Cooperative Solutions in Multi-Star Network Games Текст научной статьи по специальности «Компьютерные и информационные науки»

Contributions to Game Theory and Management, XVI, 132—144

Cooperative Solutions in Multi-Star Network Games *

Tian Lu1 and Jianpeng Zhang2

1 Yan'an University, Faculty of Mathematics and Computer Science, Yan'an, 716000, Shaanxi, China

E-mail: lutian2085@163.com 2 St.Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: 2392748644@qq.com

Abstract A model also known as multi-agent systems, namely the multistar model is considered. In a multi-agent system, a two-level game is played, the first-level is the external game, and the second-level is the internal game. An approach is proposed how to distribute the benefits to players in the first-level game and the second-level game. The characteristic functions are constructed for the multi-star model. Based on the proposed characteristic functions, the combination of the Shapley value and the proportional solution as natural optimal principle to distribute the benefits in the first-level game and in the second-level game is proposed.

Keywords: multi-agent system, the Shapley value, the proportional solution.

1. Introduction

In (Petrosyan and Bulgakova, 2015) and (Bulgakova, 2019), the star model is proposed, and the Shapley value formula is given to study its distribution of income. (Petrosyan and Bulgakova, 2020) is dedicated to the study of multi-stage games with pairwise interactions under the consideration of complete graphs, constructing characteristic functions, and calculating Shapley values. In (Bulgakova, 2019) and (Pankratova, 2018), studies on the construction of the characteristic function are carried out. (Hernandez, Munoz-Herrera and Sanchez, 2013) considered the interaction model between cooperation and coalition in network games. In (Petrosyan and Sedakov, 2019), a two-level structure of player exchanges is considered and a procedure for assigning value in two steps is proposed, demonstrates how to use Shapley values to assign values in two steps and shows the differences from the classic one-step assignment procedure. The second-level game theory is illustrated in (Zhonghao, 2012) by taking the study of the China-Korea FTA negotiation scheme as an example. In (Shapley, 1998), the Shapley value formula for n-person games is explained.

2. The Model

Define a nonzero-sum game r on the multi-star network G, where the vertices of the network are players and the edges of the network are connections between players. The game r is a family of pairwise simultaneous bimatrix games {^ij}

*This work was supported by Yan'an University, Yan'an, China. https://doi.org/10.21638/11701/spbu31.2023.09

Fig. 1. The connections of the Multi-star model

between the neighbors i,j G N, i = j. Denote by N the set of players and we

ik, ..., im}

divide N into n coalitions Si,...,Sr,...,Sn where Si = {î 1,..., î|,..., ¿m},---, Sr =

{¿i,...,î|,...,¿m},..., Sn = {¿l,...,¿ï,..., ¿m}. Sr(r = 1,2,...,n) denotes any coalition, an d |Sr| > 2.

Given the connections of each coalition Sr as Figure 1. Each coalition can cut the connection. The player who is connected to all players in his coalition and connected to players in other coalitions like player î 1, « i , called 'hub'. In a coalition, satellites other than the 'hub' are only connected to the 'hub'. And in network G, any coalition Sr is connected through the 'central players' to form a circle.

Define the elements of the set N = {j G N\{î}, îj G G} the neighbors of player ¿, where îj denotes the direct connection between the player î and j. Let player î G N play with his neighbor j G N a bimatrix game Yj with non-negative payoff matrix Aij = [aj]x=1 ,...,p;y=1 ,...,i and Bj = [^j]x=1 ,...,p;y=1 ,...,i of player î and j, respectively.

Define [P(¿, j), L(î, j)] is the strategy profile of player î and his neighbor j, where x G P(î, j), y G L(î, j) are the strategies of player î and j, respectively. Kj (x, y) > 0 is the sum of the payoff of players î, j in Yij.

