Underweighting of rare events in social interactions and its implications to the design of voluntary health applications

Abstract

Research on small repeated decisions from experience suggests that people often behave as if they underweight rare events and choose the options that are frequently better. In a pandemic, this tendency implies complacency and reckless behavior. Furthermore, behavioral contagion exacerbates this problem. In two pre-registered experiments (N_total = 312), we validate these predictions and highlight a potential solution. Groups of participants played a repeated game in one of two versions. In the basic version, people clearly preferred the dangerous reckless behavior that was better most of the time over the safer responsible behavior. In the augmented version, we gave participants an additional alternative abstracting the use of an application that frequently saves time but can sometimes have high costs. This alternative was stochastically dominated by the responsible choice option and was thus normatively irrelevant to the decision participants made. Nevertheless, most participants chose the new (“irrelevant”) alternative, providing the first clear demonstration of underweighting of rare events in fully described social games. We discuss public policies that can make the responsible use of health applications better most of the time, thus helping them get traction despite being voluntary. In one field demonstration of this idea amid the COVID-19 pandemic, usage rates of a contact tracing application among nursing home employees more than tripled when using the app also started saving them a little time each day, and the high usage rates sustained over at least four weeks.

1 Introduction

Public behavioral response to physical distancing guidelines can have huge effects on getting the coronavirus pandemic under control. For example, model-based simulations suggest that a mere 10% increase in compliance with guidelines can be more effective than isolating the entire senior population of a country for three months (Barak et al., 2020). It is thus worth trying to understand the determinants of compliance.

Compliance is likely influenced by many factors, including people’s perception of the potential risks and their system of beliefs (e.g., Glöckner et al., 2020). These factors are themselves moderated by political ideologies, religious beliefs, cultural norms, insufficient or biased information, and lacking or incoherent communication from authorities. In the current paper, however, we hold constant people’s perceptions of the risks and their beliefs and aim to shed light on another factor that can affect compliance: the basic tendencies that affect repeated decisions from experience.

Analysis of repeated decisions from experience suggests people are unlikely to naturally adopt responsible behaviors in a pandemic, even if it is in their best interest to do so (Erev et al., 2020). This unfortunate prediction is consistent with evidence that, absent supervisory enforcement, workers often fail to engage in safe behaviors like using protective gear, even if they perceive the potential risk to be higher than it truly is (Zohar & Erev, 2007). More generally, people often want and plan to behave responsibly, but experience leads them to let down their guard and become complacent. For example, in the late 1990s, most buyers of new vehicles in Israel chose to purchase more expansive car radios that have detachable panels, safety devices that reduce the risk of theft. However, after only two weeks, most people stopped detaching the panel, behaving as if they believe “it won’t happen to me” (Yechiam et al., 2006).

Learning to become complacent is predicted under the hypothesis that, in repeated experience-based decisions, people are far more sensitive to the common (high probability) experiences than they are to rare events (Erev & Roth, 2014).¹ In a pandemic, reckless behaviors lead to dire consequences (e.g., getting infected or unknowingly infecting others) relatively rarely; more often, such behaviors save time and/or effort compared to the more responsible behaviors. To make things worse, in a pandemic, even careful people are at risk when others around them behave recklessly. The tendency of many people to behave as if “it won’t happen to them” can then reduce the incentive for otherwise-careful people to behave responsibly: if responsible behavior is unlikely to keep one safe, it is not worth the extra effort.

We previously demonstrated this predicament using the “Reckless or Responsible” game shown in the top panel (“Without-App”) of Table 1 (Roth et al., 2020). This game abstracts a setting in which being reckless is better most of the time, whereas responsible behavior is better on average, but only if no other person behaves recklessly (below we provide a detailed explanation of the logic that underlies the choice of game features). The game has two Nash equilibria (choice profiles from which unilateral deviation is futile): An efficient equilibrium in which all players behave responsibly (and get 0 with certainty) and an inefficient equilibrium in which all players behave recklessly (and lose 0.22 on average). Importantly, beliefs concerning the pandemic and perception of the risks involved are held constant in this game because (a) information concerning the payoff distributions is accurate and fully disclosed and (b) participants are unaware of the framing of the choice options as reckless or responsible behaviors (the options are neutrally labelled). Experimental results showed near-universal convergence to the inefficient and dangerous equilibrium (Table 1, Exp. 1, Condition Without-App).

Table 1: The Reckless or Responsible game, with and without the possibility of App-Use.

Note. The notation x, p means x with probability p. For example, +2, 0.88; −19, 0.1; −60 otherwise means the distribution +2 with probability .88, −19 with probability .1, and −60 with probability .02. EV = expected value.

