We first describe our experimental design. Further details regarding implementation, timing, and exclusion criteria are relegated to Appendix A1.

Each subject made choices in three experimental tasks: individual lotteries (LO), Stag Hunt games (SH), and Hawk-Dove games (HD). All three tasks have a Safe option S and a two-state Risky option, R_H and R_L, so that R_L < S < R_H. The inherent nature of risk in each task differs. LO is an individual choice problem, where the (objective) probability of earning R_H, p ≡ Pr(R_H), is known before the choice is made. SH and HD are two-person, simultaneous, non-cooperative games, where the probability of earning R_H depends on the choice of another subject in the room. In SH, the probability of earning R_H is increasing in the level of risk chosen by the other subject (a coordination game where risk-taking is a strategic complement), whereas in SH it is decreasing in the level of risk chosen by the other subject (an anti-coordination game where risk-taking is a strategic substitute). The basic structure of the tasks is summarized in Table 1⁴.

To implement these three tasks, we construct the following novel design. In each round, subjects are given 100 tokens, that they must allocate between the Safe and Risky options (neutrally labeled “Option A” and “Option B” in the experiment). The computer then randomly selects a ball from an urn with 100 green and orange balls (see below). For any token allocation x (∈{0, …, 100}) to Safe and 100 − x to Risky, the payoff obtained by the subject is:

In words, each token allocated to Safe yields S100 whereas each token allocated to Risky yields either RH100 or RL100. As x decreases, the spread between the payoff if the computer draws a green and an orange ball increases. If the subject sets x = 100, she obtains S for sure. If the subjects sets x = 0, she obtains either R_H (green ball) or R_L (orange ball).

As described, for each token allocated to Risky, the probability of earning payoffs RH100 and RL100 are simply the proportion of green balls and orange balls in the computer's urn, respectively. The only difference between our three tasks LO, SH, and HD is the way in which the number of green and orange balls is determined:

In addition, in SH and HD subjects are told that their choice affects the number of green and orange balls in the urn of the participant with whom they are matched in the exact same way. That is, in SH (HD) the more tokens a subject allocates to Risky, the more (less) likely it is that the other participant earns R_H.

Figure 1 provides screenshots of the LO (top), HD (bottom left) and SH (bottom right) tasks. At the top of the screen, the subject is told the current task (neutrally labeled as “Method 1,” “Method 2,” and “Method 3,” respectively). She is also reminded how the number of green and orange balls in her urn is determined. At the center of the screen, the subject can observe the parameters of the current round. In these three tasks, S = $21, R_H = $53 and R_L = $13. At the bottom of the screen, there is a slider that the subject can use to allocate her 100 tokens across Safe and Risky. As the subject moves the slider to test different token allocations, the earnings for each ball color are calculated and presented in real-time on the screen. In all three screenshots, the subject has set x = 29. After the subject is satisfied with the allocation of tokens, she has to click the “confirm” button to submit her choice.

Our experiment has two methodological contributions that we would like to emphasize. First, the contextual presentation of the three tasks is almost identical. Only the information concerning the determination of green and orange balls is changed. Capturing the inherently different natures of risk in such a symmetric way serves an important purpose: different behavior is likely to be only in response to the meaningful differences between these tasks, rather than to superficial differences in presentation or comprehension. Second, endowing subjects with 100 tokens that can be allocated across Safe and Risky can be used to measure “interior” behavior. In lotteries, it is analogous to portfolio diversification. In games, it is analogous to allowing subjects to play mixed strategies. In both cases, it provides more information than the standard binary choice method.

Subjects played a total of 48 rounds, 16 rounds of each task all with different payoffs. The experiment was broken up into blocks of 4 consecutive rounds of the same task, and all sessions started with a LO block, which was arguably simpler. Before each block, subjects were shown a screen reminding them that a new block was starting. This screen ensured that subjects would be aware of which task (LO, SH, or HD) they were playing next. For the games, subjects were randomly and anonymously rematched after each round. For the lotteries, they were playing an individual decision problem (the exact experimental instructions are in Appendix B). To avoid learning effects, subjects did not see the behavior of their partner nor the color of the ball drawn by the computer in each round. At the end of the 48 rounds, subjects observed all their choices and those of their partners. One round was randomly drawn by the computer and the outcome in that round was used for payment. Subjects earned an average of $31, with a minimum of $1 (twice) and a maximum of $53 (three subjects). In addition to these earning, all subjects were paid a $5 show-up fee.

