A NEW ALLOCATION RULE FOR COOPERATIVE GAMES WITH HYPERGRAPH COMMUNICATION STRUCTURE Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, XII, 204-219

A New Allocation Rule for Cooperative Games with Hypergraph Communication Structure*

David A. Kosian and Leon A. Petrosyan

St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: kosyan-david@mail .ru

Abstract A new allocation rule was proposed by splitting the original game into a game between hyperlinks and games within them. A special case of cooperative game on flower type hypergraphs is investigated. The proposed allocation rule has been generalized for games with a communication structure represented by acyclic reduced hypergraphs and was illustrated by examples.

Keywords: cooperation, characteristic function, hypergraph, communication structure, allocation rule.

1. Introduction

In cl clclSSiCcll way for group N := 1,..., n of agents the economic possibilities of each subgroup are described by cooperative game (N, v), where N is a set of players and v is a characteristic function. The characteristic function shows the power of each coalition. In this paper, we assume the cooperative game with transferable utility or TU-games.

Classically in this game, we assume that each subset of players can decide to cooperate and the total payoff of this cooperation can be distributed among the players. But in many practical situations, not all players can communicate with each other due to some economic, technological or other reasons, thus some coalitions cannot be created. It is the class of TU-games with limited cooperation. The communication structure can be introduced by an undirected graph. In this way, just players who have a link between them can cooperate. These games were first studied in (Myerson, 1977), he introduced games on a graph and characterized the Shapley value (Shapley, 1953). Hereafter, games with communication structure have received a lot of attention in cooperative game theory. In (Owen, 1986) were studied games where the communication structure is a tree. The position value for games where communication structure is given by a graph was introduced in (Meessen, 1988).

But generally, the communication structure can be given by a graph or hypergraph. For example, it can be some companies or sports teams. Cooperation between two organizations is only possible if they have at least one member in both of them.

The TU-games on hypergraph were studied in (Nouweland, Borm and Tijs, 1992), they characterized the Myerson value and the position value for these games. The third value, which is called degree value for the games with hypergraph communication structure was introduced in (Shan, Zhang and Shan, 2018). Many allocation rules for TU-games with a hypergraph communication structure can be proposed

* This research was supported by Russian Science Foundation under grants No.17-11-01079.

based on some different interpretations. The Myerson value highlighting the role of the players, the position value focuses on the role of communication.

In this paper, we introduced a new allocation rule for TU-games on the hypergraph.

2. Game on subclass of hypergraph

2.1. Preliminaries

In this section, we recall some notations and definitions about TU-games and hypergraph.

TU-game is a pair (N, v). Characteristic function v :2N ^ R and v(0) = 0. We will use |S | to show the cardinality of any S G N.

Hypergraph is a pair (N, H), H Ç {H G 2N | |H| > 2}. H is some set of subsets N

2.2. Definition of the game

Let N = {1,... ,n — 1, c} be a player set. Communication possibilities described by

(N, H) which given by

H Ç {H G 2N| |H| > 2, Hj n Hk = c; j = k VHj, Hk G H}.

The interpretation of this structure is there is just one player who included in all hyperlinks and other players included just in one of them. The communication is only possible between the players in hyperlinks. It can be also interpreted as the central player has some companies with workers. An example of this hypergraph shown in fig.l

Fig. 1. An example of this liypergrapli.

Denote the numbers of hyperlink in communication structure by L. Also denote the central-player by c. Let r be a set of players which included in hyperlink Hj except player c and U - the set of their strategies. Also denote a strategy of simple-player j as uj. A strategy of player c from his set of strategies we will denote by uc G Uc. We define the payoff fonction of simple-player j in hyperlink Hi in this

hj (Uj ,uc) = Kj (Uj,uc)

where Kj — payoff of piayer j which is defined on hyperlink which include player j. The payoff function of central-player c:

hc(Ui, U2,. ..,UL,uc) = K1c(Ul,uc) + Kl(U2,uc) + • • • + KL(UL,uc)

2.3. Cooperation

Now consider the case when the players agree to cooperate. It means that they will choose their strategies to maximize the sum of their payoffs

LL

E^ = E E Km (Ui,uc)+Y, Kj (Uj,uc).

heN i=1 meHi j=1

We suppose transferable payoffs. Thus the main question is how to allocate the total payoff between players. We will do it in three steps. On the first step, we construct a new cooperative game where we consider hyperlinks as players. We will create a characteristic function for all coalitions in this game. After that we solve this game proposing some allocation rule, in this paper we use a solution with equal excess. So we get the payoffs for all hyperlinks. The second step is to allocate this payoff between the members in a hyperlink. To solve this problem we will use the proportional solution. The last step is to find the total payoff for the central player. It will be the sum of his payoffs from all hyperlinks.

