Equal Pay for All Prisoners

Abstract

- l By prisoners we mean, of course, players of the well-known Prisoner's Dilemma game (to be described presently). We shall show that there exist simple strategies for the infinitely iterated Prisoner's Dilemma that act as equalizers in the sense that all co-players receive the same payoff, no matter what their strategies are like. The Prisoner's Dilemma game, a favorite with game theorists, social scientists, philosophers, and evolutionary biologists, displays the vulnerability of cooperation in a minimalistic model (see [1] to [5]). The two players engaged in this game can choose whether to cooperate or to defect. If both defect, they gain 1 point each; if both cooperate, they gain 3 points; but if one player defects and the other does not, then the defector receives 5 points and the other player only 0. The right move is obviously to defect, no matter what the other player does. As a result, both players earn 1 point instead of 3. But if the same two players repeat the game very frequently, there exists no strategy that is best against all comers. The diversity of strategies is staggering. If we simulate on a computer populations of strategies evolving under a mutationselection regime (with mutation introducing new strategies and selection weening out those with lowest payoff), we observe a rich variety of evolutionary histories frequently leading to cooperative regimes dominated by strategies like Pavlov (cooperate whenever the opponent's move, in the previous round, matched yours) or Generous Tit For Tat (always reciprocate your opponent's cooperative move, but reciprocate only two-thirds of the defections). Remarkably, all strategies of the iterated Prisoner's Dilemma, which can be very complex and make up a huge set, obtain the same payoff against some rather simple equalizer strategies. More generally, let us consider a two-player game where both players have the same two strategies and the same payoff matrix. We denote the first strategy (row 1) by C (for 'cooperate') and the second (row 2) by D (for 'defect') and write the payoff matrix as