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.     

Algorithms for computing Nash equilibria in deterministic LQ games

In this paper we review a number of algorithms to compute Nash equilibria in deterministic linear quadratic differential games. We will review the open-loop and feedback information case. In both cases we address both the finite and the infinite-planning horizon.     

A class of differential games for which the closed-loop and open-loop Nash equilibria coincide

It is well known that, in general, Nash equilibria in open-loop strategies do not coincide with those in closed-loop strategies. This note identifies a class of differential games in which the Nash equilibrium in closed-loop strategies is degenerate, in the sense that it depends on time only. Consequently, the closed-loop equilibrium is also an equilibrium in open-loop strategies.The helpful comments of Professors Y. C. Ho, G. Leitmann, H. Y. Wan, Jr., and an anonymous referee are gratefully acknowledged.     

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.     

Optimal strategic reasoning with McNaughton functions

The aim of the paper is to explore strategic reasoning in strategic games of two players with an uncountably infinite space of strategies the payoff of which is given by McNaughton functions—functions on the unit interval which are piecewise linear with integer coefficients. McNaughton functions are of a special interest for approximate reasoning as they correspond to formulas of infinitely valued Lukasiewicz logic. The paper is focused on existence and structure of Nash equilibria and algorithms for their computation. Although the existence of mixed strategy equilibria follows from a general theorem (Glicksberg, 1952) [5], nothing is known about their structure neither the theorem provides any method for computing them. The central problem of the article is to characterize the class of strategic games with McNaughton payoffs which have a finitely supported Nash equilibrium. We give a sufficient condition for finite equilibria and we propose an algorithm for recovering the corresponding equilibrium strategies. Our result easily generalizes to n-player strategic games which don't need to be strictly competitive with a payoff functions represented by piecewise linear functions with real coefficients. Our conjecture is that every game with McNaughton payoff allows for finitely supported equilibrium strategies, however we leave proving/disproving of this conjecture for future investigations.     

Interactions between government and firms: a differential game approach

In this paper we study an extended version of the model described in Gradus (J. Econ. 81:1092–1109, 1989) in order to determine the optimal taxation policy of a government and its effects on the stock of capital goods growth as a result of the activity developed by firms. It is shown that, by introducing a wealth tax, there exists an optimal wealth tax rate for which the open-loop/feedback Nash/Stackelberg equilibria coincide, maximizing the payments for both agents (government and firms), so that the open-loop Stackelberg equilibrium becomes both time consistent and subgame perfect. Partially supported by MEC (Spain) Grants MTM2006-13468, BMF2002-03493 and project ‘Ingenio Mathematica (i-MATH)’ No. CSD2006-00032 (Consolider-Ingenio 2010).     

Infinite-Horizon Stochastic Differential Games with Branching Payoffs

In this paper, we consider infinite-horizon stochastic differential games with an autonomous structure and steady branching payoffs. While the introduction of additional stochastic elements via branching payoffs offers a fruitful alternative to modeling game situations under uncertainty, the solution to such a problem is not known. A theorem on the characterization of a Nash equilibrium solution for this kind of games is presented. An application in renewable resource extraction is provided to illustrate the solution mechanism.     

Weak and Strong Time Consistency in a Differential Oligopoly Game with Capital Accumulation

We illustrate a differential oligopoly game with capital accumulation where the accumulation dynamics of productive capacity is modelled à la Ramsey. The model is solved under the open-loop information structure, to show that it admits an open-loop Nash equilibrium which is indeed a degenerate feedback one and therefore strongly time consistent, even if, by construction, the problem under consideration is not a linear state game. We thank George Leitmann, Massimo Marinacci, Daniele Ritelli, Arsen Palestini and two anonymous referees for very useful comments and suggestions.     