^a In Condition Without-App of Exp. 2, one of the two choice options (Reckless or Responsible) was duplicated such that participants had three choice options to choose from on-screen. Choice rates displayed here are the pooled choice rates of options that reflect the same payoff distribution.

To increase the chances for getting the pandemic under control, it may thus be helpful to change the game by making the common experience from reckless behavior less attractive than the common experience from responsible behavior. Doing this at scale is challenging, but the example of China is revealing. One prominent example of a policy implemented there that made the common experience from reckless behavior less appealing than the common experience from responsible behavior is the development of health signal applications that estimate users’ risk of being infectious (which increases with reckless behaviors) and translate it to a color code (green, yellow, or red). Having a green code (which implies mostly responsible behaviors) on such applications was nearly essential for daily life, as those without it could not travel or access public facilities (Weinland, 2020).

Although effective, these types of systems may be incompatible with democratic principles and are unlikely to be adopted by most western democracies. Instead, it should be possible to offer health signal applications on a voluntary basis. How can we achieve wide adoption for such voluntary applications? We suggest that because people are sensitive to the common experiences, it may suffice to make the common experience from using the application better than the common experience from not using it. For example, consider a policy that allows two ways to enter workplaces and public facilities: showing a “green code” on a health signal application or joining a queue for people without the application. Those without applications would be required to fill certain declaration forms and pass certain physical tests, and their numbers inside certain public facilities (e.g., public transportation) could even be limited. Policies of this kind can be politically justified. Access is not denied from people without the application; it is only made slightly more cumbersome, and for valid reasons: using the application simply serves the need to monitor those who enter public facilities during a pandemic.

Here, we investigate the potential of a policy designed to decrease the prevalence of reckless behaviors while preserving people’s freedom of choice, that is, to influence choice without significantly changing the incentive structure (Thaler & Sunstein, 2008). Specifically, we keep the payoff structure of both options from the original Reckless or Responsible game unchanged, but provide an additional alternative, App-Use (Table 1). The payoff scheme of App-Use reflects the idea that when users have a green signal, their common experience from using the application is better than the common experience from either of the other alternatives, as it saves time and/or effort. However, a red signal leads to a large loss (perhaps reflecting restricted access or entering quarantine), and the application does not eliminate the risk of infection when others behave recklessly. Consequently, App-Use is never the best option on average.

Specifically, the payoff structure of Responsible has second-order stochastic dominance over that of App-Use (Hadar & Russell, 1969), meaning the cumulative distribution function (CDF) of App-Use is never under the CDF of Responsible. In practice, this means that for each set of choices of other players, App-Use has a lower mean and is more uncertain (riskier) than Responsible. This implies that any risk-averse utility-maximizing agent would prefer Responsible over App-Use (Rothschild & Stiglitz, 1970). Hence, in equilibrium, App-Use should never be selected by risk averse agents. In this sense, App-Use is an irrelevant alternative and its addition should not change the prevalence of reckless behavior (or, should change it only to the degree that the population consists of many risk loving agents). Nevertheless, we hypothesized that because participants will be more sensitive to the common experiences, they will use the application frequently and the Reckless rates will be considerably diminished.

2 Reliance on small samples and alternative predictions

Before describing the experiments and the results, it is worthwhile to mention the theoretical underpinnings of our predictions, and several alternative predictions. The prediction that people will be more sensitive to the frequent experience and therefore will tend to choose App-Use on account of Reckless is rooted in the notion that people tend to rely on small samples of past experiences (Barron & Erev, 2003; Fiedler et al., 2000; Hertwig & Pleskac, 2010; Kareev et al., 1997; Plonsky et al., 2015). Because the probability that rare events will be included in small samples is smaller than their objective probability, reliance on small samples leads to over-reliance on frequent outcomes. There are many reasons to believe people tend to rely on small samples. Reliance on small samples can be the result of cognitive limitations (Fiedler, 2000; Kareev, 2000) and can also be the result of a sophisticated attempt to provide a near optimal response under the assumption that there are patterns in the environment (Plonsky et al., 2015). Moreover, reliance on small samples was the one major common assumption of the best performing models in a series of choice prediction competitions for repeated choice, including competitions to predict choice when decisions makers can rely both on complete verbal description of the choice task and on feedback from previous choices, as is the case in our games (Erev et al., 2017; Erev, Ert & Roth, 2010; Erev, Ert, Roth, et al., 2010; Plonsky et al., 2019). Although our game setting adds complexity, we believe that simple models of reliance on small samples that ignore this complexity can still provide useful predictions of behavior in this setting (Roth et al., 2020). In the supplement, we describe one such simple model and derive its predictions for the current setting (to foreshadow our results, this simple model is surprisingly accurate).

