Indirect reciprocity provides only a narrow margin of efficiency for costly punishment

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Abstract
In human societies, altruistic behaviour can evolve when those who fail to co-operate acquire a bad reputation. But how far is it sensible to go on punishing those who don't cooperate, as punishment incurs a cost for both the punisher and the punished? Ohtsuki et al. explore the circumstances under which punishment is favoured using a game theory model in which all individuals observe the interactions between others and assess their reputation according to various social norms. They find that costly punishment can facilitate the evolution of cooperation, but only under a narrow set of parameters. As indirect reciprocity evolves to become more effective, costly punishment is rendered inefficient. In human societies, altruistic behaviour can evolve because those who fail to co-operate are lumbered with a bad reputation. This study explores the circumstances under which punishment is favoured using a game theory model in which all individuals observe the interactions between others and assess their reputation under various social norms. It is shown that punishment is only a successful strategy under a narrow set of parameters, including the relative costs of punishment and cooperation, the reliability of reputations and the spread of gossip. Indirect reciprocity1,2,3,4,5 is a key mechanism for the evolution of human cooperation. Our behaviour towards other people depends not only on what they have done to us but also on what they have done to others. Indirect reciprocity works through reputation5,6,7,8,9,10,11,12,13,14,15,16,17. The standard model of indirect reciprocity offers a binary choice: people can either cooperate or defect. Cooperation implies a cost for the donor and a benefit for the recipient. Defection has no cost and yields no benefit. Currently there is considerable interest in studying the effect of costly (or altruistic) punishment on human behaviour18,19,20,21,22,23,24,25. Punishment implies a cost for the punished person. Costly punishment means that the punisher also pays a cost. It has been suggested that costly punishment between individuals can promote cooperation. Here we study the role of costly punishment in an explicit model of indirect reciprocity. We analyse all social norms, which depend on the action of the donor and the reputation of the recipient. We allow errors in assigning reputation and study gossip as a mechanism for establishing coherence. We characterize all strategies that allow the evolutionary stability of cooperation. Some of those strategies use costly punishment; others do not. We find that punishment strategies typically reduce the average payoff of the population. Consequently, there is only a small parameter region where costly punishment leads to an efficient equilibrium. In most cases the population does better by not using costly punishment. The efficient strategy for indirect reciprocity is to withhold help for defectors rather than punishing them.