Introduction

Publicly visible norm violations may subsequently trigger more norm violations and eventually set off dynamics of normative decay and disorder. This dynamics has recently been tested in a series of field experiments where graffiti, litter, unreturned shopping carts, and illegal parking caused people to violate the same and even other norms [1]. Similarly, it has been shown that people litter if they observe others littering [2], and people lie more if they observe others lying [3], [4]. The contagiousness of disorder tends to be particularly strong if there is nobody around giving cues that show respect for social order [5].

This dynamics may be explained by a mechanism linking the perceived prevalence of a certain behavior with subjective beliefs about its common approval. People hold beliefs about the average behavior, i.e., about the “descriptive norm”, and make inferences about its appropriateness, i.e., about the “injunctive norm” [6]. In this way, occurrences of public norm violations may make people aware of a larger than initially believed prevalence of the behavior, trigger reassessments of its common approval and result in an amplification of disorder and normative decay.

However, many norm violations are not publicly visible but conducted in private. Two-timing, tax evasion, consumption of pornography, visits to prostitutes or alcohol abuse are only some of many examples where norm violations are typically concealed from others; consequently, large parts remain in the dark. Therefore, the complete rate of norm violations consists of detected and undetected norm violations. Hence, normative dynamics and normative decay are crucially dependent on actors’ subjective beliefs about the rate of undetected norm violations. If actors perceive others’ norm violations, their beliefs about the additional extent of undetected norm violations are crucial for their evaluation of the appropriateness of the behavior and their own decision to adhere or violate the norm.

If actors underestimate the complete extent of norm violations, the proposed dynamics of the above mentioned authors hold if certain conditions are met [7]. Those underestimating others’ transgressions may perceive occurrences of others’ norm violations as relatively frequent or strong if they are informed about the true extent of norm violations. As a consequence, they increase their subjective estimates about the extent of norm violations and subsequently perform more own norm violations. In a classic paper [8] sociologist Heinrich Popitz already outlined this idea of a “preventive effect of ignorance”. That means that an actor’s ignorance of other peoples’ norm violations has a deterrent effect on his or her norm-related behavior. Lifting the ‘veil of ignorance’ is expected to increase the extent of norm violations. An example is the Kinsey report [9] on sexual behavior. The publication of the report at that time had the consequence of changing sexual behaviors and norms of sexual conduct [4], [7], [10]. Of course, information on the true amount of norm violations does not always lead to an upward spiral of transgressions [7]. First, and almost trivially, potential transgressors do not always gain from norm violations. Secondly, the disclosure of norm violations is often paralleled by increasing sanctions, stigmatization and strengthening of the legitimation of the norm as in the case of child abuse by catholic priests [7]. Hence, adjustment of an actor’s underestimation of transgressions does not necessarily lead to an increase in his or her own propensity to violate the norm.

So far we have assumed that actors underestimate the amount of other peoples’ transgressions. However, what will happen to those who overestimate from the start? Will the dynamics be inverted? Will those who believe more transgressions are being committed than de facto adjust their beliefs and violate fewer norms if they hear about the true state of the world? If this consequence were true, the dynamics will crucially depend on actors’ subjective beliefs. My research question therefore asks whether information about others’ norm violations will increase transgressions in societies consisting predominantly of believers in too few transgressions and decreasing ones in those with believers in too many. The conjecture that beliefs about the underestimation or overestimation of others’ norm violations are crucial for determining whether the dynamics set off more or fewer norm violations has been raised consecutively in [11], [12], [4], [7].

The conjecture about the interaction between beliefs and behavior is supported by evidence showing that over- as well as underperformance of norm-relevant behaviors is regarded as deviant. Hence, information about norm violations of others can trigger adjustments in both directions – more or less adherence to social norms. Taking the example of norms of environmental protection, people were informed whether their energy consumption level was above or below the average of their neighborhood. Those above the average reduced their energy consumption; however, those below increased it when there was no special mention of their laudable behavior [11].