And PSl ULSl = P (î 1, ¿2)UP (î 1, ¿1 )U...UP (î 1, ¿m)UP (î 1, î?)UL(î 1, î 1 )UL(î 1, ¿3)U ... U L(î 1, ¿m) U L(î ^¿ï) is the strategy profile for coalition S1, and PSr U LSr = P(¿1 ,¿2) UP(q,¿3) U ... UP(¿1 ,îm) UP(¿1 ,î 1r+1 }) U L(î1 ,¿2) UL(î 1, ¿3) U... UL(îî,îm) U

L(i 1, i 1r 1 •*) is the strategy profile for coalition Sr, and PSn U LSn = P(i™, in) U

P (i™n, i?n) U... U P (in,c) UP (in,i1) U L(i?,in) U L(in,in)u... u L(in,c) u L(in,4n-1)) is the strategy profile for coalition Sn. Where X1 G PSl U LSl, Xr G PSr U LSr Xn G PSn U LSn are the strategy profile of player S1; Sr and Sn respectively. Wsi (X1,X2,Xn), Wsr (X ,Xrir,Xr+T) and ws„ (Xn,Xn-7,X1) are the payoff of coalition S1; Sr and Sn respectively.

3. Cooperation at the Game

Divide the game into two levels, the first level is the external game, and the second level is the internal game. Define the game inside of any coalition Sr as the second level game, and the game between coalitions as the first level game, and we call them level — II and level — I, respectively.

In this section, we consider cooperation at level — II and level — I, respectively.

3.1. Cooperation at level — II

Consider the first cooperation method in level — II, for each coalition M C Sr, the value V(M) is the maxmin value of a two-person zero-sum game, of coalition M against its complement Sr\M. We call the function V(M) characteristic function. Denote maxmin value of player i(j) in game yx with his neighbor j(i) as,

{Wj = maxxminy aj, x = 1, ...,p, y = 1,..., l, j = maxy minxjy, x = 1, ...,p, y =1,..., l.

Following, we can obtain the characteristic function V(M) for each coalition

m c sr,

5 E iesr EjeNinsr maxxy(aXjy + filJy), M = Sr,

V (M) = < 5 ^ ieM^jeN nM maxx,y (aXy + PXXy) + ieM^2keN\M , M C Sr,

( ) ^ Ej£Ni Wij, M = {i},

0, M = 0.

_ _ (1)

Suppose player i and his neighbor j choose strategies x and y respectively to maximize their joint payoff in game yx, i.e.,

Kij (x,y) = maxxy Kj (x,y)

Consider the second cooperation method in level — II, V(M) is n(n € (0,1)) times the payoff of M under the strategy that maximizes the joint payoff, coalition M against its complement Sr\M, we call the function V(M) characteristic function. And denote value of player i(j) in game yx with his neighbor j(i) as,

°ij = axjy,

\ 0 = Bij

0ji = Pxy.

Following, we can obtain the characteristic function V(M) for each coalition

M Sr

2 E iesr^2 jeNinsr maxxy (axXy + fixy), M = Sr,

V (M) = I 2 ^ ieM^jeNi nM maxx,y (axjy + Pxy) + ieM^2keNi\M 0ik, M C Sr, ( ) n Ej£Ni 0ij, M = {i},

0, M = 0.

(2)

3.2. Cooperation at level — I

Denote coalition Si play a bimatrix game YSiSj with his neighbor Sj as level — I of the game.

Consider cooperation in level — I. Because we define that each coalition Sr can cut the connection, we consider only one cooperative method in level — I.

Following, we can get the value of characteristic function V(M) of coalition M C N as follows,

V(M) = i 22 EieN^jeN, maxxy (axXy + fixXy ^ M = N, (3)

\ 2 EieM EjeMnN, maxxy (a*xXy + fixXy^ M C N.

4. The Shapley Value

In this section, we consider how to distribute the payoff in level — II and level —I respectively.

4.1. The Shapley Value of level — I

For level — I (N = {S2,..., Sr,..., Sn}), we consider only one cooperative method in level — I, we denote the Shapley value by Sh = [ShSl,..., ShSr,..., ShSn], where,

Shsr = £ <|M|- 1)(n — 'M|>! [V M — V (MAIS,})]. (4)

M CN,Sr EM '

Let fixed player Sr € M, and consider its limiting contribution V(M)—V(M\{Sr}) in coalition M. Take into account formula (3) for characteristic function V(M), we

can get,

V (M) — V (M\{Sr }) = V (Sr)+ £ £ maxxy (a^ + fijy). (5)

ieSr jeMnN,

From formulas (4) and (5), we can get the Shapley value ShSr for any player Sr as follows,

