DYNAMIC COOPERATIVE NETWORK GAMES WITH PAIRWISE INTERACTIONS Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, XIII, 95-120

Dynamic Cooperative Network Games with Pairwise

Interactions*

Mariia A. Bulgakova

St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, Universitetskiy prosp. 35, Peterhof, 198504, Russia E-mail: mari_bulgakova@mail. ru

Abstract This article is an overview of results obtained in the field of dynamic network games with pairwise interaction. The paper provides a summary and analysis of works related to two-stage and multistage nonzero-sum games based on pairwise interaction. The meaning of pairwise interaction is to consider the game as a family of games occurring on a network between pairs of players (vertices of a graph) connected to each other by an edge. The network can be set or formed in the first stage. In the paper, solutions of cooperative pairwise interaction games are also considered.

Keywords: cooperative games, network games, dynamic games, the Shapley value, pairwise interactions, time-consistency.

1. Introduction

Cooperative network games is an important part of modern game theory. Networks illustrate connections between players and their ability to communicate in coalitions. For the first time in the game theory literature, a non-cooperative form of pairwise interaction in a network was considered in (Dyer and Mohanaraj, 2011) meaning direct interactions between network neighbors. But cooperative approach is quite natural and moreover beneficial for players as it provides them with a better outcome rather than noncooperative behavior. First cooperative interpretation of pairwise interaction games was in (Bulgakova and Petrosyan, 2015). Solutions of network games with pairwise interactions were examined in detail in (Bulgakova, 2019).

Also of interest are solutions for games with pairwise interaction on specific networks. The geometric structure of the network and symmetry make it possible to simplify some complex formulas and get their simple explicit form (Petrosyan, Sedakov and Bulgakova, 2018; Bulgakova and Petrosyan, 2019a).

When cooperative behavior is investigated, it is important that players follow a cooperative agreement during the whole course of the game. If a solution of the cooperative game is time consistent, players have no reason to deviate from the accepted agreement. Conditions for strongly time-consistence of core in 3-person games with pairwise interactions was studied in (Bulgakova and Petrosyan, 2016). Also, time consistent cooperative solutions for multistage games with special payoffs wad considered in (Kuzyutin and Nikitina, 2017).

Sometimes it is difficult to determine the explicit form of the core and decide on its stability, in this case, subsets of the core are used, which can be found in various ways. For example, as in (Petrosyan and Pankratova, 2018). Also, a special subset of

* This work was supported by the Russian Science Foundation (project No.17-11-01079).

the core, the IDP-core, was considered in (Petrosian, Gromova and Pogozhev, 2018; Wolf, Zakharov and Petrosian, 2017). The adaptation of the IDP-core to games with pairwise interaction was carried out in (Bulgakova and Petrosyan, 2019b).

In this paper, we analyse and compare the results in the field of cooperative network games with pairwise interaction.

2. Two-Stage Network Games with Pairwise Interactions

Consider a model of two stage cooperative network games with pairwise interactions (Dyer and Mohanaraj, 2011), when network is not given. So, players should form a network on the first stage. These game was studied in (Bulgakova and Petrosyan, 2015). Let us explain the model.

Let N be a finite set of players, which can make decision s in two stages, |N | = n > 2. Denote as z stage of game. The game started in stage zi, where every player i G N choose to behavior b1 = (b11,..., b^J — n-dimensional vector with offers for connections to other players (Petrosyan, Sedakov and Bochkarev, 2013).

We will use following notations: Mi Ç N \ {i} — those players, whom player i G N can offer a connection, wherein a G {0,..., n — 1} is maximal number of connections for player i, which he can maintain simultaneously. If Mj = N \ {i}, it means, that player i can offer connections for all players. In case, if a = n — 1, i

For every behavior b1 there exists such subset of realized offers Qi C Mj, which satisfy following restrictions

bi = i ^ if j G Qi, m

bij \0, other, 1 j

with additional condition

Ebj < ai. (2)

jew

Condition (2) means, that number of possible connections is restricted for every player. Also, obviously, |Qi| < ai.

Connection ij is formed if and only if, bj = bj = 1. Formed connections ij create edges of network g where vertex are players, i.e., if bj = bj = 1, then there is an edge between md j in network g.

Denote by Ni (g) or simply Ni neighbours of player i in network g i.e. Ni (g) = {j G N \ {i} : ij G g}. As result of the first stage we have network g^,..., b^). After network is formed players move to stage z2(g), which depend on network. On second stage z2(g), neighbours on network play pairwise in simultaneous bimatrix games, after that players get their payoffs and the game ends. In other words, we have two stage game rzi (g), which is special case of multistage nonzero-sum games. In considered case strategy of player is a rule, which for every player determine his

bi1

bimatrix game on second stage corresponding to network, which is formed on the first stage — b2. Denote by ui = (b^b2), i G N, strategy of player i in two-stage game rzi (g). Calculate payoff of player i as hi(z2^, where (z1, z2) is a path, realized by strategy profile u = (u1(^),..., «„(■)) in game rzi (g). Because on the first stage players do not get their payoffs, payoff function in game rzi (g) with starting position z1

Ki(z1; u) = Ki(z1; ui(^),... ,un(-)) = hi(z2).

2.1. Characteristic Function

On second stage game represents a set of pairwise simultaneous bimatrix games {yij} between neighbours in network. Namely, let i G N, j G N^ Then player i (the first player) play with player j (the second player)in bimatrix game yij with payoff matrices Aij and Cj of players md j respectively.

