Cooperation and coordination in heterogeneous populations

Abstract

One landmark application of evolutionary game theory is the study of social dilemmas. This literature explores why people cooperate even when there are strong incentives to defect. Much of this literature, however, assumes that interactions are symmetric. Individuals are assumed to have the same strategic options and the same potential pay-offs. Yet many interesting questions arise once individuals are allowed to differ. Here, we study asymmetry in simple coordination games. In our set-up, human participants need to decide how much of their endowment to contribute to a public good. If a group's collective contribution reaches a pre-defined threshold, all group members receive a reward. To account for possible asymmetries, individuals either differ in their endowments or their productivities. According to a theoretical equilibrium analysis, such games tend to have many possible solutions. In equilibrium, group members may contribute the same amount, different amounts or nothing at all. According to our behavioural experiment, however, humans favour the equilibrium in which everyone contributes the same proportion of their endowment. We use these experimental results to highlight the non-trivial effects of inequality on cooperation, and we discuss to which extent models of evolutionary game theory can account for these effects. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Keywords: asymmetric game; cooperation; coordination; evolutionary game theory; inequality; threshold public goods game.

Figure 1.

Basic set-up and predictions for a threshold public goods game. (a) We consider games between two players. In the beginning, players receive some fixed endowment (indicated by yellow coins). The players then independently decide how much of their endowment to contribute to a public good. The player’s effective contribution is their contribution times their individual productivity factor (indicated by the arrows). If the sum of the players’ effective contributions exceeds a threshold, both players receive a reward. We conduct experiments for five treatments. Players are either identical in all aspects, or they differ in their endowments, or they differ in their productivities. The treatment with full equality serves as our control. (b) To gain some insight into the logic of the game, we calculate the Nash equilibria of the one-shot game (these equilibria are marked by coloured dots). Each treatment allows for many Nash equilibria. These equilibria differ in whether or not the threshold is reached, and in how much the two players contribute. For better clarity, we highlight the most extreme Nash equilibria, by depicting the players’ respective contributions (c₁, c₂) in equilibrium.

Figure 2.

Main results of the experiment. (a) For each treatment, we first compute how likely groups are successful in obtaining the reward (i.e. how likely their collective contribution matches or exceeds the threshold). Dots indicate the average success rate of each individual group, averaged over all 20 rounds of the game. Compared to the treatment with full equality, both moderate and strong endowment inequality diminish a group’s success rate. In addition, also moderate productivity inequality has a weakly negative effect. (b) We next study the groups’ cooperation dynamics over time. To this end, we consider how often groups are successful (they match or exceed the threshold), and how often they are effective (they exactly match the threshold). In all treatments, players learn to better coordinate their contributions over time. However, in the treatments with endowment inequality, individuals find it more difficult to coordinate. (c,d) We then compare the players’ contributions across the five treatments. Dots again represent average absolute and relative contributions of the two players, averaged over the 20 rounds of the game. We observe that players contribute approximately equal amounts under full equality and under productivity inequality. In the treatments with endowment inequality, player 1 contributes more than player 2 in absolute terms (but less than player 2 relative to the players’ endowment). The error bars represent 95% confidence intervals.

Figure 3.

Modelling the evolution of strategies with replicator dynamics. To explore to which extent classical game dynamics are able to recover the previous empirical patterns, we consider a variety of dynamics (see the electronic supplementary material, S3 for details). Here, we report results for replicator dynamics, assuming that individuals interpret each round as an isolated one-shot game. In particular, a player’s possible strategies are all possible contributions between zero and the player’s endowment (for a repeated game analysis where players use conditional strategies, see the electronic supplementary material, S4). (a) First, we assume that initially, all strategies are played with equal frequency. In that case, we observe that eventually, either players do not contribute at all (full equality, endowment inequality), or that only the first player contributes (productivity inequality). (b) We reconsider the outcome of replicator dynamics with an initial population that matches the empirical first-round behaviour. We observe that now the model recovers the main empirical patterns. In particular, the solution according to replicator dynamics matches the most abundant experimental outcome in round 20 (see the electronic supplementary material, figure S6).