Feedback strategies with finite memory in differential games

It is shown that, in differential games, strategies can be defined for the players in such a way that their controls depend, for each timet, on a finite section of the past of the trajectoryx(t). In particular,s-delay,r-memory strategies can be defined as in the Varaiya-Lin approach. It is shown that, for deterministic differential games with terminal payoff, the upper and lower values of the game so defined, are independent of the lengthr of the memory. Lettingr 0, a feedback strategy is obtained which depends only on the present (and infinitesimal past) of the trajectoryx(t).     

Semi-cooperative strategies for differential games

The paper is concerned with a non-cooperative differential game for two players. We first consider Nash equilibrium solutions in feedback form. In this case, we show that the Cauchy problem for the value functions is generically ill-posed. Looking at vanishing viscosity approximations, one can construct special solutions in the form of chattering controls, but these also appear to be unstable. In the second part of the paper we propose an alternative semi-cooperative pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, we prove that the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.Revised: May 2004     

Single-payoff farsighted stable sets in strategic games with dominant punishment strategies

We investigate farsighted stable sets in a class of strategic games with dominant punishment strategies. In this class of games, each player has a strategy that uniformly minimizes the other players’ payoffs for any given strategies chosen by these other players. We particularly investigate a special class of farsighted stable sets, each of which consists of strategy profiles yielding a single payoff vector. We call such a farsighted stable set as a single-payoff farsighted stable set. We propose a concept called an inclusive set that completely characterizes single-payoff farsighted stable sets in strategic games with dominant punishment strategies. We also show that the set of payoff vectors yielded by single-payoff farsighted stable sets is closely related to the strict \(\alpha \)-core in a strategic game. Furthermore, we apply the results to strategic games where each player has two strategies and strategic games associated with some market models.     

Stackelberg strategies in linear-quadratic stochastic differential games

This paper obtains the Stackelberg solution to a class of two-player stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that the players make independent noisy measurements of the initial state and are permitted to utilize only this information in constructing their controls. Furthermore, by the very nature of the Stackelberg solution concept, one of the players is assumed to know, in advance, the strategy of the other player (the leader). For this class of problems, we first establish existence and uniqueness of the Stackelberg solution and then relate the derivation of the leader's Stackelberg solution to the optimal solution of a nonstandard stochastic control problem. This stochastic control problem is solved in a more general context, and its solution is utilized in constructing the Stackelberg strategy of the leader. For the special case Gaussian statistics, it is shown that this optimal strategy is affine in observation of the leader. The paper also discusses numerical aspects of the Stackelberg solution under general statistics and develops algorithms which converge to the unique Stackelberg solution.This work was performed while the second author was on sabbatical leave at the Department of Applied Mathematics, Twente University of Technology, Enschede, Holland.     

Notions of equilibrium for differential games on intersecting game sets

We suggest new notions of conflict equilibrium and demonstrate a technique of their use for finding a solution in arbitrary game problems on a game set common for all players and especially in problems with side interests of players in the static and dynamic settings.     

Solvability of systems of partial differential equations for functions defined on nonconvex sets

We consider systems of partial differential equations with constant coefficients of the form ( R(D_x, D_y)f = 0, P(D_x)f = g), f,g ? C^￥(W),\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),, where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition R(D_x, D_y)g = 0  and  W ì \Bbb Rⁿ⁺¹R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1} is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets W\Omega implies that any localization at ￥\infty of the principle part P_m of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets W\Omega by the fundamental principle of Ehrenpreis-Palamodov.     

Mixed strategies and their application in the encounter-evasion differential games

A positional encounter-evasion differential game with geometric constraints on the players' controls, depending on the system's state, is examined. The concept of the players' mixed strategies is introduced and an alternative is proved which asserts that either the positional encounter game or the positional evasion game is always solvable. The paper continues the investigations in /1–4/.     

Remarks on saddle points in nonconvex sets

Smol'yakov's saddle point theorem is generalized to admissible sets (in the sense of Klee). Moreover, convexity of the involved sets in the theorem can be replaced by acyclicity, and continuity of the involved functions by lower or upper semicontinuity.     

Stopping games with randomized strategies

We study stopping games in the setup of Neveu. We prove the existence of a uniform value (in a sense defined below), by allowing the players to use randomized strategies. In constrast with previous work, we make no comparison assumption on the payoff processes. Moreover, we prove that the value is the limit of discounted values, and we construct ε-optimal strategies. Received: 10 May 1999 / Revised version: 18 May 2000 / Published online: 15 February 2001     

Bargaining sets in cooperative games with limited communication

Given a connected undirected graph ϕ with vertex set N, cooperative games (N, v) are considered in which players can cooperate only when the corresponding vertices form a connected subgraph in the graph ϕ. For such games, two generalizations of the bargaining set M ₁ ⁱ , which was introduced by Aumann and Maschler, are investigated.     

Advantageous strategies in differential games for a class of suboptimal control laws

This paper deals with suboptimal control policies in differential games. Sufficient conditions for advantageous strategies of either player are examined for a linear-quadratic game in which the players are constrained to use suboptimal control laws of specified form. The concept of a “bargaining matrix” is employed. Specific scalar and vector examples are included to illustrate the theory.     

Some fundamental qualifications of optimal strategies and transition surfaces in differential games

At each point of a regular region of a differential game, there are two tangent cones which are complementary to each other. The two vectograms defined by the pair of pure optimal strategies must be contained separately in the two tangent cones. The velocity vector resulting from the selection of the optimal strategies consequently must represent a semipermeable direction. These conditions, which reveal a fundamental separating property of the optimal velocity vector, are weaker than that of Isaac's main equation. Moreover they hold even on singular surfaces in the regular regions. The conditions also reveal a necessary condition for a differential game to have a regular region. Each isovalued surface in the regular region is essentially a semipermeable surface. A transition surface arises only when the resulting directed isovalued surface of the previous optimal strategies cannot have a smooth semipermeable extension at the singular surface. This observation yields a necessary condition for a transition surface to occur. The localization of transition surfaces is then possible. Finally, a jump and smooth condition of the isovalued surfaces is given.This research was supported in part by the Systems Analysis Program, The University of Rochester, under the Bureau of Naval Personnel Contract No. N00022-70-C-0076. It was also supported in part by the Center for Naval Analyses of the University of Rochester. The author is grateful to Professors R. Isaacs, M. Freimer, and H. Gerber for their helpful discussion during the research. He is especially grateful to Professor G. Leitmann, University of California at Berkeley, for his very helpful comments and suggestions.     

Existence of optimal open-loop strategies for a class of differential games

In this paper, the class of differential games with linear system equations and a quadratic performance index is investigated for saddlepoint solutions when one or both of the players use open-loop control. For each formulation of the game, a necessary and sufficient condition is obtained for the existence of an optimal strategy pair that generates a regular optimal path. For those cases where a solution exists, the unique saddle-point solution is presented. Also, relationships are established between the time intervals of existence of solutions for the various formulations of the game.This research was supported by the National Science Foundation, Grant No. GK-3341.     

Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×R ⁿ 2 ^{R n} be a set-valued map with compact values; let :R ⁿ R ^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion . We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (z(T))(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.