Infinite horizon noncooperative differential games

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value functions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.     

Infinite horizon noncooperative differential games with nonsmooth costs

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we study a class of infinite-horizon scalar games with either piecewise linear or piecewise smooth costs, exponentially discounted in time. By the analysis of the value functions, we find that results about existence and uniqueness of admissible solutions to the HJ system, and therefore of Nash equilibrium solutions in feedback form, can be recovered as in the smooth costs case, provided the costs are globally monotone. On the other hand, we present examples of costs such that the corresponding HJ system has infinitely many admissible solutions or no admissible solutions at all, suggesting that new concepts of equilibria may be needed to study games with general nonlinear costs.     

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.     

Stay-in-a-set games

There exists a Nash equilibrium (ε-Nash equilibrium) for every n-person stochastic game with a finite (countable) state space and finite action sets for the players if the payoff to each player i is one when the process of states remains in a given set of states G _i and is zero otherwise. Received: December 2000     

Generic stability of Nash equilibria for noncooperative differential games

The stability of Nash equilibria against the perturbation of the right-hand side functions of state equations for noncooperative differential games is investigated. By employing the set-valued analysis theory, we show that the differential games whose equilibria are all stable form a dense residual set, and every differential game can be approximated arbitrarily by a sequence of stable differential games, that is, in the sense of Baire’s category most of the differential games are stable.     

An Approach to Fuzzy Noncooperative Nash Games

Systems that involve more than one decision maker are often optimized using the theory of games. In the traditional game theory, it is assumed that each player has a well-defined quantitative utility function over a set of the player decision space. Each player attempts to maximize/minimize his/her own expected utility and each is assumed to know the extensive game in full. At present, it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player in these types of problems. On the other hand, in many of such complex problems, each player has a heuristic knowledge of the desires of the other players and a heuristic knowledge of the control choices that they will make in order to meet their ends.In this paper, we utilize fuzzy set theory in order to incorporate the players' heuristic knowledge of decision making into the framework of conventional game theory or ordinal game theory. We define a new approach to N-person static fuzzy noncooperative games and develop a solution concept such as Nash for these types of games. We show that this general formulation of fuzzy noncooperative games can be applied to solve multidecision-making problems where no objective function is specified. The computational procedure is illustrated via application to a multiagent optimization problem dealing with the design and operation of future military operations.     

S-modular games,with queueing applications

The notion ofS-modularity was developed by Glasserman and Yao [9] in the context of optimal control of queueing networks.S-modularity allows the objective function to be supermodular in some variables and submodular in others. It models both compatible and conflicting incentives, and hence conveniently accommodates a wide variety of applications. In this paper, we introduceS-modularity into the context ofn-player noncooperative games. This generalizes the well-known supermodular games of Topkis [22], where each player maximizes a supermodular payoff function (or equivalently, minimizes a submodular payoff function). We illustrate the theory through a variety of applications in queueing systems.Supported in part by NSF Grant MSS-92-16490, and by Columbia's Center for Telecommunications Research.     

Reductions of interval noncooperative games

Noncooperative games of a finite number of persons with interval-valued payoff functions are considered. The concept of an equilibrium situation is introduced. A reduction of such games to deterministic noncooperative games is proposed. Properties of the reduced games are discussed. Interval antagonistic and bimatrix games are examined, and illustrative examples are considered.     

Optimal pricing in a duopoly: A noncooperative differential games solution

This paper deals with a differential games model of an oligopoly ofn profit-maximizing firms competing for the same stock of customers. For the sale dynamics, it is assumed that the customers of each firm are driven away gradually by increasing product prices. Since the state variable is absent from the Hamiltonian maximizing conditions as well as from the adjoint equations, open-loop Nash solutions can be obtained. By using phase diagram analysis, for two players the behavior of the optimal pricing strategies can be characterized qualitatively. The main importance of the paper lies in the solution technique, rather than in the economic significance of the proposed model. Under the proposed assumptions, the two-point boundary-value problem resulting from the maximum principle is reduced to a terminal-value problem. It turns out that, for special salvage values of the market shares and if the planning horizon is not too short, nonmonotonic Nash-optimal price trajectories occur.Thanks are due to S. Jørgensen, A. Mehlmann, and R. Willing for helpful suggestions.     

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.     

Nonrandomized strategy equilibria in noncooperative stochastic games with additive transition and reward structure

In this paper, we study a discounted noncooperative stochastic game with an abstract measurable state space, compact metric action spaces of players, and additive transition and reward structure in the sense of Himmelberget al. (Ref. 1) and Parthasarathy (Ref. 2). We also assume that the transition law of the game is absolutely continuous with respect to some probability distributionp of the initial state and together with the reward functions of players satisfies certain continuity conditions. We prove that such a game has an equilibrium stationary point, which extends a result of Parthasarathy from Ref. 2, where the action spaces of players are assumed to be finite sets. Moreover, we show that our game has a nonrandomized (- )-equilibrium stationary point for each >0, provided that the probability distributionp is nonatomic. The latter result is a new existence theorem.     

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.     

Performance versus informativeness in linear-quadratic Gaussian noncooperative games

In this paper, we study the impact of informativeness on the performance of linear quadratic Gaussian Nash and Stackelberg games. We first show that, in two-person static Nash games, if one of the players acquires more information, then this extra information is beneficial to him, provided that it is orthogonal to both players' information. A special case is that when one of the players is informationally stronger than the other, then any new information is beneficial to him. We then show that a similar result holds for dynamic Nash games. In the dynamic games, the players use strategies that are linear functions of the current estimates of the state, generated by two Kalman filters. The same properties are proved to hold in static and feedback Stackelberg games as well.This work was partially supported by the US Air Force Office of Scientific Research under Grant No. AFOSR-82-0174.     

Randomized stopping games and Markov market games

We study nonzero-sum stopping games with randomized stopping strategies. The existence of Nash equilibrium and ɛ-equilibrium strategies are discussed under various assumptions on players random payoffs and utility functions dependent on the observed discrete time Markov process. Then we will present a model of a market game in which randomized stopping times are involved. The model is a mixture of a stochastic game and stopping game. Research supported by grant PBZ-KBN-016/P03/99.     

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.     

Interactive decisions and potential games

The aim of this contribution is an overview on Potential Games. This class of games is special, in fact we can investigate their properties by a unique function: the potential function. We consider several types of potential games: exact, ordinal, bayesian and hierarchical. Some results are generalized to multicriteria decisions.      

On differential games with a feedback nash equilibrium

This note provides a lemma on differential games which possess a feedback Nash equilibrium (FNE). In particular, it shows that (i) a class of games with a degenerate FNE can be constructucted from every game which has a nondegenerate FNE and (ii) a class of games with a nondegenerate FNE can be constructed from every game which has a degenerate FNE.The author would like to thank an anonymous referee for invaluable comments and suggestions.     

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     

On Nikaido-Isoda type theorems for discounted stochastic games

We present a class of countable state space stochastic games with discontinuous payoff functions satisfying some assumptions similar to the ones of Nikaido and Isoda for one-stage games. We prove that these games possess stationary equilibria. We show that after adding some concavity assumptions these equilibria are nonrandomized. Further, we present an example of input (or production) dynamic game satisfying the assumptions of our model. We give a closed-form solution for this game.