First step For now we consider the game where the players are hyperlinks from the given communication structure.The set of hyperlinks we will denote as H S C H is coalition from this set of hyperlinks. We define the characteristic function as follows:

V(S)= E E Kj(UM+ E Ki(Ui,uc),

i: HieS jeri i: HieS

where uc the solution of this maximization problem:

maxmai^ ^ Kj (Ui,Uc )+ E Klc(Ui ,Uc) : Hi/S jEr i: Hi/S

E J2Rj(Ui,ûc)+ Ki(Ui,uc),

i: Hi/S j e A i: Hi / S

and U the solution of:

max

Ui

E T,Kj(Ui,Uc)+ E Ki(Ui

vi: Hi e S j e ri i: Hi e S

= E E Kj(Ui,Uc)+ E KC( Ui

i: Hi e S j e ri i: Hi e S

V(H) = max max I ^^ J2 K (U ,uc) + ^ Kic(Ui Ua Ui \i: Hi e H j e ri i: Hi e H

We can interpret the values of characteristic in a following way. We suppose that central player is maximizing the total payoff of players in hyperlinks which are

not in S. Based on this, the central playerBT)™s wc strategy is chosen, assuming that in the worst case the central player will play this strategy, players from S seek to maximize their total payoff. Thus, we have determined the characteristic function for all coalitions of the hyperlinks. The next step is to find the gains for each hyperlink. For this, we use a solution with equal excess.

V(H) - £ V(Hi) _

Zhj = V(Hj) +-f-, j = 1, L.

Second step After the previous step the payoffs for each hyperlink have been received, the next step will be the distribution of this payoff between the players

L

characteristic function for coalitions of players and the optimality principle. The characteristic function in these games will be determined in accordance with the approach described in (Neumann and Morgenstern, 1994). The idea is quite simple: the characteristic function in this case shows the maximum gain that a coalition can receive, provided that all other players play against it. As an optimality principle, we take a proportional solution. Consider a game on the hyperlink Hj. We denote the set of players on hyperlink Hj, including the central one, by Nj. We assume

that vj (Nj) = £Hj. Define the value of the characteristic function for each of the

Hj

vj (i) = max min Ki(Uj ,uc).

ui uc y Uj\ui

for central-player on the same hyperlink

vj (c) = maxmin Kj (Uj,uc).

Now we can define the payoff of each simple-player on the hyperlink Hj as:

E' = j) "N > = j) ■

keNj keNj

Hj

Ej = " iw v(N,) = " iw £h Ec = £ vj(k)v(Nj)= £ vj(k)fHi.

keNj keNj

Third step We already defined the payoffs of all simple-players. The payoff of the central-player is said to be equal a sum of his payoffs on each hyperlink.

Ec = E .

j=i

This is the main idea of this work.

2.4. Example

For bettor understanding we will use this solution on the example. Consider the cooperative game with player set N = {1,2, 3,4, c} and hypergraph H1 = {1,2, c}, H2 = {3, 4, c} which is shown on fig.2 .