ShS. = £ ( ' M| — — ' M' )! [V(Sr) + £ £ maxxy(aX + j)]. M cN,Sr em ' ieSr jeM nN,

(6)

4.2. The solution of level — II

For level — II of the game (ik € Sr = {i2,..., ik,..., i^})), we define the benefits assigned to each player in the multi-star model as L = [L^, L^,..., L^], where,

Shir

Lir = —-—k- x ShS . (7)

k Y^i^i (JL Sr vv

Z^j=ir Shj

We define Shir to be the Shapley value when only coalition Sr is considered, where,

Sh. = £ < 1 M | — № — |M [V (M) — V (M\№], (8)

m cSr ,ieM '

Following(Petrosyan, Bulgakova and Sedakov, 2019), we can get the Shapley value

Shr of any player ik in the first cooperation method as follows, where j is a neighbor

r k.

¿k

of player î|.

Shik = 2[Ejew ^j + Ej=ii(maxxy(aX1yj + AX1/) - j)], k = 1, (9)

Shik = 1 [m«xxy (aa1yifc + ^¿¿iyifc ) + E jew.1 ^kj - ¿k L k = 1.

k

Following (Petrosyan, Bulgakova and Sedakov, 2019), we use the same method to derive the Shapley value Sh^ of any player î| in the second cooperation method, where j is a neighbor of player î|,

Shik = 2 ^Ejew, % j + Ej=i1 (maxxy (ax1y + Pxy ) - nj )], k = 1, Shik = 2[max^y(aX1yXk + ^Xy^) + n%x 1 - n^xixk], k =1.

(10)

From formulas (6),(8) and (9), we can get the benefit L^ of any player î| in the first cooperation method as follows,

Vffî + Ej=ii ^X1« +^X«3 ) —V(j)) Lxi = —-—3—iij-—-iij——3-x

k Ej-m^i |m«Xx, (axV +A*1/ )+V(j)—^] + |V(x1) + Ej=i1 (max,, (a,1« +,8*1/ ) —V(j))]

[V(Sr ) + E xeSi EjeMnNi maxxy(aXjy + )]},k = 1

maixi (aX1yik +^X1/ik ) +V(¿k )—^¿1 L»1 = —¿r-¿ij—¿ij-—-¿ij—¿ij-x

k E^ [maxxy (ax« +£¿1/ )+V(j)—"¿^] + |V(¿1^3^ (max,, (a,1« +M3 ) — V(j))]

[V(Sr ) + E xeSi E jeM nw maxxy (aj + ^Xy)]}, k = 1.

(11)

From formulas (6),(8) and (10), we can get the benefit L^ of any player î| in the second cooperation method as follows,

V ffî+E j=¿1(mai.»( aX1yj )—V (j))

Lxi = —¿1-—,—¿13-—-¿13—¿13-x

k Ej^^ |maXx„(axV + A.1/ ) + V(j)— n^i] + |V(¿1 HEj^ (max,„(a,1,j + £,1/ ) —V(j))]

[V(Sr) + SxeS1 EjeMnNi maxxy(Oj )]} k = 1

max,, (aX1,,^ +^X1,-k )+V(¿k )— n0»1»1

L»1 = —¿1-¿13—¿13-—-¿13—¿13-X

k Ej^^ |maXx„(a,V + ,3*1/ ) + V(j)— n^-] + |V(¿1 )+Ej=i1 (max,, (a,«3 + ,3*1/ ) —V(j))]

[V (Sr )+E eS^ EjeMnw maxxy(aXjy + ^Xy)]}, = 1.

(12)

5. The Proportional Solution

In this part, we consider the proportional solutions (Youngsub Chun, 1988) in the second cooperative method of level — II and level — I respectively.

