Tolerant indirect reciprocity can boost social welfare through solidarity with unconditional cooperators in private monitoring

Abstract

Indirect reciprocity is an important mechanism for resolving social dilemmas. Previous studies explore several types of assessment rules that are evolutionarily stable for keeping cooperation regimes. However, little is known about the effects of private information on social systems. Most indirect reciprocity studies assume public monitoring in which individuals share a single assessment for each individual. Here, we consider a private monitoring system that loosens such an unnatural assumption. We explore the stable norms in the private system using an individual-based simulation. We have three main findings. First, narrow and unstable cooperation: cooperation in private monitoring becomes unstable and the restricted norms cannot maintain cooperative regimes while they can in public monitoring. Second, stable coexistence of discriminators and unconditional cooperators: under private monitoring, unconditional cooperation can play a role in keeping a high level of cooperation in tolerant norm situations. Finally, Pareto improvement: private monitoring can achieve a higher cooperation rate than does public monitoring.

Figure 1

Two monitoring systems in indirect reciprocity. (a) Private monitoring: each potential observer can observe a game independently with the probability, q, and otherwise cannot. The actual observers privately assess a donor of the game and the non-observers never update their assessments of the donor. (b) Public monitoring: a representative device observes a game and delivers its public assessment of a donor in the game to all of players with the probability, q; otherwise, the device does not observe the game and the assessment of any player is never updated.

Figure 2

Cooperation rates and strategies in a private monitoring system with different benefits, b. Each dot reflects an average of 100 trials after 30 generations with six values of b. (a) staying, (b) simple-standing, (c) image-scoring, (d) stern-judging, and (e) shunning. The error bars represent one standard deviation of the data. In the initial population, 10% are unconditional cooperators, 10% are unconditional defectors, and 80% are discriminators. The parameter values are N = 100, q = 0.01, c = 1, e ₁ = 0.03, e ₂ = 0.03, T = 100,000, T _s = 90,000, μ = 0.001, and β = 3.

Figure 3

Cooperation rates and strategies in a private monitoring system with different mutation rate, μ. Each dot reflects an average of 100 trials after 30 generations with five values of μ. (a) staying, (b) simple-standing, (c) image-scoring, (d) stern-judging, and (e) shunning. The error bars represent one standard deviation of the data. In the initial population, 10% are unconditional cooperators, 10% are unconditional defectors, and 80% are discriminators. The parameter values are N = 100, q = 0.01, b = 3, c = 1, e ₁ = 0.03, e ₂ = 0.03, T = 100,000, T _s = 90,000, and β = 3.

Figure 4

Basin of a cooperative regime in the private monitoring system. (a) staying and (b) simple-standing. The triangles describe a simplex of the state space, S = {(x, y, z): x + y + z = N}, where x, y, and z are non-negative integers denoting the frequencies of unconditional cooperators, unconditional defectors, and discriminators, respectively. The colored dots correspond to the probability of finally reaching a cooperative regime. Here, ten trials were performed on each point, (x, y, z) ∈ S. The borders of the basin of attraction are approximately (a) z = 0.15 and (b) z = 0.4. The averages of the probabilities for all of the points to finally reach a cooperative regime are (a) 76.04% and (b) 33.37%. The parameter values are N = 100, b = 3, c = 1, e ₁ = 0.03, e ₂ = 0.03, T = 100,000, T _s = 90,000, μ = 0, and β = 3.

Figure 5

Generation lengths for keeping cooperative regimes. (a and b) staying, (c and d) simple-standing, (e and f) image-scoring. (left) μ = 0.1% and (right) μ = 1%. In each trial, a run was broken off if either the cooperation rate was smaller than 0.2 or it reaches 500 generations. The histogram data are the generation lengths of 100 trials in each case. Each initial population consists of 100% discriminators. The parameter values are N = 100, q = 0.01, c = 1, e ₁ = 0.03, e ₂ = 0.03, T = 100,000, T _s = 90,000, and β = 3.

Figure 6

Cooperation rates of stable states with private monitoring and public monitoring. (a) staying and (b) simple-standing. Each dot reflects an average of 100 trials after 30 generations with six values of benefits, b. Each initial population consists of 50% unconditional cooperators and 50% discriminators. The cooperation rate with private monitoring exceeds that with public monitoring if b 2.0 in both (a) and (b). The parameter values are N = 100, q = 0.01, c = 1, e ₁ = 0.03, e ₂ = 0.03, T = 100,000, T _s = 90,000, μ = 0, β = 3, and q = 0.01 in (a) and q = 1 in (b).

Figure 7

Cooperation rates of stable states with different mutation rate, μ. (a) staying and (b) simple-standing. The cooperation rate with private monitoring exceeds that with public monitoring if μ < 0.01 in both (a and b). The simulation settings are the same as Fig. 6 except for b = 3 and μ (a variable).