Existence of moody conditional cooperation

In our experiment, subjects played a multiplayer prisoner's dilemma in which they had to choose one action to interact with their opponents. Payoffs were set as follows: Mutual cooperation was rewarded with a payoff R for both players and mutual defection earned them nothing, while a cooperator facing a defector obtained nothing, leaving the defector with the temptation payoff T. R and T where slightly modified for each of our four group sizes so the expected earnings would roughly be the same in all cases, always keeping the ratio T/R constant to stay within the same type of game. See Methods for a detailed description of our setup.

Let us begin reporting on the results of our experiment by looking at our first question, namely the existence of moody conditional cooperators and whether it depends on the group size or not. Figure 1 shows our results on this issue. We observe that moody conditional cooperators are indeed present in all sizes, including groups of four and five players, at variance with the analysis in Ref. 34. However, this disagreement is not entirely surprising, since theoretical results for repeated games are notoriously sensitive to modeling assumptions: Thus, computational results on IMPD based on genetic algorithms38 show that the evolution of cooperation in theoretical models depends very much on the implementation details. Therefore, the fact that our experimental observations do not agree with the predictions of a very specific model based on the replicator equation is something that can be expected. On the other hand, we observe only a few players using AllD (always defect) and even less players playing AllC (always cooperate), so what we are observing may be close to a homogeneous state consisting only of moody conditional cooperators, something that is possible even in large groups for certain parameters34. In any event, Fig. 1 shows very clearly the difference between the probability of cooperating after having cooperated or having defected, highlighting the importance of relating the current action with the one in the previous round. The plot also indicates that the probability of cooperation increases with the number of cooperators in the group in the last round, for all group sizes. Cooperation when no one cooperated before is relatively large in groups of size 2 and lower for other group sizes (but similar among them). Interestingly, the increment in probability with increasing number of cooperators is similar for all groups.

Figure 1 Probability that an individual cooperates after having cooperated (squares) and after having defected (circles) in the previous round, for groups of 2 (top left), 3 (top right), 4 (bottom left) and 5 (bottom right) people. Open symbols, experimental results; full symbols, predictions from our GLMM. Error bars correspond to the standard deviation of the observations. Lines are only a guide to the eye. Full size image

Group size dependence of cooperation

Figure 2 depicts the fraction of cooperative actions as a function of the iteration of the game, demonstrating that the results for groups of size two (i.e., pairwise interactions or usual 2 × 2 IPD) are very different for the observations on the rest of groups (sizes three and higher). Pairwise interactions show very high cooperation levels with an increasing trend (see below for a discussion of similar, earlier results17,18) whereas for the rest of groups we find that cooperation decays from initially large values (around 60% or larger) much in the same way as in most Public Goods or networked IPD experiments. The fact that for groups with three subjects or more the cooperation level behaves in a similar manner is in agreement with earlier findings that the level of contributions to voluntary public goods does not depend significantly on the group size39. In addition, the low levels of cooperation we observed for groups of size three and four is consistent with the results in Public Goods experiments with up to 50 rounds40,41. In this context, our experiment, by analyzing sizes from two to five in the same experimental setup, provides evidence that there is an abrupt change in behavior in going from a two-player IPD to IMPD or public goods games with three or more participants, i.e., we could say that three is a crowd.

Figure 2 Percentage of cooperation as a function of the round for groups of 2 (top left), 3 (top right), 4 (bottom left) and 5 (bottom right) people. Open circles, experimental results; solid line, predictions from our GLMM. Full size image

The case of the two-player IPD

The results for the pairwise IPD deserve a separate discussion as they offer several interesting insights. In our experiment, participants were not informed about the number of rounds of the game, although they were given an estimate of the time duration of the procedure, so they could realize that there would be a sizable number of rounds in any event. Therefore, the ‘shadow of the future’ effect is very present. As a consequence, pairwise IPD experiments show a large level of cooperation in agreement with the observations of Ref. 19, obtained for much shorter IMPDs (an expected length smaller than 6 rounds). Interestingly, the large length of our repeated game allows us to go beyond this observation: Indeed, if we compare our observations to those reported in Ref. 37, who carried out experiments of length 12, we find an agreement for this initial part of the repeated game, as in both cases the cooperation level decreases with increasing round number. However, as the game continues in our experiment, we observe a clear trend towards increasing cooperation, punctuated by episodes of lower cooperation levels which are rapidly overcome. It is also worth mentioning in this context the (often overlooked) early experiments by Rapoport and Chammah17 and by Flood and Dresher (with only one pair of subjects, reviewed in Ref. 18) which, by running up to 300 and 100 iterations of the PD respectively, already showed that cooperative behavior could be stable, in agreement with our findings here. The similarity of the cooperation curve in Ref. 17 to ours, including the initial decrease, is indeed remarkable.

The difference in cooperation among groups does not arise from the initial propensity to cooperate, as cooperation at the first round is mostly independent of the group size (cf. Fig. 2). Instead, it is due to the behavior of the players as the repeated game progresses. This is in agreement with the type of moody conditionally cooperative strategy we found: the strategy parameters for pairwise PD, being clearly different from those of the larger groups, indicate that choosing cooperation is very likely if one cooperated in the previous step, while the probability to cooperate following a defection is relatively large, below but close to 0.5. This strategy is not the well known Tit-for-tat (TFT)5, as TFT does not depend on the player's own previous action, while Fig. 1 strongly suggests that the previous action of the player affects her next choice. Our result is also in agreement with those reported by Fudenberg et al.21, who in their treatments without noise found that when a player has cooperated in all rounds, a defection by her partner is not immediately answered with defection in a 42% of the cases, a number that is roughly similar to the ones we obtain for our moody conditional cooperators (albeit the comparison must be taken with caution as the way to characterize the behavior in both experiments is not exactly the same).

