Cooperation and self-interest: Pareto-inefficiency of Nash equilibria in finite random games

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 and 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 a 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. In games with transferable utility, the numerical payoff values may be averaged across actions (so that mixed NEs are meaningful) and added across players. In simulations of two-player games when both players have small, equal numbers of actions, as the number of actions increases, the probability that a NE (pure and mixed) attains the cooperative maximum declines rapidly; the gain from cooperation relative to the Nash high value decreases; and the gain from cooperation relative to the Nash low value rises dramatically. In the cases studied here, with an increasing number of actions, cooperation is increasingly likely to become advantageous compared with pure self-interest, but self-interest can achieve all that cooperation could achieve in a nonnegligible fraction of cases. These results can be interpreted in terms of cooperation in societies and mutualism in biology.