Forward Induction in a Backward Inductive Manner

We propose a new rationalizability concept for dynamic games with imperfect information, forward and backward rationalizability , that combines elements from forward and backward induction reasoning. It proceeds by applying the forward induction concept of strong rationalizability (also known as extensive-form rationalizability ) in a backward inductive fashion: It first applies strong rationalizability from the last period onwards, subsequently from the penultimate period onwards, keeping the restrictions from the last period, and so on, until we reach the beginning of the game. We argue that, compared to strong rationalizability, the new concept provides a more compelling theory for how players react to surprises. We show that the new concept always exists, and is characterized epistemically by (a) first imposing common strong belief in rationality from the last period onwards, then (b) imposing common strong belief in rationality from the penultimate period onwards, keeping the restrictions imposed by (a), and so on. It turns out that in terms of outcomes, the concept is equivalent to the pure forward induction concept of strong rationalizability, but both concepts may differ in terms of strategies. In terms of strategies, the new concept provides a refinement of the pure backward induction reasoning as embodied by backward dominance and backwards rationalizability . In fact, the new concept can be viewed as a backward looking strengthening of the forward looking concept of backwards rationalizability. Combining our results yields that every strongly rationalizable outcome is also backwards rationalizable. Finally, it is shown that the concept of forward and backward rationalizability satisfies the principle of supergame monotonicity : If a player learns that the game was actually preceded by some moves he was initially unaware of, then this new information will only refine, but never completely overthrow, his reasoning. Strong rationalizability violates this principle.