We chose the payoffs in order to provide substantial variation in monetary stakes and equilibrium predictions. First, define:

as a measure of the monetary stakes. For all tasks, we set Δ ∈ {10, 20, 30, 40}. In the analysis, we will refer to “low stakes” as Δ ∈ {10, 20} and “high stakes” as Δ ∈ {30, 40}. Second, given a triplet (R_L, S, R_H), the mixed-strategy Nash equilibrium of the SH game is:

where α is the probability of choosing Risky. For each Δ, we choose (R_L, S, R_H) so that α ∈ {0.2, 0.4, 0.6, 0.8}. This gives 16 combinations of stakes and mixed equilibrium predictions in SH. Finally, notice that once we fix Δ, then α is proportional to S the payoff of the Safe option.

Notice that for a given triplet (R_L, S, R_H), the mixed-strategy Nash equilibrium of HD is:

where 1−α is the probability of choosing Risky. Therefore, the same payoff-triplets as in SH provide also 16 combination of stakes (Δ ∈ {10, 20, 30, 40}) and mixed-strategy equilibria (1 − α ∈ {0.8, 0.6, 0.4, 0.2}) in HD. Last, we use the technique developed by Jessie and Kendall (2015) to select the payoffs in a way that the differences between games are only in the component that the Nash Equilibrium uses to make predictions. Table 2 provides a sample of eight games used in the experiment and Appendix A2 provides the entire list.

Finally, to create the LO tasks, we choose the payoffs (R_L, S, R_H) of the SH and HD games corresponding to the extreme mixed-strategy Nash equilibria of the games: α = 0.2 and α = 0.8. Using these payoffs, we set the lottery probability of the high payoff R_H to p = 0.2 and p = 0.8. Creating four lotteries in this way for Δ ∈ {10, 20, 30, 40} yields a total of 16 LO tasks. Table 3 provides some examples of lotteries.

Our model has three parameters (Δ, α, p) in the LO tasks and two parameters (Δ, α) in the SH and HD tasks.

Predictions in LO are standard. Fixing the other two parameters, Risky becomes more attractive as p increases (first-order stochastic increase in the risky option) and α decreases (S closer to R_L). The effect of Δ is less clear. For example, increasing Δ makes Risky more desirable when p = 0.8 and α = 0.2 and less desirable when p = 0.2 and α = 0.8.

Predictions in SH and HD are more subtle. By construction, in all 32 rounds there are two pure-strategy and one mixed-strategy equilibria. Subjects may move from one equilibrium to another, so behavior depends crucially on beliefs about the other player's action and comparative statics should be taken with a grain of salt. However, fixing the belief about the other player's constant, it seems intuitive that Risky is more attractive in both SH and HD as the sure payoff S becomes closer to R_L, that is, as α decreases. Again, the effect of changes in the spread of payoffs Δ is more nuanced and depends on the position of S.

Finally, there are also interesting differences between SH and HD. SH is a coordination game, where risk-taking behavior is a strategic complement. This means that, holding constant the belief about the opponent, a decrease in α offers the subject more incentives to take risks. Furthermore, the subject realizes that the opponent also has more incentives to take risks, reinforcing the value of playing Risky. By contrast, HD is an anti-coordination game where risk-taking behavior is a strategic substitute. As α decreases, the subject has more incentives to choose Risky but realizes that the opponent has the same incentives, which decreases the value of risk-taking. Overall, strategic considerations make comparative statics significantly easier to evaluate when incentives of players are aligned (SH) than when they are not (HD).

Figure 2 shows the evolution of cortisol levels throughout the experimental sessions in both treatments. Each dot represents the average level of salivary cortisol samples (ng/mL) taken at baseline, peak, and end of the experiment. We report minutes on the x-axis. Note that the timing of the end sample was different across sessions and we represent the average number of minutes in each treatment. The control and stress groups start with statistically indifferent levels of average cortisol (2.42 vs. 2.75; two-sided Welch t-test, p-value = 0.133). The stress group experiences a large and statistically significant increase in average cortisol (2.75 vs. 5.16; p-value < 0.001). In comparison, the control group experiences a slight and statistically significant decrease in average cortisol (2.42 vs. 2.03; p-value = 0.022). Higher cortisol levels are also observed in the stress group in the end sample (1.81 vs. 3.14; p-value < 0.001).