A;.

n

12

21 a22

\an

"1k ij a2k

ij

amk /

(3)

Cij

ij ij

c11 c12

ij ij

c21 c22

1k

cij c2k

(4)

\ cm1 cm2 • • • cmk '

apt > 0, Cpi > 0, p = 1,.. ., m, l = 1,...,k.

Constants m 'And k coincide for all md j. When we have game 7ji, i.e. player i is the second player, he has payoff matrix Cji = Aj, and player j has matrix Aji = Cjj. Denote by rS2 (g) subgame of game r, which takes place on second stage z2. Consider this game in cooperative form. Find characteristic function for every subset (coalition) S C N as lower (maxmin) value for two person zero-sum game of coalition S and coalition N \ S, based on game rS2 (g). Payoff of coalition S is

S

follows from its definition. Note, that

wj = maxmin a,.

ij p i p1'

p = 1,..., m; 1 = 1,..., k,

wj = max min

j, p =1,...,m; 1 =1,...,k.

(5)

(6)

and v(z2; S), S C N, is lower value of zero-sum game rS2 (g).

Theorem 1 (Bulgakova, 2019). Function v(z2; S) is determined by following expressions:

v(z2; |0}) = O, (7)

;(z2; M) = ^ wlj.

(8)

je Ni

;(z2; = E max(aijii + cj + ^ ]T wik.S C N,

(9)

ies jeNins

ies keNi\s

N ) = max(aPl + j

2 ^ ^ p,i

ie N je Ni

(10)

1

2

Consider a cooperative form of two-stage game rzi (g). Suppose, that players chose strategies u, i € N, that maximize their total payoff in game rzi (g), i. e.

EKj(zi; Ui,..., U„) = max Kj(zi; ui,..., w„).

u z—'

i£N i£N

Strategy profile U = (u1,..., Un) we will call cooperative behavior, and corresponding path (z1? ¿2) — cooperative path.

As previously, for coalition S C N define characteristic function v(z1; S) as maxmin value in two person zero-sum game between coalition S (maximizing player) and coalition N \ S (minimizing player). For minimazing player the best way of behavor is to not create all the connections with maximizing player (because of

S

members.

Denote by v(z1; S), S c N, lower value of zero-sum game rzi (g). Theorem 2. Function v(z1; S) ¿s defined in following way:

«(¿1; {i}) = 0, v(z1; 0) = 0, (11)

«(¿1; S) = i E E ma/(api + cpj), S C N, (12)

«(¿1; N) = vfe N) = 1 EE max(aj + cjl). (13)

ieNjeNi p'

2.2. Convex Game

Definition 1. Characteristic function is called supermodular and corresponding game called convex (Shapley, 1971), if for any coalition X c N Mid Y C N holds inequality:

v(X U Y) > v(X) + v(Y) - v(X n Y). (14)

Theorem 3 (Bulgakova, 2019). In subgame rS(g) characteristic function (8)-(10) is supermodular.

This property guarantees non-emptiness of the core and that the Shapley value belongs to the core.

2.3. Star-Network

Consider a specific structure of network, and solution of cooperative game on this network. We will study network, which contains n players, where player 1 is central player with n — 1 connections, and al 1 other n — 1 players are connected by-edge only with a central player.

The Shapley Value. Denote the set of imputations by

n

M[v(zt)] = {x = (x1,... ,Xn) : EXj = v(zt; N),xj > v(z; {i}),i € N}.

i=1

Consider as a solution of star-network the Shapley value (Shapley, 1953) and denote it as y[v(zt)] = (^1[v(zi)],..., <^n[v(zt)]), t = 1, 2, where for every player i e N

^[v(zt)]= E (|S|- ^ -|Sl)! [v(zt; S) - v(zt; S \{i})]. (15)

S C N, !

i e s

We say that the cooperative solution M[v(z1)] in the two-stage game is time consistent if for any imputation £[v(^)] e M[v(z1)] there exists an imputation £[v(z2)] e M[v(z2)] such that

e[v(z1)] = e[v(^2)], (i6)

Thus the Shapley value ^[v(z1)] is a time consistent cooperative solution if

^[v(z1)] = ^[v(z2)], (17)

otherwise the Shapley value is time inconsistent.

One has to mention that the cooperative version of the game has been developed for a general case of pairwise interactions when any (feasible) network can be formed at the first stage of the game. For this general case, we have found analytical expressions for the characteristic functions (8), (11) which are used to calculate the Shapley value by (15) and verify the corresponding time consistency condition 17. Since the calculation of the Shapley value is a challenging task for a large number of players and an arbitrary network, in the following we will show how to simplify formula 15 for a network of a special type — a star. This type of the network will provide us with an analytical expression for the Shapley value which is much easier to interpret and analyze. Nevertheless, expressions (8), (11), (15) can help us to find the Shapley value for any network at least numerically. From (15) we note that to compute a component of the Shapley value, we need to list all 2n subsets of N

capabilities to calculate n! as this number can be extremely large. In contrast, for a star network we will need only O(n) calculations and will not need to list all subsets N

Because the Shapley value belongs to the core in subgame rS(g), then its significance in this class of problems increases.

