A class of trilinear differential games

This paper characterizes a class ofN-person, general sum differential games for which the optimal strategies only depend upon remaining playing time. Such strategies can be easily characterized and determined, and the optimal play can be easily analyzed.We acknowledge the helpful comments of G. Leitmann and an anonymous referee.     

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.     

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.     

Sufficient conditions for Nash equilibria inN-person games over reflexive Banach spaces

Sufficient conditions are obtained for the existence of Nash equilibrium points inN-person games when the strategy sets are closed, convex subsets of reflexive Banach spaces. These conditions require that each player's cost functional is convex in that player's strategy, weakly continuous in the strategies of the other players, weakly lower semicontinuous in all strategies, and furthermore satisfies a coercivity condition if any of the strategy sets is unbounded. The result is applied to a class of linear-quadratic differential games with no information, to prove that equilibrium points exist when the duration of these games is sufficiently small.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

On equilibrium points of diagonaln-person games

In this paper, we compute explicitly the equilibrium points of diagonaln-person games when all of them have the same number of strategies. This number is arbitrary. A wide generalization of two-person games is immediately obtained.The author is grateful to Professor Joel Cohen who visited IMASL during the winter of 1987 and commented on the paper.     

Equilibrium points in specialn-person games

The set of Nash equilibria is computed for some generalized games. It is also studied for a subclass of standardn-person games.The authors acknowledge the support of CONICET (Consejo de Investigaciones Cientificas y Tecnicas de la Republica Argentina). The first author acknowledges the support from TWAS (Third World Academy of Sciences), Grant No. 86-33.     

Saddle point for differential games with strongly convex-concave integrand

On a fixed time interval we consider zero-sum nonlinear differential games for which the integrand in the criterion functional is a sufficiently strongly convex-concave function of chosen controls. It is shown that in our setting there exists a saddle point in the class of programmed strategies, and a minimax principle similar to Pontryagin's maximum principle is a necessary and sufficient condition for optimality. An example in which the class of games under study is compared with two known classes of differential games is given. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 725–743, November, 1997. Translated by N. K. Kulman     

On Nash equilibria inN-person games over reflexive Banach spaces

In this note, an extended version of the Ky Fan inequality is obtained for reflexive Banach spaces. As an application of the result, it is proved that Nash equilibrium points exist under weaker conditions than those of Williams (Ref. 1).     

On differential games with Markov perfect triggering

This paper is an attempt to throw some light on the issue of whether defining history-dependent state variables in a differential game does allow for a dynamic play of precommitment equilibria. We suggest the application of trigger strategies in a state-feedback context. Based on a punishment mode in Markov perfect strategies and being able to detect even past deviations followed by histories returning to equilibrium, these subgame-perfect strategies lead to the enforcement of certain outcomes by means of dynamic rules of strategic interaction. The last part of our exposition is devoted to specific game structures under which a trigger equilibrium can be used as well as a punishment mode.     

Individual and collective rationality in a dynamic Pareto equilibrium

This paper deals with the problem of establishing the conditions for individual and collective rationality when a set of players cooperate in a Pareto equilibrium. To derive such conditions one follows the approach of the theory of reachability of perturbed systems. Open-loop and closed-loop concepts are discussed and are shown to be nonequivalent.The research of the first author was supported in part by Canada Council Grant No. S-701-491 and has benefited from collaboration with the Laboratoire d'Automatique Théorique de l'Université de Paris VII, Paris, France.     

Stabilization through taxation inn-person games

In formulating solutions forn-person cooperative games, the concept of stability has played a dominant role. Although the core concept has the strongest stability, the core of a game is often empty. In this paper, the taxation system is incorporated into our framework, so that a modified solution concept, which enjoys the stability of core, can be developed. Various formulations based on principles such astaxation proportional to income andequity after tax are given.     

Formulation and analysis of combat problems as zero-sum bicriterion differential games

In this paper, we present a formulation and analysis of a combat game between two players as a zero-sum bicriterion differential game. Each player's twin objectives of terminating the game on his own target set, while simultaneously avoiding his opponent's target set, are quantified in this approach. The solution in open-loop pure strategies is sought from among the Pareto-optimal security strategies of the players. A specific preference ordering on the outcomes is used to classify initial events in the assured win, draw, and mutual kill regions for the players. The method is compared with the event-constrained differential game approach, recently proposed by others. Finally, a simple example of the turret game is solved to illustrate the use of this method.     

Absorbing set forn-person games

This study tries to modify von Neumann and Morgenstern (vN-M) solution. As with John Harsanyi in 1974, vN-M solution is viewed in a dynamic sense. The final outcome of a game not only depends on the ability of players and standards of behavior of the particular society but also upon which imputation is proposed first. An absorbing property is obtained as a result of modifying the bargaining process of Harsanyi. This modified solution concept maintains the internal stability condition of vN-M solution, and replaces their external stability condition by this absorbing property.

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Zusammenfassung Diese Arbeit zielt auf eine Modifizierung des Begriffs der von Neumann-Morgenstern (vN-M) Lösung. In Anlehnung an John Harsanyi (1974) werden vN-M-Lösungen dynamisch interpretiert. Das endgültige Ergebnis eines Spiels hängt nicht nur von der Fähigkeit der Spieler und den Verhaltensstandards einer bestimmten Gesellschaft ab, sondern auch davon, welche Imputation zunächst vorgeschlagen wird. Durch eine entsprechende Modifizierung des von Harsanyi vorgeschlagenen Verhandlungsprozesses erhält man eine Absorbtionseigenschaft. Das demgemäß modifizierte Lösungskonzept erhält die interne Stabilitätsbedingung der vN-M-Lösung und ersetzt die externe Stabilitätsbedingung durch die Absorbtionseigenschaft.

     

Mixed strategy solutions forN-person quadratic games

Mixed strategy -equilibrium points are given forN-person games with cost functions consisting of quadratic, bilinear, and linear terms and strategy spaces consisting of closed balls in Hilbert spaces. The results are applied to linear-quadratic differential games with no information and quadratic integral constraints on the control functions.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

Regular linear systems governed by systems with state, input and output delays

^** Email: hadd{at}ucam.ac.ma^*** Email: idrissi{at}ucam.ac.ma In this paper, we give a new reformulation of linear systemswith delays in input, state and output. We show that these systemscan be written as a regular linear system without delays. Thetechnique used here is essentially based on the theory recentlydeveloped by Salamon and Weiss and the shift in semigroup properties.Our framework can be applied, in particular, when the delayoperators are given by Riemann–Stieltjes integrals.     

State transformations and the derivation of Nash closed-loop equilibria for non-zero-sum differential games

In this paper the usefulness of state transformations in differential games is demonstrated. It is shown that different (admissible) state transformations give rise to different closed-loop Nash equilibrium candidates, which may all be found by solving systems of ordinary differential equations. A linear-quadratic duopoly differential game is solved to illustrate the results.     

Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with H-regular noise

Existence and uniqueness of approximate strong solutions of stochastic infinite-dimensional systems

     

Homotopic invariance of infinite-dimensional homotopic groups of Hilbert space subsets

The paper discusses some equivalent ways of construction of infinite-dimensional homotopic groups of subsets and pairs of subsets in real Hilbert spaces. In the admissible class of K ₀-continuous mappings, the homotopic invariance of the mentioned groups and their isomorphism are demonstrated in the case where the basic points belong to the same component of K ₀-linear connectivity.