Pierpaolo Battigalli

Universitŕ Bocconi

Context-Dependent Forward Induction Reasoning

(Joint work with Amanda Friedenberg)

Battigalli-Siniscalchi ("Strong Belief and Forward Induction Reasoning", JET 2002) formalize the idea of forward induction reasoning as "rationality and common strong belief of rationality" (RCSBR). Here, we study the behavioral implications of RCSBR across all type structures---we argue that, in so doing, we study the behavioral implications of context-dependent forward induction. Formally, we show that RCSBR is characterized by a solution concept we call Extensive Form Best Response Sets (EFBRS's). It turns out that the EFBRS concept is equivalent to a concept already proposed in the literature, namely Directed Rationalizability (see Battigalli-Siniscalchi, "Rationalization and Incomplete Information", Advances in Theoretical Economics, 2003). We conclude by applying the EFBRS concept to games of interest.

Giacomo Bonanno

University of California,Davis

AGM belief revision, Perfect Bayesian equilibrium and sequential equilibrium

The objective of this paper is to provide a foundation for extensive-form equilibrium concepts based on the so-called AGM theory of belief revision (developed by Alchourrón, Gärdenfors and Makinson, 1985). First we show that consistency with the AGM theory requires that the players’ ex ante beliefs and disposition to change those beliefs be represented by a total pre-order of the set of histories, which we call a “plausibility ordering”. When an information set is reached, the player’s revised beliefs are given by the set of most plausible histories among the ones that constitute this information set. Secondly, we use the plausibility ordering to provide a conceptually minimal definition of Perfect Bayesian equilibrium for general extensive games, which captures the idea of “applying Bayes’ rule whenever possible” (on and off the equilibrium paths) and show that this basic notion of Perfect Bayesian equilibrium is a refinement of subgame-perfect equilibrium. Thirdly, we provide a qualitative characterization of the notion of consistency proposed by Kreps and Wilson (as part of the definition of sequential equilibrium) in terms of a property of the plausibility ordering. Finally, we highlight the qualitative content of the independence properties implied by sequential equilibrium and use these qualitative properties to define a strengthening of the basic notion of Perfect Bayesian equilibrium and show that it is implied by, but weaker than, sequential equilibrium.

[Note: the paper that is linked to this abstract contains only a subset of the results mentioned above, namely the characterization of AGM-consistency and of Kreps-Wilson consistency. The full paper will be available later.]

Adam Brandenburger

Stern School of Business, New York University

Origins of Epistemic Game Theory

In 1935, Oskar Morgenstern wrote:
“[T]here is exhibited an endless chain of reciprocally conjectural reactions and counter-reactions…. The remedy would lie in analogous employment of the so-called Russell theory of types in logistics. This would mean that on the basis of the assumed knowledge by the economic subjects of theoretical tenets of Type I, there can be formulated higher propositions of the theory; thus, at least, of Type II. On the basis of information about tenets of Type II, propositions of Type III, at least, may be set up, etc.”
We will attempt to trace, from this promising start, the steps on the path to the development of epistemic game theory.

Alfredo Di Tillio

Bocconi University

Conditional Beliefs in Dynamic Games

(Joint work with Pierpaolo Battigalli and Dov Samet)

Most of the received literature on interactive epistemology for dynamic games uses models where each state of the world specifies the players' strategies as conjunctions of "objective" behavioral conditionals of the form "if history h occurs, then action a is chosen". The actual path that obtains at a state is the path induced by the strategies specified at that state. However, the intuitive interpretation of a strategy is that of a subjective contingent plan of action; players do not delegate their moves to devices that mechanically execute strategies, hence plans cannot be anything but beliefs of players about their own behavior. In this paper we analyze strategic reasoning in dynamic games with perfect information by means of epistemic models where states of the world describe the actual play path (not behavioral conditionals) and players’ conditional probability systems (CPS) about the path and about each other conditional beliefs. Players’ beliefs include their plans. Rational planning is a property of beliefs. Material consistency connects plans with actual choices. Material rationality is the conjunction of rational planning and material consistency. In perfect information games of depth two (the simplest dynamic games), correct belief in material rationality only implies a Nash outcome, not the backward induction one. We have to consider stronger assumptions of persistence of belief in material rationality in order to obtain backward and forward induction reasoning.

