Indirect reciprocity with private, noisy, and incomplete information

Abstract

Indirect reciprocity is a mechanism for cooperation based on shared moral systems and individual reputations. It assumes that members of a community routinely observe and assess each other and that they use this information to decide who is good or bad, and who deserves cooperation. When information is transmitted publicly, such that all community members agree on each other's reputation, previous research has highlighted eight crucial moral systems. These "leading-eight" strategies can maintain cooperation and resist invasion by defectors. However, in real populations individuals often hold their own private views of others. Once two individuals disagree about their opinion of some third party, they may also see its subsequent actions in a different light. Their opinions may further diverge over time. Herein, we explore indirect reciprocity when information transmission is private and noisy. We find that in the presence of perception errors, most leading-eight strategies cease to be stable. Even if a leading-eight strategy evolves, cooperation rates may drop considerably when errors are common. Our research highlights the role of reliable information and synchronized reputations to maintain stable moral systems.

Keywords: cooperation; evolutionary game theory; indirect reciprocity; social norms.

Fig. 1.

Under indirect reciprocity, individual actions are continually assessed by all population members. (A) We consider a population of different players. All players hold a private repository where they store which of their coplayers they deem as either good (g) or bad (b). Different players may hold different views on the same coplayer. In this example, player 2 is considered to be good from the perspective of the first two players, but he is considered to be bad by player 3. (B) In the action stage, two players are randomly chosen, a donor (here, player 1) and a recipient (here, player 2). The donor can then decide whether or not to cooperate with the recipient. The donor’s decision may depend on the stored reputations in her own private repository. (C) After the action stage, all players who observe the interaction update the donor’s reputation. The newly assigned reputation may differ across the population even if all players apply the same social norm. This can occur (i) when individuals already disagreed on their initial assessments of the involved players, (ii) when some subjects do not observe the interaction and hence do not update the donor’s reputation accordingly, or (iii) when there are perception errors.

Fig. 2.

(A–H) When individuals base their decisions on noisy private information, their assessments may diverge. Models of private information need to keep track of which player assigns which reputation to which coplayer at any given time. These pairwise assessments are represented by image matrices. Here, we represent these image matrices graphically, assuming that the population consists of equal parts of a leading-eight strategy, of unconditional cooperators (ALLC) and unconditional defectors (ALLD). A colored dot means that the corresponding row player assigns a good reputation to the column player. Without loss of generality, we assume that ALLC players assign a good reputation to everyone, whereas ALLD players deem everyone as bad. The assessments of the leading-eight players depend on the coplayer’s strategy and on the frequency of perception errors. We observe that two of the leading-eight strategies are particularly prone to errors: L6 (“stern judging”) eventually assigns a random reputation to any coplayer, while L8 (“judging’) eventually considers everyone as bad. Only the other six strategies separate between conditionally cooperative strategies and unconditional defectors. Each box shows the image matrix after $2\cdot 10^6$ simulated interactions in a population of size $N=3\cdot 30=90$. Perception errors occur at rate $\epsilon =0.05$, and interactions are observed with high probability, $q=0.9$.

Fig. 3.

Most of the leading-eight strategies are disfavored in the presence of perception errors. We simulated the evolutionary dynamics when each of the leading-eight strategies competes with ALLC and ALLD. These simulations assume that, over time, players tend to imitate coplayers with more profitable strategies and that they occasionally explore random strategies (Materials and Methods). The numbers within the circles represent the abundance of the respective strategy in the selection–mutation equilibrium. The numbers close to the arrows represent the fixation probability of a single mutant into the given resident strategy. We use solid lines for the arrows to depict a fixation probability that exceeds the neutral probability $1/N$, and we use dotted lines if the fixation probability is smaller than $1/N$. In four cases, we find that ALLD is predominant (C–F). In one case (H), the leading-eight strategy coexists with ALLD but without any cooperation. In the remaining cases (A, B, and G), we find that L1 and L7 are played with moderate frequencies, but only populations that have access to L2 (“consistent standing”) settle at the leading-eight strategy. Parameters: Population size $N=50$, benefit $b=5$, cost $c=1$, strength of selection $s=1$, error rate $\epsilon =0.05$, observation probability $q=0.9$, in the limit of rare mutations $\mu \to 0$.

Fig. 4.

Noise can prevent the evolution of full cooperation even if leading-eight strategies evolve. We repeated the evolutionary simulations in Fig. 3, but varying (A) the benefit of cooperation, (B) the error rate, and (C) the observation probability. The graph shows the average cooperation rate for each scenario in the selection–mutation equilibrium. This cooperation rate depends on how abundant each strategy is in equilibrium and on how much cooperation each strategy yields against itself in the presence of noise. For five of the eight scenarios, cooperation rates remain low across the considered parameter range. Only the three other leading-eight strategies can persist in the population, but even then cooperation rates typically remain below 70%. We use the same baseline parameters as in Fig. 3.