Stochastic Evolution Dynamic of the Rock-Scissors-Paper Game Based on a Quasi Birth and Death Process.

Qian Yu,Debin Fang,Chen Jin,Qiyu Ren,Xiaoling Zhang

doi:10.1038/srep28585

Abstract

Stochasticity plays an important role in the evolutionary dynamic of cyclic dominance within a finite population. To investigate the stochastic evolution process of the behaviour of bounded rational individuals, we model the Rock-Scissors-Paper (RSP) game as a finite, state dependent Quasi Birth and Death (QBD) process. We assume that bounded rational players can adjust their strategies by imitating the successful strategy according to the payoffs of the last round of the game, and then analyse the limiting distribution of the QBD process for the game stochastic evolutionary dynamic. The numerical experiments results are exhibited as pseudo colour ternary heat maps. Comparisons of these diagrams shows that the convergence property of long run equilibrium of the RSP game in populations depends on population size and the parameter of the payoff matrix and noise factor. The long run equilibrium is asymptotically stable, neutrally stable and unstable respectively according to the normalised parameters in the payoff matrix. Moreover, the results show that the distribution probability becomes more concentrated with a larger population size. This indicates that increasing the population size also increases the convergence speed of the stochastic evolution process while simultaneously reducing the influence of the noise factor.

Highlights

Cyclic competition is a very common phenomenon in nature and society, with the simplest and most studied model being the Rock-Scissors-Paper (RSP) game[1]
It is verified that this game has no Nash equilibrium in pure strategies and precisely one Nash equilibrium in mixed strategies, namely the strategy pair in which both players randomise uniformly, i.e., (x1⁎,Fxu2r⁎t,hxe3⁎rm) =ore(,y1t⁎h,ey2p⁎a, yyo3⁎f)f
Comparison of these results indicates that: (1) Observing Fig. (3a,3d,3g), we can see when δ = 1 or δ > 0, the heat maps appear as a target, the red points representing the states with higher distribution probability gathered around the centre of this simplex regardless of population size

Summary

Introduction

Cyclic competition is a very common phenomenon in nature and society, with the simplest and most studied model being the Rock-Scissors-Paper (RSP) game[1]. This can be characterised by three strategies: R (rock), S (scissors) and P (paper), where R excludes S, S excludes P and P excludes R. Around the 1990 s, Hofbauer[12] studied the evolutionary stability of the Nash equilibrium under the replicate dynamic and proved that the Nash equilibrium of the RSP game is just the fixed point of the replicate dynamic. The random fluctuations due to sampling effects, for instance, have to be taken into account and stochastic processes, such as the Moran process and the Wright-Fisher process, must be employed to analyse the equilibrium of RSP16,17 instead of ordinary differential equations

Methods

Results

Conclusion