The competition of assessment rules for indirect reciprocity

Abstract

Indirect reciprocity is one of the basic mechanisms to sustain mutual cooperation. Beneficial acts are returned, not by the recipient, but by third parties. Indirect reciprocity is based on reputation and status: it pays to provide help because this makes one more likely to receive help in turn. The mechanism depends on knowing the past behavior of other players, and assessing that behavior. There are many different systems of assessing other individuals, which can be interpreted as rudimentary moral systems (i.e. views on what is ‘good’ or ‘bad’). In this paper, we describe the competition of some of the leading assessment rules called SUGDEN and KANDORI by analytic methods. We show that the sterner rule KANDORI has a slight advantage in the sense that KANDORI-players have more chance to earn higher payoff than SUGDEN-players in the presence of unconditional altruists. On the other hand, we see that the unconditional altruists are eliminated in the long run and that stable polymorphisms of KANDORI and SUGDEN can subsist, but that a moral consensus is realized even in those polymorphic states: all players’ images are the same in each observer's eyes.

Introduction

In indirect reciprocity, helpful acts are returned, not by the recipient, but by third parties (Alexander, 1987, Sugden, 1986, Trivers, 1971). If Alice helps Betty, then Alice is helped in turn, not by Betty, as in direct reciprocation, but by some Conny or Claire. Indirect reciprocity has been amply documented in human populations (Camerer and Fehr, 2006, Bolton et al., 2005, Seinen and Schram, 2001, Wedekind and Milinski, 2000, Wedekind and Braithwaite, 2002). In order not to be subverted by exploiters (for instance by defectors who never help others), the help must be channelled away from them, and directed preferentially towards the helpers. For this, two requirements are needed: (a) information about previous interactions, even those in which one has not been involved; and (b) an assessment of these interactions. Thus indirect reciprocity is based on constant monitoring of the other members of the population, and on judging whether they deserve to be helped or not, or in other words whether they have a good image or not (Leimer and Hammerstein, 2001; Nowak and Sigmund, 1998a, Nowak and Sigmund, 1998b; Panchanathan and Boyd, 2003). This can be viewed as an elementary form of moral judgment. Individuals assess other players’ actions as good or bad even if they are not directly affected by them.

The most elementary way for C to assess A simply reflects whether A gave help to B or not. In the first case, A is viewed as good and in the second case as bad. But this leads to an interesting inconsistency: if C refuses to help A, then C is perceived by third parties as bad irrespective of whether the potential recipient A is good or bad. As a result, C is less likely to be helped. Acting on a moral judgment can thus be costly. This suggests that a better assessment rule should also take into account whether a refusal to help was justified or not (see Camerer and Fehr, 2006, Leimer and Hammerstein, 2001, Nowak and Sigmund, 1998a, Sugden, 1986). However, there exist several ways for doing this, and it is not clear which assessment should evolve in the long term. To give an example: should the act of helping a bad individual be considered as good or as bad?

There are many possible moral systems. How do they compare? In a first approach, we may consider three different classes of assessment rules (Brandt and Sigmund, 2004). A first-order assessment rule only takes into account whether A helps B or not. A second-order assessment rule takes also into account the image of the recipient B. A third-order assessment rule takes moreover into account the image of the donor A. A strategy in the indirect reciprocity interaction consists of an assessment rule together with an action rule telling the player which decision to take, as a donor, depending on the image of the recipient and the own image (Brandt and Sigmund, 2004, Ohtsuki and Iwasa, 2004).

Ohtsuki and Iwasa have shown that among the 4096 resulting strategies, only eight lead to a stable regime of mutual cooperation, if adopted by all members of the population. These are said to be the leading eight (Ohtsuki and Iwasa, 2004, Ohtsuki and Iwasa, 2006). Two of these strategies are based on second-order assessment, none on first-order assessment. In this context, ‘stable’ means that the corresponding population cannot be invaded by other action rules. However, this does not settle the issue whether other assessment rules can invade. In the set-up considered by Ohtsuki and Iwasa, the image of an individual is the same in the eyes of all members of the population. Clearly, this does not allow to compare different assessment rules.

If one wants to analyze the evolution of even the simplest system of morals, one has to consider the competition of several assessment rules in the population. This is what we propose to do in the present paper: we consider the two second-order assessment rules belonging to the ‘leading eight’, as well as the first-order assessment rule which only registers whether help is given or not. We find that this first-order assessment rule is eliminated (not surprisingly), and that among the second-order assessment rules, the sterner rule has a slight advantage; if it is as frequent as the other rule (or more frequent), its payoff is at least as high. Stable polymorphisms of the two second-order assessment rules exist, but interestingly, the population always converges to a state where both assessments coincide: evolution leads to moral consensus.

In the following sections, we describe the model, derive the results, and discuss both outcomes and methods.