For this example we consider that in each hyperlink players have bimatrix game between each other. It means that for simple-player j in hyper link Hi payoff function

hj (Ui,uc) = Kj (Ui,uc)= Kj (uk ,uj)+ Kj (uj ,uc),

k:uk eUi\uj

and for central player the payoff function

hc(Ui, U2,. ..,UL,uc) = Ki(Ui,uc) + Kl(U2,uc) + • • • + kL(ul, uc)

where

Ki(Ui,uc)= Ki(uk,uc)

k:ukeUi

HI H2

Fig. 2. Communication structure

Dofirio the bimatrix ganie for oach pair of linkod players. Wo write a bimatrix 2 x 2 for player md j where i chooses the row and j chooses column. We consider that ail players have the set of stratégies (A, B). For players 1 and c

/4\8 3\6\

For players 2 and c

/3\6 5\5\ V°\2 4\8^

For players 1 and 2

/6\8 6\0\ \4\3 0\6j

For players 3 and c

For players 4 and c

For players 3 and 4

8\0 6\10\ 3\6 9\3 )

5\2 8\9 7\2 6\5

0\1 10\4\ 7\0 3\8 )

First step. Firstly we find the value of characteristic function for all coalitions of hyperlinks.

V(Hi) = £ Kj(Ui,uc) + K1C(U1,uc), j e r

where uc the solution of this maximization problem:

maxmax > Kj (U2,uc) + k'2(u2,u Uc I

\3 e r2

= E Kj(U2,Uc) + Kc2(U2,«c) = 41 j e r

In this example uc = B next we find UC it is the solution of:

max i E Ko(Ui, B) + Kl(uh B) ) =33

Thus V(Hi) = 33

V(H2) = E Kj(U2,Uc) + K2(U2,Uc), j e r

where uc the solution of this maximization problem:

maxmax | Kj(UC,uc) + Kl(UC,u Uc Ul \jer

= E Kj (Ui ,Uc) + KC(Ui,Uc) = 35 j e ri

In this example uc = A next we find U2 it is the solution of:

max E Kj(U2,A) + KI(U2,A)] = 31

U \jer

Thus V(H2) = 31

V(H) = max max £ ^ Kj (Ui,uc) + ^ Kf(Ui,uc) ) = 74 * H*eU jer* i: H*eU

Now we use the solution with equal excess to get payoffs for hyperlinks

V(H) - £ V(Hi) _

Ch, = V (Hi) +-Lr-' j = L-

£H, = V(H,) + V<H> - <V'H'» + V<H'» =38

Ch, = V№) + V<H) - 'V'H+ V=36

Second step. Now we solve two cooperative game as an optimality principle we will use proportional solution. For the game on hyperlink H1 a characteristic function for players 1,2 and c

v1(1)=max min Ki(U1,uc) = max min (K1(u1, uc) + K1(u1, u2)) = 10.

u1 uc y Ui\U u1 uc,u2

v1(2)=max min Ki(U1, uc) = max min (K2(u2, uc) + K2(u1, u2)) = 6.

u2 uc U Ui\u2 u2 uc ,u1

v1(c) = max min K<1(U1, uc) = max min (Kf^u2, uc) + Kf^u1, uc)) = 11

Uc Ui Uc u1,u2

v1 (N1) = Chi =38

v1(1) 380

2 _ v (2) „,1^

= , , ^ v1(N1)

Ei2 = , , v1(N1)

v1(1) - h v1(2) - h v1(c)

v1(2)

v1(1) - h v1(2) - h v1(c)

v1 (c)

Ef = , , v1(N1)

27

228 "27

418

v1(1) + v1(2)+ v1(c^1/ "27 H2

v2(3)=max min Ki(U2,uc) = max min (K3(u3, uc) + K3(u3, u4)) = 15.

u3 uc y U,\u3 u3 uc,u4

v2(4) = max min Ki(U2,uc) = max min (K4(u4, uc) + K4(u3, u4)) = 11.

u4 uc y U2\u4 u2 uc,u3

v2(c) = maxminuc) = max min (K2(u3, uc) + K2(u4, uc)) = 8

Uo 1/^/2

Uc U2 ' ' Uc U1,U2

v2 (N2) = Ch, =36

E3 = - (3) v2(N ) = 540

E2 v2(3) + v2(4)+ v2(c) ( 2) 34

E4 = VKl v2(N2) =

4 = „ (4) 2 ) = 396

2 v1(3) + v2(4)+ v2(c) ( 2) 34

c = v2(c) v2(N ) = 288

2 vC(3) + v2 (4) + v2(c) ( 2) 34

Third step. Now we sum the payoffs of central player from each hyperlink

£ = E Ej = £ c + £ c = 288 + 418

Ec = 2- £c = £i + £2 =34 + 27

j=1

3. Generalization of the game

3.1. Preliminaries

In this part we consider the generalization of the previous game. Now we need to refresh some information about hypergraph.

