Evolutionary Voluntary Prisoner’s Dilemma Game under Deterministic and Stochastic Dynamics

Abstract

The voluntary prisoner’s dilemma (VPD) game has sparked interest from various fields since it was proposed as an effective mechanism to incentivize cooperative behavior. Current studies show that the inherent cyclic dominance of the strategies of the VPD game results in periodic oscillations in population. This paper investigated the influence of the level of individual rationality and the size of a population on the evolutionary dynamics of the VPD game. Different deterministic dynamics, such as the replicator dynamic, the Smith dynamic, the Brown-von Neumann-Nash (BNN) dynamic and the best response (BR) dynamic, for the evolutionary VPD game were modeled and simulated. The stochastic evolutionary dynamics based on quasi birth and death (QBD) process was proposed for the evolutionary VPD game and compared with deterministic dynamics. The results indicated that with the increase of the loners’ fixed payoff, the loner is more likely to remain in the stable state of a VPD game under any of the dynamics mentioned above. However, the different speeds of motion under the dynamics in the cycle dominance proved to be diverse under different evolutionary dynamics and also highly sensitive to the rationality of individuals in a population. Furthermore, in QBD stochastic dynamics, the size of the population has a remarkable effect on the possibility distribution. When the population size increases, the limited distribution of the QBD process will be in accordance with the results in the deterministic dynamics.

Keywords

• voluntary prisoner’s dilemma
• evolutionary dynamic
• quasi birth and death process