Equivalence and Invariance of the Index and Degree of Nash Equilibria

Associated with each component of the Nash equilibria of a game are its index and degree. Its index is the local degree of the displacement map whose roots are the Nash equilibria of the game. Its degree is the local degree of the projection map from the Nash graph to the space of games. We show that the index and the degree of each component are the same. Further, they are invariant to adding or deleting redundant strategies, so they depend only on the reduced normal form of the game. Applications include Kohlberg and Mertens' existence theorems for stable sets and a simple procedure for calculating the degree of a component.Journal of Economic LiteratureClassification Number: C72.     

Dynamic Interactions in Trade Policy in a Differential Game Model of Tariff Protection

This paper develops a two‐country dynamic game model of tariff protection to reconsider optimal trade policies and their implications for welfare. The authors show that an import subsidy is optimal in the feedback Nash equilibria, which results in a curious possibility that the domestic market is monopolized by the foreign firrm. However, welfare comparisons among Nash equilibria, free trade, and autarky reveal that feedback Nash equilibria involve higher welfare than both autarky and free trade, i.e. dynamic noncooperative choices of policy serve as tacit policy coordination and ensure larger trade gains relative to free trade.     

Coordination need not be a problem

In a game of common interest there is one action vector that all players prefer to every other. Yet there may be multiple Pareto-ranked Nash equilibria in the game and the “coordination problem” refers to the fact that rational equilibrium play cannot rule out Pareto-dominated equilibria. In this paper, I prove that two elements — asynchronicity and a finite horizon — are sufficient to uniquely select the Pareto-dominant action vector (in subgame perfect equilibrium play). Asynchronicity may be exogenously specified by the rules of the game. Alternatively, in a game where players choose when to move, asynchronicity may emerge as an equilibrium move outcome.     

Gains from Trade in a Differential Game Model of Asymmetric Oligopoly

This paper revisits a classical topic of trade gains in a differential game model of oligopoly in which Home and Foreign firms differ in the number and cost. After deriving the feedback Nash equilibrium, we provide examples to consider how the difference in the number of firms or costs affects gainfulness of trade. We prove that feedback strategies can result in implications for trade gains which are sharply different from the open-loop case.     

Competitive environments and protective behavior

The class of two-person competition games is introduced and analyzed. For any game in this class the set of Nash equilibria is convex and all Nash equilibria lead to the same payoff vector. Competition games are compared to other competitive environments such as unilaterally competitive games and rivalry games. Moreover, protective behavior within competitive environments is analyzed. For matrix games it is known that protective strategies profiles exactly correspond to proper equilibria. It is shown that this result can be extended to the class of unilaterally competitive games.     

Cooperation in an Overlapping Generations Experiment

Recent theoretical work shows that folk theorems can be developed for infinite overlapping generations games. Cooperation in such games can be sustained as a Nash equilibrium. But, of course, there are other equilibria. This paper investigates experimentally whether cooperation actually occurs in a simple overlapping generations game. Subjects both play the game and formulate strategies. Our main finding is that subjects fail to exploit the intertemporal structure of the game. Even when we provided subjects with a recommendation to play the grim trigger strategy, most of the subjects still employed safe history-independent strategies. Journal of Economic Literature Classification Numbers: C72, C92, D90.     

Dynamic Voluntary Provision of Public Goods: A Generalization

In this note we examine if the proposition offered by Fershtman and Nitzan (1991) and Wirl (1996) in the context of a dynamic voluntary provision model with a linear production function can be generalized to a more general CES formulation. By comparing the steady-state stocks of a public good in open-loop and feedback Nash equilibria with that under the cooperative solution, we demonstrate that their ranking among the steady-state stocks is indeed preserved under the CES framework.     

The Acid Rain Differential Game

This paper considers an acid rain differential game. Countries emit sulphur which is partly transferred to other countries. Depositions above critical loads ultimately destroy the soil. Countries face a trade-off between the costs of emission reductions and the damage to the soil due to the depletion of the acid buffers. Because of the transboundary externalities the outcome will depend on whether the countries cooperate or not. This paper presents the cooperative outcome and the open-loop and Markov-perfect Nash equilibria of the acid rain differential game. It will be shown that the depositions always converge to the critical loads but the steady-state levels of the buffer stocks differ. The theory is used to analyse the acid rain differential game for sulphur between Great Britain and Ireland. Finally, some results are given for the whole of Europe.     

The supercore for normal-form games

This paper analyzes the supercore of a system derived from a normal-form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore coincides with the set of Nash equilibria of the last game in that sequence. This result is illustrated with the characterization of the supercore for the n-person prisoner's dilemma. With regard to the mixed extension of a normal-form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria.     

