Eddie Dekel

RATIONALIZABILITY AND CORRELATED EQUILIBRIA

We discuss the unity between the two standard approaches to noncooperative solution concepts for games. The decision-theoretic approach starts from the assumption that the rationality of the players is common knowledge. This leads to the notion of correlated rationalizability. It is shown that correlated rationalizability is equivalent to a posteriori equilibrium — a refinement of subjective correlated equilibrium. Hence a decision-theoretic justification for the equilibrium approach to game theory is provided. An analogous equivalence result is proved between independent rationalizability, which is the appropriate concept if each player believes that the others act independently, and conditionally independent a posteriori equilibrium. A characterization of Nash equilibrium is also provided.

Representing Preferences with a Unique Subjective State Space

Ž. We extend Kreps’ 1979 analysis of preference for flexibility, reinterpreted by Kreps Ž. 1992 as a model of unforeseen contingencies. We enrich the choice set, consequently obtaining uniqueness results that were not possible in Kreps’ model. We consider several representations and allow the agent to prefer commitment in some contingencies. In the representations, the agent acts as if she had coherent beliefs about a set of possible future Ž. ex post preferences, each of which is an expected-utility preference. We show that this set of ex post preferences, called the subjectie state space, is essentially unique given the restriction that all ex post preferences are expected-utility preferences and is minimal even without this restriction. Because the subjective state space is identified, the way ex post utilities are aggregated into an ex ante ranking is also essentially unique. Hence when a representation that is additive across states exists, the additivity is meaningful in the sense that all representations are intrinsically additive. Uniqueness enables us to show that the size of the subjective state space provides a measure of the agent’s uncertainty about future contingencies and that the way the states are aggregated indicates whether these contingencies lead to a desire for flexibility or commitment.

Standard State-Space Models Preclude Unawareness

We show that a very broad class of models, including possibility correspondences, necessarily fail to capture very simple and intuitive implications of unawareness. We explain why standard state–space formulations suffer from this problem, illustrating the point with an example of a nonstandard state–space model which avoids the difficulty.

Sequential Voting Procedures in Symmetric Binary Elections

We explore sequential voting in symmetric two‐option environments. We show that the (informative) symmetric equilibria of the simultancous voting game are also equilibria in any sequential voting structure. In unanimity games, (essentially) the whole set of equilibria is the same in all sequential structures. We also explore the relationship between simultaneous and sequential voting in other contexts. We illustrate several instances in which sequential voting does no better at aggregating information than simultaneous voting. The inability of the sequential structure to use additional information in voting models is distinct from that in the herd‐cascade literature.

Lexicographic Probabilities and Equilibrium Refinements

This paper develops a decision-theoretic approach to normal-form refinements of Nash equilibrium and provides characterizations of (normal-form) perfect equilibrium and proper equilibrium. The approach relies on a theory of single-person decision-making that is a non-Archimedean version of subjective expected utility theory. Copyright 1991 by The Econometric Society.

Vote Buying: General Elections

We examine the consequences of vote buying, assuming this practice were allowed and free of stigma. Two parties compete in a binary election and may purchase votes in a sequential bidding game via up‐front binding payments and/or campaign promises (platforms) that are contingent on the outcome of the election. We analyze the role of the parties’ and voters’ preferences in determining the winner and the payments to voters.

Signaling future actions and the potential for sacrifice

We consider extensions of games where some players have the option of signaling future actions by incurring costs. The main result is that in a class of games, if one player can incur costs, then forwards induction selects her most preferred outcome. Surprisingly, the player does not have to incur any costs to achieve this—the option alone suffices. However, when all players can incur costs, one players attempt to signal a future action is vulnerable to a counter-signal by the opponent. This vulnerability to counter-signaling distinguishes signaling future actions from signaling types.

Recent developments in modeling unforeseen contingencies

We survey recent models of unforeseen contingencies, discussing both epistemic and decision-theoretic approaches. We also briefly comment on the hurdles which remain for applying these models to contract theory

Asset Demands without the Independence Axiom

An important application of the theory of choice under uncertainty is to asset markets, and an important property in these markets is a preference for portfolio diversification. If an investor is an expected utility maximizer, then he is risk averse if, and only if, he exhibits a preference for diversification. This paper examines the relationship between risk aversion and portfolio diversification when preferences over probability distributions of wealth do not have an expected utility representation. Although risk aversion is not sufficient to guarantee a preference for portfolio diversification, it is necessary. Quasiconcavity of the preference functional (over probability distributions of wealth), together with risk aversion, does imply a preference for portfolio diversification. Copyright 1989 by The Econometric Society.

Common knowledge with probability 1

Abstract Two people, 1 and 2, are said to have common knowledge of an event if both know it, 1 knows that 2 knows it, 2 knows that 1 knows it, 1 knows that 2 knows that 1 knows it, and so on. This paper provides a Bayesian definition of common knowledge, that is, a definition in terms of beliefs (probability measures). The main result is an equivalence between this definition and a definition in terms of the σ-fields representing 1 and 2s information. To obtain this result the conditional probabilities must be proper and the σ-fields posterior completed .