Whereas norms of energy consumption are one of the few examples where prevalence is directly measurable, the actual extent of most norm violations is unknown to both the researcher and the people in the field making their choices for pro- or antisocial behaviors. Therefore, I developed an experimental design where the prevalence of norm violations can be observed, the accuracy of beliefs measured and the offsetting dynamics of pro- and antisocial behaviors traced. This allows the understanding of the dynamics between objective information, subjective beliefs and the co-evolution of social norms.

Experimental Design

The dice experiment [13] was used, as in a previous experiment [4], to study the spread of norm violations. This follow-up study of [4] has been extended by introducing the measurement of beliefs about the extent of others’ norm violations. This experimental setup allowed subjects to violate the honesty norm under highly anonymous conditions. In the experiment, subjects performed multiple dice casts in separate booths where they were isolated from others and unable to be observed (see Fig. S1 for experimental instructions). They entered their cast numbers into a tailor-made graphical computer interface and received Swiss Francs according to the number they reported (Fig. S2). The reported number six was an exception and yielded no earnings. This design put subjects with numbers “six” and numbers lower than “five” into a moral conflict between either adhering to the honesty norm or increasing own payoffs by reporting higher payoffs than they would be entitled to.

This design is an improvement to previous designs in that it avoids deception of subjects. This enhances subjects’ trust in the experimenter and serious completion of the experimental task. A truthful and straightforward description of the experimental protocol ensures that honest reports can be regarded as subjects’ willingness to pay a price for honesty. This is a design improvement compared to experiments where subjects are deceived by predetermined dice cast by manipulated computer programs [14]. If subjects think they are being deceived by the experimenter, they may reciprocate deception and act dishonestly in response to the experimenter’s dishonesty. This would bias the results and render inconclusive results.

The sample consisted of 240 subjects, subdivided into 24 experimental sessions of groups of ten. In each period each subject cast a die twelve times and reported each number. There was one trial period and four periods with payments where one cast was randomly selected for payouts. Subjects were randomly allocated to one of three treatments. In the “info treatment”, subjects were informed about the frequency of each reported payoff (zero, one, two, three, four, and five) in their group. In addition, beliefs were elicited by asking for estimates of these frequencies before information feedback (Fig. S3, S4). Belief accuracy was incentivized with monetary payments for good estimates using a quadratic scoring rule (see Table S1 and Materials and Methods for details).

There were two control treatments. The “control belief” treatment consisted of belief elicitation without information feedback. The “control base” treatment was without belief elicitation and without information feedback. Note that the honesty threshold in this design is such that each number is reported twice and average reported payments are 2.5 Swiss Francs per round. Lying is measureable as upward deviance from this honesty threshold in terms of higher average claimed payments and higher frequencies of high reported dots (i.e., fives).

To the best of my knowledge, this is a novel design to measure subjects’ beliefs about others’ honesty in dice experiments. This improves previous dice designs where the contagiousness of lying was measured without controlling for subjective beliefs about others’ honesty [4]. In [4] it was observed that subjects who were informed about others’ lies increased their own propensity to lie compared to a control condition without information feedback. This main effect could, however, not be separated for those underestimating and those overestimating others’ dishonesty. The novel belief measures allow observing the interaction between information feedback and beliefs and can therefore contribute to the existing literature. In this way the research question whether believers in too little honesty will increase and those in too much decrease their own propensity for honest dice reports can be answered.

The implementation of multiple dice casts per subjects in each round was mainly done for two reasons. The first reason was to improve the measurement of beliefs. The elicitation of beliefs requires subjects to estimate the distribution of dice reports in their group. If there are too few dice casts to be estimated, the distribution becomes very noisy and subjects will not make as much effort in generating good estimates. Hence, elicitation of meaningful beliefs requires robust distributions to be estimated. The repetitions in the form of twelve dice casts per round yield a distribution consisting of twelve casts for each of the nine other group members, totaling in 108 dice casts from other group members per round. Accordingly, the distribution to be estimated by each subject is relatively robust. The second argument for introducing multiple dice casts per round was to ensure robust information feedback. If information feedback is driven by too much random noise, it does not give enough information about the lying behavior of others in the group. Consequently, too few dice casts per round in each group hinder studying time trends in lying over the four rounds of the experiment.

