Existence of nash equilibria for constrained stochastic games

In this paper, we consider constrained noncooperative N-person stochastic games with discounted cost criteria. The state space is assumed to be countable and the action sets are compact metric spaces. We present three main results. The first concerns the sensitivity or approximation of constrained games. The second shows the existence of Nash equilibria for constrained games with a finite state space (and compact actions space), and, finally, in the third one we extend that existence result to a class of constrained games which can be “approximated” by constrained games with finitely many states and compact action spaces. Our results are illustrated with two examples on queueing systems, which clearly show some important differences between constrained and unconstrained games.Mathematics Subject Classification (2000): Primary: 91A15. 91A10; Secondary: 90C40     

A cooperative location game based on the 1-center location problem

In this paper we introduce and analyze new classes of cooperative games related to facility location models defined on general metric spaces. The players are the customers (demand points) in the location problem and the characteristic value of a coalition is the cost of serving its members. Specifically, the cost in our games is the service radius of the coalition. We call these games the Minimum Radius Location Games (MRLG).We study the existence of core allocations and the existence of polynomial representations of the cores of these games, focusing on network spaces, i.e., finite metric spaces induced by undirected graphs and positive edge lengths, and on the ?_p metric spaces defined over R^d.     

具有联盟结构的position值

position值是图对策中著名的分支有效解, 该值充分体现了图的边在合作中的贡献, 因而也可作为网络中心性的一种测度方法。本文基于van den Brink等提出的具有联盟结构与图结构的合作对策, 将position值推广到具有联盟结构的图对策上, 提出了具有联盟结构的position值, 该值可以作为受优先联盟约束的网络中心性的一种测度方法。本文首先证明了具有联盟结构的position值可以由分割分支有效性和平衡边贡献性所唯一刻画。其次, 以跨国天然气管道网的收益分配为例, 对这个值与其他值做了比较分析。     

Existence of equilibrium stationary strategies in discounted noncooperative stochastic games with uncountable state space

This paper considers discounted noncooperative stochastic games with uncountable state space and compact metric action spaces. We assume that the transition law is absolutely continuous with respect to some probability measure defined on the state space. We prove, under certain additional continuity and integrability conditions, that such games have -equilibrium stationary strategies for each >0. To prove this fact, we provide a method for approximating the original game by a sequence of finite or countable state games. The main result of this paper answers partially a question raised by Parthasarathy in Ref. 1.     

Existence of optimal strategies in Markov games with incomplete information

The existence of a value and optimal strategies is proved for the class of two-person repeated games where the state follows a Markov chain independently of players’ actions and at the beginning of each stage only Player 1 is informed about the state. The results apply to the case of standard signaling where players’ stage actions are observable, as well as to the model with general signals provided that Player 1 has a nonrevealing repeated game strategy. The proofs reduce the analysis of these repeated games to that of classical repeated games with incomplete information on one side. This research was supported in part by Israeli Science Foundation grants 382/98, 263/03, and 1123/06, and by the Zvi Hermann Shapira Research Fund.     

Discrete-time zero-sum Markov games with first passage criteria

In this paper, we deal with two-person zero-sum stochastic games for discrete-time Markov processes. The optimality criterion to be studied is the discounted payoff criterion during a first passage time to some target set, where the discount factor is state-dependent. The state and action spaces are all Borel spaces, and the payoff functions are allowed to be unbounded. Under the suitable conditions, we first establish the optimality equation. Then, using dynamic programming techniques, we obtain the existence of the value of the game and a pair of optimal stationary policies. Moreover, we present the exponential convergence of the value iteration and a ‘martingale characterization’ of a pair of optimal policies. Finally, we illustrate the applications of our main results with an inventory system.     

Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.     

Total Reward Stochastic Games and Sensitive Average Reward Strategies

In this paper, total reward stochastic games are surveyed. Total reward games are motivated as a refinement of average reward games. The total reward is defined as the limiting average of the partial sums of the stream of payoffs. It is shown that total reward games with finite state space are strategically equivalent to a class of average reward games with an infinite countable state space. The role of stationary strategies in total reward games is investigated in detail. Further, it is outlined that, for total reward games with average reward value 0 and where additionally both players possess average reward optimal stationary strategies, it holds that the total reward value exists.     

Some Classes of Imperfect Information Finite State-Space Stochastic Games with Finite-Dimensional Solutions

Stochastic games under imperfect information are typically computationally intractable even in the discrete-time/discrete-state case considered here. We consider a problem where one player has perfect information. A function of a conditional probability distribution is proposed as an information state. In the problem form here, the payoff is only a function of the terminal state of the system, and the initial information state is either linear or a sum of max-plus delta functions. When the initial information state belongs to these classes, its propagation is finite-dimensional. The state feedback value function is also finite-dimensional, and obtained via dynamic programming, but has a nonstandard form due to the necessity of an expanded state variable. Under a saddle point assumption, Certainty Equivalence is obtained and the proposed function is indeed an information state.     

Noncooperative stochastic games under a n-stage local contraction assumption

We consider a class of noncooperative stochastic games with general state and action spaces and with a state dependent discount factor. The expected time duration between any two stages of the game is not bounded away from zero, so that the usual N-stage contraction assumption, uniform over all admissible strategies, does not hold. We propose milder sufficient regularity conditions, allowing strategies that give rise with probability one to any number of simultaneous stages. We give sufficient conditions for the existence of equilibrium and ∈-equilibrium stationary strategies in the sense of Nash. In the two-player zero-sum case, when an equilibrium strategy exists, the value of the game is the unique fixed point of a specific functional operator and can be computed by dynamic programming.     

Relations between game parameters,value, and optimal strategy spaces in stochastic games and construction of games with given solution

Results of Bohnenblust, Karlin, and Shapley and results of Shapley and Snow, concerning solutions of matrix games, are extended to the class of discounted stochastic games. Prior to these extensions, relations between the game parameters, value, and optimal stationary strategy spaces are established. Then, the inverse problem of constructing stochastic games, given the solution, is considered.     

Mixed strategy solutions for quadratic games

Mixed strategy solutions are given for two-person, zero-sum games with payoff functions consisting of quadratic, bilinear, and linear terms, and strategy spaces consisting of closed balls in a Hilbert space. The results are applied to linear-quadratic differential games with no information, and with quadratic integral constraints on the control functions.     

On the existence of almost-pure-strategy Nash equilibrium in <Emphasis Type="Italic">n</Emphasis>-person finite games

This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.     

双准均衡合作对策τ值的公理化

介绍了能准确刻画现实生活中每个参与者有三种选择的双合作对策,在此基础上研究了双合作对策的τ值,并对双准均衡合作对策的τ值进行了公理化,其中双合作对策的上向量、间隙函数、让步向量的构造是刻画其τ值的基础．     

On the existence of good stationary strategies for nonleavable stochastic games

This paper discusses the problem regarding the existence of optimal or nearly optimal stationary strategies for a player engaged in a nonleavable stochastic game. It is known that, for these games, player I need not have an -optimal stationary strategy even when the state space of the game is finite. On the contrary, we show that uniformly -optimal stationary strategies are available to player II for nonleavable stochastic games with finite state space. Our methods will also yield sufficient conditions for the existence of optimal and -optimal stationary strategies for player II for games with countably infinite state space. With the purpose of introducing and explaining the main results of the paper, special consideration is given to a particular class of nonleavable games whose utility is equal to the indicator of a subset of the state space of the game.