Asymptotic analysis of the evolutionary snowdrift game on a cycle
Date
2021-03
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Stellenbosch : Stellenbosch University
Abstract
ENGLISH ABSTRACT: Cooperation abounds in the natural world. Behaviour transcending selfishness has been witnessed among humans and in the animal kingdom throughout history. The underlying principles of this cooperation have become a focal point of study in the field of evolutionary game theory.The snowdrift game is a social dilemma in the form of a 2-player, 2-strategy game which has been used within this field in attempts to understand the precise nature of cooperation.The concept of population structure is employed in the field of evolutionary spatial game theory in attempts at investigating the occurrence and subsequent persistence of cooperation in competitive environments. Players are modelled as the vertices of a graph, representing structure amongst the players, in which pairs of players iteratively play games against each other over successive rounds if their corresponding vertices are adjacent in the graph structure. Adopting the basic learning assumption that players adopt playing strategies that mimic the best performing players in their neighbourhoods, the temporal dynamics of the (deterministic) evolutionary spatial snowdrift game(ESS) can be investigated.
The results of such an investigation are documented in this thesis. After adapting an existing mathematical model for analysing the temporal dynamics of another evolutionary spatial game to the context of the ESS, a similar analysis is conducted for the ESS played on cycle graphs.The investigation is conducted within the context of three game parametric regions in which the temporal game dynamics differ significantly from one another. For each region, the probability of persistent cooperation is determined. This requires a complete characterisation of randomly generated initial game states which lead to persistent cooperation. Bounds are also established on the fixation probabilities of the two strategies of the ESS, namely the strategies of cooperation and defection, and the relative magnitudes of these probabilities are compared for each of thethree aforementioned regions. Finally, the components of the ESS state graph, which captures all possible temporal dynamics of the ESS graphically, are enumerated in each of the parametric regions.In general, it is found that the probability of persistent cooperation increases with the order of the underlying cycle. Furthermore, in two of the three parametric regions, the strategy of cooperation is favoured above the strategy of defection, supporting the hypothesis that the strategy of cooperation dominates in the ESS played on cycles.
AFRIKAANSE OPSOMMING: Raadpleeg teks vir opsomming
AFRIKAANSE OPSOMMING: Raadpleeg teks vir opsomming
Description
Thesis (MEng)--Stellenbosch University, 2021.
Keywords
Evolutionary game theory, Snowdrift game, Cooperative behaviour, Asymptotic analysis