We construct a fully specified extensive form game that captures competitive markets with adverse selection. In particular, it allows firms to offer any finite set of contracts, so that cross-subsidization is not ruled out. Moreover, firms can withdraw from the market after initial contract offers have been observed. We show that a subgame perfect equilibrium always exists and that, in fact, when withdrawal is costless, the set of subgame perfect equilibrium outcomes may correspond to the entire set of feasible contracts. We then focus on robust equilibria that exist both when withdrawal costs are zero and when they are arbitrarily small but strictly positive. We show that the Miyazaki-Wilson contracts are the unique robust equilibrium outcome of our game. This outcome is always constrained efficient and involves cross-subsidization from low to high risk agents that is increasing in the share of low risks in the population under weak conditions on risk preferences.

Riley (1979)'s reactive equilibrium concept addresses problems of equilibrium existence in competitive markets with adverse selection. The game-theoretic interpretation of the reactive equilibrium concept in Engers and Fernandez (1987) yields the Rothschild-Stiglitz (1976)/Riley (1979) allocation as an equilibrium allocation, however multiplicity of equilibrium emerges. In this note we imbed the reactive equilibrium's logic in a dynamic market context with active consumers. We show that the Riley/Rothschild-Stiglitz contracts constitute the unique equilibrium allocation in any pure strategy subgame perfect Nash equilibrium.

JEAN-FRANQOIS MERTENS This book presents a systematic exposition of the use of game theoretic methods in general equilibrium analysis. Clearly the first such use was by Arrow and Debreu, with the "birth" of general equi librium theory itself, in using Nash's existence theorem (or a generalization) to prove the existence of a competitive equilibrium. But this use appeared possibly to be merely tech nical, borrowing some tools for proving a theorem. This book stresses the later contributions, were game theoretic concepts were used as such, to explain various aspects of the general equilibrium model. But clearly, each of those later approaches also provides per sea game theoretic proof of the existence of competitive equilibrium. Part A deals with the first such approach: the equality between the set of competitive equilibria of a perfectly competitive (i.e., every trader has negligible market power) economy and the core of the corresponding cooperative game.

The theory of competition has held a central place in economic analysis since Adam Smith. This book, written by one of the most distinguished of contemporary economic theorists, reports on a major research program to provide strategic foundations for the theory of perfect competition. Beginning with a concise survey of how the theory of competition has evolved, Gale makes extensive and rigorous use of dynamic matching and bargaining models to provide a more complete description of how a competitive equlibrium is achieved. Whereas economists have made use of a macroscopic description of markets in which certain behavioral characteristics, such as price-taking behavior, are taken for granted, Gale uses game theory to re-evaluate this assumption, beginning with individual agents and modelling their strategic interaction. A strategic foundation for competitive equilibrium shows how such interaction leads to competitive, price-taking behavior. Essential reading for graduate courses in game theory and general equilibrium.

The goal of this book is to elaborate on the main principles of the theory of the Berge equilibrium by answering the following two questions: What are the basic properties of the Berge equilibrium? Does the Berge equilibrium exist, and how can it be calculated? The Golden Rule of ethics, which appears in Christianity, Judaism, Islam, Buddhism, Confucianism and other world religions, states the following: “Behave towards others as you would like them to behave towards you." In any game, each party of conflict seeks to maximize some payoff. Therefore, for each player, the Golden Rule is implemented through the maximization of his/her payoff by all other players, which matches well with the concept of the Berge equilibrium. The approach presented here will be of particular interest to researchers (including undergraduates and graduates) and economists focused on decision-making under complex conflict conditions. The peaceful resolution of conflicts is the cornerstone of the approach: as a matter of fact, the Golden Rule precludes military clashes and violence. In turn, the new approach requires new methods; in particular, the existence problems are reduced to saddle point design for the Germeier convolution of payoff functions, with further transition to mixed strategies in accordance with the standard procedure employed by E. Borel, J. von Neumann, J. Nash, and their followers. Moreover, this new approach has proven to be efficient and fruitful with regard to a range of other important problems in mathematical game theory, which are considered in the Appendix.