5.1. The Proportional Solution of level — I

In the multi-star model, we denote the Proportional Solution by Pr = [PrS1,...,

PrS1,

..., PrS„ ], where,

PrS1 = sV(Sr)(,) x V(N), N = {S1,...,Sr,...,Sn}. (13)

Ej=S1 V (j )

According to (2), (3) and (13), we can get proportional solution PrSr for any coalition Sr (r = 1, 2,..., n) as follows,

PrSr = J T'j^NinSr maXXV ^ + ^j X 2 £ £ moxxy (aj + j)

Er=1 2 S ieSr E jeNiHSr maxxy (axy + ßxJy ) 2

EieSr EjpNiriSr maxxy(aXjy + ßXjy) 1

ieN jeN

ieSr ¿-^jeNinSr xy\ xy > Hxy) i y- y- , y + )

— ^n ^ ^ ( ij + pij S X 2 maXxy (axy + pxy ;

z^r=i ieSr jeNidSr maXxy (axy + P'xy ) ieN jeNi

(14)

5.2. The Proportional Solution of level — II

For level — II of the game (irk G Sr = {¿1, ..., ¿k,..., ¿ml)), we denote the Proportional solution by Pr = [Prir,..., Prir,..., Pr^], where,

V (¿k )

Prik = _ir k- x PrSr. (15)

k Ej=ir v(j) ( )

Following (2), (14) and (15) , we can get the Proportional solution Pr^ for player ¿r in the second cooperation method as,

Ej=iS ^¡j \Sr ei\j

Prir = -ir-1-X

k E,-^ (^ +j )+Ej-eNir \Sr ^

EieSr EjeNnsr maxxy

xxy ( ^

Pf,r = -

.je^nsr maxxy("xy+ßxy)_ 1 ^ v- max (aij + ßij ) k i

E!Ll E,ESr Ej-eNinSr maxxy (aXy X ^ieN ^jeNi maXxy (axy + ßxy )' k = 1'

E^ E^nSr maxxy« jy)ij x 1 V._„ V. max...(aij

= l iGSr ^jGNinSr maxxy (axy +ßxy )

Ej=ir +j )+Eie»,y \Sr ^ j

X 2 E ie^ EjeNi maXxy (axjy + ppxy ), k = 1.

(16)

6. Mixture of Shapley Value and Proportional Solution

In this part, we only consider the benefit distribution under the second cooperation method. First, we consider distributing benefits using Shapley value at level — I and proportional solution at level — II. Then, we distribute benefits using proportional solution at level — I and Shapley value at level — II.

6.1. The First Mixture

In this section, we consider using the Shapley values at level — I to distribute benefits, and the proportional solution at level — II to distribute benefits.

The Shapley Value of level —I For level —I (N = {S1,..., Sr,..., Sn|), we denote the Shapley value by Sh = [ShSl,..., ShSr,..., ShSn], where,

ShSr = E (|MI~ ^ -'M|>' [V(M) — V(M\{Sr})]. (17)

m cn,srem '

Let fixed player Sr G M, and consider its limiting contribution V(M)—V(M\{Sr}) in coalition M. Take into account formula (3) for characteristic function V(M), we

can get,

V(M) - V(M\{Sr}) = V(Sr)+£ ]T maxxy(aj + ßj). (18)

iesr jeM nNi

From formulas (17) and (18), we can get the Shapley value ShSr of any player Sr as follows,

Shsr = E (|M|- ^ -|M|)! [V(Sr) + E E maxxy(aj + ßj)].

m cN,sr eM ! iesr jeM nN

(19)

The Proportional Solution of level — II For level — II of the game (¿k G Sr =

{¿1,...,«k,..., ¿m})), we denote the Proportional solution by Pr = [Pr^ ,...,Prik, ...,Pri;m], where,

V (¿r )

^ = im (k;(,x sh(sr ). (20)

V(i) = n E 0ij.

ij. (21) je Ni

Shsr = E (|M|- ^ -|M|)! [V(Sr) + E E maxxy(aj + ^)].

m cN,sr eM ! iesr jeM nN

(22)

Following (20), (21) and (22) , we can get the Proportional solution Pr^ of player ¿k in the second cooperation method as,

Pr,;r =

(|M |-1)!(n-|M | )!

EmCN,SreM ire! [V(Sr ) + EieSr jeMnNi malxy (axy + i^Xy )], k = 1,

Pr,r = —^-^-X

k +j)+Ej-eNir\Sr ^

, EmcN,sreM I [V(Sr) + EieSr ^jeMnN maxœy(aXy + ^Xy)],k = 1.