The average proportion of wealth invested in Safe is 0.63 in LO, 0.53 in SH and 0.65 in HD. Results between lotteries and games are not directly comparable. By contrast, results between the two games are comparable since the 16 rounds of SH involve the same payoff triplets (R_L, S, R_H) as the 16 rounds of HD. We notice a significantly lower allocation to Safe in SH than in HD < 0.001).

We noted a slight increase in the average proportion allocated to Safe in the stress treatment in all tasks compared to the control treatment (0.64 vs. 0.63 in LO, 0.55 vs. 0.52 in SH, and 0.65 vs. 0.65 in HD). However, the differences were not statistically significant. As presented in Figure 3, the cumulative distribution functions of the average amounts allocated to Safe were also similar across treatments in all three tasks, with no statistically significant effect according the Kolmogorov-Smirnoff test (p-value = 0.31 in LO, p-value = 0.31 in SH, and p-value = 0.97 in HD). Overall, we found no evidence that stress affected behavior within each task.

The existing literature is ambiguous on this issue. Some studies have found that stress affects behavior in lotteries (Preston et al., 2007; Lighthall et al., 2009; Van Den Bos et al., 2009) whereas others found no effect of stress (von Dawans et al., 2012; Gathmann et al., 2014). Differences in responses to stress may be attributed to differences across studies in risk elicitation methods (BART, IGT, objective lotteries) and experimental procedures (presence/absence of incentives, hypothetical/real choices, different stressors). For instance, it may be that the emotional component contained in the BART experiment (anticipation of the balloon explosion and visual representation of such explosion) is responsible for shifts in behavior. Moreover, in BART and IGT subjects are typically not informed of the objective probabilities of the events. This ambiguity component may also trigger different thought processes that are differentially affected by stress (Buckert et al., 2014; Danese et al., 2017).

In Table 5 we present the differences in allocation across gender. In the control condition, females allocate significantly more to Safe than males in LO and SH but not in HD. In the stress condition we find no significant gender differences in any task.

Our data contribute to gender research in three ways. First, the fact that women take less risk in LO in the control group aligns with earlier literature (Charness and Gneezy, 2012). Second, finding males in the control group to be more cooperative in SH contributes to our understanding of gender differences in coordination games. However, we are hesitant to extrapolate about general inclinations to cooperate since, as suggested by Croson and Gneezy (2009), gender differences seem to be highly sensitive to context. Finally, since the only significant gender differences are found in the control group, we conclude that stress has the capability to diminish differences between genders.

Our next question is whether the willingness of individuals to choose Risky is correlated across tasks. On the one hand, it seems natural that subjects who are less risk-averse, that is, those who invest more in Risky in LO (individual lotteries with objective probabilities) are also expected to take more risks in games. On the other hand, this may not be necessarily true since our games have multiple equilibria, so risk-taking in SH and HD depends crucially on beliefs about the other player's behavior. Furthermore, the two games are fundamentally opposite in the optimal reaction to the other player's choice (coordination vs. anti-coordination). Table 6 presents the Pearson correlation coefficient (ρ) of the proportion allocated to Safe by individuals across tasks, both in the control (left panel) and stress (right panel) conditions.

In the control condition, the amount allocated to Safe in LO is significantly correlated with the amount allocated to Safe in SH, suggesting that risk attitude is a reasonably good predictor of behavior in the coordination game. This finding aligns with previous studies showing a correlation between LO and SH choices (Heinemann et al., 2009; Chierchia and Coricelli, 2015). By contrast, the control condition shows no significant correlation between LO and HD or between SH and HD. This may not be surprising given the previous research showing that these tasks activate different areas of the brain (Ekins et al., 2013; Nagel et al., 2014).

By contrast, in the stress condition, the amounts allocated to Safe are significantly correlated across all tasks. Correlations are also stronger, suggesting that risk-taking under stress is very similar across tasks, irrespective of the situation. This important result indicates that, even though stress did not have an effect on the overall distribution of risk taking in the population across tasks, it did affect intra-personal decisions. The result was confirmed by a set of robust regressions reported in Table 7, which suggests a stronger relationship between the amount allocated to Safe in LO, SH and HD under stress than in the control treatment. This effect will be corroborated with the trial-by-trial regression analysis.