The predictions of at least two prominent classes of models stand in stark contrast to those of reliance on small samples. First, according to prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992), rare events tend to be overweighted. Hence, relative to the stochastically-dominant option Responsible, the attractiveness of the stochastically dominated option App-Use, which includes a low-probability loss of −19, is predicted to be even lower than it objectively is. It follows that, according to prospect theory, the addition of App-Use should not influence the choice rate of the other two options and there should be no difference between the two conditions in our experiment. Yet, while cumulative prospect theory has been shown to describe risky choice behavior well in some cases (e.g., Glöckner et al., 2016; Glöckner & Pachur, 2012), it has been less successful when tasks were somewhat more complex (e.g., Payne, 2005; Sonsino et al., in press) and when decision makers received feedback on previous similar choices (e.g., Cohen et al., 2020; Erev et al., 2017).

A second class of models that would predict negligible differences between the two conditions in our experiment are belief-based models of choice in games, and particularly the popular “levels of reasoning” models (e.g., Camerer et al., 2004; Nagel, 1995; Stahl & Wilson, 1995). These models assume that decision makers form beliefs concerning the likely actions of other agents and then play best-response to these actions. In most of these models, best-response is assumed to be the strategy that maximizes the expected utility of the decision maker. As explained above, unless the decision maker is risk seeking, expected utility maximization will never predict choice of App-Use over Responsible. Hence, according to such models, App-Use will not be frequently selected and surely not more frequently than Responsible.²

3 Experiment 1

We compared choice behavior in the two games from Table 1. Condition Without-App was originally run as part of another study (Roth et al., 2020) and is reanalyzed here. Condition With-App was preregistered (https://aspredicted.org/tz6vr.pdf) and run a month later. It is analyzed here for the first time.³

3.1 Method

3.1.1 Structure of the games

The games in this experiment (Table 1) are context-free and abstract. As explained above, this provides the opportunity to investigate behavior while keeping many factors that could influence decisions in a pandemic (political ideology, risk perception etc.) fixed. The game features (the values of the payoffs and their probabilities) we use are not meant as an exact representation of the real-life consequences of behavior in pandemic-related decisions. Rather, they are used as rough relative representations of the main outcomes. Yet, while there is little meaning to the exact values chosen to be used in the games, the conceptual structure of the different distributions of possible payoffs aims to capture several basic assumptions concerning the outcomes people experience when making choices concerning their behavior in a pandemic. This sub-section details these assumptions and explains our choice of game parameters.

First, we assume that responsible behavior (e.g., properly wearing a face-and-nose mask or avoiding crowded spaces) is mildly inconvenient and thus carries a small cost in comparison with the more reckless behavior. Therefore, the frequent outcome from choice of Responsible (0) is slightly lower than the frequent outcome from choice of Reckless (+1). Second, we assume that in a small proportion of the time, reckless behavior may lead to dire events with very high costs (a natural example is getting infected and turning ill; other examples are being forced to quarantine after contacting someone who turns out to be infected or unintentionally transmitting the virus to others). Thus, with small probability (2%), choice of Reckless leads to a large negative payoff (−60). Third, we assume that due to the nature of the pandemic, people who behave responsibly cannot protect themselves if others around them behave recklessly. Hence, choice of Responsible may (with small probability) also lead to very large negative payoff (−60) if another agent chooses Reckless. Fourth, we assume that using a health-signal application does not protect the user from the consequences of other reckless agents. Therefore, choosing App-Use also leads (with 2% chance) to very large negative payoff (−60) if another agent chooses Reckless. Fifth, we assume that a green code on a health-signal application provides a small benefit that outweighs the small costs of responsible behavior (relative to reckless behavior). Therefore, the frequent outcome from choosing App-Use (+2) is slightly higher than the frequent outcome from choosing Reckless (+1). Finally, we assume that a red code on a health signal application is relatively rare because those who choose to use it usually behave responsibly (had they been frequently reckless, their color code would have been frequently red thus invalidating the advantage of using the app), and that a red code carries a relatively large cost (but not as large as getting infected or similar dire events). Thus, in relatively rare cases (10% of the time), choosing App-Use provides a medium-size loss (−19).

3.1.2 Procedure and design

The experiment was a 4-person repeated game programmed using OTree (Chen et al., 2016). Participants were first briefed and read the instructions, which included a complete description of the payoff structure and an attention check⁴ (see the supplement for complete instructions). The instructions also stated that the more points participants accumulate, the higher their chances of winning a bonus payment at the end of the experiment (see below).