Now formalize construction of star-network on the first stage of game. Suppose following, let M1 = N\{1}, a1 = n — 1 Mid Mi = {1} ai = 1 for i = 1. To maximize total payoff, players should chose on the first stage following behaviors:

b1 =/(0, 1,..., 1), i = 1 /i o\

6> = \(1, o,..., 0),i = 1. (18)

Behavior (18) formed a star network on the first stage with central player 1, (fig. 1), where |N1| = n — 1 Mid |Ni | = 1, i = 1. For star-network calculate the characteristic function in view of specific structure of network. Transform expressions (8-10) corresponding to a star-network:

v(z2; {0}) = 0, (19)

( E ,i = 1

v(z2; {i})=< j=i (20)

Wn, i = 1.

Fig. 1. Star-network

Y, maxp,i(apl + ) + wjfc, S C N, i G S,

v(z2; S) = jewns k£Ni\s (21)

0, i G S.

Similarly, we transform the expressions (11)-(13)

«(¿i; {i}) = 0, v(zi; 0) = 0, (22)

f ^ maxp,i(apj + cpj), S C N, i G S, v(zi; S) = J jewins pl pl (23)

[ 0, i G S.

The network has central symmetry and formula (15) can be simplified.

Proposition 1 (Petrosyan, Sedakov and Bulgakova, 2018). For star-network with centraI player 1, components of the Shapley value y[v(zt)], t = 1, 2, has the form:

... vfe {1}) + E(mw - {j}))

.........i = 1,

{«}) + mii - w1*] , i = 1.

(24)

where

max (apj + bj), if i and j are neighbors,

p = 1, . . . , m p p

I = 1 , . . . , k

0, otherwise.

(25)

Time-Consistency of the Shapley Value in a Star-Network Consider time-consistency of Shapley value in a star-network. Remind that the Shapley value )] is time-consistent cooperative solution, if

^(¿i)]= )]. (26)

Proposition 2 (Petrosyan, Sedakov and Bulgakova, 2018). For star-network with centraI player 1, the Shapley value is time-consistent, when w1 = w* 1 for any

i=1

Conditions for time-consistency w^ = w*1, i = 1, from (2) for star-network can

i=1

have similar minimal guaranteed payoff (maxmin value) in bimatrix game yu which

they play as neighbours. If at least for one player j wj = wj1, then the Shapley value will be time-inconsistent. However, in the latter case, a cooperative agreement may be reached at the expense of IDP — imputation distribution procedure. Remind the definition:

Definition 2. Function ft% i G N called imputation distribution procedure (IDP) x G M (see Petrosyan and Danilov, 1979), if

xi = fti + ft2, i G N. (27)

Proposition 3. For star-network with centraI player 1, time-consistent IDP ft = for the Shapley value has the form:

2 E Ki - w"] ' i = 1

ftl Hj i=i . (28)

2 [w1i - wii] ' i = 1 and,

1 E [w1i + m1i - ' i = 1'

fti2 H 2 i=i (29)

1 Ki + mii - wii] > i =1-

From an economic perspective, it is preferable for players to have a nonnegative IDP, such that decomposing the Shapley value over two game stages, each player receives her nonnegative payments at either of the two game stages. However the IDP of the Shapley value mentioned in the latter proposition can be nonnegative

w11i = wii1 i = 1

consistency of the Shapley value which has already been discussed.

Three examples below demonstrate that the Shapley value being an imputation in a cooperative two-stage game with pairwise interactions can be both time consistent and time inconsistent despite players' asymmetry in their neighbors. The first example show the time consistency of the Shapley value in an important class of bimatrix games.

Prisoner's Dilemma Consider the case, when n players play the same bimatrix game 7 with their neighbors, i.e., Aij = A Cj = C for all i G N j G Ni where

A = CT =( hl 0>) , 0 <a<6. \a + ha y

h

a a data

transmission problem in a network whose nodes have to "coordinate" their actions trying to achieve certain performance (for example, the number of packets sent, a profit from sending packets).

For the given game using (5), (6), (25) we obtain: wj = wjj = a and

2b, if i and j Me neighbors, 0, otherwise,

for any two players i, j e N.

To find the Shapley value f[v(z2)], we first determine characteristic function v(z2; S) for all S C N. Following (8), we obtain

{2b(n — 1), S = N,

2b(|S| — 1) + (n — |S|)a, S C N, 1 e S, ( .

|S|a, S C N, 1 e S, (31)

0, S = 0.

Using the formula for the Shapley value (24) adapted to the star network, we obtain

n1

f1[v(^)] = 2 [a + 2b — a] = b(n — 1), fi [v(^2)] = 1 [a + 2b — a)] = b, i = 1.

Similarly, to find the Shapley value f [v(z1)], we determine characteristic function v(z1; S) for all S C N. Following (11), we have

f 2b(n — 1), S = N, v(z1; S) = ^ 2b(|S| — 1), S C N, 1 e S, (32)

[ 0, S C N, 1 e S or S = 0.

Again, using the formula for the Shapley value (24) adapted to the star network, we get f [v(z1)]:

2b(n — 1) , . f1[v(z1)] = —2— =b(n —1),

fi[v(z1)] = f" = b, i = 1

Comparing f [v(z1 f [v(z2)], we note that they coincide and hence the Shapley value is time consistent. What is interesting, this result does not depend upon specific values of a Mid b. The only restriction which has to be imposed is 0 < a < b, and it is natural for this class of games.

The time consistency of the Shapley value can also be shown with the use of Proposition 2 instead of applying direct calculations. Indeed, we see that w*1 = wji = a for each i = 1 what implies its time consistency.

