Abstract: The evolution of boundedly rational rules for playing normal form games is studied within stationary environments of stochastically changing games. Rules are viewed as algorithms prescribing strategies for the different normal form games that arise. It is shown that many of the “folk results” of evolutionary game theory, typically obtained with a fixed game and fixed strategies, carry over to the present environments. The results are also related to some recent experiments on rules and games.
Abstract: We show that every N-player $$K_{1} times ... times K_{N}$$ game possesses a correlated equilibrium with at least $$prod_{i=1}^{N} K_i -1 - sum_{i=1}^{N} K_i (K_i -1)$$ zero entries. In particular, the largest N-player K × ... × K games with unique fully supported correlated equilibrium are two-player games.
Abstract: We construct an uncoupled randomized strategy of repeated play such that, if every player plays according to it, mixed action profiles converge almost surely to a Nash equilibrium of the stage game. The strategy requires very little in terms of information about the game, as players' actions are based only on their own past payoffs. Moreover, in a variant of the procedure, players need not know that there are other players in the game and that payoffs are determined through other players' actions. The procedure works for finite generic N-player games and is based on appropriate modifications of a simple stochastic learning rule introduced by Foster and Young [Theoretical Economics, 2006, 1, 341–367].
Abstract: Equivalence classes of normal form games are defined using the discontinuities of correspondences of standard equilibrium concepts like correlated, Nash, and robust equilibrium, or risk dominance and rationalizability. Resulting equivalence classes are fully characterized and compared across different equilibrium concepts for 2 × 2 games; larger games are also studied. It is argued that the procedure leads to broad and game-theoretically meaningful distinctions of games as well as to alternative ways of representing, comparing and testing equilibrium concepts.
Abstract: We construct a family of retaliatory equilibria for the Japanese ascending auction for multiple objects and show that, while it is immune to many of the tacitly collusive equilibria studied in the literature, it is not entirely immune when some bidders are commonly known to be interested in a specific object.
Abstract: We consider two ascending auctions for multiple objects, namely, an English and a Japanese auction, and derive a perfect Bayesian equilibrium of the Japanese auction by exploiting its strategic equivalence with the survival auction, which consists of a finite sequence of sealed-bid auctions. Thus an equilibrium of a continuous time game is derived by means of backward induction in finitely many steps. We then show that all equilibria of the Japanese auction induce equilibria of the English auction, but that many collusive or signaling equilibria of the English auction do not have a counterpart in the Japanese auction.
Abstract: We extend Kohlberg and Mertens' [Econometrica, 1986, 54, 1003–1037] structure theorem on the Nash correspondence to show that its graph is not only homeomorphic to the underlying space of games, but that the homeomorphism extends to the ambient space of games times strategies, thus implying the graph is unknotted. This has several consequences for dynamics whose rest points are Nash equilibria. In particular, a homotopy property is established that allows to determine whether Nash equilibria or components of Nash equilibria—based directly on their local geometry on the graph—may be stable under a wide class of dynamics.
Abstract: We introduce a framework for analyzing Bertrand–Edgeworth equilibria in finite Arrow–Debreu exchange economies. A key feature is the way trade takes place. There are two main stages. In the first stage agents simultaneously choose prices and quantities of commodities they want to sell; in the second stage they enter the markets as buyers, sequentially, and choose quantities of commodities they want to buy. We view the framework as providing a realistic, entirely decentralized abstraction of how trade occurs on numerous markets, especially for consumer goods and services, and show that resulting equilibria are Walrasian if and only if initial endowments are already Pareto efficient.
Abstract: In an example where some bidders have superadditive values, we characterize the equilibria of a simultaneous ascending auction and compare the revenue and efficiency generated with ones generated by the sequential, the one-shot simultaneous, and the Vickrey–Clarke–Groves auctions.
Abstract: Two basic properties concerning the dynamic behavior of competitive equilibria of exchange economies with complete markets are derived essentially from the fact that the Walras correspondence has no knots.
Abstract: We show a fundamental property of dynamics whose zeros are essentially the Nash equilibria of underlying games; namely, the indices of zeros coincide with the degrees of the projection from the graph of the Nash correspondence onto the underlying space of games. This is important since it implies that for a wide class of dynamics local stability properties of zeros are determined by the geometry of the Nash correspondence, providing further links between learning or evolutionary game theory, the theory of equilibrium refinements, and the geometry of Nash equilibrium. The result extends beyond general n-player games e.g. to Walrasian equilibrium theory.