After agreeing to participate, participants usually had to wait several minutes until four consenting participants were available to form a group. When a group formed, participants completed demographics data and proceeded to face a choice between the two (Condition Without-App) or three (Condition With-App) options presented in Table 1 for 60 rounds. Following each choice, participants received feedback concerning both their obtained and forgone payoff(s) in that round (see Figures S1-S2 in the supplement for screenshots).

The group advanced to the next round only after all its members submitted a choice. To guarantee that the experiment advanced smoothly, participants had to submit a choice within 10 seconds (20 seconds in the first 3 rounds). Participants knew that if they failed to submit a choice in time, their payoff in the round would be reduced by 2 points, and the system would auto-submit a choice for them. Unbeknownst to participants, auto-submissions repeated the choice the participants made in the previous trial (except in the first round when a Responsible choice was auto-submitted). In addition, participants knew that if they “make most of the choices on their own”, they will get an additional $1 bonus.⁵ Hence, participants were incentivized to make an active choice both in each trial and across all trials.

The total number of points participants accumulated in the 60 rounds were converted to a probability to win an additional $1 bonus.⁶

3.1.3 Participants

Participants were MTurk workers who were compensated $1. Additional bonuses were provided as explained in the Procedure section. We analyzed the data of 48 participants (12 groups, 18 female, M _age = 41) in Condition Without-App and 68 participants (17 groups, 25 female, 31 male, M _age = 47) in Condition With-App. We preregistered that after data exclusions, Condition With-App should include at least 12 groups. To guarantee sufficient exposure of the study on the platform (so the study runs smoothly), each session included several groups and the final group number was higher.

As preregistered, we excluded all groups in which more than 20% of the group choices were auto-submitted (9 groups in Without-App and 8 groups in With-App). In nearly every instance, such high auto-submission rate reflected a case in which participants failed to make more than a single choice (likely due to server or computer errors). Using a different cutoff for exclusion (e.g., more than 5% auto-submissions) does not meaningfully affect the results. We also discarded one group in Condition With-App that mistakenly included a participant who participated in the other condition.

3.2 Results and discussion

We analyze only rounds in which participants actively submitted a choice (i.e., not including auto-submissions).⁷ Table 1 shows the mean aggregate choice rates for each option in each condition. As preregistered, the main variable of interest was the mean group choice rate of the Reckless option (hereafter Reckless-rate), reflecting how often participants choose the option corresponding with the dangerous and inefficient equilibrium. The Reckless-rate was 90.6% (SD = 8.0) in Condition Without-App and 15.9% (SD = 9.3) in Condition With-App. This difference (M _diff = 74.7%, 95% CI [68.0, 81.3]) is significant at any significance level, t(25.9) = −23.1 (one-sided t-test), d = −8.5, 95% CI [−6.1, −10.9], reflecting, as predicted, a major decrease in choice of the Reckless alternative when App-Use is available.⁸

Notably, the Reckless-rates in Condition Without-App were all above 75%, suggesting every group tended to converge to the inefficient equilibrium. Conversely, in Condition With-App, Reckless-rates were all below 31%, suggesting no group converged to this dangerous equilibrium. Figure 1a shows the proportion of Reckless choices for individual participants by condition. It highlights that the (group) Reckless-rates reported above reflect the individual Reckless choices well. Specifically, in Condition Without-App nearly all participants chose Reckless very often whereas in Condition With-App most participants chose Reckless only rarely.⁹

Figure 1: Individual choice rates and individual disaster rates in Experiment 1. The two games (“Without-App” and “With-App”) are shown in Table 1. (a) Violin and dot plots of the individual Reckless rates with and without the application option. The red X marks the mean group Reckless rate. Dots are binned to nearest 0.01 (b) Violin and dot plots of the individual disaster rates (loss of 60) with and without the application option. The red X marks the median disaster rate in each condition. Dots are binned to nearest 0.001. The disaster rates are the proportion of disasters out of the total number of trials in which participants made a choice on their own (did not auto-submit) and so the rates sometimes do not divide exactly by 60 (the number of trials).

Because in Condition With-App many participants avoided the reckless option, they also experienced fewer “disasters” (i.e., they obtained fewer losses of 60 points abstracting, e.g., an infection) than in Condition Without-App. The mean group disaster rate was a highly skewed variable for which the assumption of normality was rejected by a Shapiro-Wilk test (p = 0.012). Hence, we compared the distributions of group disaster rates using Wilcoxon Rank Sum test. The medians for conditions Without-App and With-App were 1.48% (IQR = [1.19, 2.43]) and 0.83% (IQR = [0, 1.25]) respectively, a significant difference (p = .003, effect size r = 0.52, one sided test). Figure 1b shows that the individual disaster rates reflect the group disaster rates well.