Example In the next example we demonstrate the time inconsistency of the Shapley value. Consider a 4-person game with N = {1, 2,3,4} in which players form a star network under a cooperative agreement (see Fig. 2). Let simultaneous bimatrix games y12 , y13 and y14 be defined by means of the following payoff matrices:

( . C (2, 2) (3, 0) C (3,1) (4, 2)

(A12, C12) = (5,1) (1, 2), (A13, C13) = (6, 2) (2, 3),

(A C )=(1, 3)(3, 2) (A14,C14)= (6, 6) (4,1).

To compute the Shapley values f [v(z1)] and f [v(z2)], we use the corresponding formulas (9), (11) for characteristic functions v(z2; •) and v(z1; •), respectively, and

Fig. 2. A star with four players

the simplified formula (24). Hence we get:

wi2 = 2 wi3 = 3 wi4 = 4

w2i = 1, w3i =2, w4i = 3, (33)

mi2 = 6, mi3 = 8, mi4 = 12,

and therefore

v(zi; {1})=0, ; {1}) = 9,

v(zi; {2})=0, ; {2}) = 1,

v(zi; {3})=0, ; {3}) = 2, (34)

v(zi; {4})=0, ; {4}) = 3, v(zi; N) = 26, v(z2; N) = 26.

Thus the Shapley values are given by

^[v(zi)] = (13, 3, 4, 6),

)] = (29/2, 5/2, 7/2,11/2).

We observe that the Shapley value ^[v(zi)] in the two-stage game differs from the Shapley value ^[v(z2)] in the one-stage game starting at the second stage. This means time inconsistency of the Shapley value. Since y2[v(z2)] = 5/2 < [v(zi)] = 3

that players do not receive payoffs at the network formation stage). Similarly, it is also true for player 3: y3[v(z2)] = 7/2 < ^3[v(zi)] = 4 and player 4: y4[v(z2)] = 11/2 < ^4[v(zi)] = 6. However introducing a time-consistent IDP of the Shapley value )] over two stages determined with the use of Proposition 3 for the star

network, we obtain

fti = -3/2, fti = 1/2, ft3 = 1/2, ft4 = 1/2, ft2 = 29/2, ft2 = 5/2, ft32 = 7/2, ft42 = 11/2,

and therefore implementing it, the cooperation of players will be sustainable. Thus

ftii fti2 i G N

^i[v(zi)] in two stages which is exactly her cooperative payoff prescribed by the Shapley value ^[v(zi)].

2.4. The Core in Two-Stage Three-Person Game

Consider three-person-game with pairwise interactions on any network and take as solution the core. Find conditions for strongly time-consistency (Petrosyan, 1995) of core in this game.

Define the core C(z) c Mv in game r and suppose, that for every zi5 z2, C (z) = 0.

Definition 3. The core in game ^ is set of imputations x = (xi,..., xn), which satisfied following conditions:

for all S C N, and

y^xi > v(z i; S)

ies

N

y^xi = v(z i; N).

(35)

(36)

For subgame rz2 we have following values of characteristic function:

v(z2; 0) = 0,v(Z2; {1}) = w ^ + w 12,v(z2; {2}) = + w23,v(z2; {3}) = w33 + w33, v(Z2; {12}) = max(ap2 + c^2) + w^ + w^

v(Z2; {13}) = max(apf + cPf) + w ^ + wis, p,l

v(Z2; {23}) = maxfaj23 + cp3) + w22 + w^ p,l

v(z2; N) = max(ap2 + bp;1) + max^ + bp]) + max(apf + bp2). p,l p,l p,l

Imputation x belongs to the core C(z2), when following inequalities holds:

x 1 + x2 > v(z2; {12}), x i + x3 > v(^2; {13}), x2 + x3 > v(z2; {23}),

x i > v(Z2; {1}), (37)

x2 > v(^2; {2}), x3 > v(^2; {3}), x i + x2 + x3 = v(z2; N).

In view of (37) and values of characteristic function, we have:

x i + x2 > max(ap2 + cp2) + w23 +

3,

^23,

(38)

x i + x3 > max(ap3 + cp3) + w i 2 + pl

x2 + x3 > max(ap3 + cp3) + w22 + w33, pl pl pl

x i + x2 + x3 = v(z2; N).

To simplify the above expressions, use following notations:

Ei2 = npax(ap2 + cp2), Di = W23 + w iз,

E3 = mpax(ap3 + cp3)' D2 = w i 2 + w3з,

E23 = mpix(«p3 + cp3^ D3 = w22 + w33.

Then system of inequalities (38), which defines the structure of the core C(z2) can be rewritten in following way:

x i + x2 > Ei 2 + D i, x i + x3 > Ei3 + D2, x2 + x3 > E23 + D3, x i + x2 + x3 = v(z2; N).

1

Consider the core C(zi) of two-stage game r

x i + x2 > v(zi; {12}), x i + x3 > v(zi; {13}), x2 + x3 > v(zi; {23}), x i + x2 + x3 = v(z i; N). Corresponding to (40) and characteristic function, we have:

x i + x2 > max(ap2 + cp2), x i + x3 > max^ + cp3),

x2 + x3 > mpax(ap3 + cp3^

x i + x2 + x3 = v(Z i; N)).

Using above notations, we will have followings:

x i + x2 > Ei2, x i + x3 > Ei3, x2 + x3 > E23, 5i + x2 + x3 = v(z i; N).

