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An isotone pure strategy equilibrium exists in any game of incomplete information in which (1) each player i's action set is a finite sublattice of multi-dimensional Euclidean space, (2) types are multidimensional and atomless, and each player's interim expected payoff function satisfies two "non-primitive conditions" whenever others adopt isotone pure strategies: (3) single-crossing in own action and type and (4) quasisupermodularity in own action. Similarly, given that (134) and (2') types are multi-dimensional (with atoms) an isotone mixed strategy equilibrium exists. Conditions (34) are satisfied in supermodular and log-supermodular games given affiliated types, and in games with independent types in which each player's ex post payoff satisfies (a) supermodularity in own action and (b) non-decreasing differences in own action and type. These results also extend to games with a continuum action space when each player's ex post payoff is also continuous in his and others' actions
The first volume of this wide-ranging Handbook contains original contributions by world-class specialists. It provides up-to-date surveys of the main game-theoretic tools commonly used to model industrial organization topics. The Handbook covers numerous subjects in detail including, among others, the tools of lattice programming, supermodular and aggregative games, monopolistic competition, horizontal and vertically differentiated good models, dynamic and Stackelberg games, entry games, evolutionary games with adaptive players, asymmetric information, moral hazard, learning and information sharing models.
We consider games with incomplete information a la Harsanyi, where the payoff of a player depends on an unknown state of nature as well as on the profile of chosen actions. As opposed to the standard model, players' preferences over state--contingent utility vectors are represented by arbitrary functionals. The definitions of Nash and Bayes equilibria naturally extend to this generalized setting. We characterize equilibrium existence in terms of the preferences of the participating players. It turns out that, given continuity and monotonicity of the preferences, equilibrium exists in every game if and only if all players are averse to uncertainty (i.e., all the functionals are quasi--concave). We further show that if the functionals are either homogeneous or translation invariant then equilibrium existence is equivalent to concavity of the functionals.
Consider a symmetric 2-player game of complete information. Consider an arbitrary Bayesian extension of that game with payoff-irrelevant types, independent random matching, and anonymity (private types). We show that, in this setting, while strategies in a Bayesian Nash equilibrium of that game can differ across types, aggregate play in any such equilibrium must coincide with a symmetric Nash equilibrium of the complete information game. This justifies the interpretation of certain data, including many laboratory experiments, as arising from a symmetric equilibrium, even when asymmetric equilibria exist and, in addition, subjects may be heterogeneous.
This book develops the central aspect of fixed point theory – the topological fixed point index – to maximal generality, emphasizing correspondences and other aspects of the theory that are of special interest to economics. Numerous topological consequences are presented, along with important implications for dynamical systems. The book assumes the reader has no mathematical knowledge beyond that which is familiar to all theoretical economists. In addition to making the material available to a broad audience, avoiding algebraic topology results in more geometric and intuitive proofs. Graduate students and researchers in economics, and related fields in mathematics and computer science, will benefit from this book, both as a useful reference and as a well-written rigorous exposition of foundational mathematics. Numerous problems sketch key results from a wide variety of topics in theoretical economics, making the book an outstanding text for advanced graduate courses in economics and related disciplines.
We define belief-free equilibria in two-player games with incomplete information as sequential equilibria for which players' continuation strategies are best-replies, after every history, independently of their beliefs about the state of nature. We characterize a set of payoffs that includes all belief-free equilibrium payoffs. Conversely, any payoff in the interior of this set is a belief-free equilibrium payoff.