Global analyses of evolutionary dynamics and exhaustive search for social norms that maintain cooperation by reputation

Abstract

Reputation formation is a key to understanding indirect reciprocity. In particular, the way to assign reputation to each individual, namely a norm that describes who is good and who is bad, greatly affects the possibility of sustained cooperation in the population. Previously, we have exhaustively studied reputation dynamics that are able to maintain a high level of cooperation at the ESS. However, this analysis examined the stability of monomorphic population and did not investigate polymorphic population where several strategies coexist. Here, we study the evolutionary dynamics of multiple behavioral strategies by replicator dynamics. We exhaustively study all 16 possible norms under which the reputation of a player in the next round is determined by the action of the self and the reputation of the opponent. For each norm, we explore evolutionary dynamics of three strategies: unconditional cooperators, unconditional defectors, and conditional cooperators. We find that only three norms, simple-standing, Kandori, and shunning, can make conditional cooperation evolutionarily stable, hence, realize sustained cooperation. The other 13 norms, including scoring, ultimately lead to the invasion by defectors. Also, we study the model in which private reputation errors exist to a small extent. In this case, we find the stable coexistence of unconditional and conditional cooperators under the three norms.

Introduction

Richard Alexander (1987) said that indirect reciprocity is “a consequence of direct reciprocity occurring in the presence of interested audiences”. The audiences repeatedly evaluate members in a society and judge who deserves help. Those who gain a good reputation receive donation from others while those who gain a bad reputation miss help. Cooperative act is passed from person to person via reputation. Hence, having good reputation or status is of great importance in indirect reciprocation (Fehr, 2004). Recently, much empirical work has been done to reveal the nature of indirect reciprocity and reputation formation in humans (e.g. Bolton et al., 2005; Milinski et al., 2001, Milinski et al., 2006; Wedekind and Milinski, 2000; Wedekind and Braithwaite, 2002). For a recent review on indirect reciprocity, we refer to Nowak and Sigmund (2005).

In theory, Nowak and Sigmund, 1998a, Nowak and Sigmund, 1998b investigated how indirect reciprocity works among individuals. Nowak and Sigmund (1998b) introduced binary reputation, either good or bad, to represent the social status of players. Individuals repeatedly play a Prisoner's Dilemma game with others, each time recruiting a different opponent from the society. There are two strategic choices in this game, cooperation or defection. Those who cooperate pay cost c for their opponent to receive benefit b. Those who choose defection pay nothing. Players do not interact with the same person more than once. According to the result of the game, observers assign a new reputation to players. The way to attach reputation is as follows. Those who cooperated in the previous interaction receive a good reputation. In contrast, those who refused to help others in the previous round gain a bad reputation. This rule of assigning reputation, or namely the “norm”, is called “scoring” (Brandt and Sigmund, 2004; Ohtsuki and Iwasa, 2004). Nowak and Sigmund (1998b) showed that under scoring conditional cooperators who help only good individuals are resistant to defectors, because they selectively give help only to cooperative members. Reputation works as media for community enforcement (Kandori, 1992).

The simplest as it is, scoring has a critical shortcoming in it. It cannot distinguish sanction from selfish defection. Scoring assigns a bad reputation to conditional cooperators who refused to help a bad person as punishment. Therefore, no conditional cooperators are motivated to give cheaters a penalty, which obviously leads to a triumph of defectors. Previous theoretical works confirmed that scoring is not able to sustain cooperation under errors (Panchanathan and Boyd, 2003, Ohtsuki, 2004, Ohtsuki and Iwasa, 2004) without an additional mechanism, such as growing social networks (Brandt and Sigmund, 2005).

Following this result, Ohtsuki and Iwasa, 2004, Ohtsuki and Iwasa, 2006 searched for combinations of a norm and a behavioral strategy that can maintain cooperation, among a huge number of possibilities. According to their formulation, a norm judges whether an observed action is good or bad taking the following three components into account: (i) the action of the focal player (cooperation or defection), (ii) the reputation of the opponent (good or bad), and (iii) the reputation of the focal player (good or bad). This type of norm is called third-order assessment (Brandt and Sigmund, 2005). Also, a player's behavioral strategy that prescribes the action (to cooperate or to defect) toward an opponent is conditional on (a) the reputation of the opponent (good or bad), and (b) the reputation of the focal player (good or bad). In this framework, Ohtsuki and Iwasa (2004) asked under which norm which behavioral strategy becomes an evolutionarily stable strategy (ESS) that realizes cooperation at a high level even under a small amount of errors. As a result, they found eight combinations of a norm and a behavioral strategy, called the “leading eight”, which were characterized in the subsequent paper (Ohtsuki and Iwasa, 2006).

While exhaustive ESS analyses have been completed by Ohtsuki and Iwasa, 2004, Ohtsuki and Iwasa, 2006, their analysis is restricted to the invasibility of the equilibrium dominated by a single strategy. No works have conducted a global analysis of evolutionary dynamics of strategies over all possible norms. One of the aims of the present paper is to obtain a complete classification of evolutionary dynamics over all possible norms. Here, we focus on the norms that are based on second-order assessment instead of third-order assessment (Brandt and Sigmund, 2005). That is, a norm in this category specifies whether a player is good or bad based on (i) the action of the focal player (cooperation or defection), and (ii) the reputation of the opponent (good or bad), but without using the previous reputation of the focal player. Similarly, we consider behavioral strategies that are conditional only on the reputation of the opponent (good or bad), but not on the reputation of the self. By such a simplification, we can reduce the total number of norms from 256 to 16 and the total number of behavioral strategies from 16 to 4, making the exhaustive examination of global behavior feasible. Out of all 16 conceivable norms we find that three norms, called “simple-standing”, “Kandori”, and “shunning”, realize sustained cooperation.

We also study the effect of private reputation errors in evaluating others in indirect reciprocation. If players individually and independently commit errors they result in having different opinions on the same person. Hence, reputation is not public information but can be a private opinion (that is, we must consider not only who is good but also who thinks who is good). Here, we study the effect of private reputation errors under a very simple assumption. The reputation of all the members in the population is determined publicly, but there is a small chance that each player makes a mistake in interpreting the reputation of others. We show that the existence of such private reputation errors induces the stable coexistence of unconditional and conditional cooperators under the three norms mentioned above.