In the theory of dynamic games, it is often assumed that players reason about all stages in the game. In this paper we relax this assumption, by allowing players to only reason about some stages in the game not all. We take the forward induction concept of common strong belief in rationality (Battigalli and Siniscalchi (2002)) and adapt it to this framework. We also present an algorithm that yields precisely those conditional beliefs and strategies that are allowed by the concept we propose.
Introduction always verify whether for every opponent there exists a strategy which (a) is compatible with the history that has been observed so far, and (b) is optimal for that opponent at every stage of the game.
The aim of this paper is to relax this assumption, by allowing the players in a dynamic game to only reason about some stages in the game not all. Relaxing this assumption seems important for several reasons. First, it may be cognitively too demanding for a player to reason about all stages of the game, especially when the game is large. It may also be that some stages in the game seem more relevant than other stages, and that a player deliberately chooses to only reason about these stages in the game. Finally, there are some reasoning concepts in game theory that require a player to only actively reason about some stages of the game not all. Take, for instance, the backward induction concepts of subgame perfect equilibrium (Selten (1965)), sequential equilibrium (Kreps and Wilson (1982)) and common belief in future rationality (Perea (2014), see also Baltag, Smets and Svesper (2009) and Penta (2009)). These concepts either implicitly or explicitly assume that players are completely forward looking, in the sense that they only critically reason about the stages that lie ahead, and not about choices that have been made in the past.
Our formal approach is that we assume, for every player i; and every history h where player i has to move, some collection Fi(h) of stages, representing the stages that player i reasons about when he nds himself at h: We then take Battigalli and Siniscalchis (2002) notion of common strong belief in rationality as a bechmark model, and adapt this concept to the assumption that player i; at h; only reasons about the stages in Fi(h):
More precisely, let F be the reasoning mapping, which assigns to every player i; and to every history h where player i is active, the collection Fi(h) of histories that player i reasons about when he is at h. We say that player i strongly believes in the opponentsrationality relative to F if at every history h where player i is active, player i believes whenever possible that his opponents choose rationally at all stages in Fi(h), but not necessarily at stages not in Fi(h): That is, player i always believes whenever possible that his opponents choose optimally at those stages he actually reasons about.
If we choose F such that it always includes all histories so if we assume that players always reason about all stages then this de nition reduces to Battigalli and Siniscalchis (2002) notion of strong belief in rationality. If, on the other hand, Fi(h) only includes histories that follow h that is, if players only reason about future stages at any point in time then we obtain Pereas (2014) notion of belief in the opponents future rationality. But our de nition allows for much more variants, as we are completely exible in choosing the stages that players reasons about.
Suppose now we do not only require that players strongly believe in the opponentsrationality relative to F; but in addition also impose that players whenever possible believe that the opponents themselves also strongly believe in their opponents rationality relative to F: This is called two-fold strong belief in rationality relative to F: In a similar vain we can recursively de ne k-fold strong belief in rationality relative to F , for every k: We say that a player expresses common strong belief in rationality relative to F if he expresses k-fold strong belief in rationality relative to F; for every k:
This concept, which is at the heart of our paper, contains forward induction reasoning and backward induction reasoning as special cases. Indeed, if we choose F such that it always contains all histories, then common strong belief in rationality relative to F is equivalent to plain common strong belief in rationality, as de ned by Battigalli and Siniscalchi (2002), which is a forward induction concept. If we choose F such that Fi(h) only contains stages that follow h; then the concept reduces to common belief in future rationality, as developed in Perea (2014), which is a typical backward induction concept. Here, by forward induction we mean a type of reasoning in which a player whenever possible tries to nd a rational explanation for the choices that his opponents have made in the past. Backward induction, in contrast, represents a type of reasoning in which a player does not actively reason about the opponentspast choices, but instead focuses completely on the game that lies ahead. But our concept allows for many other interesting cases as well, by choosing di¤erent options for the F:
We then proceed by o¤ering an algorithm, which we call the iterated conditional dominance procedure relative to F , which delivers as an output precisely those conditional beliefs and those strategy choices that are possible under common strong belief in rationality relative to F: The algorithm starts by designing for every history h in the game, and for every player i who is active at h; the full decision problem for i at h: This full decision problem consists of (a) the possible strategies that i can choose at h; and (b) the possible strategy combinations that his opponents can choose at h: We refer to (a) as the possible decisions, and to (b) as the possible beliefs that i can hold at h: In every further step of the algorithm, we keep at every history h only those beliefs that whenever possible prescribe non-eliminated decisions for the opponents at all histories in Fi(h): Subsequently, we keep at h only those decisions that are optimal for a non-eliminated belief at h: We keep removing beliefs and decisions in this way until no further eliminations are possible. The algorithm is shown to always terminate within nitely many steps, and is relatively easy to apply.
If we choose F such that it always contain all histories, then the algorithm is equivalent to Shimoji and Watsons (1998) iterated conditional dominance procedure, which characterizes Battigalli and Siniscalchis concept of common strong belief in rationality. On the other hand, if Fi(h) only includes histories that follow h; then the algorithm reduces to Pereas (2014) backward dominance procedure, which characterizes the concept of common belief in future rationality.