Evolutionary Stable Strategies in Interacting Communities

In many social networks or biological systems, several populations may be influenced by interactions with not only individuals of their own communities but also with other communities in the network. Moreover, the inter-community interactions can affect the outcome of a community in addition to the strategies followed inside the community itself. However, throughout the variety of research works in this topic, classical model of evolutionary games rather assumes individuals are equally likely to interact with any other member of the population and the success of any individual depends only on the frequency with which all strategies are represented in the population. Instead, we devote in this paper a novel mathematical tool to model the evolution of a population composed of several communities where the interaction between them is non-uniform. The main objective of our work is to model these interactions using evolutionary games and to find the evolutionary stable strategies (ESS). Hence, we show through the study of two communities and pairwise interactions that when taking into account the non-uniform interaction, any mixed Nash equilibrium is not an ESS wherein all communities are robust against a deviation of a fraction of the population from both communities which may wish to deviate. Moreover, the same mixed equilibrium is an ESS under some conditions when we consider the fitness of the whole population instead of the fitness of each community.