An Empirical Game-Theoretic Analysis of the Dynamics of Cooperation in Small Groups

Many models of the evolution of cooperation have shown the importance of direct reciprocity (for example "tit for tat" strategies) or alternatively indirect reciprocity (conspicuous altruism based on a reputation or "image score"). In the latter case many models make the implicit assumption that group sizes are large relative to the expected number of interactions, which makes their analysis more tractable in several ways, not least by allowing us to ignore any strategic interaction between the direct and indirect classes of reciprocation strategy. However, in smaller groups the possibility arises that both classes of strategy will play a role in determining the equilibrium behaviour. Therefore we introduce a replicator dynamics model which incorporates both direct and indirect reciprocity, and use simulation and numerical methods to quantitatively assess how the level of cooperation in equilibrium is affected by changes in the group size and the frequency with which other group members are encountered. Our analysis shows that, for intermediate group sizes, direct reciprocity persists in equilibrium alongside indirect reciprocity. In contrast to previous simulation studies, we provide a sound game-theoretic underpinning to our analysis, and examine the precise conditions which give rise to a mix of both forms of reciprocity.