We further made an unplanned analysis comparing the group Responsible-rates between conditions. These rates were 9.4% (SD = 8) and 22.8% (SD = 10.1) in conditions Without-App and With-App respectively, revealing an unexpected significant difference (t(26.6) = 3.99, p < .001 two tailed, d = 1.44, 95% CI [0.58, 2.31]). This may be explained by the existence of participants who chose to be responsible (in either condition) as long as other participants in their group were not Reckless. Because only few people chose Reckless in Condition With-App (most chose App-Use), such participants could then safely choose to be Responsible, thus increasing the Responsible-rate in this condition.

Qualitatively, there were little effects of learning within the task. The main changes within games were a decrease in Responsible behavior and an increase in Reckless behavior. In Condition Without-App, experience increased choice of Reckless behavior from an average of 85.4% in the first 10 rounds of the game to an average of 93.4% in the last 10 rounds of the game. In Condition With-App, the increase was more modest, from an average of 13.5% in the first 10 rounds to an average of 17.6% in the last 10 rounds. This latter increase resulted from a decrease in Responsible choices (from 25.3% in the first 10 to 19.8% in the last 10 rounds), whereas the choice rate of the stochastically dominated action, App-Use, changed very little throughout the game. The full learning curves are provided in Figure S3 in the supplement.

Note that, for simplicity, we chose the parameters of App-Use to reflect a suboptimal design for a health signal application: Participants could not condition their choices on the signals. Users experiencing a “red code” could not avoid the utility loss it entailed (i.e., they could not condition their choice on their own signal). This property of the payoff function can reflect, for example, an application that automatically notifies the authorities when a user receives a red code. An app that generates a completely private signal would likely be even more attractive. Moreover, participants could not choose to avoid contact with participants that could present a a green code, a strategy that reduces the risk of infections and increases further the incentive to use the application. Despite these suboptimal design choices, most people chose App-Use. Nevertheless, one may argue that our results are driven by other experimental design choices we made. Experiment 2 was designed to test this claim.

4 Experiment 2

To increase our confidence that the results of Experiment 1 were not due to an immaterial design choice we made, we designed and pre-registered (https://aspredicted.org/i4bm7.pdf) a conceptual replication with the following changes. First, we increased the time limit for each choice (before an auto-submission is made) from 10 to 30 seconds. This increased our confidence that choices were deliberate and thoughtful. Consequently, and to keep the maximal length of the experiment fixed, we reduced the number of rounds from 60 to 20. Second, Condition Without-App also consisted of three rather than two alternatives, with the added third alternative a duplicate of one of the original ones (Responsible or Reckless). All groups in a session (and all members of each of those groups) got the same duplicated alternative. This change allows for a safer comparison of the choice rates in the With-App and the Without-App conditions, since they each now included the same number of alternatives. Finally, we added an attention check at the end of the 20 rounds in which participants were shown a screen with five options and were told the payoff each option was going to generate in the next round (with certainty). Then, they were asked to choose one of these options.¹⁰

4.1 Method

4.1.1 Participants

After exclusions, we analyzed data of 116 participants in Condition Without-App and of 80 participants (20 groups, 30 female, 41 male, M _age = 35.5) in Condition With-App.¹¹ Among those in the With-App condition, 48 participants (12 groups, 11 female, 29 male, M _age = 35.7) had two identical Responsible buttons and 68 participants (17 groups, 24 female, 34 male, M _age = 35.4) had two identical Reckless buttons. We planned (and pre-registered) somewhat smaller sample sizes, particularly for Condition With-App with two Reckless alternatives, but ended up with these sample sizes due to an error and the need to run multiple groups in parallel for the session to run smoothly on MTurk. Restricting the analysis to the sample sizes stated in the pre-registration does not change the results in any way.

4.1.2 Procedure

The procedure was identical to that in Experiment 1, with the changes listed above.

4.2 Results

We first discuss the difference between conditions, when pooling the two sub-groups of the Without-App condition, that is, regardless of which option was duplicated. As shown on the right-hand column in Table 1, in condition Without-App, the Reckless-rate was 73.8% (SD = 9.0), whereas in Condition With-App it was only 13.9% (SD = 9.3). This difference (M _diff = 59.9%, 95% CI [54.5, 65.3]) is highly significant (t(39.9) = −22.4 one-sided, d = −6.56, 95% CI [−5.1, −8.0]), reflecting, as predicted, a major decrease in choice of the Reckless alternative when App-Use is available, even when Condition Without-App includes three choice options and participants have plenty of time to make a choice.¹²

As in Experiment 1, the high reckless-rate in Condition Without-App and the low Reckless-rate in Condition With-App reflected all groups in the condition well. The lowest group Reckless-rate in condition Without-App was 59.4%, and the highest group reckless-rate in Condition With-app was 35%. Figure 2a shows these rates also reflect the individual choice rates well: In Condition Without-App most participants chose Reckless very often whereas in Condition With-App most participants chose Reckless rarely.

