Cognitive heterogeneity as a solution to the second-order free-rider problem

Costly peer punishment is thought to be a key solution to sustaining human cooperation and is frequently observed in Prisoner’s Dilemma experiments even in one-shot anonymous interactions. However, compared with those who do not punish, punishers get lower payoffs. Thus, the punishment of free riders constitutes a second-order free rider problem. We propose that cognitive heterogeneity — formalized through the Cognitive Hierarchy framework — offers a plausible account for this pattern. In this framework, non-strategic (step-0) players may punish simply as a result of random choice over the available strategies, whereas higher-level reasoners never punish but may cooperate in response to the perceived threat of punishment. This implies a non-monotonic relationship between the level of cooperation and the population’s average reasoning depth(τ). When τ is very low, most individuals are step-0 players who choose strategies uniformly at random; as a result, about 1/3 of people will defect. When τ is very high, nearly all agents are higher-level reasoners who correctly anticipate that no rational player will incur the cost of punishment; in this case, the outcome converges to the Nash equilibrium of universal defection. The highest cooperation rate emerges only at intermediate values of τ, where stochastic punishment by a minority of step-0 players creates a credible threat that induces rational cooperation among higher-level types.