# On the Usefulness of Cooperation inN Person Games

Authors: Mikhail Sergeevich Nikolskii; Aboubacar Moussa
Journal: Journal of Applied Mathematics
 (2016)

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## Abstract

TheN 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 the N 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.


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## Body

## 1. Introduction

In the game theory (see, e.g., [1–3]), much attention is given to the cooperative game theory of N 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 N 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 N players. In the second part, N players games with perturbing factors of an ill-intentioned destructive player are considered. Here we studied the feasibility of joining (cooperation) the N 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, [6–9].


## 2. Usefulness of Cooperation inN Person Game

LetN⩾3; in the Euclidean arithmetical spaces Rk1,…,RkN  (k1⩾1,…,kN⩾1)  (k1,…,kN—dimensions of the spaces) fixed nonvoid compact set X1,…,XN, respectively, and on X1×⋯×XN we define the continuous scalar functions f1(x1,…,xN),…,fN(x1,…,xN). We denote I={1,2,…,N}. In the considered game the ith player chooses the vector xi∈Xi and strives to maximize his payoff fi(x1,…,xN), i∈I. The players make the choice of vectors x1,…,xN independently of each other. As we know from the game theory (see, e.g., [1–3]) each ith player can guarantee a payoff (1)γi=maxxi⁡minx1,…,xi-1,xi+1,…,xN⁡fix1,…,xN,i∈I,where x1∈X1,…,xN∈XN (in this formula, it is considered that if i=1 the minimization is carried out over x2,…,xN and if i=N the minimization is carried out over x1,…,xN-1), if he chooses a vector xi∗∈Xi according to the requirement(2)γi=minx1,…,xi-1,xi+1,…,xN⁡fix1,…,xi-1,xi∗,xi+1,…,xN,i∈I,where x1∈X1,…,xN∈XN (in this formula, it is considered that if i=1 the minimization is carried out over x2,…,xN for x1=x1∗ and if i=N the minimization is carried out over x1,…,xN-1 for xN=xN∗). It is assumed that the payoffs fi(x1,…,xN), i∈I, are measured in the same physical units (in economic applications, e.g., payoffs fi(x1,…,xN), i∈I, are usually measured in monetary units). In this kind of assumption, the value (3)∑i=1Nfix1,…,xNhas also a physical sense. If N players are joined together in a union (coalition), then acting concertedly (i.e., choosing (x1,…,xN) in X1×⋯×XN) they can use the quantity (4)γN+1=maxx1,…,xN⁡∑i=1Nfix1,…,xN,where x1∈X1,…,xN∈XN. It is easy to prove that (see (1))(5)γN+1⩾γ1+γ2+⋯+γN.If(6)γN+1>γ1+γ2+⋯+γN,then the joining of the players in a union (coalition) is advantageous to all players, since the positive quantity(7)Δ=γN+1-γ1+γ2+⋯+γNcan be distributed in the form of positive addition to guaranteed payoffs γ1,γ2,…,γN. To know how actually this distribution is reasonable to do, see, for example, [10, c. 181–186].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.


