Random multi-player games

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 can not 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 $p$, and by three players with the complementary probability $1-p$. We show the existence of equilibria in pure and mixed strategies depending on the probability $p$, on a concrete example of the duel-truel game. We find a range of $p$ values for which the game has a mixed equilibrium and the proportion of players in each strategy depends on the particular value of $p$. We prove that each of these mixed equilibrium points are 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 the probability $p$ and also on the mean degree of the network.