Resonance-like cooperation due to transaction costs in the prisoner's dilemma game

Cooperation is omnipresent in the evolution of social species. In human societies, people voluntarily associate and cooperate with each other to exchange payments, which results in the inclusion of transaction costs during the process. This paper applies transaction costs to the spatial prisoner's dilemma game to better understand the evolution of cooperation by introducing an active–passive mechanism. In particular, a player who actively proposes the game should pay a transaction costs, while the passive responder pays nothing. Using Monte Carlo simulations and pair-approximation analysis, we investigate the frequency of cooperators in the stationary state for different values of the transaction cost, which maintain the same trend. It is found that there exists an optimal value of the transaction cost at which cooperation is optimally promoted. For small or large values of the transaction cost, the promotive effect decreases, and the evolution of cooperation may be impaired if compared to the absence of the active–passive mechanism. Finally, we explain the emergence of large clusters and theoretically confirm the existence of the optimal transaction cost. The mechanism of transaction costs enhancing cooperation resembles a resonance-like phenomenon, which may be helpful in understanding the cooperative behavior induced by the different behaviors between individuals in society.