Random multi-player games

Abstract

The study of evolutionary games with pairwise local interactions has been of interest to many different disciplines. Also, local interactions with multiple opponents had been considered, although always for a fixed amount of players. In many situations, however, interactions between different numbers of players in each round could take place, and this case cannot be reduced to pairwise interactions. In this work, we formalize and generalize the definition of evolutionary stable strategy (ESS) to be able to include a scenario in which the game is played by two players with probability [Formula: see text] and by three players with the complementary probability [Formula: see text]. We show the existence of equilibria in pure and mixed strategies depending on the probability [Formula: see text], on a concrete example of the duel–truel game. We find a range of [Formula: see text] values for which the game has a mixed equilibrium and the proportion of players in each strategy depends on the particular value of [Formula: see text]. We prove that each of these mixed equilibrium points is ESS. A more realistic way to study this dynamics with high-order interactions is to look at how it evolves in complex networks. We introduce and study an agent-based model on a network with a fixed number of nodes, which evolves as the replicator equation predicts. By studying the dynamics of this model on random networks, we find that the phase transitions between the pure and mixed equilibria depend on probability [Formula: see text] and also on the mean degree of the network. We derive mean-field and pair approximation equations that give results in good agreement with simulations on different networks.