We then compared the correlation coefficients across conditions by assessing statistical significance of the Fisher's r to z transformations. We found that the correlation between LO and SH are not significantly different between control and stress conditions. By contrast, correlations between LO and HD and between SH and HD respectively are significantly different (with respective p-values of 0.040 and 0.012). This result further supports the finding that subjects under stress make choices that are more similar across tasks than subjects in the control treatment.

A possible explanation for this result is that subjects under stress (and only those subjects) exhibit contextual blindness, that is, they ignore the context that distinguishes these three tasks. Indeed, LO measures an individual's propensity to take risks which has no social context. SH captures a tension between risk and cooperation whereas HD captures a tension between risk and aggression. The experiment was designed so that these contexts were the only difference between tasks. Table 6 reveals that the behavior of stressed subject when faced with an objective probability over earnings was strongly and positively correlated with their behavior when faced with a strategic opponent, even if games were opposite in nature. For control subjects there was only a relationship between LO and SH. In other words, control subjects responded more to the differing contexts than stressed subjects. One implication is that the choices of subjects under stress are generally more predictable: knowing the average amount a subject invests into Safe in any one task provides significant information about behavior in the other two.

We also ran OLS regressions of the trial-by-trial amounts allocated to Safe for each game and in each condition. We used as regressors the individual average amount allocated to Safe in LO (which captures the risk attitude of each individual), and dummies for stakes (1 = High stakes), for the position of S relative to R_L and R_H (α), and for gender (1 = Male). We constructed a fixed effect model by including a dummy variable for each individual. The results are compiled in Table 8.

In the Control condition, the average allocation to Safe in SH is predicted by the behavior in LO, but the average allocation in HD is not. In the Stress condition, the average amounts allocated to Safe in both SH and HD are highly predicted by behavior in LO. These regressions further confirm the contextual blindness result⁷. We also notice that gender has no explanatory power and that the allocation to the safe choice is increased for high stakes, but only in the Stress condition.

To better assess the significance of the effect of stress in HD, we ran a regression of the trial-by-trial amounts allocated to Safe in HD in both conditions on the same regressors as before as well as the individual difference in cortisol between baseline and peak (ΔCortisol) and an interaction term between that measure and the average allocation to Safe in LO⁸. For comparison, we ran the same regression for SH as well. This exercise tests directly whether the coefficients of the average allocation to Safe in LO in the previous table are significantly different across treatments. The results are reported in the first two columns of Table 9. The absence of a significant interaction in the case of SH confirms that the amount allocated to Safe in LO does not predict differentially behavior in SH across conditions. By contrast, the interaction term is significant in the case of HD, the contribution of the amount allocated to Safe in LO to behavior in HD differs across conditions. We finally ran a full regression over both games using a dummy variable for our games (1 = SH). The results are reported in the last column of Table 9. The fact that the three way interaction between the average allocation to Safe in LO, the treatment and the increase in cortisol is significant indicates that the interaction between Safe in LO and Stress is significantly different across games. The regression also shows a subtle interaction between cortisol increase and games: subjects who exhibit a higher increase in cortisol level tend to increase more their investment to Safe in HD.

The fact that stress does not have any visible effect on aggregate behavior (Section 4.1) but reduces gender differences (Section 4.2) and impacts the relationship between tasks (Section 4.3) is puzzling. We therefore decided to study in more detail the behavior of individuals across the three tasks.

We conducted a cluster analysis in each condition to group subjects according to their average allocation to Safe in each task. We retained a model-based clustering method to identify the clusters present in our population. A wide array of heuristic clustering methods are commonly used but they typically require the number of clusters and the clustering criterion to be set ex-ante rather than endogenously optimized. Mixture models, on the other hand, treat each cluster as a component probability distribution. Thus, the choice between numbers of clusters and models can be made using Bayesian statistical methods (Fraley and Raftery, 2002). We implemented our model-based clustering analysis with the Mclust package in R (Fraley and Raftery, 2006). We considered ten different models with a maximum of nine clusters each, and retained the cluster combination that yielded the minimum Bayesian Information Criterion (BIC). In the Control condition, the best model consisted of three clusters (C1, C2, and C3). In the Stress condition, four different clusters best summarized behavior (S1, S2, S3, and S4). Table 10 summarizes the descriptive statistics in each cluster. Figure 4 provides a visual representation of the clusters across treatments⁹.

