EXISTENCE OF NASH EQUILIBRIUM IN DIFFERENTIAL GAME APPROACH TO FORMATION CONTROL

2018 ◽  
Vol 33 (4) ◽  
Author(s):  
Hossein B. Jond ◽  
Vasif V. Nabiyev ◽  
Nurhan G. Ozmen ◽  
Dalibor Lukas
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Formation Control

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.

scholarly journals Feedback Nash Equilibrium of N Agents Fishery Problem

2021 ◽  
Vol 14 (5) ◽  
pp. 78
Author(s):  
Letian Jiao ◽  
Haitao Chen
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Feedback Nash Equilibrium ◽  
Common Resource ◽  
Resource Stock

This paper is built on the fundamental of Jorgensen and Sorge considering a differential game about fishery problem. In reality, the exploiters can be many because of the non-excludability of common resource. Thus, we expand the former two players model to N players model and we find more different equilibriums in N players scenario. Through this, we want to find some guidance for the changing of common resource stock. Further to control overexploitation.

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 A stochastic differential reinsurance game

2010 ◽  
Vol 47 (2) ◽  
pp. 335-349 ◽  
Author(s):  
Xudong Zeng
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Payoff Function ◽  
The Other ◽  
Insurance Companies ◽  
Stochastic Differential Game ◽  
Claim Size ◽  
Expected Payoff ◽  
Exit Probability ◽  
Risk Of Exposure

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.

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