Nash refinement of equilibria

A method for choosing equilibria in strategic form games is proposed and axiomatically characterized. The method as well as the axioms are inspired by the Nash bargaining theory. The method can be applied to existing refinements of Nash equilibrium (e.g., perfect equilibrium) and also to other equilibrium concepts, like correlated equilibrium.The authors thank the reviewers for their comments, which led to an improvement of the paper.     

Inequalities and Nash equilibria

This paper concentrates on the problem of the existence of equilibrium points for non-cooperative generalized N-person games, N-person games of normal form and their related inequalities. We utilize the K-K-M lemma to obtain a theorem and then use it to obtain a new Fan-type inequality and minimax theorems. Various new equilibrium point theorems are derived, with the necessary and sufficient conditions and with strategy spaces with no fixed point property. Examples are given to demonstrate that these existence theorems cover areas where other existence theorems break down.     

Nash equilibria on topological semilattices

A function ${u : X \to \mathbb{R}}$ defined on a partially ordered set is quasi-Leontief if, for all ${x \in X}$ , the upper level set ${\{x\prime \in X : u(x\prime) \geq u(x)\}}$ has a smallest element; such an element is an efficient point of u. An abstract game ${u_{i} : \prod^{n}_{j=1} X_j \to \mathbb{R}, i \in \{1, \ldots , n\}}$ , is a quasi-Leontief game if, for all i and all ${(x_{j})_{j \neq i} \in \prod_{j \neq i} X_{j}, u_{i}((x_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ is quasi-Leontief; a Nash equilibrium x* of an abstract game ${u_{i} :\prod^{n}_{j=1} X_{j} \to \mathbb{R}}$ is efficient if, for all ${i, x^{*}_{i}}$ is an efficient point of the partial function ${u_{i}((x^{*}_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ . We establish the existence of efficient Nash equilibria when the strategy spaces X _i are topological semilattices which are Peano continua and Lawson semilattices.     

On the geometry of Nash equilibria and correlated equilibria

It is well known that the set of correlated equilibrium distributions of an n-player noncooperative game is a convex polytope that includes all the Nash equilibrium distributions. We demonstrate an elementary yet surprising result: the Nash equilibria all lie on the boundary of the polytope.We are grateful to Francoise Forges, Dan Arce, the editors, and several anonymous referees for helpful comments. This research was supported by the National Science Foundation under grant 98–09225 and by the Fuqua School of Business.The use of correlated mixed strategies in 2-player games was discussed by Raiffa (1951), who noted: it is a useful concept since it serves to convexify certain regions [of expected payoffs] in the Euclidean plane. (p. 8)Received: April 2002 / Revised: November 2003     

Nash equilibria in all-optical networks

We consider the problem of routing a number of communication requests in WDM (wavelength division multiplexing) all-optical networks from the standpoint of game theory. If we view each routing request (pair of source-target nodes) as a player, then a strategy consists of a path from the source to the target and a frequency (color). To reflect the restriction that two requests must not use the same frequency on the same edge, conflicting strategies are assigned a prohibitively high cost.Under this formulation, we consider several natural cost functions, each one reflecting a different aspect of restriction in the available bandwidth. For each cost function we examine the problem of the existence of pure Nash equilibria, the complexity of recognizing and computing them and finally, the problem in which we are given a Nash equilibrium and we are asked to find a better one in the sense that the total bandwidth used is less. As it turns out some of these problems are tractable and others are NP-hard.     

Nash equilibria in random games

We consider Nash equilibria in 2‐player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m × n payoff matrices, it runs in time O(m²nloglog n + n²mloglog m) with high probability. Our result follows from showing that a 2‐player random game has a Nash equilibrium with supports of size two with high probability, at least 1 − O(1/log n). Our main tool is a polytope formulation of equilibria. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Paths to constrained Nash equilibria

We propose and analyze a primal-dual, infinitesimal method for locating Nash equilibria of constrained, non-cooperative games. The main object is a family of nonstandard Lagrangian functions, one for each player. With respect to these functions the algorithm yields separately, in differential form, directions of steepest-descent in all decision variables and steepest-ascent in all multipliers. For convergence we need marginal costs to be monotone and constraints to be convex inequalities. The method is largely decomposed and amenable for parallel computing. Other noteworthy features are: non-smooth data can be accommodated; no projection or optimization is needed as subroutines; multipliers converge monotonically upward; and, finally, the implementation amounts, in essence, only to numerical integration.     

Efficient Nash equilibria on semilattices

This contribution introduces the so-called quasi-Leontief functions. In the framework and the language of tropical algebras, our quasi-Leontief functions are the additive functions defined on a semimodule with values in the semiring of scalars. This class of functions encompasses as a special case the usual Leontief utility function. We establish the existence of efficient Nash equilibria when the strategy spaces are compact and pathconnected topological semilattices.     