### 2.1. Case A

The continuous functionsf1,f2,…,fN have on X1×X2×⋯×XN a separated form: (8)f1x1,x2,…,xN=f11x1+f12x2+⋯+f1NxN,f2x1,x2,…,xN=f21x1+f22x2+⋯+f2NxN,⋮fNx1,x2,…,xN=fN1x1+fN2x2+⋯+fNNxN,where the functions f1i(xi),f2i(xi),…,fNi(xi) are continuous on Xi, i∈I.In what follows, let us agree to write operationsmaxx1,…,maxxN instead of the operations maxx1∈X1,…,maxxN∈XN, respectively. Similarly, we write minx1,…,minxN instead of the operations minx1∈X1,…,minxN∈XN, respectively. Investigated inequality (6) can be rewritten as (9)maxx1f11x1+f21x1+⋯+fN1x1+maxx2f12x2+f22x2+⋯+fN2x2+⋯+maxxNf1NxN+f2NxN+⋯+fNNxN>maxx1f11x1+minx1f21x1+⋯+minx1fN1x1+maxx2f22x2+minx2f12x2+minx2f32x2+⋯+minx2fN2x2+⋯+maxxNfNNxN+minxNf1NxN+minxNf2NxN+⋯+minxNfN-1NxN.We consider separately for i∈I the inequality(10)maxxif1ixi+f2ixi+⋯+fNixi⩾maxxifiixi+minxif1ixi+⋯+minxifi-1ixi+minxifi+1ixi+⋯+minxifNixi.In this formula, it is considered that the right-hand side if i=1 has the form(11)maxx1f11x1+minx1f21x1+⋯+minx1fN1x1and if i=N has the form(12)maxxNfNNxN+minxNf1NxN+⋯+minxNfN-1NxN.It is easy to prove the following lemma.Lemma 1.
Under assumptions (8) and (9) for each i∈I inequality (10) holds.
Note that, in general, for eachi∈I inequality (10) is not necessarily fulfilled strictly. However one has the following.Lemma 2.
For a giveni∈I, there exists a point xi∗ of the set argmaxxi⁡fii(xi), which does not belong to at least one of the sets argminxi⁡fji(xi), where j∈I and is not equal to i. Then inequality (10) holds in the strict sense.Proof.
It is evident that for a giveni∈I the left-hand side of inequality (10) for xi=xi∗ is greater than or equal to the value(13)f1ixi∗+f2ixi∗+⋯+fNixi∗,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 symbolsargmaxxi⁡ωji(xi) and argminxi⁡ωji(xi) denote, respectively, the sets of points of maximum and minimum of the function ωji(xi) on Xi.From the above we get the following.Theorem 4.
If at least for one numberi∈I the conditions of Lemma 2 are realized, then one has strict inequality (9) and, therefore, strict inequality (6).


### 2.2. Case B (Subcase of Case A)

The setsX1,X2,…,XN, convex compact, and the continuous functions f1(x1,…,xN),f2(x1,…,xN),…,fN(x1,…,xN) have on X1×X2×⋯×XN linear form(14)f1x1,…,xN=a11,x1+a12,x2+⋯+a1N,xN,f2x1,…,xN=a21,x1+a22,x2+⋯+a2N,xN,⋮fNx1,…,xN=aN1,x1+aN2,x2+⋯+aNN,xN,where a1i,a2i,…,aNi are fixed vectors in Rki, i∈I; the symbol ·,· denotes the standard scalar product in Rk1,…,RkN, respectively. As the functions f1(x1,…,xN),f2(x1,…,xN),…,fN(x1,…,xN) have a separated form (cf. (8)), you can use the results that we obtained in Case A. In considered linear Case B, strict inequality (10) can be rewritten for each i∈I in the form (15)maxxia1i+a2i+⋯+aNi,xi>maxxiaii,xi+minxia1i,xi+minxia2i,xi+⋯+minxiai-1i,xi+minxiai+1i,xi+⋯+minxiaNi,xi.In what follows we need some auxiliary information.In the Euclidean spaceRm (m⩾1) we denote σm={v∈Rm:|v|=1}, where |v| mean standard length of the vector v. The following will be useful.Definition 5.
A nonvoid convex compact setK⊂Rm (m⩾2) with nonvoid interior is called an S-set, if(1)
for anyψ∈σm in K there exists only one vector v(ψ), maximizing on v∈K the scalar product v,ψ;(2)
for each boundary pointv0 of the compact set K there exists only one support hyperplane, passing through the point v0.Remark 6.
Here and further, we use some notions of convex analysis (see, e.g., [11, 12]). We note that for each ψ∈σm the point v(ψ) belongs to the boundary of the set K. Using the terminology of convex analysis, we can say that the S-set is a strictly convex set and also a convex body.In [4] is proved the following.Lemma 7.
Let the setK⊂Rm  (m⩾2) be an S-set, and let vectors p and q be some nonzero vectors in Rm, such that(16)1pp≠1q-q.Then,(17)vp,-q<v-q,-q.Lemma 8.
For a giveni∈I, the set Xi⊂Rki  (ki⩾2) is an S-set, and the vectors a1i,a2i,…,aNi are nonzero and the vector (1/|aii|)aii is not equal to at least one of the vectors (18)1aji-aji,where j∈I and j is not equal to i. Then for this number i one has strict inequality (15).Proof.
Under the above assumptions, for a giveni∈I, the vector xi(aii) that is maximizing the function xi,aii on xi∈Xi is uniquely defined. Thus,(19)argmaxxixi,aii=xiaii.Let the vector (1/aii)aii be not equal to the vector (1/aki)(-aki), where k∈I and k is not equal to i. From Lemma 7 we obtain the following inequality: (20)xiaii,-aki<xi-aki,-aki,where xi(-aki) means the maximizer of the function xi,-aki for xi∈Xi.
We can show that(21)minxixi,aki=-xi-aki,-aki.From relations (20) and (21) we obtain the inequality(22)xiaii,aki>minxixi,aki.Hence it follows that (23)xiaii∉argminxixi,aki.From the above and Lemma 2, it follows that strict inequality (15) holds.On the basis of the above, we obtain the following theorem.Theorem 9.
If for at least one numberi∈I the conditions of Lemma 8 are realized, then for the functions fj(x1,…,xN), j=1,…N, one has strict inequality (9) and hence strict inequality (6).Remark 10.
The payoff functions, having a separated form (see (8)), are often considered in the game theory (see, e.g., [10]).


