A new approach toN-person,nonzero-sum,linear differential games

It is shown that there exist equilibrium strategies forn-person, nonero-sum, linear differential games if the cost to each player is convex. The approach used is believed to be novel, and is based on a theorem of Fan.This research was supported by the National Research Council of Canada under Grant No. A-7790.     

Guaranteed control in differential games with ellipsoidal payoff

A class of antagonistic linear differential games (DGs) in a fixed time interval with ellipsoidal payoff functional is considered. This class of DGs includes problems which assume both rigid constraints on the players' controls and requirements to minimize control expenses. Other known classes of differential games, such as linear DGs with a quadratic performance index and linear DGs with ellipsoidal terminal sets and admissible sets of controls for the players, considered in Kurzhanskii's ellipsoidal technique, are limiting cases of DGs of this class. The concept of a u-strategic function, which expresses the property of u-stability for ellipsoidal functions, is introduced. An effective algorithm is presented for computing a u-strategic function, based on Kurzhanskii's ellipsoidal technique. The main result of this paper is that a guaranteed positional strategy for player u is defined by a certain explicit formula in terms of a u-strategic function. The proof of this result is based on a viability theorem for differential equations.     

On pursuit problems in controlled distributed systems

In the present paper, we study the problem of transferring a system from one state into another state in the presence of a player trying to prevent the occurrence of this transfer in the system. The system dynamics is described by partial differential equations whose right-hand sides contain the player controls in additive form. In a similar setting, the problem was solved in several papers, but it has not been considered for the case in which various constraints are imposed on the controls of the players. Here, in contrast to several other papers, we consider games in the entire scale of spaces H _r , r ≥ 0. We propose a new approach for completing the pursuit under various constraints on the player controls.     

Quitting games – An example

Quitting games are multi-player sequential games in which, at every stage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; each player i then receives a payoff r _S ⁱ, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is zero.? We exhibit a four-player quitting game, where the “simplest” equilibrium is periodic with period two. We argue that this implies that all known methods to prove existence of an equilibrium payoff in multi-player stochastic games are therefore bound to fail in general, and provide some geometric intuition for this phenomenon. Received: October 2001     

Convex semi-infinite games

This paper introduces a generalization of semi-infinite games. The pure strategies for player I involve choosing one function from an infinite family of convex functions, while the set of mixed strategies for player II is a closed convex setC inR ⁿ. The minimax theorem applies under a condition which limits the directions of recession ofC. Player II always has optimal strategies. These are shown to exist for player I also if a certain infinite system verifies the property of Farkas-Minkowski. The paper also studies certain conditions that guarantee the finiteness of the value of the game and the existence of optimal pure strategies for player I.Many thanks are due to the referees for their detailed comments.     

Preferred coalitions in cooperative differential games

There are many interesting situations which can be described by anN-person general-sum differential game. Such games are characterized by the fact that the strategy of each player depends upon reasonable assumptions about the strategies of the remaining players; and, thus, these games cannot be considered asN uncoupled optimal control problems. In such cases, we say that the game is not strictly competitive, but involves a mutual interest which makes it possible for all of the players to reduce their costs by cooperating with one another, provided the resulting agreement can be enforced. When cooperation is allowed and there are more than two players, there is always the question of whether all possible subcoalitions will be formed with equal ease. This work considers the situation in which a particular subcoalition is preferred. A theory of general-sum games with preferred coalitions is presented, together with constructive examples of alternative approaches which are unsatisfactory.     

Algorithms for non-convexS n -games

We considerS-games in which the setS is the parametric curve{(p ₁ (t),, p _n(t)): t [0, 1]} and thep _i(t) are real polynomials. These games will be referred to asS _n -games. Two iterative algorithms are given in the case ofn = 2. One gives linear convergence to an optimal of player I, and the other gives monotone convergence. In the case of arbitraryn we give an algorithm of the cutting plane family converging to the value. The principal features of this algorithm are that the hyperplanes arise intrinsically as tangents to an associated concave function which is not in general differentiable, and the linear subproblems arise as matrix games. Because of the latter property inactive constraints are in principle automatically dropped. We give a game theoretic convergence proof. This algorithm may be used multiply for bounding optimals of player I.     