With respect to introducing multiple dice casts per subject, it obviously needs a good compromise between guaranteeing enough robustness for eliciting meaningful beliefs and meaningful information feedback on the one hand and allowing for enough randomness for ensuring anonymity for lying on the other hand. This compromise is critical since to cast high numbers twelve times is far more unlikely than it is to throw a high number once. As a result, the identification of liars on the individual level becomes easier the more repetitions there are. All in all, I believe that the choice of twelve repetitions per round in groups of ten subjects represents a good compromise between robustness and anonymity.

Results

Figure 1 shows the main results. The upper panels (A–C) show the trend in mean payment claims. The lower panels (D–F) display the number of reported fives over the four payment periods. The left column (panels A and D) shows the general trend for all types subdivided into info (solid black line), control belief (dotted red line) and the control base treatment (dashed green line). The x-axes in the upper panels depict the treatment averages of subjects’ average reported payment claims per round. The “honesty threshold” of 2.5 is denoted by the dashed horizontal black lines. The x-axes in the lower panels depict the treatment averages of subjects’ numbers of reported fives per round. The five was taken, because this is the highest possible payout per cast. Here, the “honesty threshold” of two (12⋅1/6) is also denoted by a dashed horizontal black line.

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Figure 1. Trend of reported payment claims in means (panels A–C) and fives (panels D–F).

Error bars show adjusted 95% confidence intervals such that non-overlapping intervals refer to treatment differences with p≤5% (see Materials and methods for calculations of adjustments). Underestimators hold beliefs below and overestimators above reported claims in their group at respective periods.

https://doi.org/10.1371/journal.pone.0077878.g001

First of all, there is substantial lying in all three treatments. This demonstrates the usefulness of the dice experiment for studying violations of the honesty norm. With respect to mean claimed payments (panel A), people claim on average about 35% more than what they are entitled to: the average claim is about 3.3 CHF per period compared to the honesty threshold of 2.5 CHF. With respect to the number of claimed fives (panel D), people claim about 230% more than what they are entitled to: on average, there are more than four claimed “fives” compared to the honesty threshold of two. All deviances from honesty are highly significant at the 0.1% level. This is shown by confidence intervals of 99.9% of differences between dice reports and honesty thresholds in Fig. S5 in the appendix.

In a first step, the main effect of information feedback was estimated (panel A and D). This was done by computing the trends in dishonesty over the four periods regardless of beliefs about others’ honesty. This analysis can be regarded as a replication of [4] with more data per subject. In [4] the authors used one dice report per subject before and one after information feedback. The present study uses twelve dice reports before information feedback and three times twelve dice reports after information feedback, i.e., considering three periods with twelve reports each.

Fig. 1A shows the main effect of information feedback for all subjects (i.e., regardless of their beliefs) in terms of reported means and Fig. 1D, in terms of reported fives. It can be seen that subjects lie slightly more in subsequent periods if they are informed about others’ reported payment claims. In the information treatment, subjects claim roughly between 0.10 and 0.14 higher average payouts and between 0.1 and 0.4 more fives compared to the control treatments. However, these differences are statistically not significantly different from both control treatments (Table S2). This means that the main “broken windows effect” in this study for all types of people is weak. Information of others’ transgressions triggers actors to increase their own transgressions only modestly. The direction of the effect is similar as in [4] and can be replicated. Yet, the magnitude of the effect is statistically weaker than in [4]. Other recent lab and online experiments yield a similar result of a small and non-significant increase of cheating in the feedback group as reported in [15].