Model

To take into account that our data contains repeated measures on each subject of a binary variable, we resorted to the development a GLMM as follows. Let y ijt be the response of the subject i in group j at time t. Let y ijt = 1 if this subject cooperates at time t and 0 otherwise for all i, j and t. Then y ijt ~ Bernoulli(p ijt ). By the nature of the experiment, the subjects are nested in groups. Thus, a model needs to take into account the nested structure of the data and the repeated measures on the subjects.

Our concern with respect to dependency is the repeated measures on the same subject. First, the observations on the same subject are correlated just because they are decisions of the same person. This is also known as within subject variability. Second, the observations close in time, on the same actor, are more likely to be highly correlated as opposed to the observations further apart. We interpret this as latent generosity with a time component. Third, another source of variation is the latent component of the individual reaction to the number of cooperative actions observed in the group in the previous round. We can interpret this as latent reciprocity. These latent effects then measure “between-subject” variability.

Before introducing the model we finally chose as the best for our data, let us point out that, in alternative specifications, we checked for effects of major and gender, without finding any significant effect. Most importantly, we tested the dependence on whether the group was manipulated by the computer or not, again finding no differences (see Methods below). With these inputs, we finally proposed the following model:

where p it is the probability of cooperation of subject i at time t and the factors that affect it are as follows: χ(size il ) is the characteristic function corresponding to the group size of subject i, that is, χ(size il ) = 1 if subject i played in a group of size l and 0 otherwise; LagCoop it is the number of cooperative actions received by subject i at time t − 1; LagAction it is equal to 1 if the subject cooperated in the previous round and 0 otherwise and β l and , l = 2…5 and are the parameters of the fixed effects. On the other hand α i is the latent cooperativeness of each subject and γ i is her latent reciprocity (the individual random variation in the response to perceived cooperation). Individual latent effects follow normal distributions: α ~ N(0,Σ), where , where is the identity matrix and analogously γ ~ N(0, Σ γ ), where, . In addition, we have the repeated measure structure modeled as AR(1) structure through the ξ it term, where , where u is a vector of random variables with variance σ u . That is, there is a random component on the left hand side of the model which measures the “within subject” variability. The structure of the covariance matrix for this effect is given by a symmetric matrix, R, whose (ij)-entry is .

Model results

The model captures well the observations from the experiment, as can be seen from the comparison between the experimental data and the model predictions in Figs. 1 and 2. The agreement is particularly good for the cooperation level, as this magnitude can be obtained directly from the model, whereas there are small discrepancies in the slope of the conditional cooperation lines, mostly for the highest cooperative contexts. These discrepancies can be understood as a consequence of the fact that the estimation of these lines is an indirect product of the model. Another feature that is confirmed is the clear dependence on the players' own previous action, their ‘moodiness’, an aspect to which we come back below.

We first discuss the latent factors in the model. The corresponding variance components estimated within our model are represented in Table 1. The corresponding p–values are obtained by applying the log-likelihood ratio significance test (LRT) on the boundary of variance parameter space as in Refs. 43 and 44. From this table, a very interesting result which could not be seen from our analysis so far arises: While there is substantial heterogeneity in baseline attitudes to cooperation, the attitudes to reciprocal altruism are more uniform. To put it more formally, the variation between the individual latent effect, that is, the generosity, is three times larger than the variation of the between-individual reciprocity random effects (γ). Hence, we can conclude that individuals, while differing greatly in generosity, are closer in reciprocity, i.e., they enter the game with a naturally diverse predisposition to cooperate, but once they are playing, the way they answer to a given number of cooperative partners is similar among players. This is a remarkable finding insofar as there have been reports of the importance of heterogeneous behavior in related experiments30,45, but here we are able to identify for the first time the aspect for which heterogeneity is more relevant, namely the a priori cooperative predisposition of the subjects.

Table 1 Results for the variance of the random effects. Shown are the estimates, their standard error and the log-likelihood ratio (LRT) p-value assessing their significance. From top to bottom, the table shows the results for the generosity, the reciprocity and the two parameters of the AR(1) formalism Full size table

Turning now to the fixed effects, the predicted values for the corresponding parameters are presented in Table 2. The estimates and their p-values give us the individual significance levels. The type 3 tests collect the information on overall significance of the effects. Based on the Table 2, we have size, LagCoop and LagAction as highly significant covariates at 1% significance level). Other relevant results include, for instance, the fact that the size of the groups is important for cooperative attitudes. As Table 2 shows, the parameter for the baseline cooperative attitude in a group of size 2 is larger and statistically different from all the others. In turn, the baseline cooperative attitude is not statistically different between sizes 3 and 5. The conditional cooperation declines monotonically with group size, although the differences become smaller as size increases and the coefficient is still statistically different from zero even at the largest size. This is an interesting point that might be useful to understand why cooperation is more fragile in large groups, which could in turn explain why social groups often evolve punishment strategies directed solely at deviators, as in Ref. 10. Finally, the result that LagAction is relevant points to the dependence of actions on what occurred at the previous round. In this respect, it is important to mention that we also tried other models in which dependence on two previous time steps was included and we found that this was not significant. Therefore, the dependence on the player's own previous choice is enough to capture the results of the experiment, a finding that is in agreement with earlier work20,21.