Pure Nash equilibriums and independent dominating sets in evolutionary games on networks

Recently, a new model was developed in evolutionary game dynamics, extending the standard replicator equation to a finite set of players interacting on some network of connections. Players are modeled as the subpopulations of a multipopulation game, connected on an arbitrary graph, allowed to change their strategies over time and to adopt mixed strategies. A relation between the stability of the mixed equilibrium with the topology of the network was proved to depend on the eigenvalues of the graph's adjacency matrix multiplied by a scalar. The present paper studies the pure (strict) Nash equilibrium, (S)NE, of these games and their connection to the network. Specifically, we present necessary and sufficient conditions for a pure steady state in a coordination or anti-co ordination game to be a (strict) Nash equilibrium. A significant application of the model concerns finding the conditions for the onset of pure cooperative or pure defective behavior of players. In this direction, our main result provides conditions in an anti-co ordination game such that a pure steady state of the system is a Nash equilibrium if and only if the set of players that cooperate or the set of players that defect are independent dominating sets.