However, what happens if beliefs about others’ dishonesty are taken into account? Will believers in too little dishonesty of others increase their own dishonesty and will believers in too much reduce it? If this were the case, the main effect would conceal the underlying dynamics in panels A and D because it works in the opposite direction for under- and overestimators of others’ dishonesty. Therefore, in a second step, the trends in dishonesty were calculated separately for those believing in too little and those in too much dishonesty of others. Since beliefs were not measured in the control base treatment, trends are only separated between control belief and info treatments.

Panels B, C, E and F of Fig. 1 show the results when types are differentiated by under- and overestimators. It can be seen that the results change substantially, confirming the interaction effect very clearly. “Underestimators” are defined such that they hold beliefs below and “overestimators” above reported payment claims in their group at respective periods. Per round with twelve dice casts, underestimators in periods after information feedback report 0.44 CHF more in average payments than underestimators in the control belief treatment (Fig. 1B, Table 1A). With respect to the number of reported fives, underestimators in the info treatment report 1.3 more fives than underestimators in the control belief treatment (Fig. 1E, Tab. 1B). Both interactions are highly significant at the 0.1% level.

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Table 1. Linear regression models of treatment differences between info and belief control treatments.

https://doi.org/10.1371/journal.pone.0077878.t001

The same interaction holds for overestimators. Actors overestimating the extent of lying subsequently adjust their reports downwards compared to the control belief treatment. Per round with twelve dice casts, overestimators in periods after information feedback claim 0.72 CHF less in average payments than overestimators in the control belief treatment (Fig. 1C, Table 1A). With respect to the number of reported fives, overestimators claim 3.5 fewer fives in the info condition compared to overestimators in the control belief condition (Fig. 1E, Table 1B). Both reductions are statistically significant at the 0.1% level. The differences between info and control belief treatments for over- and underestimators are substantial, taking into account that honest subjects would report an average of 2.5 Swiss Francs per period and two occurrences of the highest payout “five”.

Note that the discussed percentage comparisons refer to the average effects over periods 2 to 4, where information feedback was given in the info treatment and withheld in the control belief treatment. The reported percentages in the text can be computed from Table 1. This is done by referring to the intercept as baseline, which represents the mean claimed payments (model A) or average number of reported fives (model B) for overestimators in the control belief treatment. The percentages for the other three groups can be calculated by taking the differences from the baseline with respect to main and interaction effects.

Another way of quantifying the strength of the interaction is to compare the proportion of liars in different treatments. One way of doing this is to compute the expected proportion of the highest payoff five of a fair die (which is 1/6), compare it to the empirically reported proportion of fives π and adjust it for liars who actually threw a five but would have lied in case of lower cast numbers (i.e., multiply by 6/5). This yields an estimate of the proportion of liars λ = (π−1/6)⋅6/5 (see also [13]). The proportion of reported fives for over- and underestimators in the information and control belief treatments can be calculated from Table 1.

The proportion of liars in the population of underestimators is more than twice as large in the info as in the control treatment: there are 25.6% underestimating liars in the info and 12.7% underestimating liars in the control treatment. Note that these percentages refer again to periods 2 to 4, demonstrating that underestimators increase their lying if they are informed about the extent of lying in their group. Moreover, the proportion of liars in the population of overestimators decreases by less than half if they are informed about the extent of liars in their group. There are 56.3% overestimating liars in the control condition and 21.8% overestimating liars in the info treatment.

This shows that lying can either be more than halved or more than doubled depending on subjective beliefs and whether information feedback is provided. This clearly demonstrates that the observed interaction is substantial and gives rise to the conclusion that the direction of the dynamics towards either decay or stabilization of social order is strongly contingent on actors’ subjective beliefs regarding the extent of norm violations in the population. Furthermore, it can be noted that the dynamics become stronger over time in the sense that treatment differences increase over time (Fig. 1), giving additional weight to the conclusion.

