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

Qian Yu,Xiaoyan Wen,Ran Chen

doi:10.3390/e17041660

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.

Highlights

The emerging trend of cooperation among selfish individuals, which seem to prefer defection, has stimulated interest from researchers in the fields of biology and social science
The simulation results of the four deterministic dynamics and stochastic dynamic were demonstrated
We have investigated the influence of the level of individual rationality and the size of a population on the evolutionary dynamics of the voluntary prisoner’s dilemma (VPD) game

Summary

Introduction

The emerging trend of cooperation among selfish individuals, which seem to prefer defection, has stimulated interest from researchers in the fields of biology and social science. In the original PD game, each player can choose to be a cooperator (C) or a defector (D). Former players get a payoff T (temptation to defect), while the latter only get payoff S (sucker’s payoff). Mutual cooperation makes both players get the payoff R (reward for mutual cooperation), and mutual defection gives every player the payoff P (punishment for mutual defection). The condition R > P implies that mutual cooperation yields higher payoff than mutual defection, while T > R and P > S imply that each player will get higher payoff if he or she chooses to be a defector, regardless of what choice the opponent makes in a single round of the game. We require 2R > (T + S), because if the condition did not hold, cooperation would be eliminated, because players would be able to gain more by alternatively exploiting each other rather than cooperating

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Results

Conclusion