Galton-Watson games

We address two-player combinatorial games whose graph of positions is a directed Galton–Watson tree. We consider normal and misère rules (where a player who cannot move loses or wins, respectively), as well as an “escape game” in which one designated player loses if either player cannot move. We study phase transitions for the probability of a draw or escape under optimal play, as the offspring distribution varies. Across a range of natural cases, we find that the transitions are continuous for the normal and misère games but discontinuous for the escape game; we also exhibit examples where these properties fail to hold. We connect the nature of the phase transitions to the length of the game under optimal play. We establish inequalities between the different games. For instance, the draw probability is no smaller in the misère game than in the normal game.