# The emergence of cooperation in spatial evolutionary games played on graphs

Thesis (MEng)--Stellenbosch University, 2018.

Thesis

ENGLISH ABSTRACT: The principle of cooperation pervades our society and the natural habitat in which we function. In the classic Darwinian framework of evolution, however, individuals rather tend to compete with one another because of a perceived fitness advantage, while cooperation requires altruistic behaviour. Hence, the emergence of cooperation is paradoxical. This leads to the following interesting question: How can cooperation emerge in a world of egoists without the interference of central authority? In game theory, the well-known prisoner's dilemma is often employed as a simplified hypothetical context in which to study cooperation and the factors that enable its persistence. Past studies have shown that cooperation may be a viable strategy if the prisoner's dilemma is placed within an evolutionary framework. In evolutionary game theory, games are repeated and players with bounded rationality and limited knowledge of these games are given the opportunity to learn and adapt their strategies iteratively. In such a context, one mechanism that enables the persistence of cooperation is the structure of interaction between players. A mathematical framework is proposed in this thesis for the prisoner's dilemma within an evolutionary game theoretic context, called the Evolutionary Spatial Prisoner's Dilemma (ESPD). This game is analysed on relatively simple graph structures in order to investigate the effect of various spatial player arrangements on the emergence of persistent cooperation. More specifically, analytical means (void computer aid) are employed to establish conditions for, and the likelihood of, persistent cooperation among players of the ESPD on a circulant graph, a natural extension of a cycle for which an analysis of the ESPD has already been analysed. The objective is to determine how the extension of each player's cyclic neighbourhood affects the likelihood of persistent cooperation when players are arranged in a cyclic topology. It is found that as players extend the sizes of their neighbourhoods from two to four players, the probability of the emergence of persistent cooperation decreases. A further analysis is carried out (this time with the aid of a computer) to investigate the conditions for, and the likelihood of, persistent cooperation in the ESPD on small toroidal grid graphs. The objective of this second analysis is to determine how the order of the underlying graph affects the likelihood of persist cooperation. It is found that for certain (pay-o value) parameter combinations, the probability of cooperation persisting increases as the order of the underlying graph increases, while for other parameter combinations this probability decreases.

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