HIERARCHIES OF CONDITIONAL BELIEFS and

The New Palgrave Dictionary of Economics, 2008

Economics Letters, 2003

Following Tan and Werlang [Journal of Economic Theory 45 (1988) 370], we consider games as collections of decision problems, in which the uncertainty facing any player is the strategy choice of the other players, their beliefs about the other players' strategy choice, and so on. A distinctive feature of our approach is that we model players' beliefs as infinite hierarchies of compact possibility sets, rather than probability measures as in the Bayesian setup. Within this framework, we derive an axiomatic foundation for the point-rationalizability concept proposed by Bernheim [Econometrica 52 (1984) 1007].

Games

Epistemic game theory and the systems of logic that support it are crucial for understanding rational behavior in interactive situations in which the outcome for an agent depends, not just on her own behavior, but also on the behavior of those with whom she is interacting. Scholars in many fields study such interactive situations, that is, games of strategy. Epistemic game theory presents the epistemic foundations of a game's solution, taken as a combination of strategies, one for each player in the game, such that each strategy is rational given the combination. It considers the beliefs of the players in a game and shows how, along with the players' goals, their beliefs guide their choices and settle the outcome of their game. Adopting the Bayesian account of probability, as rational degree of belief, it yields Bayesian game theory. Epistemic game theory, because it attends to how players reason strategically in games, contrasts with evolutionary game theory, which applies to non-reasoning organisms such as bacteria. Logic advances rules of inference for strategic reasoning. It contributes not just standard rules of deductive logic, such as modus ponens, but also rules of epistemic logic, such as the rule going from knowledge of a set of propositions to knowledge of their deductive consequences, and rules of probabilistic reasoning such as Bayesian conditionalization, which uses probabilities conditional on receiving some new evidence to form new non-conditional probabilities after receiving exactly that new evidence. Perea [1] offers an overview, and Weirich [2] shows how principles of choice support solutions to games of strategy. The papers in the special issue came in response to the journal's call for papers. Diversity of perspectives was a goal. The papers include four by economists, one by computer scientists, three by philosophers, and one by a psychologist. They display a variety of approaches to epistemic game theory and logic. The following paragraphs briefly describe the topics of the papers, grouped according to discipline and within a discipline according to date of publication.

2009

We analyze situations of dynamic strategic interaction with incomplete information where certain restrictions on players'initial or conditional beliefs are "transparent", that is, not only the restrictions hold, but there is common belief in conditional on every history. Most applied models of asymmetric information are covered as a special case whereby pins down the probabilities initially assigned to states of nature. But the abstract analysis also allows for transparent restrictions on beliefs about behavior, e.g. restrictions induced by the context behind the game. Let E be the event (in the canonical universal space) that restrictions are transparent. Technically, E is a self-evident event. As such, any assumption about players'rationality and beliefs has to be taken in conjunction with E . We capture E -based forward induction reasoning with the following epistemic assumptions: (a) players are rational and E holds, (b) there is "common strong belief" of (a) (cf. Battigalli and Siniscalchi, J.Econ. Theory, 2002). We prove that the set of outcomes consistent with the foregoing assumptions is obtained by a reduction procedure calledrationalizability (Battigalli and Siniscalchi, Adv. Econ. Theory, 2003). This result is not trivial because the "strong belief" operator used to represent forward induction reasoning is non monotone.

Bulletin of Economic Research, 2001

Economic Theory, 2002

This paper is written as an introduction to epistemic logics and their game theoretic applications. It starts with both semantics and syntax of classical logic, and goes to the Hilbert-style proof-theory and Kripke-style model theory of epistemic logics. In these theories, we discuss individual decision making in some simple game examples. In particular, we will discuss the distinction between beliefs and knowledge, and how false beliefs play roles in game theoretic decision making. Finally, we discuss extensions of epistemic logics to incorporate common knowledge. In the extension, we discuss also false beliefs on common knowledge.

International Journal of Game Theory, 1985

Abstract: A formal model is given of Harsanyi's infinite hierarchies of beliefs. It is shown that the model doses with some Bayesian game with incomplete information, and that any such game can be approximated by one with a finite number of states of world.

Theory and Decision, 1992

International Game Theory Review, 2007

Game-theoretic solution concepts describe sets of strategy profiles that are optimal for all players in some plausible sense. Such sets are often found by recursive algorithms like iterated removal of strictly dominated strategies in strategic games, or backward induction in extensive games. Standard logical analyses of solution sets use assumptions about players in fixed epistemic models for a given game, such as mutual knowledge of rationality. In this paper, we propose a different perspective, analyzing solution algorithms as processes of learning which change game models. Thus, strategic equilibrium gets linked to fixed-points of operations of repeated announcement of suitable epistemic statements. This dynamic stance provides a new look at the current interface of games, logic, and computation.

Formal Models of Agents, 1999

In this paper we introduce and formalise dynamic belief hierarchies. We give a formal de nition of beliefs, which requires that they are coherent; that is, that each belief is jointly consistent with every other belief which the agent considers at least as reliable. Thus an agent's beliefs cannot be de ned independently, but only with reference to the agent's existing belief hierarchy. We then show how preferential entailment can be used, in conjunction with the rationality constraints on beliefs, to formalise the rational revision of beliefs and belief hierarchies. We then discuss the relationship between our theory and the AGM theory of belief revision. Finally we discuss resource-bounded reasoning.