A study of the dynamics of multi-player games on small networks using territorial interactions

Abstract

Citation: Broom, M., Lafaye, C., Pattni, K. & Rychtar, J. (2015). A study of the dynamics of multi-player games on small networks using territorial interactions.. Journal of Mathematical Biology, doi: 10.1007/s00285-015-0868-1 This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract Recently, the study of structured populations using models of evolutionary processes on graphs has begun to incorporate a more general type of interaction between individuals, allowing multi-player games to be played among the population. In this paper, we develop a birth-death dynamics for use in such models and consider the evolution of populations for special cases of very small graphs where we can easily identify all of the population states and carry out exact analyses. To do so, we study two multi-player games, a Hawk-Dove game and a public goods game. Our focus is on finding the fixation probability of an individual from one type, cooperator or defector in the case of the public goods game, within a population of the other type. We compare this value for both games on several graphs under different parameter values and assumptions, and identify some interesting general features of our model. In particular there is a very close relationship between the fixation probability and the mean temperature, with high temperatures helping fitter individuals and punishing unfit ones and so enhancing selection, whereas low temperatures give a levelling effect which suppresses selection. Permanent

Evolving multiplayer networks: Modelling the evolution of cooperation in a mobile population

“…In [9] it is assumed that the fitness contribution of individual I n only depends upon those individuals that it can directly interact with. The direct group (or simply the group) of individual I n , denoted G n (m), is the set of individuals that are present with it in the same place for population distribution m and is defined as follows Table 2.…”

“…In [9], the replacement weight contribution is independent of time and history, and depends only upon direct groups. This implies that the mean replacement weight is invariant over time and is as follows…”

“…A more general framework that can be used is that of [11] where it is possible to consider multiplayer interactions in groups of any size, depending upon various factors like the population's history, whilst keeping the beneficial aspects of evolutionary graph theory. More recently this framework has been used to model different kinds of multiplayer behaviour [13,9,12]. In this paper, we extend this work to consider a population of mobile individuals, focusing on a specific multiplayer game, a public goods game [6,7,25,55].…”

“…One potential problem with this is that individuals could spend more time with some of their neighbours, less with others and some time alone. The framework of [11] solves this problem as shown in [13,9] using a simple model where individuals are confined to their neighbourhood but are still allowed to form groups of different sizes. The framework, though, is capable of handling much more complex movement behaviour [1,2,21] where individuals make a choice of where to move given the information they have at hand.…”

The Territorial Raider game and graph derangements

“…In this paper, we provide a new criterion for the existence of graph derangements. We adapt the Territorial Raider game (see for example [2,3,4,5]) and establish a one-to-one correspondence between a derangement of a graph and a strict Nash equilibrium of the Territorial Raider game. This will prove that a simple, finite, undirected, connected graph G admits a derangement if and only if a Territorial Raider game played on G has a strict Nash equilibrium.…”