Discussion

The reported results put recent findings about normative dynamics into perspective. Information about norm violations of others does not per se trigger subsequently more norm violations. The mechanism is contingent on actors’ subjective beliefs. Since some norm violations are visible and others go undetected, the dynamics depends on whether actors under- or overestimate the complete rate of norm violations.

If most subjects begin by underestimating transgression of others, it is likely to be a sign that they are normatively oriented in the beginning and project this normative orientation onto others. If they do not hear otherwise, they stick to this belief. Nonetheless, if they hear that they underestimate, they will adjust upward, norm violations will spread in the system, they may become less normatively oriented and adjust subsequently, triggering normative decay up to a society mainly consisting of liars. However, the reversed process is likely for subjects who overestimate from the beginning. They are likely to project that others are also not normatively oriented and stick to that if they do not hear otherwise. If they are informed, and as a result hear otherwise, they scale down their beliefs about cheating to a substantial degree. As a consequence, the honesty norm gains support, and honesty may spread up to a society mainly consisting of moralists.

The experimental findings discard the alternative explanation that the norm-stabilizing effect of underestimating transgressions is solely due to a sanctioning mechanism. It has been argued that this effect occurs because information about others’ norm violations serves as clue to the likelihood or severity of sanctions to be expected [3]. In this line of reasoning, underestimators adjust their beliefs about the probability or severity of sanctions after having learnt about the true prevalence of norm violations. Consequently, underestimators interpret others’ norm violations as an indication that sanctions are less likely or less severe than they originally thought – and increase their own transgressions. This mechanism explains the effect of information about others’ transgressions on increased own transgressions as a selfish, forward-looking reaction of underestimators on their updated beliefs about sanctions. Yet, in the experimental design sanctions have been completely excluded because individual norm violations have not been observable by the experimenters. Therefore, it can be concluded that there is a ‘pure’ effect of the spread of norm violations that is not generated by a change in the perception of sanctions.

The presented interaction between actions and beliefs for the case of lying also provides novel insights for the basic debate about their general interrelation. There is a lively literature on the temporal order of the two – do beliefs determine behavior or does behavior determine beliefs? The first line of reasoning is often advocated by economists and may be called reaction theory, the second one, advocated by psychologists, projection theory [16]. In social dilemmas, reaction theory refers to conditional cooperation. In this line of reasoning, cooperative behavior is a reaction on the actor’s belief that at least a certain fraction of the population will cooperate. It has been shown, for example, that a substantial proportion of actors condition their degree of cooperativeness on their belief that enough others also cooperate [17] and discontinue cooperation if they are informed about a critical fraction of freeriders [18]. Projection theory argues for a reversed causal path: actors project their own cooperative (or non-cooperative) intentions onto others and expect them to be similar to themselves [19], [20]. In this way, cooperative intentions trigger an alignment between cooperative behavior and cooperative beliefs so that beliefs may rather be the consequence of post-hoc rationalizations of behavior than its origin [21].

The data suggests a combination of reaction and projection mechanisms. Subjects without information feedback show a constant lying pattern, which is strongly correlated with their beliefs. Underestimators start with comparably few lies and seem to project their lying behavior onto the lying behavior of others since they retain their honesty level throughout the game. Conversely, overestimators start with comparably many lies and also seem to project their lying behavior onto others, retaining their dishonesty level. The dynamics in the treatment with information feedback, however, supports reaction theory. Informed underestimators react on their updated belief and adjust their lying behavior upwards. Similarly, informed overestimators react on their updated belief and adjust downwards. This combined mechanism of projection and reaction has also been supported in a recent study on cooperation in an asymmetric social dilemma [22] and it is likely to have played a role in field experiments on broken windows [1], [5].