The reduction of hypergraph (N, H) is called hypergraph (N, H ) which is obtained from the original by removing all hyperlinks that are completely contained in other hyperlinks. Hypergraph is called reduced if it is equivalent to its reduction, that is, it does not have a hyperlink inside other hyperlinks.

A simple cycle with length s in hypergraph (N, H) is a sequence

(Ho, no, H1,..., HS1,nS1,Hs),

where H0,..., Hs1 different hyperlinks, hyperlink Hs coincides with H0, n0,..., ns-1 different vertexes, and ni e Hi n Hi+1 for all i = 0,..., s — 1.

A first definition of acyclicity for hypergraphs was given in Berge, 1989. A hypergraph is acyclic if its incidence graph is acyclic.

3.2. Definition of the game

In this part, we will construct the game where communication structure defined by-acyclic reduced hypergraph (N, H). An interpretation of this communication structure can be that there are managers who work with companies and each company-has workers who work just on them.

Let N := {1,... ,n — m,c1,..., cm} be a set of players. Denote the numbers of hyperlinks in communication structure by L as before. The players which included just in one hyperlink we will call simple-players, other will be called complex-players. To construct the game we need to introduce new notations.

Let ui is strategy of simple-player i from the set of his strategies Ui. Also denote as ucj a strategy of complex-player j from the set of his strategies Ucj. The set of

Hi Ui

Uic j

in hyperlink ^denote as Kj (Ui, Uf), and the payoff function for com plex-player j

in hyperlink ^denote by Kc (Ui, Ui?). Now we can define the total payoff function

j

payoff is equal to

hj = Kj(Ui, uc) j

hcj = E Kcc (Ui,U9)

i:cj e Hi

3.3. Cooperation

Now we consider a cooperative game where the players agree to choose their strategies together to maximize the total sum of theirs payoffs. The total sum is equal:

n-m m L L

+ £ hCi = ££ Kj(ui,u?) + ^ £ k? (Ui,uc)

i=l i=l i=l j e Hi i=l j:cj e Hi

Firstly we consider the cooperative game where players are hyperlinks. We define a characteristic function for any coalition of hyperlinks, after that we use an allocation rule and get payoff for each hyperlink. Next step is consider L cooperative games and get payoff for each player. Finally we find a total payoffs for any complex-player cis ci sum from his payoffs from each hyperlink in which he exist.

First step We consider the cooperative game with hyperlinks as players. To define the characteristic function for all coalitions we need to introduce new notations. Let ri be a set of simple-players in hyperlink Hi3 rcc is a set of complex-players in hyperlink H^ Also denote a set of hyperlinks which include complex-player j as Bcj. For any coalition S we will make a partition on each hyperlink in S of complex-

Hi

f

S uic

n f n

uc , Ucc U Uic = Ucc. The set of hyperlinks we denote as H S C H is coalition from this set of hyperlinks. Now we can define the characteristic function for all coalitions of hyperlinks as follows

V(S)= £ T,Kj(Ui,u?fUn)+ £ £Kf(Ui,Uf,ur)

i:Hi e S j e ri i:Hi e S j e rc

where U^ the solution of this maximization problem:

maxmaxl £ £ Kj (U^Uf )+ £ £ Kcc (Ui,Uf )

yi:Hi/S j e ri i:Hi/S j G rc

= £ £ Kjm,Uf )+ £ £ KCj (Ui, Uf )

i:Hi/S j G ri i:Hi/S j G rc

and Ui and Uf the solution of:

max max £ £ Kj (Ui, UC ,Uf )+ £ £ Kf (Ui,Uf ,Uf )

Uc i \i:Hi G S j G ri i:Hi G S j G rc

= £ £Kj(Ui,Uf ,Uf )+ £ £ KC(Ui,Ucf,Ucn)

i:Hi G S j G ri i:Hi G S j G rc

for the grand coalition we have:

V (H) = maxmax £ £ Kj (Ui,Utc, )+ £ £ Kf (Ui, Uf)

i \i:Hi G H j G ri i:Hi G H j G rc

As allocation rule here we use Shapley value. Notice that in this step we can use any allocation rule from classic cooperative theory.