(40)

(41)

(42)

Strongly Time-Consistency

Definition 4. The core C(z 1 ) is strongly time-consistent in game r (Petrosyan, 1995), if

1. C(Zi) = 0, C(Z2)= 0

2. For every imputation x G C(z1 ) there exists such IDP | = (|1;|2), that x = I1 + I2 and

C(Zi ) D I1 © C(Z2 ). Here symbol © defines as a G Rn, B C Rn,then a © B = {a + b : b G B}. In accordance to (42) we get:

' # + + i2 + > E12,

11 + $ + |3 + ^ > E13, (43)

12 + + 13 + > E23.

For strongly time-consistency these inequalities must satisfies with following additional restrictions:

+ > E12 + D1,

+ > E13 + D2, (44)

+ ^ > E23 + D3.

Fix then for strongly time-consistency we should fulfil conditions (44) for |2. And |2 must satisfy (43). Also, from v(Z2; N) = v(Z1; N), follows I] + |2 + 13 = 0-If conditions (43) satisfy under minimal values , l2, I3 from conditions (44), then they satisfy for all other values. We have:

—13 + E12 + D1 > E12,

—12 + E13 + D2 > E13, (45)

—Ii + E23 + D3 > E23.

Thus, we have conditions for strongly time-consistency of the core C(Z1) in game

r.

Theorem 4 (Bulgakova and Petrosyan, 2016). Suppose, that following inequal-

ities are fulfilled:

ft? < D ft? < D2 fti1 < D?

(46)

(i.e there exists ft1 which satisfied (4)), then the core C(z1) is strongly time-consistent.

3. Multistage Games with Pairwise Interactions

In (Bulgakova and Petrosyan, 2019a), the results obtained for the two-stage games are generalized to the case of multistage games.

Let abstract space Z is given and called space of states. In every state z G Z n-person nonzero-sum game r(z) is determined. This is network game with network g(z), where players are vertexes and edges are connections (as in two-stage case). Game r(z) is a set of simultaneous pairwise bimatrix games {Yij (z)} between neighbours by network, i G N, j G N, i = j.

Let i G N, j G N, i = j. Then i plays with j in bimatrix game Yj(z) with payoff matrices A2j(z) and C2j(z), for players md j, respectively.

Aij (z)

( aï?(z ) «21 (z)

v «mm i(z ) «m 2(z)

12(z) ^22(z )

Cij (z)

/j ) c21(z )

ci2(z) c22(z )

\cmm 1(z) cmm 2(z)

jj

a1r ij ,

(z)

(47)

«mr(z ) )

(z)

c2Jr(z

cmr(z) )

(48)

apj(z) > 0, cPj(z) > 0, p =1,

q =

1,

, r, i, j G N.

Constants m and k coincide for all ¿Mid j. When we have game Yj (z), i-e. player i is the second player, he has payoff matrix Cj (z) = AT (z), and player j has matrix Aj (z) = Cj (z). To simplify the following expressions we suppose, that m and r coincide for all i and j and all z.

Consider the strategy of player i in game r (z). It is a vector «¿(z) = (u^z ),..., uj (z),... u"(z)), where uj is strategy of player i in bimatrix game Yij (z). So, we have, that strategy of player i is a vector consisting from rows (pure strategies), chosen by him in bimatrix games Yij (z)- Denote by u(z) = («i(z),..., un(z)) strategy profile in game r (z). Strategy of player j is a vector consisting from columns in corresponding bimatrix games Yij (z). Define payoff of player i in game r(z) in following way:

Ki(z) =

ij i J

^ «Uj (z)U (z) (z) = 1,j=i j

Consider game r(z) in cooperative form. As in two-stage case, introduce characteristic function v(S; z), S C N as lower (maxmin) value in two-person zero-sum

game between S and N \ S, based on game r(z)

maxmin apjq (z), p =l,...,m; q = 1,.. ., r,

pq

maxmin cpjq(z), p =1,...,m; q =1,...,r.

qp

Lemma 1. Characteristic function v(S; z) is defined in the following way

v({0}; z) = 0, (49)

v({i}; z)= £ Jij(z), (50)

jeN,j=i

v(S; z) = 1 £ £ max(aj(z) + cj(z)) + £ £ ^(z), S C N, (51)

ies jes,j=i p'q ies jeN\s

(z ) =

4 (z ) =

v(N; z) = ^ npax(apjq(z) + cj(z)).

ieN jeN'j=i

Suppose, that in state z e Z in game r(z) players choose the strategy profile: ui(z) = (wl(z),..., u™(z)). Then the game moves to the state z', where game r(z') takes place, with simultaneous bimatrix games with matrices, depended from strate-

z z z

in this state. We can introduce function T : Z x "=1 Ui ^ Z by formula:

z' = T(z; ui(z),u2(z),... ,un(z)).

Multistage game G(z) develops in following way. Game G(z1) starts in state z1. In state z1 game r(z^ is take place, players choose their strategies u1(z1), u2(z1),..., «„(z^, then they pass to the state z2 = T(z1; u1(z1),u2(z1),..., un(z1)). In state zk players play game r(zk), choose strategies u1(zk), u2(zk),..., un(zk) and pass to the state zk+1 = T(zk;u1(zk),u2(zk),..., un(zk)). Game ends on stage I in state z^. Then, after choosing strategies on every stage of game path z1, z2,..., zk,..., z^ is realized. Strategy in this multistage game «(•) = u{(z)}, is a set of players' strategies, defined in every state z e Z. It follows from above, that any strategy profile «(•) = {^(0,... ,«„(•)} defines only one path, and, therefore a payoff for every player, as sum of his payoffs in games, along the realized path.

