Ranking Friends ∗

We investigate the scope for cooperation within a community engaged in repeated reciprocal interactions. Players seek the help of others and approach them sequentially according to some fixed order, that is, a ranking profile. We study the ranking profiles that correspond to the social structures that are most effective in sustaining cooperation in equilibrium, that is, social structures that support full cooperation in equilibrium under the largest set of parameters. These are the profiles that spread the costs of helping others equally among the members of the community. We show that, generically, these socially optimal ranking profiles correspond to Latin squares – profiles in which each player appears in a given position exactly once in the list of each other player. In addition, we study equilibria with bilateral enforcement in which only the victims punish noncooperating deviators. We show that the Latin squares in which every two players rank each other at the same position characterize the social structures that sustain cooperation for the widest range of parameters in this case. ∗We are grateful to Nageeb Ali, David Kreps, and Brian Rogers for stimulating comments and useful suggestions. †Graduate School of Business, Stanford University. E-mail: [email protected]. ‡Kellogg School of Management, Northwestern University. E-mail: [email protected].