(V )= S 15 1 !( N 1 " 1)! (V (SU H) - V (S))

sçu\Hi !

Second step From the previous step we get payoffs for each hyperlink in our hypergraph. Now we consider a cooperative game on each hyperlink with players which included in it. It means that now we have L independent cooperative games. As an optimality principle we use proportional solution. We denote the set of players on hyperlink Hj by No-. Assume that vo (No- ) = Define the value of the characteristic function for each simple players on hyperlink Hj as

vo («) = max min K®(U,, Uoc).

u Uc U Uj\u

for complex-player i on the same hyperlink

vj (ci) = max min Kf* (U, Uf). uci U,c\uci U U, j J j

Hj

o V ( i / A \ v ( i )

= tO) " (N ) = "sor ^.

fceNj fceNj

The payoff of complex-player i on the hyperlink Ho- we will define by

(c

(Ci) (c

Efi = "Ej) " (N ) = "Ej) ^.

keN, keN,

Third step Now we get total payoffs for each simple-player. The total payoff for each complex-player is the sum of his payoffs from each hyperlink in which it is included.

Efi = E Ei

j:H, EBc*

3.4. Example

Consider the cooperative game with player set N = {1, 2,3,4, c1,c2} and hypergraph H1 = {1, 2, c1}, H2 = {3, c1; c2}, H3 = {c2, 4} which is shown on fig.3.

For this example we consider that in each hyperlink players have bimatrix game

i Hj

is

hi(Uj ,U,c) = Ki(Uj ,Ujc)= E K i(uk ,ui)+ E Ki(ui,uck),

k:uk EU, \u* k:uck EU,

and for central player the payoff function

hCj = E Kf' (Ui,Uic)

i:c, EHi

HI H2

Fig. 3. Communication structure

where

KjCi (Uj ,UjC)= ]T KCi (uk,uCi)+ E KCi (uCi ,uck)

k:ukEU, k:uck EU,c\uci

Define the bimatrix game for each pair of linked players. We will write a bimatrix 2 x 2 for player md j where i chooses the row and j chooses column. We consider that all players have a set of strategies (A, B). c1

/4\8 3\6\ \1\3 5\6;

c1

f3\6 5\5\ 0\2 4\8

For players 1 and 2

(6\8 6\0\ 4\3 0\6

c1

/8\0 6\10\ 3\6 9\3

c2 c1

/5\2 8\9\ 7\2 6\5

c2

For players 4 and c2

0\1 10\4 7\0 3\8

/1\4 2\7 U\0 3\5

First step. Firstly we will find the value of the characteristic function for all coalitions of hyperlinks. In coalition S = [Hi], Uf = U£ = (uf) then the value of characteristic function of this coalition is equal

V(Hf) = £ Kj(UuUf) + £ kc (Uf,Uf)

j e A j e rc

maxmax I £ £ Kj (Ui ,Uf)+ £ £ Kj (U^Uf) Ui i \i:Hi/S jeri i:Hi/S jerc

= £ £ Kj (Ui,Uf)+ £ £ Kj (Ui, Uf) = 50

i:Hi/S jeri i:Hi/S jerc

from this we get Uf = (Uci) = B

£ Kj(UuUf)+ £ Kj(UuUf)

max m K(UUUC ^ ^

1 \j e A j e rf

= £ Kj(U, Of) + £ k?3' (u, Uf) = 33 j e ri j e rc

V(H ) = 33

In coalition S = [H2], U2 = U2 = (uci, uc2) then the value of characteristic function of this coalition is equal

v(H2) = £ Kj(u2,Uf ) + £ K2Cj(U2,û

j e A j e rc

maxma^ £ £ Kj(Ui,Uf )+ £ £ Kf (Ui,Uf )

Uc i l:Hi/S jeri i:Hi/S jerc

= £ £ Kj (Ui,Uf )+ £ £ K22j m,Uf ) = 44

i:Hi/S je ri i:Hi/S jerc

from this we get U2 = (U21, U22 ) = (A, B)

max I £ Kj (U2, Uf ) + £ Kf (U2, Uf )

2 \j e A j e rc

= £ K(U Uf ) + £ Kf (U2, Uf ) = 31

j e A j e rc

Thus we get V(H2) = 31.