Figure 2: Individual reckless rates and individual expected disaster rates in Experiment 2. The two games (“Without-App” and “With-App”) are shown in Table 1. (a) Violin and dot plots of the individual Reckless rates with and without the application option. The red X marks the mean group Reckless rate. Dots are binned to nearest 0.01 (b) Violin and dot plots of the expected individual disaster rates with and without the application option. The red X marks the median disaster rate in each condition. Dots are binned to nearest 0.00125.

Because in this experiment, each group played only 20 rounds, even if participants chose Reckless in every round, their probability to experience at least one disaster was only 1−(1−0.02)²⁰ = 0.33. Hence, comparing the experienced disaster-rates between conditions (as in Experiment 1) is not a good measure for how safely groups were playing. Instead, and as pre-registered, we compared the conditions’ expected disaster-rates (the number of rounds in which at least one person in the group played Reckless, multiplied by the probability for a disaster in such case, 0.02). Again, because the Shapiro-Wilk test rejected the assumption of normality (p < .001), we used the Wilcoxon Rank Sum test. The medians for conditions Without-App and With-App were 2.0% (IQR = [2.0, 2.0]) and 1.0% (IQR = [0.38, 1.24]) respectively, a significant difference (p < .001 one-sided, effect size r = 0.86). Figure 1b shows that the individual expected disaster rates reflect the group expected disaster rates well. Note that in Condition Without-App, rounds in which no person chose Reckless were extremely uncommon, hence nearly all expected reckless-rates were exactly 2% (the probability of disaster when at least one person chooses reckless).

The Responsible-rate in Condition With-App was 25.5% (SD = 15.6), similar to that found in Experiment 1. Yet, the Responsible-rate in Condition Without-App was 26.2% (SD = 9.0), which was much higher than that found in Experiment 1. Hence, unlike in Experiment 1, in this experiment, we did not observe a difference between Responsible rates in the two conditions and it is unclear how robust that result is.

Moving to analysis of the two sub-groups in Condition Without-App, we found relatively little differences. In the 12 groups that were presented with two identical “responsible” options (and one reckless option), the mean Reckless-rate was 70.5% (SD = 9.0), whereas in the 18 groups that were presented with two identical “reckless” options (and one responsible option), the mean Reckless rate was 76.1% (SD = 8.5). This difference is marginally significant (t(23) = 1.72, p = .099). Hence, there may be some effect to providing participants with “more opportunities” to choose Responsible, an effect that can be explained by some random error that is uniformly distributed among available options. Yet, even with two Responsible options and only one Reckless option, the Reckless-rate is much greater than the Reckless-rate in Condition With-App.

The most apparent effect of learning within-task emerged in the Without-App condition in which the mean Reckless-rate increased from 59.8% (SD = 15.7) in the first five trials to 79.7% (SD = 12.2) in the last five trials of the task. In contrast, in Condition With-App there was very little learning effects over time. In particular, the mean choice rate for App-Use was relatively stable (increased from 57% in the first 5 trials to 61% in the last 5 trials), indicating that participants did not learn to avoid the stochastically dominated action with experience. Full detailed learning curves are given in Figure S4 in the supplement.

5 General discussion

Compliance with physical distancing and similar guidelines is influenced by many factors including political orientation (Painter & Qiu, 2020), socio-economic status (Wright et al., 2020), and news consumption source (Simonov et al., 2020). Yet, because compliance with guidelines consists of many small repeated experience-based decisions, we believe that other relevant factors include people’s basic decision making tendencies in repeated decisions from experience.

Unlike description-based one-shot decisions without feedback, in which prospect theory predicts people behave as if they overweight rare (low probability) events, research on repeated decisions with feedback has documented a robust tendency to behave as if rare events are underweighted (and common experiences are overweighted). Such bias was documented in basic repeated decisions with partial (Barron & Erev, 2003), complete (Camilleri & Newell, 2011), and biased feedback (Plonsky & Teodorescu, 2020), as well as in more complex settings of repeated choice like two-stage decisions (Roth et al., 2016), investment decisions (Taleb, 2007), market entry games (Erev, Ert & Roth, 2010), and in animal choice (Shafir et al., 2008). In a pandemic, underweighting rare events likely implies reckless behaviors. Moreover, the problem is exacerbated by the fact that reckless behaviors are “contagious”: they make it less attractive for otherwise-careful people to behave responsibly (Erev et al., 2020).

