Stability in negotiation games and the emergence of cooperation†

Consider a two–player game in which each player contributes a costly resource to the common good of the pair. For such contests, the Nash equilibrium contribution, x⊃*, is one for which neither player can increase its pay–off by unilaterally altering its contribution from x⊃*. We study an elaboration of this game, which allows the players to exchange x–offers back and forth in a negotiation phase until they converge to a final pair of contributions, x◯⊂1 and x◯⊂2. A significant feature of such negotiation games, hitherto unrecognized, is the existence of a set of neutrally stable equilibrium points in negotiation phase space. To explore the long–term evolutionary outcome of such games, we simulate populations containing various mixtures of negotiation strategies and, contrary to previous results, we often find convergence to a contribution that is more cooperative than the Nash equilibrium. Mathematical analysis suggests why this might be happening, and provides a novel and robust explanation for cooperation, that negotiation can facilitate the evolution of cooperative behaviour.