Repeated Games with Bonuses

This paper deals with 2-player zero-sum repeated games in which player 1 receives a bonus at stage t if he repeats the action he played at stage t−1. We investigate the optimality of simple strategies for player 1. A simple strategy for player 1 consists of playing the same mixed action at every stage, irrespective of past play. Furthermore, for games in which player 1 has a simple optimal strategy, we characterize the set of stationary optimal strategies for player 2.     

The polynomial hierarchy and a simple model for competitive analysis

The multi-level linear programs of Candler, Norton and Townsley are a simple class of sequenced-move games, in which players are restricted in their moves only by common linear constraints, and each seeks to optimize a fixed linear criterion function in his/her own continuous variables and those of other players. All data of the game and earlier moves are known to a player when he/she is to move. The one-player case is just linear programming.We show that questions concerning only the value of these games exhibit complexity which goes up all levels of the polynomial hierarchy and appears to increase with the number of players.For three players, the games allow reduction of the ₂ and ₂ levels of the hierarchy. These levels essentially include computations done with branch-and-bound, in which one is given an oracle which can instantaneously solve NP-complete problems (e.g., integer linear programs). More generally, games with (p + 1) players allow reductions of _p and _p in the hierarchy.An easy corollary of these results is that value questions for two-player (bi-level) games of this type is NP-hard.The author's work has been supported by the Alexander von Humboldt Foundation and the Institut fur Okonometrie und Operations Research of the University of Bonn, Federal Republic of Germany; grant ECS8001763 of the National Science Foundation, USA; and a grant from the Georgia Tech Foundation.     

Static search games played over graphs and general metric spaces

We define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure. The game involves two players, the searching player and the concealing player, and is played over a metric space. Each player simultaneously chooses to deploy at a point in the space; the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position, the concealing player receiving a payoff of 1 otherwise. The concepts of dominance and equivalence of strategies are examined in the context of this game, before focusing on the more specific case of the game played over a graph. Methods are presented to simplify the analysis of such games, both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph. Lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs.     

Differential games with mixed leadership: The open-loop solution

This paper introduces the notion of mixed leadership in nonzero-sum differential games, where there is no fixed hierarchy in decision making with respect to the players. Whether a particular player is leader or follower depends on the instrument variable s/he is controlling, and it is possible for a player to be both leader and follower, depending on the control variable. The paper studies two-player open-loop differential games in this framework, and obtains a complete set of equations (differential and algebraic) which yield the controls in the mixed-leadership Stackelberg solution. The underlying differential equations are coupled and have mixed-boundary conditions. The paper also discusses the special case of linear-quadratic differential games, in which case solutions to the coupled differential equations can be expressed in terms of solutions to coupled Riccati differential equations which are independent of the state trajectory.     

Nonsmooth semipermeable barriers, Isaacs’ equation, and application to a differential game with one target and two players

We study the target problem which is a differential game where one of the players aims at reaching a target while the other player aims at avoiding this target forever. We characterize the victory domains of the players by means of geometric conditions and prove that the boundary of the victory domains is a nonsmooth semipermeable surface, i.e., is a solution (in a weak sense) of the Isaacs equation: sup _u inf _v 〈f (x, u, v),p〉 = 0, wheref is the dynamic of the system,u andv are the respective controls of the players, andp is a normal to the boundary of the victory domains at the pointx.     

An operator solution of stochastic games

A class of zero-sum, two-person stochastic games is shown to have a value which can be calculated by transfinite iteration of an operator. The games considered have a countable state space, finite action spaces for each player, and a payoff sufficiently general to include classical stochastic games as well as Blackwell’s infiniteG _δ games of imperfect information. Research supported by National Science Foundation Grants DMS-8801085 and DMS-8911548.     