Nash equilibria of finitely repeated games

Under weak conditions, any feasible and individually rational payoff vector of a one-shot game can be approximated by the average payoff in a Nash equilibrium of a finitely repeated game with a long enough horizon.     

Nash equilibria in competitive project scheduling

We consider the problem of scheduling activities of a project by a firm that competes with another firm that has to perform the same project. The profit that a firm gets from each activity depends on whether the firm finishes the activity before or after its competitor. It is required to find a Nash equilibrium solution or show that no such solutions exist. We present a structural characterization of Nash equilibrium solutions, and a low order polynomial algorithm for the problem.     

Nash equilibria for non-binary choice rules

We prove the existence of equilibria in games with players who employ abstract (non-binary) choice rules. This framework goes beyond the standard, transitive model and encompasses games where players have non-transitive preferences (e.g., skew-symmetric bilinear preferences).      

Nash equilibria in optimal reinsurance bargaining

We introduce a strategic behavior in reinsurance bilateral transactions, where agents choose the risk preferences they will appear to have in the transaction. Within a wide class of risk measures, we identify agents’ strategic choices to a range of risk aversion coefficients. It is shown that at the strictly beneficial Nash equilibria, agents appear homogeneous with respect to their risk preferences. While the game does not cause any loss of total welfare gain, its allocation between agents is heavily affected by the agents’ strategic behavior. This allocation is reflected in the reinsurance premium, while the insurance indemnity remains the same in all strictly beneficial Nash equilibria. Furthermore, the effect of agents’ bargaining power vanishes through the game procedure and the agent who gets more welfare gain is the one who has an advantage in choosing the common risk aversion at the equilibrium.     

Uncoupled automata and pure Nash equilibria

We study the problem of reaching a pure Nash equilibrium in multi-person games that are repeatedly played, under the assumption of uncoupledness: EVERY player knows only his own payoff function. We consider strategies that can be implemented by finite-state automata, and characterize the minimal number of states needed in order to guarantee that a pure Nash equilibrium is reached in every game where such an equilibrium exists.     

Finitely additive and epsilon Nash equilibria

We prove the existence of a mixed strategy Nash equilibrium in normal form games when the space of mixed strategies consists of finitely additive probability measures. It is then proved that from this result an existence result for epsilon equilibria with countably additive mixed strategies can be obtained. These results are applied to the classic Cournot game.     

On the existence of strong Nash equilibria

This paper investigates the existence of strong Nash equilibria (SNE) in continuous and concave games. It is shown that the coalition consistency property introduced in the paper, together with concavity and continuity of payoffs, permits the existence of SNE in games with compact and convex strategy spaces. We also characterize the existence of SNE by providing necessary and sufficient conditions. We suggest an algorithm for computing SNE. The results are illustrated with applications to economies with multilateral environmental externalities and to the static oligopoly model.     

Nash equilibria of discontinuous non-zero-sum two-person games

The paper investigates two classes of non-zero-sum two-person games on the unit square, where the payoff function of Player 1 is convex or concave in the first variable. It is shown that this assumption together with the boundedness of payoff functions imply the existence of -Nash equilibria consisting of two probability measures concentrated at most at two points each.This research project No. 211589101 was supported by KBN Grant under contract 664/2/91.     

An ordinal well-posedness property for Nash equilibria

The aim of this article is to give, for Nash equilibria, a well-posedness criterion in the form of an ordinal property. This property is important for games because it captures the case when players' decisions depend on preferences and not on a special choice of a utility function. The ordinal characteristics of this well-posedness criterion comes from considering value-bounded approximate equilibria.     

Pairwise-stability and Nash equilibria in network formation

Suppose that individual payoffs depend on the network connecting them. Consider the following simultaneous move game of network formation: players announce independently the links they wish to form, and links are formed only under mutual consent. We provide necessary and sufficient conditions on the network link marginal payoffs such that the set of pairwise stable, pairwise-Nash and proper equilibrium networks coincide, where pairwise stable networks are robust to one-link deviations, while pairwise-Nash networks are robust to one-link creation but multi-link severance. Under these conditions, proper equilibria in pure strategies are fully characterized by one-link deviation checks. We thank William Thomson, an associate editor and two anonymous referees for their suggestions that led to substantial improvements. We also thank Sjaak Hurkens, Bettina Klaus, Jordi Massó and Giovanni Neglia for helpful conversations. The first author gratefully acknowledges the financial support from the Spanish Ministry of Education and FEDER through grant SEJ2005-01481/ECON, the Fundación BBVA and the Barcelona Economics Program of XREA. The second author is grateful to the Netherlands Organization for Scientific Research (NWO) for its support under grant VIDI-452-06-013.