Finding Equilibria in Games of No Chance

We consider finding maximin strategies and equilibria of explicitly given extensive form games with imperfect information but with no moves of chance. We show: 1. A maximin pure strategy for a two-player extensive form game with perfect recall and no moves of chance can be found in time linear in the size of the game tree. In contrast, it is known that this problem is NP-hard for games with chance moves. 2. All pure Nash equilibrium outcomes of a two-player general-sum extensive form game with perfect recall and no moves of chance can be enumerated in time linear in the size of the game tree. In contrast, finding a behavior Nash equilibrium is known to be PPAD-hard. 3. Finding an optimal behavior strategy for a one-player game of no chance in extensive form without perfect recall is NP-hard. In contrast, finding an optimal pure strategy in such a game can be done in linear time. 4. Determining whether an equilibrium in behavior strategies exists in a two-player zero-sum extensive form game of no chance without perfect recall is NP-hard.