Institute for Mathematical Behavioral Sciences

The relative merits of cooperation and self-interest in an ensemble of strategic interactions can be investigated by using finite random games. In finite random games, finitely many players have finite numbers of actions. Players receive independently and identically distributed (iid) random payoffs with continuous distribution functions. In each realization, players are shown the values of all payoffs and then choose their strategies simultaneously. Noncooperative self-interest is modeled by Nash equilibrium (NE). Cooperation is advantageous when the NE is Pareto-inefficient. In ordinal games, the numerical value of the payoff function gives each player's ordinal ranking of payoffs. For a fixed number of players, as the number of actions of any player increases, the conditional probability that a pure strategic profile is not pure Pareto-optimal, given that it is a pure NE, apparently increases, but is bounded above strictly below 1. With an increasing number of actions, cooperation is apparently increasingly likely to become advantageous compared with pure self-interest, but self-interest can achieve all that cooperation could achieve in a non-negligible fraction of cases. These results can be interpreted in terms of cooperation in societies and mutualism in biology. The fly in the ointment is that word "apparently". An inequality of monotonicity, while true numerically, remains unproved. I hope to recruit someone to prove the result.