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In this paper we introduce a novel framework that allows us to model games with players who reason about the opponents’ rationality only in some part of the game tree. We refer to this type of bounded rationality as limited focus. In particular, players try to rationalize their opponents’ moves only at the histories they focus on, i.e., formally, they strongly believe in their opponents’ rationality in these particular histories only. Our main result characterizes the strategy profiles that can be played under rationality and common strong belief in rationality by means of a simple elimination procedure, for every specification of the players’ focus. Finally, we present several special cases and applications of our framework, and we discuss how it differs from other forms of bounded perception such as unawareness.

Battigalli (1997) has shown that in dynamic games with perfect information and without relevant ties, the forward induction concept of extensive-form rationalizability yields the backward induction outcome. In this paper we provide a new proof for this remarkable result, based on four steps. We first show that extensive-form rationalizability can be characterized by the iterated application of a special reduction operator, the strong belief reduction operator. We next prove that this operator satisfies a mild version of monotonicity, which we call monotonicity on reachable histories. This property is used to show that for this operator, every possible order of elimination leads to the same set of outcomes. We finally show that backward induction yields a possible order of elimination for the strong belief reduction operator. These four properties together imply Battigalli's theorem.

Pearce’s (Econometrica 52:1029-1050, 1984) extensive-form rationalizablity (EFR) is a solution concept embodying a best-rationalization principle (Battigalli, Games Econ Behav 13:178-200, 1996; Battigalli and Siniscalchi, J Econ Theory 106:356-391, 2002) for forward-induction reasoning. EFR strategies may hence be distinct from backward-induction (BI) strategies. We provide a direct and transparent proof that, in perfect-information games with no relevant ties, the unique BI outcome is nevertheless identical to the unique EFR outcome, even when the EFR strategy profile and the BI strategy profile are distinct.

For dynamic games we consider the idea that a player, at every stage of the game, will always believe that his opponents will choose rationally in the future. This is the basis for the concept of common belief in future rationality, which we formalize within an epistemic model. We present an iterative procedure, backward dominance, that proceeds by eliminating strategies from the game, based on strict dominance arguments. We show that the backward dominance procedure selects precisely those strategies that can rationally be chosen under common belief in future rationality if we would not impose (common belief in) Bayesian updating.

In this chapter we focus on the epistemic concept of common belief in future rationality (Perea [37]), which describes a backward induction type of reasoning for general dynamic games. It states that a player always believes that his opponents will choose rationally now and in the future, always believes that his opponents always believe that their opponents choose rationally now and in the future, and so on, ad infinitum. It thus involves infinitely many conditions, which might suggest that this concept is too demanding for real players in a game. In this chapter we show, however, that this is not true. For finite dynamic games we present a finite reasoning procedure that a player can use to reason his way towards common belief in future rationality.

Within an epistemic model for two-player extensive games, we formalize the event that each player believes that his opponent chooses rationally at all information sets. Letting this event be common certain belief yields the concept of sequential rationalizability. Adding preference for cautious behavior to this event likewise yields the concept of quasi-perfect rationalizability. These concepts are shown to (a) imply backward induction in generic perfect information games, and (b) be non-equilibrium analogues to sequential and quasi-perfect equilibrium, leading to epistemic characterizations of the latter concepts. Conditional beliefs are described by the novel concept of a system of conditional lexicographic probabilities.

In this paper we explore game-theoretic reasoning in dynamic games within the framework of belief revision theory. More precisely, we focus on the forward induction concept of ‘common strong belief in rationality’ (Battigalli and Siniscalchi (2002) and the backward induction concept of ‘common belief in future rationality’ (Baltag et al. 2009; Perea 2014). For both concepts we investigate whether the entire collection of selected belief revision policies for a player can be characterized by a unique plausibility ordering. We find that this is indeed possible for ‘common strong belief in rationality’, whereas this may be impossible in some games for ‘common belief in future rationality’.

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents’ utility functions as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player, at information set h, should not change his belief about an opponent’s relative ranking of two strategies s and s′ if both s and s′ could have led to h, and (3) the players’ initial beliefs about the opponents’ utility functions should agree on a given profile u of utility functions. Common belief in these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given game tree with observable deviators and a given profile u of utility functions, every properly point-rationalizable strategy is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees with observable deviators and all profiles of utility functions. We provide an algorithm that can be used to compute the set of persistently rationalizable strategies for a given profile u of utility functions. For generic games with perfect information, persistent rationalizability uniquely selects the backward induction strategy for every player.

In this paper, we study two-person bargaining problems represented by a space of alternatives, a status quo point, and the agents' preference relations on the alternatives. The notion of a family of increasing sets is introduced, which reflects a particular way of gradually expanding the set of alternatives. For any given family of increasing sets, we present a solution which is Pareto optimal and monotonic with respect to this family, that is, it makes each agent weakly better off if the set of alternatives is expanded within this family. The solution may be viewed as an expression of equal-opportunity equivalence as defined in Thomson [Soc. Choice Welf. 11 (1994) 137–156]. It is shown to be the unique solution that, in addition to Pareto optimality and the monotonicity property mentioned above, satisfies a uniqueness axiom and unchanged contour independence. A noncooperative bargaining procedure is provided for which the unique backward induction outcome coincides with the solution.

This paper presents a formal epistemic framework for dynamic games in which players, during the course of the game, may revise their beliefs about the opponents' utility functions. We impose two key conditions upon the players' beliefs: (a) throughout the game, every move by the opponent should be interpreted as a rational move, and (b) the belief about the opponents' relative utilities between two terminal nodes should only be revised if you are sure that the opponent has decided to avoid one of these nodes. Common belief about these events leads to the concept of persistent rationalizability. It is shown that persistent rationalizability implies the backward induction procedure in generic games with perfect information. We next focus on persistently rationalizable types having beliefs with "minimal complexity", resulting in the concept of minimal rationalizability. For two-player simultaneous move games, minimal rationalizability is equivalent to the concept of Nash equilibrium strategy. In every outside option game, as defined by van Damme (1989), minimal rationalizability uniquely selects the forward induction outcome.

In Chapter 3 we have discussed some restrictions on assessments which, roughly speaking, reflect the requirement that players should believe that opponents act optimally. In the concepts of Nash equilibrium and normal form perfect equilibrium, this requirement is only imposed at the beginning of the game, whereas in other concepts such as backward induction, subgame perfect equilibrium and quasi-perfect equilibrium, it is imposed at every stage of the game. The concepts discussed so far did not explicitly involve the belief system, however, since explicit restrictions were put on the behavioral conjecture profile only. In this chapter we discuss some rationality criteria which impose explicit conditions on the belief system as well. We consider two types of restrictions. The first type, which we refer to as consistency conditions, should guarantee that the behavioral conjectures and the beliefs in a given assessment should not contradict one another. For instance, the beliefs held by a player at an information set should be in accordance with his beliefs held at earlier information set, which, in turn, should be compatible with the behavioral conjecture held by this player about the opponents’ behavior. Conditions of the second type are similar to those presented in Chapter 3, as they require players to attach positive probability only to those opponents’ strategies that constitute sequential best responses. By imposing this requirement at every information set we obtain the notion of sequential rationality. If we impose it only at those information sets that are not avoided by the initial conjectures, the weaker notion of weak sequential rationality results.