The evolutionary spatial prisoner's dilemma on a cycle

Ap Burger,M Van Der Merwe,Jh Van Vuuren

doi:10.5784/29-1-126

Abstract

In this paper we consider the Evolutionary Spatial Prisoner's Dilemma (ESPD) in which players are modelled by the vertices of a cycle representing a spatial or organisational structure amongst the players. During each round of the ESPD every pair of adjacent players in the cycle play a classical prisoner's dilemma against each other, and they update their strategies from one round to the next based on the perceived success achieved by the strategies of neighbouring players during the previous round. In this way players are able to adapt and learn from each other's strategies as the game progresses without being able to rationalise good strategies. We characterise all steady states of the game as well as the structures of those initial states that lead to the emergence of persistent substates of cooperation over time. We finally determine analytically (i.e. without using simulation) the probability that the game's states will evolve from a randomly generated initial state towards a steady state which accommodates some form of persistent cooperation. More specifically, we show that there exists a range of game parameter values for which the likelihood of the emergence of persistent cooperation increases to almost certainty as the length of the cycle increases.

Highlights

The Prisoner’s Dilemma (PD) may be attributed to a 1950 lecture by Albert W Tucker [10] and is the archetypal example of a two-person zero-sum game in classical game theory
In this paper we presented an asymptotic analysis of the Evolutionary Spatial Prisoner’s Dilemma (ESPD) on a cycle
We showed that interesting structures of persistent cooperation are possible if the sum of the temptationto-defect parameter and the punishment parameter, a+b, in (1) is not too large

Summary

Introduction

The Prisoner’s Dilemma (PD) may be attributed to a 1950 lecture by Albert W Tucker [10] and is the archetypal example of a two-person zero-sum game in classical game theory. The static games of classical game theory, such as the PD described above, are repeated and players are afforded the possibility of adapting and learning good strategies as a result of achieving high pay-off values as the game progresses, rather than having to decide on a rigid strategy beforehand. The objective in this paper is to establish analytically the likelihood that a randomly generated initial state will result in the ESPD terminating in a steady state where the strategy of cooperation is able to persist in some structural form from one round to the if the players are arranged cyclically (i.e. the underlying graph is a cycle). This paper may be viewed as an extension of that work

The game dynamics

Automorphism classes of game states

The state graph of the ESPD

Steady states of game states

The probability of persistent cooperation

Conclusion