On the Usefulness of Cooperation in N Person Games

The N person games in which each player maximizes his payoff function are considered. We have studied an interesting question for the cooperative game theory about the usefulness of uniting theN players in a union. The aim of such cooperation is for each player to get a positive increase to his guaranteed payoff. We have obtained some effective sufficient conditions under which the joining of the players in union is useful for each player. The linear case, specially, is being considered. In the second part of the paper, we have studied the question about the usefulness of cooperation of the N players in the presence of the (N + 1)th player, an ill-intentioned destructive player, whose whole aim is not to win but to harm each player individually, and also the union of these players, for example, global terrorism. It should be noted that the considered situation in the second part is related to A. V. Kryazhimskiy’s talk delivered in the summer of 2014. We obtain constructive conditions under which the union of the players is beneficial in this situation as well.


Introduction
In the game theory (see, e.g., [1][2][3]), much attention is given to the cooperative game theory of  person.In [4,5], we considered the games of two and three persons from the point of view of the usefulness of combining them in an alliance in order to get additional dividends.In the first part of this paper, we considered  person games in terms of usefulness of uniting players in union in which the choice of strategies is made in concert in order to maximize the sum of payoffs of the  players.In the second part,  players games with perturbing factors of an ill-intentioned destructive player are considered.Here we studied the feasibility of joining (cooperation) the  players in union to counter the possible troubles from the ill-intentioned destructive player.For the different aspects of the theory of cooperation see, for example, [6][7][8][9].
There arises an interesting question for the game theory and its applications of finding constructive conditions on the elements of the game considered by us under which strict inequality (6) holds.We consider two cases, in which it is possible to indicate such conditions.
In what follows, let us agree to write operations max x 1 , . .., max x  instead of the operations max x 1 ∈ 1 , . . ., max x  ∈  , respectively.Similarly, we write min x 1 , . . ., min x  instead of the operations min x 1 ∈ 1 , . . ., min x  ∈  , respectively.Investigated inequality (6) can be rewritten as We consider separately for  ∈  the inequality In this formula, it is considered that the right-hand side if  = 1 has the form max and if  =  has the form It is easy to prove the following lemma.
Lemma 1.Under assumptions ( 8) and ( 9) for each  ∈  inequality (10) holds.Note that, in general, for each  ∈  inequality (10) is not necessarily fulfilled strictly.However one has the following.Lemma 2. For a given  ∈ , there exists a point x *  of the set arg max x    (x  ), which does not belong to at least one of the sets arg min x    (x  ), where  ∈  and is not equal to .Then inequality (10) holds in the strict sense.
Proof.It is evident that for a given  ∈  the left-hand side of inequality (10) for x  = x *  is greater than or equal to the value where it is not difficult to see that it is greater than or equal to the right side of inequality (10).Further, by using the hypotheses of Lemma 2, it is easy to prove that this value is strictly greater than the right-hand side of inequality (10).
From what has been said, Lemma 2 follows.
Remark 3. The symbols arg max x    (x  ) and arg min x    (x  ) denote, respectively, the sets of points of maximum and minimum of the function   (x  ) on   .
From the above we get the following.

Theorem 4.
If at least for one number  ∈  the conditions of Lemma 2 are realized, then one has strict inequality (9) and, therefore, strict inequality (6).
In the Euclidean space R  ( ⩾ 1) we denote   = {k ∈ R  : |k| = 1}, where |k| mean standard length of the vector k.The following will be useful.Definition 5. A nonvoid convex compact set  ⊂ R  ( ⩾ 2) with nonvoid interior is called an -set, if (1) for any  ∈   in  there exists only one vector k(), maximizing on k ∈  the scalar product ⟨k, ⟩; (2) for each boundary point k 0 of the compact set  there exists only one support hyperplane, passing through the point k 0 .

Remark 6.
Here and further, we use some notions of convex analysis (see, e.g., [11,12]).We note that for each  ∈   the point k() belongs to the boundary of the set .Using the terminology of convex analysis, we can say that the -set is a strictly convex set and also a convex body.
In [4] is proved the following.
where  ∈  and  is not equal to .Then for this number  one has strict inequality (15).
Proof.Under the above assumptions, for a given  ∈ , the vector x  (a  ) that is maximizing the function ⟨x  , a  ⟩ on x  ∈   is uniquely defined.Thus, arg max Let the vector (1/|a  |)a  be not equal to the vector (1/|a  |)(−a  ), where  ∈  and  is not equal to .From Lemma 7 we obtain the following inequality: where x  (−a  ) means the maximizer of the function ⟨x  , −a  ⟩ for x  ∈   .We can show that From relations (20) and ( 21) we obtain the inequality Hence it follows that x  (a  ) ∉ arg min From the above and Lemma 2, it follows that strict inequality (15) holds.
On the basis of the above, we obtain the following theorem.

Usefulness of Cooperation in 𝑁 Person Game in Presence of an Ill-Intentioned Destructive Player
In this part, we briefly studied some more general game models than in the first part which considers the existence of the ( + 1)th player, an ill-intentioned destructive player.The statement of the considered problem here was motivated by A. V. Kryazhimskiy's talk.We consider the  person game in a form almost classical, where  ⩾ 3. The payoff function of the th player is a continuous function where x 1 ∈  1 , . . ., x  ∈   , y ∈  and  1 , . . .,   ,  are nonvoid compact sets in the corresponding finite dimensional Euclidean spaces.The th player selects x  ∈   within order to maximize The vector y ∈  is selected by the ill-intentioned destructive player, whose objectives are not to win, but to harm the players.We want to prove that sometimes it is advantageous for the players to be united in a coalition and fight together with the possible actions of the ill-intentioned destructive player.
We will study the game, for which max Here and further in similar inequalities x 1 ∈  1 , . . ., x  ∈   , y ∈ .
Note that when inequality (26) is fulfilled, it is advantageous for the players to be united in a coalition in order to get greater payoff through bargaining than in a two-person zero-sum game.In fact, in inequality ( 26 (33) Note that the fulfillment of condition (33) is independent of the choice of the functions  11 (x 1 , . . ., x  ),  21 (x 1 , . . ., x  ), . . .,  1 (x 1 , . . ., x  ).Thus, it follows from the above that, for strict inequality (28) to be fulfilled, it is sufficient that relation (33) is fulfilled.

Conclusion
This article relates to the theme of the theory of  person games.In this theory, a distinction of noncooperative and cooperative games is made.The formulations of considered problems are related to the cooperative games theory (see, e.g., [1][2][3]).Note that the problems we are studying are directly related to the study of the properties of the characteristic function of  person cooperative games theory, where all the players formed a union.The specific participants of the union may be firms, states, and so forth.In the article we have considered the general case of nonlinear payoff functions in the separable form and a subcase of these functions in the linear form.Effective sufficient conditions which ensure the usefulness of the union of all the players in coalition have been obtained.A method of studying ( + 1)-person games has also been proposed in which  players are active and the (+1)th player is an ill-intentioned destructive player, whose aim is not to win, but to harm others.