A Course in Game Theory presents the main ideas of game theory at a level suitable for graduate students and advanced undergraduates, emphasizing the theory's foundations and interpretations of its basic concepts. The authors provide precise definitions and full proofs of results, sacrificing generalities and limiting the scope of the material in order to do so. The text is organized in four parts: strategic games, extensive games with perfect information, extensive games with imperfect information, and coalitional games. It includes over 100 exercises.

A Course in Game Theory

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Convex Analysisの二,三の進展について

This paper reviews the literature on gender differences in economic experiments. In the three main sections, we identify robust differences in risk preferences, social (other-regarding) preferences, and competitive preferences. We also speculate on the source of these differences, as well as on their implications. Our hope is that this article will serve as a resource for those seeking to understand gender differences and to use as a starting point to illuminate the debate on gender-specific outcomes in the labor and goods markets.

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Gender Differences in Preferences

This paper develops a class of recursive, but not necessarily expected utility, preferences over intertemporal consumption lotteries An important feature of these general preferences is that they permit risk attitudes to be disentangled from the degree of intertemporal substitutability Moreover, in an infinite horizon, representative agent context these preference specifications lead to a model of asset returns in which appropriate versions of both the atemporal CAPM and the intertemporal consumption-CAPM are nested as special cases In our general model, systematic risk of an asset is determined by covariance with both the return to the market portfolio and consumption growth, while in each of the existing models only one of these factors plays a role This result is achieved despite the homotheticity of preferences and the separability of consumption and portfolio decisions Two other auxiliary analytical contributions which are of independent interest are the proofs of (i) the existence of recursive intertemporal utility functions, and (ii) the existence of optima to corresponding optimization problems In proving (i), it is necessary to define a suitable domain for utility functions This is achieved by extending the formulation of the space of temporal lotteries in Kreps and Porteus (1978) to an infinite horizon framework A final contribution is the integration into a temporal setting of a broad class of atemporal non-expected utility theories For homogeneous members of the class due to Chew (1985) and Dekel (1986), the corresponding intertemporal asset pricing model is derived

Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework

This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.

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Prediction, learning, and games

An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom

Hierarchies of Beliefs and Common Knowledge

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.

Rationalizability and correlated equilibria

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.

Representing preferences with a unique subjective state space

Conventional Bayesian theory of choice under uncertainty, subjective expected utility theory, fails to satisfy the properties of admissibility and existence of well-defined conditional probabilities; weakly dominated acts may be chosen, and the usual definition of conditional probabilities applies only to nonnull events. This paper develops a non-Archimedean variant of subjective expected utility where decisionmakers have lexicographic beliefs. This generalization can be made to satisfy admissibility and yield well-defined conditional probabilities and at the same time allow for "null" events. Copyright 1991 by The Econometric Society.

Eddie Dekel

Papers

An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom

Hierarchies of Beliefs and Common Knowledge

Rationalizability and correlated equilibria

Representing preferences with a unique subjective state space

Lexicographic probabilities and choice under uncertainty