In coalition S = {H3}, U3 = Uf = (uc2) then the value of characteristic function of this coalition is equal

v(H3) = E Kj(t/s, Uf) + E K3Cj (U3, Uf)

j E A j E

m^max! £ Ekj(Ui,Uf)+ E EKf (Ui, Uf)

Ui i \i:Hi/S j E A i:Hi/S j E ric

= E E Kj №,Uf)+ e E KCj №,Uf ) = 74

i:Hi /S j E ri i:Hi /S j E ric

from this we get Uf = (Uc2 ) = (B)

max I E Kj (Us, Uf) + E Kf (Us, Uf)

3 \, j E A j E rc

= E Kj(/, Uf) + E KiCj (Us, Uf) = 9

j E A j E r3c

Thus we get V(H3) = 9.

In coalition S = {H1, H2}, Uf/ = Uf = (U1), Uf = (U2), Uf = (uCl) then the value of characteristic function of this coalition is equal

V (S ) = V ({H1,H2})= E EKj (ti,Uf ,Uf)+ E EKf (Ui,Uf ,Uf

i:Hi E S j E ri i:Hi ES j E r?

moa™xl E EKj (Ui,Uf)+ E EKf (Ui, Uf)

Uc i :Hi/S j E ri i:Hi/S j E A<=

E E kj№,Uf)+ E E Kf №,Uf) = 9

-Hi/S j E ri i:Hi /S j E rc

From this we get U2 = (uc2) = B

max max E E Kj (Ui,Uf, Uf)+ E E Kf (Ui, Uf, Uf)

Uc i \i:Hi ES j E ri i:Hi E S j E A<=

= E E Kj (///,Uf ,Uf)+ e E Kf (Ui,Uf ,Uf) = 74

i:Hi ES j E A i:HiE S j E ric

Thus we get V({H1, H2}) = 74.

In coalition S = {H2, H3}, U3c/ = U3c = (U3), Uf = (U1), Uf = (uc2) then the value of characteristic function of this coalition is equal

V(S) = V({H2,H3})= E E^'OW ,Uf) + E E^' ,U?

e S j e r e S j e ric

maxma^ j(U, U?" )+ E E K? (Ui, U?" )

U 1 ;:Hi/S j e A i:Hi/S j e r?

= E E Kj)+ e E K* №,Uf ) = 4i

i:Hi/S j e r i:Hi/S j e

From this we get Uf" = (uCl ) = A

max max E E Kj (Ui, Uf', Uf")+ E E K? (Ui, u/ , Uf)

Uc i \i:Hi E S j E ri i:Hi E S j E rc

= E E Kj№,ucf,Uf)+ e E Kf (Ui,u/,uf) = 40

i:Hi E S j E ri i:Hi E S j E rf

Thus we get V({H2, H3}) = 40.

In coalition S = {H1, H3} Uf = Uf = (uCl), Uf = U3c = (uc2) then the value of characteristic function of this coalition is equal

V(S) = V({Hi,Hs})= E E^'oW ,Uf) + E EKf (Ui,uf ,Uf")

i:Hi e S j e ri i:Hi e S j e rc

maxma^ ^Kj(Ui, U?" )+ E E K*(Ui, Uf)

i ;:Hi/S j e ri i:Hi/S j er?

= E E Kj(t/î,Uf )+ e E Kf №,Uf ) = 35

i:Hi/S j e ri i:Hi/S j e

From this we get Uf" = (uc2 ) = B, and Uf" = (uCl ) = B

ma™ E E Kj(Ui,UCf ,Uf")+ E EKf (Ui,Uif,Uc")

Ui i \i:Hi E S j E ri i:Hi E S j E rc

= E E Kj№,Uicf,Uf)+ e E Kr(Ui,u/,Uf) = 42

i:Hi E S j E ri i:Hi E S j E rf

Thus we get V({H, H3}) = 42.

