scholarly journals A differential game of N persons in which there is Pareto equilibrium of objections and counterobjections and no Nash equilibrium

2021 ◽  
Vol 57 ◽  
pp. 104-127
Author(s):  
V.I. Zhukovskii ◽  
Yu.S. Mukhina ◽  
V.E. Romanova
Keyword(s):  
Nash Equilibrium ◽  
Differential Games ◽  
Explicit Form ◽  
Differential Game ◽  
Linear Quadratic ◽  
Equilibrium Situation ◽  
Pareto Equilibrium ◽  
Noncooperative Differential Games

A linear-quadratic positional differential game of N persons is considered. The solution of a game in the form of Nash equilibrium has become widespread in the theory of noncooperative differential games. However, Nash equilibrium can be internally and externally unstable, which is a negative in its practical use. The consequences of such instability could be avoided by using Pareto maximality in a Nash equilibrium situation. But such a coincidence is rather an exotic phenomenon (at least we are aware of only three cases of such coincidence). For this reason, it is proposed to consider the equilibrium of objections and counterobjections. This article establishes the coefficient criteria under which in a differential positional linear-quadratic game of N persons there is Pareto equilibrium of objections and counterobjections and at the same time no Nash equilibrium situation; an explicit form of the solution of the game is obtained.

scholarly journals Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 132
Author(s):  
Valery Y. Glizer
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Quadratic Functional ◽  
Linear Quadratic ◽  
Feedback Controls ◽  
Cheap Control ◽  
Optimal Values ◽  
Definition Of ◽  
Zero Sum ◽  
Equilibrium Sequence

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

ON A CLASS OF NASH EQUILIBRIA WITH MEMORY STRATEGIES FOR NONZERO-SUM DIFFERENTIAL GAMES

2000 ◽  
Vol 02 (02n03) ◽  
pp. 173-192 ◽  
Author(s):  
JEAN MICHEL COULOMB ◽  
VLADIMIR GAITSGORY
Keyword(s):  
Nash Equilibrium ◽  
Feedback Control ◽  
Differential Games ◽  
Differential Game ◽  
Nash Equilibria ◽  
Payoff Function ◽  
Feedback Controls ◽  
Memory Strategies ◽  
Payoff Functions ◽  
With Memory

A two-player nonzero-sum differential game is considered. Given a pair of threat payoff functions, we characterise a set of pairs of acceptable feedback controls. Any such pair induces a history-dependent Nash δ-equilibrium as follows: the players agree to use the acceptable controls unless one of them deviates. If this happens, a feedback control punishment is implemented. The problem of finding a pair of "acceptable" controls is significantly simpler than the problem of finding a feedback control Nash equilibrium. Moreover, the former may have a solution in case the latter does not. In addition, if there is a feedback control Nash equilibrium, then our technique gives a subgame perfect Nash δ-equilibrium that might improve the payoff function for at least one player.

scholarly journals To the individual stability of Pareto equilibrium of objections and counterobjections in a coalition differential positional 3-person game without side payments

2021 ◽  
Vol 13 (1) ◽  
pp. 89-101
Author(s):  
Владислав Иосифович Жуковский ◽  
Vladislav Zhukovskiy ◽  
Константин Николаевич Кудрявцев ◽  
Konstantin Kudryavtsev ◽  
Лидия Владиславна Жуковская ◽  
...  
Keyword(s):  
Explicit Form ◽  
Linear Quadratic ◽  
Side Payments ◽  
Pareto Equilibrium ◽  
The Individual ◽  
Person Game

The notion <<individual stability>> of Pareto equilibrium of objections and counter objections in one differential linear-quadratic 3-person game without side payments is used. The explicit form corresponding equilibrium is found.

scholarly journals Disturbance attenuation problem using a differential game approach for feedback linear quadratic descriptor systems

2015 ◽  
Vol 25 (4) ◽  
pp. 445-462 ◽  
Author(s):  
Muhammad Wakhid Musthofa
Keyword(s):  
Differential Games ◽  
Differential Game ◽  
Information Structure ◽  
Disturbance Attenuation ◽  
Descriptor Systems ◽  
Linear Quadratic ◽  
Feedback Information ◽  
Zero Sum

Abstract This paper studies the H∞ disturbance attenuation problem for index one descriptor systems using the theory of differential games. To solve this disturbance attenuation problem the problem is converted into a reduced ordinary zero-sum game. Within a linear quadratic setting the problem is solved for feedback information structure.

scholarly journals Mean-field linear-quadratic stochastic differential games in an infinite horizon

2021 ◽  
Author(s):  
Xun Li ◽  
Jingtao Shi ◽  
Jiongmin Yong
Keyword(s):  
Nash Equilibrium ◽  
Differential Games ◽  
Closed Loop ◽  
Infinite Horizon ◽  
Mean Field ◽  
Riccati Equations ◽  
Stochastic Differential Games ◽  
Open Loop ◽  
Linear Quadratic ◽  
Algebraic Riccati Equations

This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. Existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is  characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite time horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

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