(23)

6.2. The Second Mixture

In this section, we consider using the proportional solution at level — I to distribute benefits, and the Shapley value at level — II to distribute benefits.

The Proportional Solution of level — I In the multi-star model, we denote the Proportional Solution by Pr = [PrSl, ...,PrSr, ...,PrSn], where,

Prsr = gV(Sr)( ,) x V(N), N = {Si,...,Sr,...,S„}. (24)

Ej=Si V (j )

According to formulas (2), (3) and (19), we can get proportional solution PrSr of any coalition Sr (r = 1,2,..., n) as follows,

22 E¿esr EieNi.nSr maxxy (aXcy + pxy) 1

Pr„ - _2 ^S_xy ■ ■ _ x - > V max (a^ + Bn )

P'Sr - ^n I ^ ^ ( ij + ßij s x a Z^ maXxy(axy + Pxy;

z^r=l 2 i£Sr jENidSr maxxy (a xy + Pxy ) ieN jeNi

_ Eiesr EjeNins1 maxxy(aXjy + ßXjy) 1

^n ^ ^ ( ij + pij) x 2 ^ max-y(a % + p%)

z^r=l ieSr jENiHSr maxxy (axy + Pxy ) ieN jeNi

(25)

The solution of level—II For level-// of the game (irk G Sr = {il,..., irk,..., ¿ml)), we define the benefits assigned to each player in the multi-star model as L = [Lij,Li2,..., Lim], where,

Shir

Lik = im k x PrSr. (26)

2^j=ir Shj

Shik - 2[V(il) + £j=ir (maxxy(aÏJ + fèyj) - V(j))], k - 1, Shik - 1 [maxxy(a^ + ßxiyifc) + V(irk) - n%ik], k - 1.

(27)

Following (26) and (27), we can get the benefit Li^ of any player ¿k as follows, where j is a neighbor of player iTr.

V (ir HE ,-=-r (maxxy (aX1Hj)-V (j))

Lir = —-irj—irj-—-irj—irj-x

k Ejmir [maxxy (axyj + 13xV' ) + V(j)-^,'] + [V(ir )+Ej=-r (max.y (axyj + ,Sx1yj )-V(j))]

[_^eSr Eje^iHSr maxxy (aXy )_ l y y- x (aij + pij )] k = 1

[ E!Li E,£Sr E^nSr maxxy (aXjy + j* 2 ^ieN^jeN- maXxy ("xy + Pxy ^ k = 1,

maxxy (aX1yifc +^X1yifc )+V(ik )-n®-r-r

Lir = —--rj--rj-L-k--rj--rj-X

k Ej=-r [maxxy (axV + ,Sx1yj ) + V(j)-^,'] + [V(iL)+Ej=-L (maxxy (ajJ + fix\3 )-V(j))]

[_E-6Sr EjEW-nSr maxxy (a'xj + ^y )_ l ^ max (aij + pij )] . = ,

[E!Li E-eSr E3eN.nSr maxxy(a j + ^) X 2 ^ieN ^eN- maXxy("xy + Pxy^ k = 1.

(28)

7. Example

Consider the case, each coalition Sr has m players, when ml players in any coalition Sr play the Prisoner's Dilemma game with their neighbors, i.e., Aj = A, Bij = B for all i G Sr, j G Ni. And m2 players in any coalition Sr play the Battle of the Sexes game with their neighbors, i.e. Aj = C, Bj = D for all i G Sr, j G Ni. And ml +m2 = m+1. We define that each coalition Sr plays the Prisoner's Dilemma game with his neighbors. Given the set of players playing Prisoner's Dilemma game in each group Sr as Ml and the set of players playing the Battle of the Sexes game as M2. Where

A = BT =( \ ^ , 0 < a < b. Va + 6 ay

C = id 0) • D = (C0). 0 < c < d.

For coalition M C Sr: we define that in each coalition M, there are m3 players playing the Prisoner's Dilemma game, and m4 players playing the Battle of the Sexes game. Where m3 + m4 = |M| + 1, ir G M and m3 + m4 = |M|, ir G M .

