Galton–Watson games

We consider two-player combinatorial games in which the graph of positions is random and perhaps infinite, focusing on directed Galton-Watson trees. As offspring distribution varied, a game can undergo phase transition, probability draw under optimal play becomes positive. study nature transitions occur for normal rules (where player unable to move loses game) misere wins), as well an escape one tries force end while other prolong it forever. For instance, Poisson$(\lambda)$ distribution, tree infinite with positive soon $\lambda>1$, but has draws if only $\lambda>e$. The three generally have different critical points; certain assumptions are continuous discontinuous game, we also discuss cases where opposite possibilities occur. connect behaviour quantities such expected length play. establish inequalities relating each other; at least great game.