I n

W» = £ E aj {zk ^ (zfc) (zk). k=1j=1j=i

Note, that set of all paths in multistage game G(z) is finite. Hence, the set of states is also finite. Denote this set by Z C Z Consider special case, when v(N; z) is the same for all z e Z. Introduce function w(S), S C N:

w(S) = max v(S; z).

Define also characteristic function V(S; zk) for multistage game G(zk), which starts in state zk. Function V(S; zk) can be calculated using the following analogue of Bellman equation:

V (S; zfc_i) = max [ min (V Kf-1 (ui,...un)+ V (S; zfc))] =

«¿.¿eSuj,jeN\s i^S

= max [ min (V (ui,...u„) + V (S; T(zfc_i; u(zfc_i))))j;

ui ,i£S Uj ,j£N\S i^S

V(S; zi) = v(S; zi).

Define

W(S; zfc ) = (1 - k +1)w(S) where I is number of stages in game G(zi).

The following inequality takes place (see Petrosyan and Pankratova, 2018):

W(S, zk) > V(S, zfc), S C N.

Remind that one-stage game r(z) is convex and characteristic function v(S; z), SCN

v(X U Y; z) > v(X; z) + v(Y; z) - v(X n Y; z). Take the maximum by z eZ in the left and right sides of the inequality max v(X U Y; z) > max v(X; z) + max v(Y; z) — max v(X n Y; z).

z z z z

Since maxz v(S; z) = w(S), we have:

w(X U Y) > w(X) + w(Y) — w(X n Y). The function w(S) is supermodular.

Lemma 2. Function w(S) in game r(z) ¿s supermodular.

3.1. Optimality Principle

Define the set of all imputations MW in game G(zi) as

n

Mw = {x = (n,...,x„)^ xi = W (N; zi), > W ({i}; zi), i e N}.

¿=i

And by optimality principle any subset of this set MW.

Choose in one-stage game r(z) as optimality principle an analogue of the core — the set G(w(S)), containing all imputations x = (xi,... ,xn), which satisfy conditions:

Exi > w(S), S C N,

ies

N

= w(N).

i=i

In similar way define set C(W(S; z)) in multistage game G(z) as set of imputations, which satisfy following conditions:

> W(S; z), S C N, (52)

ie s

N

X>i = W(N; z). (53)

i=1

Theorem 5. For any x e C(W(S; zk)),x = (x1;..., xn) and any k = 1,l, the following equality holds:

xi = (I — k + 1)xi, where xi e (7(w(S)), i = 1, n.

3.2. Strongly Time-Consistency

Suppose, that players chose strategies «(•) = (u1(^),..., «„(•)), which maximize the total payoff in G(z1):

EKi(z1; «1,... ,«„) = maxV^ Ki(z1; «1,.. .,«„)•

ieN ieN

The path (z1; Z2,..., Z^), corresponding to this strategies is called cooperative path (z1 = z1). Rewrite definition of IDP (2) for the game under consideration.

Definition 5. Vector P% i e N is called imputation distribution procedure (IDP) if for x e MW (see Petrosyan and Danilov, 1979),

i

xi = Y, Pk, i e N. (54)

k=1

Definition 6. Optimality principle C(W(S; z1)) is strongly time-consistent in game G(z1) (see Petrosyan, 1995), if

1. C(W(S; Zk))= 0, k = M

2. For every imputation x e (7(W(S; Z^) there exists such IDP P = (P1, p2,..., P^), such that

k

]TPj © C(W(S; Zk+1)) C C(W(S; Z1)), k = M—1. j=1

Here symbol © means, that if a e Rn, B C Rn, then a © B = {a + b : b e B}.

Proposition 4. Optimality principle (7(W(S; Z1)), k = 1,1 in game G(Z1) ¿s strongly time-consistent.

3.3. General Case

Now consider general case, when values of v(N; z) do not coincide for different z. Introduce function L(zk) is joint payoff of maximal coalition N in state zk e z, where z is cooperative path in game G(zi).

L(Zk ) = 2 V (aij (Zk) + j (Zk)). (55)

ieN,jeN,i=j

Suppose, that w(S) < min2fc L(zk), S = N. And consider function w(S; zk), w(S; zk) = w(S), w(N, zk) = L(zk). Let x be any imputation in game G(zi). Consider as optimality principle an analogue of the core in game G(zk) called G(W(S; zk)); the following set

> (I — k +1)w(S) = W(S; zk), S C N, S = N, (56)

ies

Ni

= $3 L(zk) = W^(N; zk). (57)

i=i t=k

Suppose, that all G(W(S; zk)) = 0.

Definition 7. Optimality principle G(W(S; zi)) is strongly time-consistent in game G(zi), if

1). G(W(S; zk)) = 0, k = M

2). For every imputation x e G(W(S; zi)) there exists such IDP ft = (fti,..., ftl), x = j=i ftj that

k

]Tftt © G(W(S; zk+i)) C G(W(S; zi)), k = M. j=i

Proposition 5. Optimality principle G(W(S; zi)) ¿s strongly time-consistence.

