treatment of “Self-interested behavior” compared to the “Control” treatments. The focus on self-interest leads to higher variance because the presented solution legitimizes a higher demand from proposers.

We found a tendency toward a higher frequency of rejections in the loss domain compared to the gains domain. This finding is in line with previous research on responders' behavior (Camerer et al., 1993). For future research, we suggest analyzing subjects' behavior in that subjects also use norm focus to determine their expectations or limits of acceptability in the loss domain compared to the gains domain. With this, the responders' behavior could be analyzed in more detail (see, e.g., Neumann et al., 2018b). The results show that behavior reacts with norm focusing and with the domain of gains and losses when making an offer in the Ultimatum Game.

In our study, the losses occurred by chance for the participants; that is, they were caused by what may be perceived as bad luck and were not the result of their own efforts. Following Weiner's attribution theory (Weiner, 1985, 2005; Weiner et al., 2011), this may have had an influence on the subjects' behavior. The subjects could therefore have assumed that they (like the other subjects, i.e., the responders) were not responsible for the loss and therefore behaved more prosocially (cf. Fong, 2001; Weiner et al., 2011).

Besides these limitations, the ultimatum game experiments took place only in Germany, so it is a single country subject pool. Furthermore, it is predominantly a student-subject pool (Bader et al., 2021). Due to cultural differences, the proposition and the behavior of the responder can differ from country to country (Oosterbeek et al., 2004). The results of the ultimatum game are not fully transferable to countries other than Germany.

The different behavior in the loss domain compared to the gains domain toward more selfishness in the loss domain, due to higher proposer demands and a tendency toward a higher amount of responder rejections, can be explained by the loss aversion aspect of prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1986; i.e., “losses loom larger”), which leads to higher proposer demands. Other approaches, such as the loss attention model, can be also applied as an explanation for why losses lead to a higher sensitivity compared to gains (Yechiam and Hochman, 2013a,b). This psychological model also indicates that losses are perceived differently, and in the distribution of losses, it provides an explanation for the deviating behavior seen in the comparison.

To conclude, losses change the behavior of participants in the ultimatum game as compared to gains by making the more self-oriented. In addition, it is possible to trigger the participants behavior by focusing them on a norm. Yet, the latter effect is rather weak, and it remains unsure if it would remain stable in real situations with more costly losses. These results are in accordance with Hobbes (1996[1651]) fundamental remark on the problem. Firstly, he also states that humans tend to be selfish if they interact in situation with losses. Secondly, Hobbes advises to encounter this by combining (unstable focusing on) norms with social sanctions to enforce compliance to norms.¹⁰

To take the example of the introduction out, in particular, the scarcity of the gases and thus its rising costs will lead to more self-oriented behavior in the distribution of these costs, and (focusing on) norms, probably will not overcome the problem.

In any case, we recommend further research investigating the distribution of losses, especially investigations into responder behavior to explain their perception and behavior regarding the distribution of losses.