TAILIEUCHUNG - The study of the set of nash equilibrium points for one two-players game with quadratic payoff functions

In this paper, the general two-players game on the square with quadratic payoff functions is considered. We have studied the problem of determination of the set of Nash equilibrium points, and here we present a constructive graphical method for determination of the required set, which we have developed. | Yugoslav Journal of Operations Research 25 (2015), Number 2, 291–297 DOI: THE STUDY OF THE SET OF NASH EQUILIBRIUM POINTS FOR ONE TWO-PLAYERS GAME WITH QUADRATIC PAYOFF FUNCTIONS Mikhail Sergeevich NIKOLSKII Steklov Mathematical Institute, Russian Academy of Sciences, 119991, Moscow-RUSSIA, Gubkina str., 8. mni@ Aboubacar MOUSSA Department of Mathematics and Computer Science, Faculty of Sciences, Abdou Moumouni University, BP: 10662 Niamey-NIGER moussa@ Received: September 2013 / Accepted: April 2014 Abstract: In this paper, the general two-players game on the square with quadratic payoff functions is considered. We have studied the problem of determination of the set of Nash equilibrium points, and here we present a constructive graphical method for determination of the required set, which we have developed. Keywords: Two-players game, Nash equilibrium, quadratic payoff functions. MSC: 91A05. 1. INTRODUCTION We consider a general two-players game (see, for example [1]-[7]) on the square with quadratic payoff functions. Here, the problem of determination of the set of Nash equilibrium points is studied, and our newly developed constructive graphical method for determination of the required set is presented. We construct a general example of our game in which the set of Nash equilibrium points contains a non-zero length segment . Note that there are many articles (see, for example [1]-[7], etc.) devoted to properties of the set of Nash equilibrium. Our paper provides an useful illustrative 292 . Nikolskii, A. Moussa / The Study of the Set of Nash Equilibrium Points material for the game theory, granting frequent use of quadratic criteria in this theory. 2. PROBLEM FORMULATION We consider a two-players game on the square K = [0, 1] × [0, 1]. The payoff function of the 1st player is f (x, y) = a1 x2 + a2 xy + a3 y2 + a4 x + a5 y, (1) where x, y ∈ K, ai are arbitrary fixed numbers, and a1 , 0. The payoff function of

• Toán học
• Two players game
• Nash equilibrium
• Quadratic payoff functions
• Quadratic payoff functions
• Constructive graphical method