Strong stability of Nash equilibria in load balancing games

We study strong stability of Nash equilibria in load balancing games of m(m 2)identical servers,in which every job chooses one of the m servers and each job wishes to minimize its cost,given by the workload of the server it chooses.A Nash equilibrium(NE)is a strategy profile that is resilient to unilateral deviations.Finding an NE in such a game is simple.However,an NE assignment is not stable against coordinated deviations of several jobs,while a strong Nash equilibrium(SNE)is.We study how well an NE approximates an SNE.Given any job assignment in a load balancing game,the improvement ratio(IR)of a deviation of a job is defined as the ratio between the pre-and post-deviation costs.An NE is said to be aρ-approximate SNE(ρ1)if there is no coalition of jobs such that each job of the coalition will have an IR more thanρfrom coordinated deviations of the coalition.While it is already known that NEs are the same as SNEs in the 2-server load balancing game,we prove that,in the m-server load balancing game for any given m 3,any NE is a(5/4)-approximate SNE,which together with the lower bound already established in the literature yields a tight approximation bound.This closes the final gap in the literature on the study of approximation of general NEs to SNEs in load balancing games.To establish our upper bound,we make a novel use of a graph-theoretic tool.     

Various characterizations of convex fuzzy games

This note enlarges the literature on convex fuzzy games with new characterizing properties of such games besides the increasing average marginal return property, namely: the monotonicity of the first partial derivatives, the directional convexity and forC ²-functions the non-negativity of the second order partial derivatives.     

Bimatrix games have quasi-strict equilibria

Received April 21, 1995 / Revised version received June 12, 1998 Published online November 24, 1998     

On weakly correlated and correlated equilibria in discontinuous games

This paper treats of non-zero-sum discontinuous games with compact Hausdorff strategy spaces. It is assumed that the payoff function of each player in the game is bounded, Borel measurable and is upper semicontinuous on his strategy space, for all fixed actions of the remaining players. It is shown that for each ε>0, such games possess weakly correlated ε-epuilibria introduced by Moulin and Vial as extension of correlated equilibria in the sense of Aumann. An upper semicontinuous came having weakly correlated equilibria and no correlated equilibria is discussed in details.     

Characterization of correlated equilibria in stochastic games

A general communication device is a device that at every stage of the game receives a private message from each player, and in return sends a private signal to each player; the signals the device sends depend on past play, past signals it sent, and past messages it received.  An autonomous correlation device is a general communication device where signals depend only on past signals the device sent, but not on past play or past messages it received.  We show that the set of all equilibrium payoffs in extended games that include a general communication device coincides with the set of all equilibrium payoffs in extended games that include an autonomous correlation device. A stronger result is obtained when the punishment level is independent of the history. Final version July 2001     

A procedure for finding Nash equilibria in bi-matrix games

In this paper we consider the computation of Nash equilibria for noncooperative bi-matrix games. The standard method for finding a Nash equilibrium in such a game is the Lemke-Howson method. That method operates by solving a related linear complementarity problem (LCP). However, the method may fail to reach certain equilibria because it can only start from a limited number of strategy vectors. The method we propose here finds an equilibrium by solving a related stationary point problem (SPP). Contrary to the Lemke-Howson method it can start from almost any strategy vector. Besides, the path of vectors along which the equilibrium is reached has an appealing game-theoretic interpretation. An important feature of the algorithm is that it finds a perfect equilibrium when at the start all actions are played with positive probability. Furthermore, we can in principle find all Nash equilibria by repeated application of the algorithm starting from different strategy vectors.This author is financially supported by the Co-operation Centre Tilburg and Eindhoven Universities, The Netherlands.     

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.     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

Nash equilibria of Cauchy-random zero-sum and coordination matrix games

We consider zero-sum games (A,  − A) and coordination games (A,A), where A is an m-by-n matrix with entries chosen independently with respect to the Cauchy distribution. In each case, we give an exact formula for the expected number of Nash equilibria with a given support size and payoffs in a given range, and also asymptotic simplications for matrices of a fixed shape and increasing size. We carefully compare our results with recent results of McLennan and Berg on Gaussian random bimatrix games (A,B), and describe how the three situations together shed light on random bimatrix games in general.     

Two hardness results for core stability in hedonic coalition formation games

We establish NP-completeness of two problems on core stable coalitions in hedonic games. In the first problem every player has only two acceptable coalitions in his preference list, and in the second problem the preference structures arise from the distances in an underlying metric space.     

Population monotonic solutions on convex games

The Dutta-Ray solution and the Shapley value are two well-known examples of population-monotonic solutions on the domain of convex games. We provide a new formula for the Dutta-Ray solution from which population-monotonicity immediately follows. Then we define a new family of population-monotonic solutions, which we refer to as “sequential Dutta-Ray solutions.” We also show that it is possible to construct several symmetric and population-monotonic solutions by using the solutions in this family. Received September 1998/Revised version: December 1999     

General sum games with joint chance constraints

We consider an $n$-player non-cooperative game with continuous strategy sets. The strategy set of each player contains a set of stochastic linear constraints. We model the stochastic linear constraints of each player as a joint chance constraint. We assume that the row vectors of a matrix defining the stochastic constraints of each player are independent and each row vector follows a multivariate normal distribution. Under certain conditions, we show the existence of a Nash equilibrium for this game.     

The nucleolus is not aggregate-monotonic on the domain of convex games

We show that the nucleolus is not aggregate-monotonic on the domain of convex games, and that this lack of monotonicity holds even if there are as few as four agents. Received May 1999/Revised version December 1999     

有强法式的体上矩阵

设D为除环,A∈Dn×n,则可用初等变换将λI-A化简为对角阵A= diag(1,…,1,φ1,…,φr),其中(?)i为D上首1多项式并且φ1|…|φr.如果这个对角阵A在形状上是唯一的,则称A是有强法式的矩阵.本文应用中心原子因子与初等因子给出了体上有强法式的矩阵的本质刻画,给出了体上矩阵有强法式的一些充要条件.     

A coalition formation value for games in partition function form

The coalition formation problem in an economy with externalities can be adequately modeled by using games in partition function form (PFF games), proposed by Thrall and Lucas. If we suppose that forming the grand coalition generates the largest total surplus, a central question is how to allocate the worth of the grand coalition to each player, i.e., how to find an adequate solution concept, taking into account the whole process of coalition formation. We propose in this paper the original concepts of scenario-value, process-value and coalition formation value, which represent the average contribution of players in a scenario (a particular sequence of coalitions within a given coalition formation process), in a process (a sequence of partitions of the society), and in the whole (all processes being taken into account), respectively. We give also two axiomatizations of our coalition formation value.     

On existence of undominated pure strategy Nash equilibria in anonymous nonatomic games: a generalization

In this paper, we generalize the exitence result for pure strategy Nash equilibria in anonymous nonatomic games. By working directly on integrals of pure strategies, we also generalize, for the same class of games, the existence result for undominated pure strategy Nash equilibria even though, in general, the set of pure strategy Nash equilibria may fail to be weakly compact. Received August 2001     

Stability in coalition formation games

In the context of coalition formation games a player evaluates a partition on the basis of the set she belongs to. For this evaluation to be possible, players are supposed to have preferences over sets to which they could belong. In this paper, we suggest two extensions of preferences over individuals to preferences over sets. For the first one, derived from the most preferred member of a set, it is shown that a strict core partition always exists if the original preferences are strict and a simple algorithm for the computation of one strict core partition is derived. This algorithm turns out to be strategy proof. The second extension, based on the least preferred member of a set, produces solutions very similar to those for the stable roommates problem. Received August 1998/Final version June 20, 2000