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

Indirect reciprocity 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. 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. 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 regionwhere 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. Human societies are organized around cooperative interactions. But why would natural selection equip selfish individuals with altruistic tendencies? This question has fascinated evolutionary biologists for decades. One answer is given in terms of direct reciprocity. There are repeated encounters between the same two individuals: I help you, and you help me. More recently, indirect reciprocity has emerged as a more general model: I help you, and somebody helps me. Indirect reciprocity is based on reputation. People monitor the social interactions within their group. Helping others establishes the reputation of being a helpful individual. Natural selection can favour strategies that help those who have helped others. The consequences for widespread cooperation are enormous. Direct reciprocity is like an economy based on the exchange of goods, whereas indirect reciprocity resembles the invention of money. The money that feeds the engines of indirect reciprocity is reputation. For direct reciprocity, my strategy depends on what you have done to me; for indirect reciprocity, my strategy also depends on what you have done to others. Direct and indirect reciprocity are mechanisms for the evolution of cooperation. Punishment refers to an action that implies a cost for the punished person. Costly punishment means that the punisher also pays a cost for exercising punishment. In certain experimental situations costly punishment has been called ‘altruistic punishment’, because the punishers cannot expect any material gain from their action. In reality, however, most punishment actions among humans are associated with the expectation of a delayed material gain; they are therefore not altruistic. The suggested idea for the evolution of cooperation is that people might be more willing to cooperate under the threat of punishment. However, we note that costly punishment is not a separatemechanism for the evolution of cooperation but a form of direct or indirect reciprocity. If I punish you because you have defected with me, then I use direct reciprocity. If I punish you because you have defected with others, then indirect reciprocity is at work. In the setting of direct reciprocity, punishment is a form of retaliation. For indirect reciprocity, punishment works through reputation and also includes third-party actions, which means that observers of an interaction are willing to punish defectors at a cost to themselves. Therefore, any discussion of the evolution of costly punishment brings us immediately into the framework of direct or indirect reciprocity. In general, the reputation score could be a continuous variable, but here we consider a simple model with binary reputation. People have either a good reputation (G) or a bad reputation (B). At times, two random players are chosen from the population, one in the role of donor, the other in the role of recipient. The donor can either cooperate (C), defect (D) or punish (P). Cooperation means the donor pays a cost c, and the recipient gets a benefit b. Punishment implies that the donor pays a cost a and the recipient incurs a cost b. For defection there is no cost and no benefit. The interaction between the donor and the recipient is observed by the other members of the population (Fig. 1). The reputation of the donor is updated according to a social norm. First-order assessment depends only on the action of the donor; for example, cooperation leads to a good reputation, whereas defection leads to a bad reputation. Second-order assessment depends both on the action of the donor and the reputation of the recipient: for example, it could be deemed ‘good’ to cooperate with a good recipient but ‘bad’ to cooperatewith a bad recipient.Herewe study social norms that use secondorder assessment. The donor has three possiblemoves (C,D or P) and the recipient has one of two reputations (G or B). There are therefore six combinations and 25 64 social norms with second-order assessment. All detailed calculations are shown in the Supplementary Information. Any interaction leads to either a good or a bad reputation for the donor. We assume that this process of reputation updating is subject to errors. There may be wrong observations or the spread of false rumours. With probability m an incorrect reputation is assigned and adopted by all. In the simplest model, everyone has the same opinion of everyone else. There are no private lists of reputation. Triggering a wrong reputation affects everyone equally. The parameter q5 12 2m quantifies the ability of the population to distinguish between good