In the Control condition, the majority of the subjects (C1) exhibited the typical behavior: they invested similar proportions in the Safe asset in LO and HD and less in SH, suggesting large homogeneity across subjects in this treatment. A few individuals (C2) were an extreme version of this typical play, with overly risky behavior in SH. Finally, a minority of all female subjects (C3) allocated significantly more to Safe in LO and HD, but especially in SH. This group was responsible for the gender effect detected in LO and SH in the control condition.

In the Stress condition, there were three main clusters (S4 consists of 3 outliers), similar to the clusters obtained in the control condition. Cluster S1 was the analog of C1, while S2 was similar to C2, except for a safer proportion of choices in SH. However, half of the subjects were now grouped in S3, a cluster similar to C3. These subjects allocated a large fraction of their endowment to Safe in all tasks. S3 had also the particularity that allocations were extremely similar across tasks (69.1–74.7% with low standard errors). These subjects were responsible for strengthening the relationship between tasks. Moreover, there was no gender supremacy in that cluster, causing the gender effect observed in the control condition to disappear under stress.

Result 2. Aggregate behavior is similar across treatments whereas individual choices are affected by stress. A significant fraction of participants in the stress condition are subject to contextual blindness, choosing a similar allocation independently of the task.

In Table 11 we report the average reaction time (RT) in seconds separated by task and treatment.

Making choices took more time under stress across all tasks, although the effect was mostly due to HD. We also found that RT were longer in HD compared to SH irrespective of the treatment (p < 0.001), consistent with the idea that the anti-coordination game is more complex to evaluate than the coordination game.

As reflected in Table 12, risky options with expected value below the safe alternative (EV < S) were quickly discarded. Subjects took significantly more time to choose when the expected value of the risky option was equal (EV = S) or greater (EV>S) than the safe option (t-test, p-value < 0.01 for all paired comparisons in Control and Stress treatments). For the more complex lotteries (EV>S), subjects took slightly more time under stress, although not significantly so.

Table 13 presents the reaction times in SH and HD as a function of the parameters of the games, α and Δ.

In SH, we found that RT were shorter for higher α: shortest at α = 0.8 and longest at α = 0.4 in both conditions (t-tests of difference, p < 0.01 in both conditions). We also found that RT were longer in high stakes than in low stakes rounds (t-test of difference, p < 0.001 in Control and p = 0.012 in Stress). The trend was identical in HD, with shortest RT at α = 0.8 and longest at α = 0.4 in the control group and α = 0.2 in the stress group (t-tests of difference, p < 0.001 in both conditions). RT were also longest in high stakes trials (t-test of difference, p < 0.001 in both groups). It is unclear why α significantly affects reaction times in the games. In both SH and HD, increasing α makes the safe option relatively more valuable. It is plausible that Safe becomes easier to evaluate as it becomes more attractive, resulting in a quicker response. As for stakes, we conjecture that subjects find the decision to be more important (hence, more worthy of attention) when, other things being equal, the set of payoffs is more spread out. In any case, the consistency of the reaction time comparative statics across games and conditions is remarkable and deserves further investigation. Finally, in SH there was no effect of stress. In HD, there was an increase in RT under stress only when α = 0.2 (p = 0.030) and when stakes were high (p = 0.015), suggesting an interaction between game complexity and difficulty to evaluate alternatives. It is also consistent with studies showing that stress affects working memory and executive decision-making. High levels of cortisol have been associated with more errors in card sorting tasks meant to measure executive functioning (McCormick et al., 2007) as well as O-span and backwards digit-span tasks meant to measure working memory (Schoofs et al., 2009). While our finding reflects the intuition behind results showing stressed subjects performing worse on more complicated tasks (Schoofs et al., 2009), our contribution shows that more complicated decisions also take longer (in our setting, there are no right or wrong decisions). This finding illustrates an important difference between how stressed subjects reach decisions in strategic games vs. in working memory or executive functioning tasks.

We then conducted a mixed effect OLS regression to better analyze the contribution of each effect to reaction times in both games. For both SH and HD, we regressed reaction times on a Treatment dummy (1 = Stress), a Gender dummy (1 = Male), a Stakes dummy (1 = High stakes), and dummies identifying the level of α in each round. The results are reported in Table 14. They confirm the effect of high stakes and α levels reported above. Stress and gender did not have significant effects.

Result 3. Reaction times are higher in the conceptually more difficult game HD, in the more complex rounds of LO, when stakes are high and when the safe option is intrinsically less attractive in SH and HD. Stress (weakly) increases reaction times in those cases.

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.