Songzi Du

Stanford University

Correlated Equilibrium via Hierarchies of Beliefs

(Joint work with Songzi Du)

We study a formulation of correlated equilibrium in which every player conditions his actions on his hierarchies of beliefs about the play of the game (belief on what other players will do, on what other players believe others will do, etc.). Our formulation can be thought of as a purification requirement based on hierarchies of beliefs. For any finite, complete information game, we are able to exactly characterize the strategic implications of correlated equilibria, both subjective and objective, in which players condition their actions on their hierarchies of beliefs. The characterizations are independent of type space and rely on a novel iterated deletion procedure. We show that "most" (objective) correlated equilibrium distributions can be obtained conditioning on hierarchies of beliefs; but interestingly, for generic two-person games, any non-degenerate mixed-strategy Nash equilibrium cannot be obtained. Therefore, we can purify "most" public randomizations, but not private randomizations, via hierarchies of beliefs.

Spyros Galanis

University of Southampton

Admissibility and Event-Rationality

(Joint work with Paulo Barelli)

Brandenburger et al. (2008) establish epistemic foundations for admissibility, or the avoidance of weakly dominated strategies, by using lexicographic type structures and the notion of rationality and common assumption of rationality (RCAR). Their negative result that RCAR is empty whenever the type structure is complete and continuous suggests that iterated admissibility (IA) requires players to have prior knowledge about each other, and therefore is a strong solution concept, not at the same level as iterated elimination of strongly dominated strategies (IEDS). We follow an alternative approach, using standard type structures and the notion of event-rationality. We characterize the set of strategies that are generated under event-rationality and common belief of event-rationality (RCBER) and show that, in a complete structure, it con- sists of the strategies that are admissible and survive iterated elimination of dominated strategies (Dekel and Fudenberg (1990)). By requiring that agents believe that themselves are E-rational at each level of mutual belief we construct and characterize RCBeER and show that in a complete structure it generates the IA strategies. Contrary to the negative result in Brandenburger et al. (2008), we show that RCBER and RCBeER are nonempty in complete, continuous and compact type structures, therefore providing an epistemic criterion for IA.

Joseph Halpern

Cornell University

A Logical Characterization of Iterated Admissibility

(Joint work with Rafael Pass)

Brandenburger, Friedenberg, and Keisler provide an epistemic characterization of iterated admissibility (i.e., iterated deletion of weakly dominated strategies) where uncertainty is represented using LPSs (lexicographic probability sequences). Their characterization holds in a rich structure called a complete structure, where all types are possible. I give a logical characterization of iterated admisibility that involves only standard probability and holds in all structures, not just complete structures. A stronger notion of strong admissibility is then defined. Roughly speaking, strong admissibility is meant to capture the intuition that ``all the agent knows'' is that the other agents satisfy the appropriate rationality assumptions. Strong admissibility makes it possible to relate admissibility, canonical structures (as typically considered in completeness proofs in modal logic), complete structures, and the notion of ``all I know''.

Ziv Hellman

Hebrew University of Jerusalem

Almost Common Priors

What happens when priors are not common? We show that for each type profile over a knowledge space, we can associate a non-negative value epsilon that we term the prior separation of of the space, and that there exist priors that are epsilon-almost common priors. The significance of these definitions is that if a space has epsilon prior separation, then under common knowledge the extent of possible disagreement of the players with respect to a random variable f is bounded by epsilon times the sup-norm of f. The results indicate that the geometry of the posteriors always imposes bounds on disagreement, extending no betting results under common priors. They also indicate that as more information is obtained, and partitions are refined, the extent of common knowledge disagreement decreases.

Willemien Kets


Do You Think About What I Think you Think? Finite Belief Hierarchies in Games

This paper models players with limited depths of reasoning. It
does so by constructing finite belief hierarchies. A key feature is that players' language is too coarse to conceive of higher levels than their own. The type space I construct embeds the universal type space with infinite hierarchies. As in the standard framework, a type corresponds to a belief over other players' types. However, players with limited depth of reasoning have a coarser language to ``talk'' about other players' types than more sophisticated players. Unlike in models of cognitive hierarchies or k-level reasoning, a player can believe that another player is at least as sophisticated as she is.

Martin Meier

Institut fur Hohere Studien, Wien, and Instituto de Análisis Económico, CSIC

Dynamic Unawareness and Rationalizable Behavior

(Joint work with Aviad Heifetzy and Burkhard C. Schipper)

We de ne generalized extensive-form games which allow for mutual unawareness of actions. We extend Pearce's (1984) notion of extensive-form (correlated) rationalizability to this setting, explore its properties and prove existence. We define also a new variant of this solution concept, prudent rationalizability, which refines the set of outcomes induced by extensive-form rationalizable strategies. We apply prudent rationalizability to the analysis of verifiable communication with unawareness. Finally, we define the normal form of a generalized extensive-form game, and characterize in it extensive-form rationalizability by iterative conditional dominance.