## 2.1. Case A

The continuous functionsf1,f2,…,fN have on X1×X2×⋯×XN a separated form: (8)f1x1,x2,…,xN=f11x1+f12x2+⋯+f1NxN,f2x1,x2,…,xN=f21x1+f22x2+⋯+f2NxN,⋮fNx1,x2,…,xN=fN1x1+fN2x2+⋯+fNNxN,where the functions f1i(xi),f2i(xi),…,fNi(xi) are continuous on Xi, i∈I.In what follows, let us agree to write operationsmaxx1,…,maxxN instead of the operations maxx1∈X1,…,maxxN∈XN, respectively. Similarly, we write minx1,…,minxN instead of the operations minx1∈X1,…,minxN∈XN, respectively. Investigated inequality (6) can be rewritten as (9)maxx1f11x1+f21x1+⋯+fN1x1+maxx2f12x2+f22x2+⋯+fN2x2+⋯+maxxNf1NxN+f2NxN+⋯+fNNxN>maxx1f11x1+minx1f21x1+⋯+minx1fN1x1+maxx2f22x2+minx2f12x2+minx2f32x2+⋯+minx2fN2x2+⋯+maxxNfNNxN+minxNf1NxN+minxNf2NxN+⋯+minxNfN-1NxN.We consider separately for i∈I the inequality(10)maxxif1ixi+f2ixi+⋯+fNixi⩾maxxifiixi+minxif1ixi+⋯+minxifi-1ixi+minxifi+1ixi+⋯+minxifNixi.In this formula, it is considered that the right-hand side if i=1 has the form(11)maxx1f11x1+minx1f21x1+⋯+minx1fN1x1and if i=N has the form(12)maxxNfNNxN+minxNf1NxN+⋯+minxNfN-1NxN.It is easy to prove the following lemma.Lemma 1.
Under assumptions (8) and (9) for each i∈I inequality (10) holds.
Note that, in general, for eachi∈I inequality (10) is not necessarily fulfilled strictly. However one has the following.Lemma 2.
For a giveni∈I, there exists a point xi∗ of the set argmaxxi⁡fii(xi), which does not belong to at least one of the sets argminxi⁡fji(xi), where j∈I and is not equal to i. Then inequality (10) holds in the strict sense.Proof.
It is evident that for a giveni∈I the left-hand side of inequality (10) for xi=xi∗ is greater than or equal to the value(13)f1ixi∗+f2ixi∗+⋯+fNixi∗,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 symbolsargmaxxi⁡ωji(xi) and argminxi⁡ωji(xi) denote, respectively, the sets of points of maximum and minimum of the function ωji(xi) on Xi.From the above we get the following.Theorem 4.
If at least for one numberi∈I the conditions of Lemma 2 are realized, then one has strict inequality (9) and, therefore, strict inequality (6).


