On Nash Equilibria for Noncooperative Games Governed by the Burgers Equation

Existence of a Nash equilibrium in a noncooperative game governed by the one-dimensional Burgers equation, proposed in the case of pointwise controls in Ref. 1, is proved under data qualifications that guarantee the diffusion term in the Burgers’ equation to be dominant enough with respect to the uniform convexity of the payoffs. This work was partly supported by Grants 201/03/0934 (GA čR) and MSM 0021620839 (MšMT čR). Inspiring discussions with Angel M. Ramos are acknowledged.     

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.     

Nash Equilibria of Risk-Sensitive Nonlinear Stochastic Differential Games

This paper considers a class of risk-sensitive stochastic nonzero-sum differential games with parametrized nonlinear dynamics and parametrized cost functions. The parametrization is such that, if all or some of the parameters are set equal to some nominal values, then the differential game either becomes equivalent to a risk-sensitive stochastic control (RSSC) problem or decouples into several independent RSSC problems, which in turn are equivalent to a class of stochastic zero-sum differential games. This framework allows us to study the sensitivity of the Nash equilibrium (NE) of the original stochastic game to changes in the values of these parameters, and to relate the NE (generally difficult to compute and to establish existence and uniqueness, at least directly) to solutions of RSSC problems, which are relatively easier to obtain. It also allows us to quantify the sensitivity of solutions to RSSC problems (and thereby nonlinear H-control problems) to unmodeled subsystem dynamics controlled by multiple players.     

Sampled-Data Nash Equilibria in Differential Games with Impulse Controls

Journal of Optimization Theory and Applications - We study a class of deterministic two-player nonzero-sum differential games where one player uses piecewise-continuous controls to affect the...     

Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes

In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results for the unknown quadratic payoffs to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

     

New Method to Characterize Subgame Perfect Nash Equilibria in Differential Games

In this paper, we present a method for computing Nash equilibria in feedback strategies. This method gives necessary and sufficient conditions to characterize subgame perfect equilibria by means of a system of quasilinear partial differential equations. This characterization allows one to know explicitly the solution of the game in some cases. In other cases, this approach makes a qualitative study easier. We apply this method to nonrenewable resource games.     

On Coincidence of Feedback Nash Equilibria and Stackelberg Equilibria in Economic Applications of Differential Games

The scope of the applicability of the feedback Stackelberg equilibrium concept in differential games is investigated. First, conditions for obtaining the coincidence between the stationary feedback Nash equilibrium and the stationary feedback Stackelberg equilibrium are given in terms of the instantaneous payoff functions of the players and the state equations of the game. Second, a class of differential games representing the underlying structure of a good number of economic applications of differential games is defined; for this class of differential games, it is shown that the stationary feedback Stackelberg equilibrium coincides with the stationary feedback Nash equilibrium. The conclusion is that the feedback Stackelberg solution is generally not useful to investigate leadership in the framework of a differential game, at least for a good number of economic applications This paper was presented at the 8th Viennese Workshop on Optimal Control, Dynamic Games, and Nonlinear Dynamics: Theory and Applications in Economics and OR/MS, Vienna, Austria, May 14–16, 2003, at the Seminar of the Instituto Complutense de Analisis Economico, Madrid, Spain, June 20, 2003, and at the Sevilla Workshop on Dynamic Economics and the Environment, Sevilla, Spain, July 2–3, 2003. The author is grateful to the participants in these sessions, in particular F.J. Andre and J. Ruiz, for their comments. Five referees provided particularly helpful suggestions. Financial support from the Ministerio de Ciencia y Tecnologia under Grant BEC2000-1432 is gratefully acknowledged.     

Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

We introduce a new mean field kinetic model for systems of rational agents interacting in a game-theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. An application of the presented theory to a social model (herding behavior) is discussed.     

Noncooperative Differential Games

This paper, having a tutorial character, is intended to provide an introduction to the theory of noncooperative differential games.     

Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t _f. In this paper, we study the behavior of feedback Nash equilibria for t _f.In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.     

