The evolution of antisocial punishment in optional public goods games

Abstract

Cooperation, where one individual incurs a cost to help another, is a fundamental building block of the natural world and human society. It has been suggested that costly punishment can promote the evolution of cooperation, with the threat of punishment deterring free-riders. Recent experiments, however, have revealed the existence of 'antisocial' punishment, where non-cooperators punish cooperators. While various theoretical models find that punishment can promote the evolution of cooperation, these models a priori exclude the possibility of antisocial punishment. Here we extend the standard theory of optional public goods games to include the full set of punishment strategies. We find that punishment no longer increases cooperation, and that selection favours substantial levels of antisocial punishment for a wide range of parameters. Furthermore, we conduct behavioural experiments, showing results consistent with our model predictions. As opposed to an altruistic act that promotes cooperation, punishment is mostly a self-interested tool for protecting oneself against potential competitors.

Figure 1. Using the full punishment strategy set, anti-social punishment is common and punishment does not promote cooperation

Time-averaged frequencies of each strategy and transition rates between homogeneous populations, (a) without punishment, (b) when cooperators can punish defectors, and (c,d) with the full set of punishment strategies. A strategy X-Y₁Y₂Y₃ is defined by a public goods game action (X: C=cooperator, D=defector, L=loner), a punishment decision taken towards cooperators (Y₁: N = no action, P = punish), defectors (Y₂: N or P) and loners (Y₃: N or P). Transition rates ρ are the probability that a new mutant goes to fixation multiplied by the population size. We indicate neutral drift (ρ = 1, dotted lines), slow transitions (ρ = 11.8, thin lines), intermediate transitions (ρ = 25.7, medium lines) and fast transitions (ρ = 38.7, thick lines). Transitions with rates less than 0.1 are not shown. Parameter values are N = 100, n = 5, r = 3, c = 1, γ = 0.3, β = 1 and σ = 1. In panels c and d, strategies which punish others taking the same public goods action are included in the analysis, but not pictured because they are strongly disfavored by selection and virtually non-existent in the steady state distribution. For clarity, transitions with ρ < 10 are not shown in panel d.

Figure 2. Robustness of results across parameter sets

Shown are the results of 100,000 numerical calculations using N = 100, n = 5, c = 1 and randomly sampling from uniform distributions on the intervals 1 < r < 5, 0 < σ < (r-1)c, 0 < γ < 5, and γ < β < 5γ. Results are shown for (a) the no-punishment strategy set, (b) the restricted punishment strategy set where only cooperators can punish loners, and (c) the full punishment strategy set.

Figure 3. Inverse relationship between cooperation and anti-social punishment across parameters

(a) The steady state frequency of cooperation and anti-social punishment from 5,000 random parameter sets is shown. An inverse relationship is clearly visible: when antisocial punishment is rare, cooperation (and pro-social punishment) are common. (b) To explore this relationship, we vary r from (σ-c) to 5 for various values of σ, γ, and β, fixing N=100, n=5, and c=1. We see that increasing r always increases cooperation while decreasing anti-social punishment. We also see that when β is small, the range of cooperation and anti-social punishment values is large, whereas values are tightly constrained when β is larger. Achieving high levels of cooperation and low levels of anti-social punishment requires both large expected returns on public investment (large r) and symbolic punishments (small β).

Figure 4. Two behavioral experiments are consistent with model predictions

(a) In Experiment 1, subjects play an optional public goods game followed by a compulsory public goods game. The average fraction contributed in the compulsory game is significantly lower among subjects who opt out of the optional game (Rank-sum, N=73, p=0.006). Thus as predicted, loners contribute less in compulsory games. (b) In Experiment 2, subjects play an optional game followed by a compulsory game with costly punishment. Subjects indicate how much (0, 1 or 2) they would punish each possible contribution level in the compulsory game. Subjects who opt out of the optional game invest significantly more in punishing those that contribute the maximal amount (Rank-sum, N=196, p=0.003). Thus as predicted, loners engage in more anti-social punishment. All games are one-shot interactions among 4 players, with contributions to the public good multiplied by 2. Punishment technology is 3:1. Results are robust to various controls and alternate methods of analysis; see Supplementary notes for details.