Example Consider an example with N = 3 k = 3, i.e. 3-person three-stage game

zi zi

zi

the first type. In state zi every player i e N choose to strategy ui(zi). If all uj(zi) = 1,i e N,j e N \ {i}, then players pass to the state z2, where they play bimatrix games with same payoff matrices of first type. If at least one of the components uj(zi) = 2, i e N,j e N \ {i}, then in state z2 players play bimatrix games with the second type of matrices. Similarly, the transition to the state z3: if all uj(z2) = 1, i e N, j e N \ {i}, then players in state z3 use first type of matrices. If at least one of the components uj (z2) = 2, i e N, j e N \ {i}, then in state z3 players use matrices of the second type. Matrices of the first type are:

A'2 <*> = (ll) ' C'2<z)=( 91) , AiS(z)= (107 ) ,

C,3(z)= (l^, AsM = ( f2) , C23(z)= (

Matrices of the second type are:

7 , ( 8 5 ) ~ , . (10 13 ) ~ . . (8 12 )

Al12(z) = y 10 16y ' C712(z) = ( 6 6 ) ' A713(z) ={9 10 J '

^(z) ^ C 11 12) ' , C23(z.

z1

players should to be in touch with all neighbours throughout the game. Strategies of players:

U1(z1) = (0, 2,1), U2(z1) = (1,0, 2), us(z1) = (2, 2,0), U1(z2) = (0, 2,1), U2(z2) = (2,0,1), us(z2) = (2,1,0),

U1(z3) = (0, 2,1), U2(z3) = (2,0,1), U3(z3) = (2,1,0).

Calculate the values wij (z):

w12(z0 = 1, ^22(z1) = 1, w13(z1) = 1 w33(z1) = 0, ^23(z1) = 2, ^33(z1) = 1

^2^2) = 10, w^^) = 6, ^3^2) = 9, ^33 (z2 ) = 10,

w23(z2) = ^33(z2) = 6.

Values wj (z3) will coincide with wj (z2) or with wj (z1), since there are only two types of matrices in the game.

Calculate the values of characteristic functions v(S; z), w(S; zk):

S {1} {2} {3} {12} {13} {23} {123}

v(S Z1) 2 3 1 21 20 27 62

v(S Z2) 19 15 16 40 40 36 66

v(S Z3) 19 15 16 40 40 36 66

w(S Z1) 19 15 16 40 40 36 58

w(S Z2) 19 15 16 40 40 36 66

w(S Z3) 19 15 16 40 40 36 66

z1

the new state, which depends from this choice. In every state players have only two alternatives: after choosing strategies, pass to the game with first type of payoff matrices, or pass to the second type of payoff matrices.

Numbers 1 and 2 above the arrows on Fig. 4 indicate what type of matrices the players will play in the next state. Calculate the values of function L in nodes zk:

L(z1) = 58 L(z1) = 58 L(z1) = 62 L(z|) = 66 L(zf) = 66 L(zf) = 62 l(z|) = 66

Fig. 4. Tree of all possible states of game.

Cooperative path in game G(z^ is z = (z^z^zf) = (zi,z2,z3). Calculate the

G( z1 )

S {1} {2} {3} {12} {13} {23} {123}

V(S; *3) 19 15 16 40 40 36 66

V(S; z2) 38 30 32 80 80 72 132

V(S; zi) 40 33 33 101 100 99 190

W(S; z3) 19 15 16 40 40 36 66

W(S; z2) 38 30 32 80 80 72 132

W(S; zi) 57 45 48 120 120 108 190

Condition w(S, zk) < min2fc L(zk), S = N holds:

max w(S, zk) =40 < mmin L(zk) = 58.

Consider an imputation x e C(W(S; zi))

xi > 57, x2 > 45, x3 > 48, xi + x2 > 120, xi + x3 > 120, x2 + x3 > 108, xi + x2 + x3 = 190.

And any imputation £ e (7(W(S; z2))

£i > 38, £2 > 30, £3 > 32, £i + £2 > 80, £i + £3 > 80, £2 + £3 > 72, £i + £2 + £3 = 132.

(58)

(59)

ftk ft

1,2, 3).

(fti, ft2, ft3) an imputation a e C(w(S; zi)^ (ftk = a, k

«i > 19, «2 > 15, «3 > 16, ai + «2 > 40,

ai + «g- 40, «2 + «3 > 36, «i + «2 + «3 = 58.

Construct the imputation X = « + £

(60)

x1 > 57, X2 > 45, X3 > 48,

X1 + X2 > 120, (61)

X1 + X3 > 120, X2 + X3 > 108, Xi + X2 + X3 = 190.

It follows from last inequality, that X e (7(W(S ; z1)), which proves strongly time-consistency of C(W(S ; Z1)).

4. Nonzero-Sum Games with Pairwise Interactions

In this part we will consider general nonzero-sum game with pairwise interactions. Previously we considered only bimatrix games.

Consider (Bulgakova and Petrosyan, 2019b) multistage nonzero-sum game with finite number of stages. As before, on the first stage players chose their behaviors and formed a network. On next stages simultaneous non-zero sum games will be played. Payoffs depend on controls, chosen on each current stage. Players can change the network on every stage except the first, deleting some connections.

4.1. The Model

In every state zk e Z, k > 0 players can change the network by deleting some connections, so we denote the network as g(zk), to show the network dependency on state.

Denote as Ni(g(z0)) neighbours of player i in network g(z0), i. e. Ni(g(zo)) = {j e N \{i} : ij e g(zo)}.