For coalition M C N: we define that there are m5 pairs of neighbors in coalition M. And any group Sr G M has m-6 neighbors in coalition M.

1. Consider the first cooperation method

To find the shapley value, we fist determine charscteristic function V(M) for all

m c sr.

V (M )

We use formula (3) to find the characteristic V(M) of coalition M C N, N =

^^ Sr, Sn}.

• M C N

{26n(mi — 1) + (d + c)n(m2 — 1) + 2bn =

= n(2bmi + (d + c)(m2 — 1), M = N,

2b|M|(m1 — 1) + (d + c)|M|(m2 — 1) +26ms, M C N,

0, M = 0.

We use formula (6) to find the Shapley values ShSr of any group Sr.

2b(m1 — 1) + (d + c)(m2 — 1), M = Sr ,

2b(m3 — 1) + (d + c)(m4 — 1)+

+ (m1 — m3)a + 2a, M C Sr, ¿1 G M,

am3, M c Sr, ¿1 G m,

0, M = 0, m c m1 c Sr

Sh

Sr = E

m cN,Sr eM

(|M | — 1)!(n — |M |),

[2b(mi — 1) + (d + c)(m2 — 1) + 2bme ]

We use formula (9) to find the Shapley values Shj of player j.

n,

Shj = 2 E

wew,-

+ E

¿=j

(maij

,(< + j ) — "¿j )]

Sh,

= 1 [a(m1 — 1) + 2a + (2b — a)(m1 — 1) + (d + c)(m2 — 1)] = a + b(mi — 1) + 1 (d + c)(m2 — 1),

= 2 [maxxy («X1yj + ^Xy ) + Eiew, j — "¿j] = 2 [2b + a — a] = b,

Shj = 2 [

= 1 [(d + c)]

(«xV + AXy

) + E i

¿eNj j

— "¿1 j]

¿1,

j G Mi C Sr, j = ¿i,

j G M2 C Sr, j = ¿i.

We use formula (11) to find the solution Lj for each player j.

a+6(m 1 -1)+1 (d+c)(m2-1)

Lj =

a+26(mi-1) + (d+c)(m2-l) X Shsr , b x ShSr,

X Shsr

¿r ¿1,

a+2b(m 1 -1) + (d+c)(m2-1)

_d+c_

2[a+2b(m i -1) + (d+c)(m2-1)]

J = ¿1 j G M1 C Sr j = ¿1, J G M2 C Sr j =

2. Conider the second cooperation method

To find the shapley value, we fist determine charscteristic function V(M) for all M Ç Sr. • M Ç Sr

' 2b(m1 — 1) + (d + c)(m2 — 1), M = Sr,

2b(m3 — 1) + (d + c)(m4 — 1) + n(m1 — m3)b+ +n(m2 — m4)d + 2^b, M C Sr,

V(M) = ir1 G M,

nbm3 + ncm4, M C Sr,

M,

0,

1

M = 0.

We use formula (3) to find the value of characteristic V(M) of coalition M C N, N = {Si,...,Sr,...,Sn}.

MN

V(M)

2bn(m1 — 1) + n(d + c)(m2 — 1) + 2bn = = 2bnm1 + n(d + c)(m2 — 1), M = N,

2b|M|(m1 — 1) + (d + c)|M|(m2 — 1) +2bm5, M C N,

0,

M = 0.

(1)The Shapley Value

We use formula (6) to find the Shapley values ShSr of any group Sr.

She

E

m cN,sr eM

(|M | — 1)!(n — |M |),

[2b(m1 — 1) + (d + c)(m2 — 1) + 2bme]

We use formula (10) to find the Shapley values Shj of player j.

Shj = 1 Eiew, j + Ei=j(maxxy(jy + ßxy) — wij)]

= 2 [nb(m1 — 1) + nd(m2 — 1) + 2nb

+(2b — nb)(m1 — 1) + (d + c — nc)(m2 — 1)]

= 2b(m1 + n — 1) + (nd — nc + d + c)(m2 — 1),

Shj = 2 [maxxy (aXyj + ßxyj ) + EieNj j — 1 j]

= 2 [2 b + n« — nb]

b +

n(a-b)

Shj = 2 [maxxy (aXyj + ßXyj ) + EieN3- j — 1 j] = 2 [(d + c) + nc — nd]

_ d+c+n(c-d)

= 2 ,

j = ¿1,

j G M1 C Sr j = ¿1,

j G M2 C Sr j = ¿1.

n,

We use formula (12) to find the solution Lj of each player j.