For the grand coalition H the value of characteristic function is equal

V(H)=maxmax E E Kj(Ui, U?, )+ E E Kfj (Ui, U?) ) =83

i U:Hi eH je ri i:H eH je r?

In this example at this step we will use the solution with equal excess as an optimality principle.

V(H) - £ V(Hi) _

in,=v(Hi)+-LH—, j = 1>L-

H = V(Hi) + V(H) - (V(Hi) + V(H)+ V(H3)) = 36.(3)

V (H) 3 - (V(H1) + V(H2)- V(H3))

V (H) 3 - (V(H1) + V(H2) + V(H3))

ÎH2 = V(H2) + ^ v v 3 v ^ ' v = 34.(3)

U = V(H) + V(H) - (V(H1)+ V™ + V(H)) = 12.(3)

Second step. Now we need to solve three cooperative game as an optimality principle we will use proportional solution. For the game on hyperlink H1 a characteristic function for players 1,2 and c1

vj (i) = max min Kl(U0 ,UC). U U|U Uj\п* 0

vj (ci ) = max min Kji ( U0 ,UC ).

uci Uc\u^U Uj 0 0

v1 (1) = 10, v1(2) = 6,v1(c1) = 11

v1(N1)= =36.(3)

1 = v1 (1) v1(N ) = 363.(3)

1 v1(1) + v1 (2)+ v1(c1) ( 1) 27

x __v1(2)_ ^ _ 218

E2 = v1(1) + v1(2) + v1(c1)v (N1) = "27"

1 = v1(c1) 1(N ) = 399.(6)

Eci v1(1) + v1(2) + v1(c1) (N1) 27

H2 c1 c2

v2 (3) = 15,v2(c1) =8,v2(c2 ) = 11

v2(N2)= Ch2 =34.(3)

E2 = v2(3) v2(N ) = 515

E3 v2(3) + v2(c2) + v2(c1 ) ^ (N2) 34

E2 = v2(c1) 2(N ) = 274.(6)

ci v2(3) + v2(c2) + v2(d) ( 2) 34

E2 = v2(c2) 2(N ) = 377.(6)

C2 v2(3) + v2(c2) + v2(d) ( 2) 34

H3 4 c2

v3(4) = 3,v3(c2) =5 v3(N3) = £нз = 12.(3)

C2 v3 (4) + v3(c2) ( 3) 8

Third step. Now we need to sum the payoffs of players c^d c2 E V- Ej E1 + E2 399.(6) , 274.(6)

EC1 = = ECi + ECi -

eBc1

E = V^ Ej = E2 + E3

EC2 = EC2 = EC2 + Ec

j Hj e Bc2 So we get the imputation

363.(3)

Ei —

27

515

E3 "IT 'E4

E _ 399.(6) , 274.(6) E

27 1 34

377.(6) . 61.(6)

! 34 ' 8

218

_ 27

37

_ 8

377.(6) . 61.(6)

C1 27 34 ' C2 34

4. Conclusion

A cooperative game with a hypergraph communication structure is proposed. The two-level cooperation in this class of games is considered. A new approach for the definition of the characteristic function for coalitions of hyperlinks is introduced. For a two-level cooperation structure, a new allocation rule is proposed. Examples of hypergraph games are presented.

References

Myerson R. B. (1977). Graphs and cooperation in games. Math. Oper. Res., 2, 225-229. Shapley, L. S. (1953). A value for n-person games. Annals of Math. Studies, 28, 307-317. Owen, G. (1986). Values of graph-restricted games. SIAM J. Alg. Disc. Meth. 7, 210-220. Meessen, R. (1988). Communication games, Master's thesis. Department of Mathematics.

University of Nijmegen, the Netherlands (in Dutch), van den Nouweland, A., Borm, P., Tijs, S. (1992). Allocation rules for hypergraph communication situations. Int. J. Game Theory, 20, 255-268. Shan, E., G. Zhang, X. Shan (2018). The degree value for games with communication

structure. Int. J. Game Theory, 47, 857-871. Von Neumann, J., Morgenstern, O. (1994). Theory of Games and Economic Behavior.

Princeton: Princeton University Press. Berge, C. (1989). Hypergraphs. North-Holland, Mathematical library, pp. 1-3, 155-162.