  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.