But the same tendency that implies problematic behavior also gives rise to a possible solution. If experiences that follow reckless behavior become frequently worse than those that follow more responsible alternatives, we can expect a dramatic reduction in reckless behavior. To do this in scale, policymakers can give people the option to use voluntary health signal applications designed to both discourage reckless behaviors and make people’s lives simpler and more convenient. Our analysis suggests that, if using the application frequently saves time and effort, it will get significant traction, even if, because of infrequent poor experiences, on average using the application is a bad idea.

A clear limitation of our design is that we use abstract experimental games to parallel real-world behavior that is surely much more complex. Our analysis does not prove that reckless behavior in a pandemic is driven by high sensitivity to frequent outcomes. Yet, it does imply that such a tendency may be a sufficient condition for the emergence of reckless behavior. Therefore, even if policy makers would find ways to deal with other factors that drive reckless behavior (social norms, biased information, etc.), they may not be enough: policies may have to consider people’s heavy reliance on the frequent outcomes of the possible actions. Similarly, our analysis does not guarantee that changing the incentive structure in the ways we propose will solve the behavioral challenges in a pandemic. Yet, it raises a potential solution. In addition, it may be unclear whether our proposed policy change can realistically map to the real world. For example, can shorter wait times in queue really be (slightly) more rewarding than the advantages of reckless behavior? To increase our confidence that our experimental analysis may generalize to the real world, we next describe a field demonstration of a policy that takes into account people’s presumed sensitivity to frequent outcomes and leads to compelling results.

5.1 Field demonstration

To demonstrate that this analysis is not purely theoretical or limited to abstract games, we documented a policy change that took place in a nursing home in Israel on the last quarter of 2020, amid the COVID-19 pandemic (see details in the supplement.). The management of the nursing home sought to get its employees to use contact tracing applications while at work so that there could be an accurate and swift epidemiological investigation and reaction in case of an infection. Unfortunately, simply asking the employees to voluntarily use such applications proved to be ineffective. After observing that workers who enter to the nursing home at the beginning of a shift form queues to get their temperature checked and answer a few symptom questions, we advised the management to integrate an application that automatically registered the users’ temperature and allowed them to enter the nursing home faster with a much shorter queue. The management chose to use Tamara®, a commercial contact tracing application. Figure 3 shows the number of employees who used this application from within the nursing home over nine weeks: five weeks prior and four weeks after the policy change. Before the policy change, and despite repeated marketing efforts and appeals by the management aimed to increase uptake, the average daily number of users was only 29.5 (SD = 30, approximately 19% of employees). Yet, after the policy change that implied users of the app save a few minutes when entering the nursing home, the average daily number of users more than tripled, to 113.8 (SD = 17.3, approximately 74% of employees). Moreover, the number of active users after the change remained relatively stable over time. Notably, these numbers likely reflect a lower bound on the rate of employees that would have desired to use the app because the app was incompatible with some types of phones (e.g., some nursing home employees do not use smartphones for religious reasons).

Figure 3: Nursing home employee users of a contact tracing application across time. The red dashed line marks the day of change in entrance policies to the nursing home such that using the app saved employees a few minutes each day.

For unknown reasons, prior to the policy change, there were two days in which there was a significant increase in usage rates (see Figure 3). These may be due to a local initiative by a worker or a security guard that asked arriving employees to download and use the app on these days. Importantly, these increases, unlike the increase in usage after the policy change, did not persist over time. Note also that regardless of their reasons, these increases suggest that many employees had installed the application prior to the policy change, which implies that the main effect of this change was on the usage rate — a daily repeated decision—rather than on the installation rate — a one-shot decision.

Although contact tracing applications are different from health code applications described earlier in this paper, the implicit decision task faced by the employees in the nursing home is similar to that faced by participants in our experiments. In both cases, the decisions that imply safer behavior were highly sensitive to the frequent experience. Another example of the importance of the frequent experience comes from the difference between the application Tamara, endorsed by the management of the nursing home, and the application HaMagen, that the management of the nursing home initially asked employees to install and use. HaMagen is Israel’s national contact tracing application that was available for use since early in the pandemic. Unlike Tamara, the contact data of HaMagen never transfers to an external server (it is strictly the users’ choice whether the data can be used for an epidemiological investigation). Hence, choosing HaMagen over Tamara arguably may lead to better outcomes on average. Yet, after trying it for a short period, it was revealed that HaMagen consumes more battery power and has a high false alarm rate, hence the employees’ frequent experience from using it was bad. In contrast, adding Tamara to the employees’ choice set, alongside the policy change, provided employees with a responsible alternative with frequent positive experiences.

