Adaptive exit facilitates the evolution of cooperation in the spatial prisoner's dilemma game with punishment

¹ School of Statistics and Mathematics, Yunnan University of Finance and Economics - Kunming 650221, China
² Interdisciplinary Research Institute of data science, Shanghai Lixin University of Accounting and Finance Shanghai 201209, China

^(b) E-mail: shi_lei65@hotmail.com (corresponding author)

Received: 25 September 2022
Accepted: 5 January 2023

Abstract

The enforcement of costly punishment can facilitate cooperation. However, it is unclear how punishment would influence cooperation if individuals are free to exit an interaction. In this work, individuals are allowed to exit with a certain probability, which is adaptively adjusted based on the difference between expectations and payoffs. According to simulation results, adaptive exit can significantly improve cooperation even when punishment is weak; moreover, there is an optimal level of aspiration that causes adaptive exit to best promote the evolution of cooperation. Because of reciprocity, cooperative individuals maintain the lowest level of exit probability, allowing the relationship between them to be maintained, and the interaction between cooperative individuals and defectors is affected by adaptive exit processing, which is related to aspiration levels. Results show that cooperation prevails under moderate aspiration levels, because the exit mechanism protects cooperative individuals from exploitation by defectors; however, the exit mechanism does not prevent the invasion of cooperators by defectors when the aspiration level is low. The negative impact of sanctions causes defectors to increase their exit probability, and the exit of the defectors prevents nearby punishers from exploitation as well as reduces the implementation of punishment, thereby easing the punishment burden on the punisher. Therefore, the adaptive exit mechanism helps to eliminate second-order free-riding at a lower aspiration level. The study identifies the significant effects of exit on punishment, and highlights the important role of their association in the evolution of cooperation.