## 2.2. Case B (Subcase of Case A)

The setsX1,X2,…,XN, convex compact, and the continuous functions f1(x1,…,xN),f2(x1,…,xN),…,fN(x1,…,xN) have on X1×X2×⋯×XN linear form(14)f1x1,…,xN=a11,x1+a12,x2+⋯+a1N,xN,f2x1,…,xN=a21,x1+a22,x2+⋯+a2N,xN,⋮fNx1,…,xN=aN1,x1+aN2,x2+⋯+aNN,xN,where a1i,a2i,…,aNi are fixed vectors in Rki, i∈I; the symbol ·,· denotes the standard scalar product in Rk1,…,RkN, respectively. As the functions f1(x1,…,xN),f2(x1,…,xN),…,fN(x1,…,xN) have a separated form (cf. (8)), you can use the results that we obtained in Case A. In considered linear Case B, strict inequality (10) can be rewritten for each i∈I in the form (15)maxxia1i+a2i+⋯+aNi,xi>maxxiaii,xi+minxia1i,xi+minxia2i,xi+⋯+minxiai-1i,xi+minxiai+1i,xi+⋯+minxiaNi,xi.In what follows we need some auxiliary information.In the Euclidean spaceRm (m⩾1) we denote σm={v∈Rm:|v|=1}, where |v| mean standard length of the vector v. The following will be useful.Definition 5.
A nonvoid convex compact setK⊂Rm (m⩾2) with nonvoid interior is called an S-set, if(1)
for anyψ∈σm in K there exists only one vector v(ψ), maximizing on v∈K the scalar product v,ψ;(2)
for each boundary pointv0 of the compact set K there exists only one support hyperplane, passing through the point v0.Remark 6.
Here and further, we use some notions of convex analysis (see, e.g., [11, 12]). We note that for each ψ∈σm the point v(ψ) belongs to the boundary of the set K. Using the terminology of convex analysis, we can say that the S-set is a strictly convex set and also a convex body.In [4] is proved the following.Lemma 7.
Let the setK⊂Rm  (m⩾2) be an S-set, and let vectors p and q be some nonzero vectors in Rm, such that(16)1pp≠1q-q.Then,(17)vp,-q<v-q,-q.Lemma 8.
For a giveni∈I, the set Xi⊂Rki  (ki⩾2) is an S-set, and the vectors a1i,a2i,…,aNi are nonzero and the vector (1/|aii|)aii is not equal to at least one of the vectors (18)1aji-aji,where j∈I and j is not equal to i. Then for this number i one has strict inequality (15).Proof.
Under the above assumptions, for a giveni∈I, the vector xi(aii) that is maximizing the function xi,aii on xi∈Xi is uniquely defined. Thus,(19)argmaxxixi,aii=xiaii.Let the vector (1/aii)aii be not equal to the vector (1/aki)(-aki), where k∈I and k is not equal to i. From Lemma 7 we obtain the following inequality: (20)xiaii,-aki<xi-aki,-aki,where xi(-aki) means the maximizer of the function xi,-aki for xi∈Xi.
We can show that(21)minxixi,aki=-xi-aki,-aki.From relations (20) and (21) we obtain the inequality(22)xiaii,aki>minxixi,aki.Hence it follows that (23)xiaii∉argminxixi,aki.From the above and Lemma 2, it follows that strict inequality (15) holds.On the basis of the above, we obtain the following theorem.Theorem 9.
If for at least one numberi∈I the conditions of Lemma 8 are realized, then for the functions fj(x1,…,xN), j=1,…N, one has strict inequality (9) and hence strict inequality (6).Remark 10.
The payoff functions, having a separated form (see (8)), are often considered in the game theory (see, e.g., [10]).