Sufficiency and game structure in Nash open-loop differential games

This note deals with the following problem inN-player Nash open-loop differential games: Which conditions on the Hamiltonians will simplify the verification of the sufficient conditions? In the class of games with Hamiltonians linear in the state, and where the state is separated from the control of playeri in the Hamiltonian of this player, the first-order and second-order conditions for an extremum of the Hamiltonian of playeri also constitute a set of sufficient conditions.     

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.     

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.     

Optimal allocation of drug control efforts: A differential game analysis

The present paper considers a dynamic nonzero-sum game between drug dealers and the authorities. Although the game is neither linear-quadratic nor degenerate, in the sense that the closed-loop equilibria coincide with the open-loop equilibria, we are able to calculate explicitly a stationary feedback Nash equilibrium of that game. In a numerical example, we determine the optimal allocation of governmental efforts between treatment and law enforcement minimizing the total discounted cost stream in the equilibrium. Moreover, we provide sensitivity analyses with respect to the efficiency parameters of both competitors. Our results show that a farsighted authority should attack the drug problem from the demand side and put much effort in treatment measures and the improvement of the efficiency of the treatment.This research was supported by the Austrian Science Foundation under Contract P9112-SOZ. We would like to thank A. Luhmer and E. J. Dockner for helpful comments.     

一种n人静态博弈纯策略纳什均衡存在性判别法

本首先给出了n人静态博弈纯策略纳什均衡存在的充要条件。然后给出n人静态博弈纯策略纳什均衡存在性的一种判别方法。最后在判别纯策略纳什均衡存在的条件下，给出判定该静态博弈存在多少纯策略纳什均衡以及哪些纯策略组合是纯策略纳什均衡(解)的方法。     

Infinite Player Noncooperative Games with Vector Payoffs Under Relative Pseudomonotonicity

In this paper we consider the Nash equilibrium problem for infinite player games with vector payoffs in a topological vector space setting. By employing new concepts of relative (pseudo)monotonicity, we establish several existence results of solutions for usual and normalized vector equilibria. The results strengthen existence results for vector equilibrium problems, which were based on classical pseudomonotonicity concepts. They also extend previous results for vector variational inequalities and finite player games under relative (pseudo)monotonicity.     

Coordinate Transformations and Derivation of Open-Loop Nash Equilibria

Leitmann (Ref. 1) introduced coordinate transformations to derive global optima of a class of dynamic optimization problems. We present applications of this method to derive open-loop Nash equilibria for finite-time horizon differential games. The method of coordinate transformations is especially useful in cases where the original game does not satisfy the global curvature conditions normally imposed in sufficient optimality conditions.     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.     

N-person differential games governed by semilinear stochastic evolution systems

In this paper the problem ofN-person infinite-dimensional stochastic differential games governed by semilinear stochastic evolution control systems is discussed. First the minimax principle which is the necessary condition for the existence of open-loop Nash equilibrium is proved. Then the necessary and sufficient conditions of open-loop and closed-loop Nash equilibrium for linear quadratic infinite-dimensional stochastic differential games are derived.     

可行策略对应的图像拓扑下广义博弈Nash平衡的稳定性

以往关于广义博弈Nash平衡的稳定性的研究,均利用可行策略映射之间的一致度量.现考虑在更弱的度量下,利用可行策略映射图像之间的Hausdorff距离定义度量.在此弱图像拓扑下,证明了广义博弈空间的完备性,以及Nash平衡映射的上半连续性和紧性,进而得到广义博弈Nash平衡的通有稳定性.即在Baire分类的意义下,大多数的广义博弈都是本质的.     

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.     

Remarks on sensitive equilibria in stochastic games with additive reward and transition structure

A class of stochastic games with additive reward and transition structure is studied. For zero-sum games under some ergodicity assumptions 1-equilibria are shown to exist. They correspond to so-called sensitive optimal policies in dynamic programming. For a class of nonzero-sum stochastic games with nonatomic transitions nonrandomized Nash equilibrium points with respect to the average payoff criterion are also obtained. Included examples show that the results of this paper can not be extented to more general payoff or transition structure.