On Existence of Nash Equilibria of Games with Constraints on Multistrategies

In this paper, sufficient conditions are given, which are less restrictivethan those required by the Arrow–Debreu–Nash theorem, on theexistence of a Nash equilibrium of an n-player game {₁, . . . , Y_n,f₁, . . . , f_n} in normal form with a nonempty closedconvex constraint C on the set Y=i Y_i of multistrategies. Theith player has to minimize the function fi with respect to the ithvariable. We consider two cases.In the first case, Y is a real Hilbert space and the loss function class isquadratic. In this case, the existence of a Nash equilibrium is guaranteedas a simple consequence of the projection theorem for Hilbert spaces. In thesecond case, Y is a Euclidean space, the loss functions are continuous, andf_i is convex with respect to the ith variable. In this case, the techniqueis quite particular, because the constrained game is approximated with asequence of free games, each with a Nash equilibrium in an appropriatecompact space X. Since X is compact, there exists a subsequence of theseNash equilibrium points which is convergent in the norm. If thelimit point is in C and if the order of convergence is greater than one,then this is a Nash equilibrium of the constrained game.     

Identification of Efficient Subgame-Perfect Nash Equilibria in a Class of Differential Games1

We present a method for the characterization of subgame-perfect Nash equilibria being Pareto efficient in a class of differential games. For that purpose, we propose a new approach based on new necessary and sufficient conditions for computing subgame-perfect Nash equilibria.     

Open and Closed Loop Nash Equilibria in Games with a Continuum of Players

In this paper, the problem of relations between closed loop and open loop Nash equilibria is examined in the environment of discrete time dynamic games with a continuum of players and a compound structure encompassing both private and global state variables. An equivalence theorem between these classes of equilibria is proven, important implications for the calculation of these equilibria are derived and the results are presented on models of a common ecosystem exploited by a continuum of players. An example of an analogous game with finitely many players is also presented for comparison.     

Limit Consistent Solutions in Noncooperative Games

Strong and limit consistency in finite noncooperative games are studied. A solution is called strongly consistent if it is both consistent and conversely consistent (Ref. 1). We provide sufficient conditions on one-person behavior such that a strongly consistent solution is nonempty. We introduce limit consistency for normal form games and extensive form games. Roughly, this means that the solution can be approximated by strongly consistent solutions. We then show that the perfect and proper equilibrium correspondences in normal form games, as well as the weakly perfect and sequential equilibrium correspondences for extensive form games, are limit consistent.     

Approximation of Noncooperative Semi-Markov Games

An approximation of a general V-ergodic semi-Markov game with Borel state space by discrete-state space strongly-ergodic games is studied. The standard expected ratio-average criterion as well as the expected time-average criterion are considered. New theorems on the existence of ∊-equilibria are given.Communicated by D. A. CarlsonThe authors thank an anonymous referee for constructive comments. This work is supported by MEiN Grant 1P03A 01030.     

Constructions of Nash Equilibria in Stochastic Games of Resource Extraction with Additive Transition Structure

A class of N-person stochastic games of resource extraction with discounted payoffs in discrete time is considered. It is assumed that transition probabilities have special additive structure. It is shown that the Nash equilibria and corresponding payoffs in finite horizon games converge as horizon goes to infinity. This implies existence of stationary Nash equilibria in the infinite horizon case. In addition the algorithm for finding Nash equilibria in infinite horizon games is discussed     

A Qualitative Characterization of Symmetric Open-Loop Nash Equilibria in Discounted Infinite Horizon Differential Games

The local stability, steady state comparative statics, and local comparative dynamics of symmetric open-loop Nash equilibria for the ubiquitous class of discounted infinite horizon differential games are investigated. It is shown that the functional forms and values of the parameters specified in a differential game are crucial in determining the local stability of a steady state and, in turn, the steady state comparative statics and local comparative dynamics. A simple sufficient condition for a steady state to be a local saddle point is provided. The power and reach of the results are demonstrated by applying them to two well-known differential games.     

A Strong Equilibrium in Noncooperative Games

A new concept of game equilibrium is proposed, which makes it possible to find a unique solution in a wide class of noncooperative games in pure strategies. Its place in the hierarchical sequence of previously known equilibria is defined. Examples of static and differential games are used to demonstrate the procedure that finds this new equilibrium.     

Redefinition of Belief Distorted Nash Equilibria for the Environment of Dynamic Games with Probabilistic Beliefs

In this paper, a new concept of equilibrium in dynamic games with incomplete or distorted information is introduced. In the games considered, players have incomplete information about crucial aspects of the game and formulate beliefs about the probabilities of various future scenarios. The concept of belief distorted Nash equilibrium combines optimization based on given beliefs and self-verification of those beliefs. Existence and equivalence theorems are proven, and this concept is compared to existing ones. Theoretical results are illustrated using several examples: extracting a common renewable resource, a large minority game, and a repeated Prisoner’s Dilemma.