Furthermore, the study sheds a novel light on the debate about peer effects in social dilemmas. It is often argued that peer effects are biased towards people’s self-interest, triggering asymmetric peer effects in social dilemmas. For example, people often exhibit a self-serving bias in their perception of their own cooperativeness [23] and are more averse towards envy than guilt [24]. In a novel study, the possible confounding between peer effects and strategic incentives in earlier studies has been ruled out by design [22]. The authors show in the context of a gift-exchange game that agents revise their level of reciprocity between the principal’s offered wage and their returned effort contingent on information about another unrelated agent’s effort. These peer effects are shown to be strongly asymmetric: Agents only make downward adjustments which are in their own self-interest and do not increase their effort if the other agent has shown higher effort compared to their own. In contrast, my study does not reveal such an asymmetry with predominantly self-interested adjustments. Honesty revisions occur in both directions – actors exhibit more but also less lying after information feedback, based on whether they under- or overestimated the extent of lying in the population.

One may reason whether the above discussed symmetry of self-serving and self-harming honesty adjustments may be caused by the monetary incentives for correct beliefs. There is evidence that the measurement of incentivized beliefs can have consequences for the measurement of cooperative behavior. If people receive money for accurate beliefs, they behave more cooperatively compared to setups without such premiums [25]. One explanation for this may be that subjects aim for a specific minimum payment as their final earnings for participating in the experiment. Hence, subjects without monetary compensation for accurate beliefs may compensate this relative “mental” loss by more freeriding in order to increase their earnings. However, if this were true for my setup, compared to a version without incentivizing beliefs, there would be higher honesty levels for both kinds of subjects, over- and underestimators. Therefore, this argument does not explain why overestimators decrease their lying about as much as underestimators increase theirs. In addition, it has been shown that monetary incentives for beliefs increase belief accuracy [25]. A likely reason is that subjects take belief elicitations more seriously when getting paid for accuracy. Hence, incentivized beliefs yield more accurate belief measures and thus have more predictive power. This gives more weight to the conclusion that information feedback in my study yields behavioral adjustments in both directions – more and less honesty. Subsequent studies should explore this issue in more detail and also investigate whether the kind of dilemma may explain the differences between both studies.

The group sizes of under- and overestimators in the analysis deserve some further discussion. The fraction of under- and overestimators varies over time and experimental conditions; however, it is always sufficiently large for robust estimations. Over all, there are more under- than overestimators, and the fraction of underestimators varies between 44% and 89%. Initially, most subjects underestimate the extent of norm violations. With respect to reported means, there are 70% underestimators in the info and 73% in the control belief condition. Similarly, there are 83% initial underestimators of reported fives in both conditions. Over time, the fraction of underestimators decreases in the info condition, whereas it remains at a roughly constant level between 72% and 84% in the belief control condition. Detailed time trends of the fraction of under- and overestimators and respective confidence intervals are given in the (Fig. S6).

The causal interpretation of the differential effects for under- and overestimators deserves some caution, however. It has to be noted that the groups are determined by subjects’ behaviors during the course of the lying experiment and not via random assignment. Hence, it cannot be ruled out that effects of third, unmeasured variables may mitigate some of the observed dynamics. Nevertheless, it seems hardly feasible to use random assignment instead since it is not possible to assign subjective beliefs externally. Furthermore, it can be doubted that such strong and straightforward interactions as presented here may be completely driven by other, unmeasured mechanisms.

In summary, the findings show that it is important to take subjective beliefs about undetected norm violations of others into account. Such beliefs can serve as strong mediators for disorder effects on norm conformity. The projection of own norm violations on others and their offsetting dynamics into both normative decay and norm conformity with respect to beliefs deserves further study.

Materials and Methods

Supporting Information

Acknowledgments

I thank Christian Schulz for developing the computer program for the laboratory experiment, including graphical interface and network capabilities. I thank Silvana Jud and Stefan Wehrli for help in recruiting subjects and conducting the experiments. I acknowledge support by Fabian Winter with whom I jointly programmed the statistical code of the adjusted confidence intervals. Andreas Diekmann has given considerable comments to the design of the study and the revision of the manuscript as well as discussed the different stages of this research with me. I also acknowledge helpful comments by Siegwart Lindenberg and Michael Mäs. I thank Claudia Jenny for proofreading.