Математика UDC 517.9 The Usefulness of Cooperation in Two-Person Games with Quadratic Payoff Functions on the Rectangle M. Aboubacar Department of Mathematics and Computer Science Faculty of Sciences, Abdou Moumouni University BP: 10662 Niamey-NIGER In this paper we study an important question for the game theory of two players, about essentiality of such games. <...> The investigation is carried out in a particular case, in the class of games with quadratic payoff functions on a rectangle. <...> The essentiality in two players’ games means that by joining the two players in union, both players can get positive additions to guaranteed payoff. <...> Thus, the joining of the two players in union, in general, may be useful and sometimes (in case of absence of essentiality) useless. <...> In applications, for example in analysis of the economic activities of firms or countries, the question of usefulness of the union acquires a lot of interest. <...> In the general game theory, the question of essentiality of games is given a little attention at the moment. <...> Apparently, this is due to the difficulty of this problem in the general case. <...> Note that the games with quadratic payoff functions are frequently used in game theory for modeling different kinds of processes being investigated, for example, in mathematical economics. <...> Key words and phrases: two-person game, cooperation, usefulness, strategy, quadratic payoff functions. 1. <...> Introduction In game theory (see, for example [1–4]) a lot of attention is given to N person cooperative game theory. <...> The payoff function of the 2nd player is (2) where (x, y) ∈ K, c, d are arbitrary fixed non-zero numbers. <...> By selecting x ∈ [p, q] the first player strives to maximize his payoff f(x, y). <...> By selecting y ∈ [r, s] the second player strives to maximize his payoff g(x, y). <...> Similarly, the second player can guarantee a payoff g(x, y), min x∈X if he selects a vector y0 ∈ Y from the condition γ2 = min x∈X g(x, y0). (6) Generally speaking, from physical point of view, payoffs f(x, y), g(x, y) can be measured in different physical units. <...> We shall assume that payoffs f(x, y), g(x, y <...>