25(mi+n-1) + (nd-nc+d+c)(m2-1)

2b(mi+n — 1) + (nd—nc+d+c+ j =

d + c + n (c — d)

)(m2-1) + (mi-1)[b+

n(a —b)

x ShSr,

b+

n(a —b)

= J 2b(mi+V-1) + (Vd-Vc+d+c+ d+c+2(c —d) )(m2-1) + (mi-1)[b+ > ] X ShSr '

j S' e M1 c Sr, j =

d + c+n(c — d) 12!

2b(mi+n-1) + (nd-nc+d+c+

j e M2 c Sr, j = i1.

d + c + n(c — d)

)(m2-1) + (mi-1)[b+

n(a —b)

x ShSr,

(2)The Proportional Solution

We use formula (14) to find the Proportional solution Prsr of any group Sr.

]

2

]

2

Prs = 2b(m1 - 1\+(d + c)((m2 - \ x [2bnm1 + (d + c)n(m2 - 1)] Sr n[2b(m1 - 1) + (d + c)(m2 - 1)] [ 1 + ( + ) ( 2 )]

= 2bm1 + (d + c)(m2 - 1)

We use formula (16) to find the Proportional solution Prj of each player j.

n(mi — 1)6+n(m2 — 1)d+2nb

nbm3+ncm4+n(mi — 1)b+n(m2 — 1)d+2nb x

[2bm1 + (d + c)(m2 - 1)], _nb_-

Pr • = nbm'3+ncm4+n(mi—1)b+n(m2 — 1)d+2nb '

j ' [2bm1 + (d + c)(m2 - 1)],

nc

j = ¿'^

j e M1 c Sr,j = ¿1,

nbm3+ncm4+n(mi — 1)b+n(m2 — 1)d+2nb

x[2bm1 + (d + c)(m2 - 1)], j e M2 C Sr, j = ¿1,

Simplifying the above formula, we get the Proportional solution Prj of each player j as follows.

(mi — 1)b+(m2 —1)d+2b

bm3 + cm4 + (mi — 1)b+(m2 b —1)d+2b

Prj = < bm3 + cm4 + (mi — 1)b+(m2 c —1)d+2b

bm3 + cm4 + (mi—c1)b+(m2 —1)d+2b X [2bm1 + (d + c)(m2 - ^

¿r ¿1,

e M1 c Sr

= ¿1,

e M2 c Sr

J = ¿1,

(3)The Shapley Value and the Proportional Solution

We use formula (19) to find the Shapley values ShSr of any group Sr.

Shsr = £ (M|- 1)!(," l)! [2b(m1 - 1) + (d + c)(m2 - 1) + 2bm6]

n!

m cN,sr eM

We use formula (23) to find the Proportional solution Prj of each player j.

(mi-1)b+(m2-1)d+2b

bm3+cm4+(mi — 1)b+(m2 — 1)d+2b

[2b(mi - 1) + (d + c)(m2 - 1) + 26me], j = ¿1, b

Pr • = bm3+cm4 + (mi —1)b+(m2 —1)d+2b

X

[2b(m1 - 1) + (d + c)(m2 - 1) + 2bm6], j G M1 C Sr, j = ¿1,

_c_X

bm3+cm4 + (mi — 1)b+(m2 — 1)d+2b

[2b(m1 - 1) + (d + c)(m2 - 1) + 2bm6], j G M2 C Sr, j = ¿1,

(4)The Proportional Solution and the Shapley Value

We use formula (25) to find the Proportional solution Prsr of any group Sr

Prsr

2b(m1 - 1) + (d + c)(m2 - 1) n[2b(m1 - 1) + (d + c)(m2 - 1)] 2bm1 + (d + c)(m2 - 1)

x [2bnm1 + (d + c)n(m2 - 1)]

We use formula (28) to find the Shapley values Shj of player j.

