On equilibria in games with imperfect recall

We generalize the modified multiselves approach of Piccione and Rubinstein to (multiplayer) games of imperfect recall. Four solution concepts are introduced: the multiselves agent equilibrium, the multiselves Nash equilibrium, the multiselves sequential equilibrium, and the multiselves perfect equilibrium. These modified equilibrium notions satisfy two important properties not fulfilled by the original ones. First, they always exist: every finite extensive game has at least one multiselves equilibrium of each type. Second, they form a strict hierarchy: every multiselves perfect equilibrium is a multiselves sequential equilibrium, every multiselves sequential equilibrium is a multiselves Nash equilibrium, and every multiselves Nash equilibrium is a multiselves agent equilibrium—but not conversely. Finally, in games of perfect recall, the multiselves equilibrium notions reduce to their original counterparts.