Backward Induction in Games without Perfect Recall

The equilibrium concepts that we now think of as various forms of backwards induction, namely subgame perfect equilibrium (Selten, 1965), perfect equilibrium (Selten, 1975), sequential equilibrium (Kreps and Wilson, 1982), and quasi-perfect equilibrium (van Damme, 1984), are explicitly restricted their analysis to games with perfect recall. In spite of this the concepts are well defined, exactly as they defined them, even in games without perfect recall. There is now a small literature examining the behaviour of these concepts in games without perfect recall. We argue that in games without perfect recall the original definitions are inappropriate. Our reading of the original papers is that the authors were aware that their definitions did not require the assumption of perfect recall but they were also aware that without the assumption of perfect recall the definitions they gave were not the “correct” ones. We give definitions of two of these concepts, sequential equilibrium and quasi-perfect equilibrium, that identify the same equilibria in games with perfect recall and behave well in games without perfect recall.