## 3. Usefulness of Cooperation inN 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(N+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 theN person game in a form almost classical, where N⩾3. The payoff function of the ith player is a continuous function (24)fix1,…,xN,y,i∈I,where x1∈X1,…,xN∈XN, y∈Y and X1,…,XN, Y are nonvoid compact sets in the corresponding finite dimensional Euclidean spaces. The ith player selects xi∈Xi within order to maximize (25)fix1,…,xN,y,i∈I.The vector y∈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(26)maxx1,…,xNminy∑i=1Nfix1,…,xN,y>maxx1minx2,…,xN,yf1x1,…,xN,y+maxx2minx1,x3,…,xN,yf2x1,…,xN,y+⋯+maxxNminx1,…,xN-1,yfNx1,…,xN,y.Here and further in similar inequalities x1∈X1,…,xN∈XN, y∈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) we use the notion of the characteristic function from cooperative game theory. Our study is greatly simplified if (27)f1x1,…,xN,y=f11x1,…,xN+f12y,f2x1,…,xN,y=f21x1,…,xN+f22y,⋮fNx1,…,xN,y=fN1x1,…,xN+fN2y,where the functions f11,f21,…,fN1 are continuous on X1×X2×⋯×XN, while the functions f12,f22,…,fN2 are continuous on Y.By using (27) relation (26) can be rewritten in the form(28)maxx1,…,xNf11x1,…,xN+f21x1,…,xN+⋯+fN1x1,…,xN+minyf12y+f22y+⋯+fN2y>maxx1minx2,x3,…,xNf11x1,…,xN+maxx2minx1,x3,…,xNf21x1,…,xN+⋯+maxxNminx1,…,xN-1fN1x1,…,xN+minyf12y+minyf22y+⋯+minyfN2y.Note that the following inequality holds:(29)maxx1,…,xNf11x1,…,xN+f21x1,…,xN+⋯+fN1x1,…,xN⩾maxx1minx2,x3,…,xNf11x1,…,xN+maxx2minx1,x3,…,xNf21x1,…,xN+⋯+maxxNminx1,…,xN-1fN1x1,…,xN.Proof of Inequality (29).
We have(30)f11x1,…,xN+f21x1,…,xN+⋯+fN1x1,…,xN⩾minx2,x3,…,xNf11x1,…,xN+minx1,x3,…,xNf21x1,…,xN+⋯+minx1,…,xN-1fN1x1,…,xN.Applying to both parts of the operation maxx1,x2,…,xN⁡ we obtain the required inequality.
From (29) it follows that for the fulfillment of inequality (28) it is sufficient to guarantee the inequality (31)minyf12y+f22y+⋯+fN2y>minyf12y+minyf22y+⋯+minyfN2y.Note that in the general case (32)minyf12y+f22y+⋯+fN2y⩾minyf12y+minyf22y+⋯+minyfN2y.Therefore, for inequality (31) to be realized, it is necessary that inequality (32) must become strict.
It is proved by contradiction that strict inequality (31) holds if (33)argminy∈Yf12y∩argminy∈Yf22y∩⋯∩argminy∈YfN2y=⌀.Note that the fulfillment of condition (33) is independent of the choice of the functions f11(x1,…,xN),f21(x1,…,xN),…,fN1(x1,…,xN). Thus, it follows from the above that, for strict inequality (28) to be fulfilled, it is sufficient that relation (33) is fulfilled.


## 4. Conclusion

This article relates to the theme of the theory ofN 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., [1–3]). Note that the problems we are studying are directly related to the study of the properties of the characteristic function of N 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 (N+1)-person games has also been proposed in which N players are active and the (N+1)th player is an ill-intentioned destructive player, whose aim is not to win, but to harm others.



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