Shj = 2 Eiew, j + E(maxxy+ ) - Wij) +(2b - nb)(m1 - 1) + (d + c - nc)(m2 - 1)] = 2b(m1 + n - 1) + (nd - nc + d + c)(m2 - 1),

Shj = 2 [maxxy («X1jj + ) + EieWj j - ] = b + ^^,

Shj = 2 [maxxy («X1yj + ) + E^- j - ]

_ d+c+n(c—d)

= 2 ,

j = ¿1,

j G M1 C Sr ,j = ¿1,

j G M2 C Sr ,j = ¿1.

We use formula (5.39) to find the solution Lj of each player j.

_25(mi+n-1) + (nd-nc+d+c)(m2-1)_

2b(mi+n-1) + (nd-nc+d+c+ d+c+2(c-d) )(m2 - 1) + (mi - 1)[b+ ]

b+ nV)

x Prsv, j = ¿1,

2b(mi+n—1) + (nd—nc+d+c+ d+c+2(c—d) )(m2 — 1) + (mi — 1)[b+ ^ ] X PrSr ,

j G M1 C Sr, j = ¿1,

d+ c+n (c— d)

2b(mi+n— 1) + (nd—nc+d+c+

j G M2 C Sr, j = ¿1.

d+c+n(c— d) 2

)(m2—1)+(m1—1)[b+

n(a —b)

x PrSr,

8. Conclusion

In (Petrosyan, Bulgakova and Sedakov, 2019), a two-stage network game with pairwise interactions is considered and the characteristic function is constructed. At the same time, a special network star model is constructed, and the benefit distribution problem of the star model is solved by using the Shapley value.

Based on the star model in literature (Petrosyan, Bulgakova and Sedakov, 2019), I respectively build a multi-star model, also called multi-agent systems. A two-level game played in the multi-agent system. The multi-star model consists of n star coalition, and each star coalition has m players.

]

In the multi-star model, I also constructed the characteristic function in two different ways of cooperation. Then, based on the characteristic function, I calculated the Shapley values and proportional solutions of the multi-star model. And take the Prisoner's Dilemma Game and the Battle of the Sexes Game as examples to verify the results of the multi-star model.

References

Bulgakova, M. A. (2018). About construction of characteristic function in game with pair-wise interactions. Proceedings of international conference Control Processes and Stability, 2018, 53-58.

Bulgakova, M. A. (2019). Solutions of network games with pairwise interactions. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 15(1), 147-156. Bulgakova, M. A. and L. A. Petrosyan (2015). The Shapley value for the network game with pairwise interactions. 2015 International Conference "Stability and Control Processes" in Memory of V. I. Zubov (SCP). Bulgakova, M. A. and L. A. Petrosyan (2020). Multistage Games with Pairwise Interactions

on Complete Graph. Automation and Remote Control, 81, 1519-1530. Chun, Y. (1988). The proportional solution for rights problems. Mathematical Social Sciences, 15(3), 231-246. Hernandez, P., Munoz-Herrera, M. and A. Sanchez (2013). Heterogeneous network games:

Conflicting preferences. Games and Economic Behavior, 79, 56-66. Pankratova, Y. B. and L. A. Petrosyan (2018). About new characteristic function for multistage dynamic games. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 14, 316-324. Petrosyan, L.A., Bulgakova, M. A. and A. A. Sedakov (2019). The Time-Consistent Shapley Value for Two-Stage Network Games with Pairwise Interactions. In: Song, J., Li, H., Coupechoux, M. (eds) Game Theory for Networking Applications. EAI/Springer Innovations in Communication and Computing. Springer, Cham. https://doi.org/10.1007/978-3-319-93058-9_2 Petrosyan, L., Sedakov, A. (2019). Two-Level Cooperation in Network Games. In: Avrachenkov, K., Huang, L., Marden, J., Coupechoux, M., Giovanidis, A. (eds) Game Theory for Networks. GameNets 2019. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-030-16989-3_5 Shapley, L. S. (1988). A value for n-person games. The Shapley Value Essays in Honor of

Lloyd S. Shapley. Ed. by A. E. Roth, 31-40. Zhonghao, P. (2012). Research on China-Korea FTA Negotiation Scheme Based on Second-Level Game Theory.