A value for games withn players andr alternatives

We study value theory for a class of games called games withn players andr alternatives. In these games, each of then players must choose one and only one of ther alternatives. A linear, efficient value is obtained using three characterizations, two of which are axiomatic. This value yields an a priori evaluation for each player relative to each alternative.     

Partizan octal games: Partizan subtraction games

A subtraction gameS=(s ₁, ...,s _k)is a two-player game played with a pile of tokens where each player at his turn removes a number ofm of tokens providedmεS. The player first unable to move loses, his opponent wins. This impartial game becomes partizan if, instead of one setS, two finite setsS _L andS _R are given: Left removes tokens as specified byS _L, right according toS _R. We say thatS _L dominatesS _R if for all sufficiently large piles Left wins both as first and as second player. We exhibit a curious property of dominance and provide two subclasses of games in which a dominance relation prevails. We further prove that all partizan subtraction games areperiodic, and investigatepure periodicity.     

Computational complexity of winning strategies in two-person polynomial games

One considers two-person games, with players called I and II below. In order, they choose natural numbers, for example, for length 4, I chooses x₁, II chooses x₂. I chooses x₃, II chooses x₄. Then I wins if P(x₁,x₂,x₃,x₄)=0.Here P is a polynomial with integer coefficients. An old theorem of von Neumann and Zermelo shows that such a game is determined, i.e., there exists a winning strategy for one player or the other but not necessarily a computable winning strategy or one computable in polynomial time. It will be shown that there exists a game of polynomial type of length 4 for which there do not exist winning strategies for either player which are computable in polynomial time.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 69–73, 1991.     

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.     

Analyzing n-player impartial games

Combinatorial game theory is the study of two player perfect information games. While work has been done in the past on expanding this field to include n-player games we present a unique method which guarantees a single winner. Specifically our goal is to derive a function which, given an n-player game, is able to determine the winning player (assuming all n players play optimally). Once this is accomplished we use this function in analyzing a certain family of three player subtraction games along with a complete analysis of three player, three row Chomp. Furthermore we make use of our new function in producing alternative proofs to various well known two player Chomp games. Finally the paper presents a possible method of analyzing a two player game where one of the players plays a completely random game. As it turns out this slight twist to the rules of combinatorial game theory produces rather interesting results and is certainly worth the time to study further.     

A sufficiency condition for coalitive pareto-optimal solutions

In ak-player, nonzero-sum differential game, there exists the possibility that a group of players will form a coalition and work together. If allk players form the coalition, the criterion usually chosen is Pareto optimality whereas, if the coalition consists of only one player, a minmax or Nash equilibrium solution is sought.In this paper, games with coalitions of more than one but less thank players are considered. Coalitive Pareto optimality is chosen as the criterion. Sufficient conditions are presented for coalitive Pareto-optimal solutions, and the results are illustrated with an example.     

Importance Sampling,Large Deviations,and Differential Games

A standard heuristic in the area of importance sampling is that the changes of measure used to prove large deviation lower bounds give good performance when used for importance sampling. Recent work, however, has shown that a naive implementation of the heuristic is incorrect in many situations. The present paper considers the simple setting of sums of independent and identically distributed (iid) random variables, and shows that under mild conditions asymptotically optimal changes of measure can be found if one considers dynamic importance sampling schemes. For such schemes, the change of measure applied to an individual summand can depend on the historical empirical mean of the simulation (i.e. the system state). In analyzing the asymptotic performance of importance sampling, we show that the value function of a differential game characterizes the optimal performance, with player A corresponding to the choice of change of measure and player B arising due to a large deviations analysis. From this perspective, the traditional implementation of the heuristic is shown to correspond to player A choosing a control that is fixed, regardless of the system state or player B's choice of control. This leads to an "open loop" control for player A, which is suboptimal except in special cases. Our final contribution is a method, based on the Isaacs equation associated with the differential game, for constructing dynamic schemes. Numerical examples are presented to illustrate the results.