Andres Perea

Maastricht University

Belief in the opponents' future rationality

For dynamic games we consider the idea that a player, at every stage of the game, believes that his opponents will choose rationally in the future. Not only this, we also assume that players, throughout the game, believe that their opponents always believe that their opponents will choose rationally in the future, and so on. This leads to the concept of common belief in future rationality, which we formalize within an epistemic model. Our main contribution is to present an easy elimination procedure, backwards dominance, that selects exactly those strategies that can rationally be chosen under common belief in future rationality. The algorithm proceeds by successively eliminating strategies at every information set of the game. More specifically, in round k of the procedure we eliminate at a given information set h those strategies for player i that are strictly dominated at some player i information set h′ weakly following h, given the opponents' strategies that have survived at h′ until round k.

Miklos Pinter

Corvinus University of Budapest

The non-existence of a universal topological type space

The concept of types was introduced by Harsányi (1967-68). In the literature there are two approaches for formalizing types, type spaces: the purely measurable and the topological models. In the former framework Heifetz and Samet (1998) showed that the universal type space exists and later Meier (2001) proved that it is complete. In this paper we examine the topological approach and conclude that there is no universal topological type space in the category of topological type spaces.

Michael Trost

Albert Ludwig University of Freiburg

On the Equivalence of Iterated Application of a Choice Rule and Common Belief of Applying that Rule

In this paper solution concepts originating from an iterated application of some choice rule (IACR) are analyzed from a decision-theoretic perspective. The question is whether the solutions generated by IACR coincide with the solutions resulting from choice rule following behavior and common belief of that. In general, this equivalence does not hold, but we specify conditions on choice rules which ensure it. The conditions are positive, internal, conditional and marginal consistency. Prominent examples of choice rules satisfying those conditions are strict undominance in pure acts, strict undominance in mixed acts and Börgers' undominance concept. Furthermore, by providing examples, it is established that our result is weak in the sense that none of those conditions can be canceled without breaking up the above epistemic characterization of IACR.

Elias Tsakas

Maastricht University

A reasoning approach to introspection and unawareness

(Joint work with Olivier Gossner)

We introduce and study a unified reasoning process which allows to represent the beliefs of both a fully rational agent and an unaware one. This reasoning process endows introspection and unawareness with natural properties. The corresponding model for the rational or boundedly rational agents is easy both to describe and to work with, and the agent's full system of beliefs has natural descriptions using a reduced number of parameters.

Chih Chun Yang

University of Rochester

Weak Assumption and Admissibility

This paper examines the epistemic foundation of iterative admissibility. Brandenburger, Friedenberg, and Keisler [Econometrica 76(2008), 307-352] propose the notion of rationality and common assumption of rationality (RCAR) to characterize iterative admissibility (IA). They show that when players are not indifferent, RCAR is empty in a complete and continuous type structure. We propose a notion of weak assumption as an extention of knowledge. We show that rationality and common weak assumption of rationality (RCWAR) is nonempty in a complete, continuous and compact type structure. Moreover, the outcome of RCWAR is IA.

Akira Yokotani

University of Rochester

The Belief Hierarchy Representation of Harsanyi Type Spaces with Redundancy

In the standard Bayesian formulation of games of incomplete information, some types may represent the same hierarchy of beliefs over the given set of basic uncertainty. Such types are called redundant types. Redundant types present an obstacle to the Bayesian analysis because Mertens-Zamir approach of embedding type spaces into the universal type space can only be applied without redundancies. Also, because redundant structures provide different Bayesian equilibrium predictions (Liu, 2009), their existence has been an obstacle to the universal argument of Bayesian games. In this paper, we show that every type space, even if it has redundant types, can be embedded into a space of hierarchy of beliefs by adding an appropriate payoff irrelevant parameter space. And, for any type space, the parameter space can be chosen to be the space {0,1}. Moreover, Bayesian equilibrium is characterized by this ``augmented\'\' hierarchy of beliefs. In this process, we show that the syntactic characterization of types by Sadzik (2007) is essentially the same as whether or not they can be mapped to the same ``augmented\'\' hierarchy of beliefs. Finally, we show that the intrinsic correlation in Brandenburger-Friedenberg (2008) can be interpreted as a matter of redundant types and we can obtain their results in our framework.