Walrasian analysis via two-player games

We associate to any pure exchange economy a game with only two players, regardless of the number of consumers. In this two-player game, each player represents a different role of the society, formed by all the individuals in the economy. Player 1 selects feasible allocations trying to make Pareto improvements. Player 2 chooses an alternative from the wider range of allocations that are feasible in the sense of Aubin. The set of Nash equilibria of our game is non-empty and our main result provides a characterization of Walrasian equilibria allocations as strong Nash equilibria of the associated society game.     

A Non-cooperative Approach to Bankruptcy Problems

We propose an elementary game form that allows to obtain the allocations proposed by any acceptable bankruptcy rule as the unique payoff vector of the corresponding Nash equilibria.     

Properties and applications of dual reduction

The dual reduction process, introduced by Myerson, allows a finite game to be reduced to a smaller-dimensional game such that any correlated equilibrium of the reduced game is an equilibrium of the original game. We study the properties and applications of this process. It is shown that generic two-player normal form games have a unique full dual reduction (a known refinement of dual reduction) and all strategies that have probability zero in all correlated equilibria are eliminated in all full dual reductions. Among other applications, we give a linear programming proof of the fact that a unique correlated equilibrium is a Nash equilibrium, and improve on a result due to Nau, Gomez-Canovas and Hansen on the geometry of Nash equilibria and correlated equilibria.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

Endogenous private information structures

We formally incorporate the option to gather information into a game and thus endogenize the information structure. We ask whether models with exogenous information structures are robust with respect to this endogenization. Any Nash equilibrium of the game with information acquisition induces a Nash equilibrium in the corresponding game with an exogenous structure. We provide sufficient conditions on the structure of the game for which this remains true when ‘Nash’ is replaced by ‘sequential’. We characterize the (sequential) Nash equilibria of games with exogenous information structures that can arise as a (sequential) Nash equilibrium of games with endogenous information acquisition.     

The number of pure Nash equilibria in a random game with nondecreasing best responses

We randomly draw a game from a distribution on the set of two-player games with a given size. We compute the distribution and the expectation of the number of pure-strategy Nash equilibria of the game conditional on the game having nondecreasing best-response functions. The conditional expected number of pure-strategy Nash equilibria becomes much larger than the unconditional expected number as the size of the game grows.     

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

A dynamic model of fiscal reconstruction

This paper investigates dynamic properties of fiscal reconstruction by analyzing the infinite duration differential game among interest groups with a framework of voluntary acceptance of tax burden. By comparing the first-best solution of fiscal reconstruction, the open-loop Nash equilibrium path, and the feedback Nash equilibrium path, we highlight the free-riding behavior of interest groups in the process of fiscal reconstruction. We derive explicitly the target levels of government debt and primary expenditures, and the adjustment speed of fiscal reconstruction under the respective solutions. We also examine the impacts of consumption taxes on their adjustment speeds as well as long-run outcomes of the respective paths.     

Bertrand price competition with differentiated commodities

This develops a general equilibrium, differentiated commodity version of Bertrand price competition. We study two, related market games in which buyers as well as sellers announce both quantities and prices. In the first game, buyers' strategies are artificially restricted. The Nash allocations of this game will be nearly competitive, provided that the commodities supplied by sellers are sufficiently similar. In the second game, the restriction on buyers' strategies is relaxed and a stronger solution criterion, called local perfection, is invoked. The locally perfect equilibria of the unrestricted game coincide the Nash equilibria of the restricted game.     

On price taking behavior in a nonrenewable resource cartel–fringe game

We consider a nonrenewable resource game with one cartel and a set of fringe members. We show that (i) the outcomes of the closed-loop and the open-loop nonrenewable resource game with the fringe members as price takers (the cartel–fringe game à la Salant, 1976) coincide and (ii) when the number of fringe firms becomes arbitrarily large, the equilibrium outcome of the closed-loop Nash game does not coincide with the equilibrium outcome of the closed-loop cartel–fringe game. Thus, the outcome of the cartel–fringe open-loop equilibrium can be supported as an outcome of a subgame-perfect equilibrium. However the interpretation of the cartel–fringe model, where from the outset the fringe is assumed to be price taker, as a limit case of an asymmetric oligopoly with the agents playing Nash–Cournot, does not extend to the case where firms can use closed-loop strategies.     

Competitive behavior in market games: Evidence and theory

We explore whether competitive outcomes arise in an experimental implementation of a market game, introduced by Shubik (1973) [21]. Market games obtain Pareto inferior (strict) Nash equilibria, in which some or possibly all markets are closed. We find that subjects do not coordinate on autarkic Nash equilibria, but favor more efficient Nash equilibria in which all markets are open. As the number of subjects participating in the market game increases, the Nash equilibrium they achieve approximates the associated competitive equilibrium of the underlying economy. Motivated by these findings, we provide a theoretical argument for why evolutionary forces can lead to competitive outcomes in market games.