5.2 Theoretical and wider practical implications

The observation that in our experiments App-Use is chosen far more than the other alternatives, and particularly more than Responsible that stochastically dominates it, is consistent with the idea that people behave as if they underweight rare events. To our knowledge, this is the first demonstration of such a pattern in fully described social games. Of course, we demonstrated this pattern in only a single game, and there could be other explanations for the observed behavior. For example, behavior is also consistent with choice of the option with the highest positive outcome. Future research should investigate how robust this tendency is in games, and examine alternative explanations. Nevertheless, considering the fact that underweighting of rare events in repeated choice settings is a highly robust phenomenon and considering the success of models of reliance on small samples (which imply underweighting of rare events) in describing and predicting behavior in such settings, we feel the observed pattern of results is likely a reflection of underweighting of rare events. As a result, we also believe that the pattern will hold under small changes in the values of game features, as long as the general structure of the payoff distributions is similar to those we consider here. Indeed, in the supplement, we demonstrate that as long as App-Use remains frequently better than other alternatives, the predictions of the naïve sampler model are very robust to changes in the payoff structure.

If indeed people behave as if they underweight rare events in similar games and our results are robust, this finding is theoretically important for several reasons. First, it has been suggested that in completely described social games people may put more focus on getting to fair and efficient outcomes, but when they have to learn the games from experience, this focus will be diminished (Erev & Haruvy, 2016; Hertwig & Erev, 2009). Our results show that the effects of experience trumps those of description when both information prompts are available in social games, consistent with findings from individual decisions from experience (Erev et al., 2017; Jessup et al., 2008; Lejarraga & Gonzalez, 2011; Weiss-Cohen et al., 2016; Yechiam et al., 2005).

Second, the commonplace analysis of behavior in games focuses on the expected outcomes (or utilities) choice profiles generate. For example, entries in cells of normal-form games almost always include at most a single number per player (e.g., Camerer, 2011). Our study adds to the small literature demonstrating that this practice can mask important regularities and highlighting the importance of considering the distribution of potential payoffs rather than their expected values/utilities (Avrahami et al., 2005; Bereby-Meyer & Roth, 2006; Budescu et al., 1990; Kunreuther et al., 2009; Levati et al., 2009; Rapoport et al., 1992; Schulze & Newell, 2015).

Finally, many models of behavior in games assume that people make the best response to the beliefs they have concerning the strategies of other people (Brown, 1951; Camerer et al., 2004; Stahl & Wilson, 1995). Such models cannot predict a preference for stochastically dominated options which are never a best response from a normative standpoint. Other mainstream models of behavior in games that assume some form of reinforcement learning (Camerer & Hogarth, 1999; Erev & Roth, 1998; Fudenberg & Levine, 1998) may sometimes predict choice of stochastically dominant options, but are very unlikely to predict underweighting of rare events on the long term.¹³ In contrast, models that assume reliance on small samples with underweighting of rare events may better capture behavior in many games (Erev, Ert & Roth, 2010). Future research should examine the likely properties of such models and how they differ from models that assume reliance on small samples to capture individual behavior.

From a practical perspective, the observation that people tend to choose options that are better most of the time can facilitate development of solutions to many other social issues. For example, we can use similar ideas to promote vaccinations for SARS-CoV-2. Those who get vaccinated can receive a certificate that will act similarly to the “green code” on a health signal application discussed above. That is, while both people with and without the certificate will be able to access public facilities and workplaces, not carrying a certificate will incur a small cost in time and effort. While this cost may not get all people to vaccinate (particularly those high on vaccine hesitancy), we predict that the small frequently experienced costs will help increase vaccine uptake among most people.

Another example for the possible use of these ideas in a different domain concerns food waste. To reduce food waste, it is desirable to get people to purchase smaller quantities of food that has short expiration dates. One way to do so is to promote purchase of smaller bundles or product batches: If one buys only two yogurts rather than a bundle of eight, there’s a better chance the yogurts are consumed by their expiration date. It is sensible to assume people prefer large bundles and batches to avoid having products run out. Yet, our analysis suggests that as long as these experiences would be sufficiently infrequent, making the purchase of smaller bundles better most of the time (e.g., by providing subsidies/lowering taxes or by providing consumers who buy them designated queues) would get many to people to choose buying them.