1. Department of Value and Decision Science, Tokyo Institute of Technology, Tokyo 152-8552, Japan
2. PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan
3. Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan
4. Program for Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA

Correspondence to: Hisashi Ohtsuki^1,2 Correspondence and requests for materials should be addressed to H.O. (Email: ohtsuki.h.aa@m.titech.ac.jp).

Abstract

Indirect reciprocity^{1, 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 reputation^{5, 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 behaviour^{18, 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.

1. Department of Value and Decision Science, Tokyo Institute of Technology, Tokyo 152-8552, Japan
2. PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan
3. Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan
4. Program for Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA

Correspondence to: Hisashi Ohtsuki^1,2 Correspondence and requests for materials should be addressed to H.O. (Email: ohtsuki.h.aa@m.titech.ac.jp).

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