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 ofbackwards 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

Backward induction reasoning beyond backward induction

Backward Induction is a fundamental concept in game theory. As an algorithm, it can only be used to analyze a very narrow class of games, but its logic is also invoked, albeit informally, in several solution concepts for games with imperfect or incomplete informa-tion (Subgame Perfect Equilibrium, Sequential Equilibrium, etc.). Yet, the very meaning of ‘backward induction reasoning’ is not clear in these settings, and we lack a way to apply this simple and compelling idea to more general games. We remedy this by introducing a solution concept for games with imperfect and incomplete information, Backwards Rational-izability, that captures precisely the implications of backward induction reasoning. We show that Backwards Rationalizability satisfies several properties that are normally ascribed to backward induction reasoning, such as: (i) an incomplete-information extension of subgame consistency (continuation-game consistency); (ii) the possibility, in finite horizon games, of being computed via a tractable backwards procedure; (iii) the view of unexpected moves as mistakes; (iv) a characterization of the robust predictions of a ‘perfect equilibrium’ notion that introduces the backward induction logic and nothing more into equilibrium analysis. We also discuss a few applications, including a new version of peer-confirming equilibrium (Lipnowski and Sadler (2019)) that, thanks to the backward induction logic distilled by Backwards Rationalizability, restores in dynamic games the natural comparative statics the original concept only displays in static settings

Backward induction reasoning beyond backward induction

Backward Induction is a fundamental concept in game theory. As an algorithm, it can only be used to analyze a very narrow class of games, but its logic is also invoked, albeit informally, in several solution concepts for games with imperfect or incomplete informa-tion (Subgame Perfect Equilibrium, Sequential Equilibrium, etc.). Yet, the very meaning of ‘backward induction reasoning’ is not clear in these settings, and we lack a way to apply this simple and compelling idea to more general games. We remedy this by introducing a solution concept for games with imperfect and incomplete information, Backwards Rational-izability, that captures precisely the implications of backward induction reasoning. We show that Backwards Rationalizability satisfies several properties that are normally ascribed to backward induction reasoning, such as: (i) an incomplete-information extension of subgame consistency (continuation-game consistency); (ii) the possibility, in finite horizon games, of being computed via a tractable backwards procedure; (iii) the view of unexpected moves as mistakes; (iv) a characterization of the robust predictions of a ‘perfect equilibrium’ notion that introduces the backward induction logic and nothing more into equilibrium analysis. We also discuss a few applications, including a new version of peer-confirming equilibrium (Lipnowski and Sadler (2019)) that, thanks to the backward induction logic distilled by Backwards Rationalizability, restores in dynamic games the natural comparative statics the original concept only displays in static settings