Equilibria in Finite Games with Imperfect Recall

We generalize traditional equilibrium concepts for finite games in extensive form with behavioral strategies so that they apply to all games, including games of imperfect recall. Adapting and augmenting previous de finitions (in particular, by Piccione and Rubinstein, and by Battigalli), we de fine four notions: Distributed Agent Equilibrium (DAE), Distributed Nash Equilibrium (DNE), Distributed Sequential Equilibrium (DSE), and Distributed Perfect Equilibrium (DPE). We show that, in a precise sense, these extend the classical equilibrium notions: (a) they form a strict inclusion hierarchy (e.g., every DNE is a DAE but not necessarily vice versa, and so on up the hierarchy), (b) every game has a DPE (and therefore also a DSE, DNE and DAE), and (c) in the subclass of games of perfect recall, DAE, DNE, DSE and DPE collapse, respectively, to agent equilibrium, Nash equilibrium, sequential equilibrium, and perfect equilibrium. In service of these results we introduce several novel notions - including the distributed agent form and phantom strategies - which may be interesting in their own right