When network g(z0) is formed, the game passed to the state z1(g(z0)), which is determined by network g(z0). In state z1(g(z0)) players can delete some connections, which were formed early. So, network g(z0) in general will change to g(z1) and we may have new set of neighbours Ni(g^)). On network g(z1) players play the simultaneous nonzero-sum game r(z1).

On the second stage z1 player i, i = 1,n, chose control yi(z1) = (bi(z1),xi(z1)) from set of controls Yi, unlike the first stage z0, contains an additional com-

ponent xi(z1) — behavior in game r(z1^. Where bi(z1) is vector with components 0 or 1, defined in following way:

bij (zi) =

1, save connection j, 0, ^^^^^e connection j,

(62)

i. e. player on second stage can delete some connections but has no abilities to create new connections. Component xj(zi) of control yi(zi) = (6i(zi),xi(zi)) is behavior of player i in game r(zi) chosen from set Xi(zi), in state zi.

Let y(zi) = (yi(zi),...,yn(zi)) be a strategy profile in game r(zi). Player's payoff i in game r(zi) is:

Hi(zi) = h (yi(zi),yj ^i^

jeNi(S(z 1))

where g(zi) is network, corresponding to strategy profile y(zi). Functions hj(xj(zi),xj-(zi)) > 0 are given for all i e N and all pairs ij, i. e. all edges of network g(zi) and all possible states z e Z.

Players i e N chose controls (yi(zk-i),...,yn(zk_i)) in state zk-i e Z in game r(zk-i^. Result of this choice is transition to state zk, where r(zk) is played, with payoffs hj(xj(zk), xj(zk)), depended on controls, chosen in state zk. We may defined a transition T : Z x Yi x Y2 x ... x Yn ^ Z by formula

zk = T (zfc_i; yi(zfc_i),y2(zfc_i),...,y„(zfc_i)), k =1,1. (63)

The, function T uniquely defines state zk, which follows after state zk_i, if players chose controls yi(zfc_i), y2(zfc_i),... ,y„(zfc_i).

Consider multistage game G(z), which develops in following way. Game G(z0) starts in state z0. In state z0 network g(z0) is formed, after that players pass to state z^. In state zk_i players chose controls yi(zk_i), y2(zk_i),..., yn(zk_i), play game r(zfc_i) and pass to state zfc = T(zfc_i;yi(zfc_i),y2(zfc_i),... ,yn(zfc_i)). Game ends on stage I + 1 in state z^. Thus, after choosing controls on every state of the game the path z0, zi,..., zk,..., zl is realized.

State zk is called acceptable, if there exists the sequence of controls and the sequence of states z0, zi,... zk, k < I generated by it, defined by formula (63), such that zk = z.

Strategy in multistage game: yi(-), i e N, is a rule, which for every acceptable state z associates components bj(z),xj(z) of control in this state, i. e. the choice of connections for deleting, and choice of behavior xj(z) in game r(z). It the follows from above that any strategy profile y(-) = {yi(^),...,yn(0} defines only one path in game, and, hence payoff of every player as sum of his payoffs in games, realized along the path.

i

^MOHE Yl hj(yj(zk),yj(zk)).

k=i ¿£N¿(3(2))

The set of all paths in game G(z) is finite. So, the set of all acceptable states also is finite. Denote it as Z c Z.

Suppose, that players choose controls yj(z), i e N, that maximize their joint

G(z)

i i ^2 ^2 Hj(yi(zk),..., y„(zk)) = ma^^ Hj(yi(zk),..., y„(zk)). (64)

k=i j£N ^ k=i j£N

Strategy profile y = (yi,..., yn) is called cooperative behavior in game G(z), and path, corresponding to controls y^z), i e N, (z0, zi,..., zl) is called cooperative

z0 = zz0

Consider one-stage game r(z) in any state z e Z in cooperative form and defined characteristic function v(S; z), S c N, for every subset (coalition) S c N in following way:

v(0; z) = 0, v({i}; z ) = 0,

v(f • z) = / hi(xi(z); x(z)) + hj (xj(z); xi (z))^f j e Ni(g(z)),

v({ij}; z 0, other, (b5j

v(S;z E hi(xi(z); xj(z ^

ies jeNi(g(z))nS

v(N;z) = E E hi(xi(z); xj(z)),

ieN jeNi(g(z))

where xi(z),xj(z) calculated corresponding to (64).

We see that to calculate values of characteristic function, we should find cooperative behavior in game G(z0) and after that calculate v(S; zk), k = 1,1 as payoffs, under assumption, that players choose cooperative behavior as components of their controls.

Defind the characteristic function V(S; zk) in multistage game G(zk), which zk S

(y(z0), y(z1),..., y(z;)) in I — k + 1 stages, starting from k:

i i

V (S; zk ^Ev(S;zr ^EE E hi(xi(zr ),xj(zr)),

r = k T=kie,SjeNi (g(zr ))ns

V(S; zi) = v(S; zi). The following theorem holds:

Theorem 6. Characteristic function v(S; z) in gam e r (z) is super modular.

4.2. The Shapley Value

Consider as a solution of game r(z) the Shapley value = (^1[vj,..., yn[v]), (15).

Calculate the difference [v(S; z) — v(S \ {i}; z)]:

[v(S;z) — v(S \{i});z ]= (hi(xi (z),xj(z)) + hj (xj(z ),xi(z))).

jeNi(g(z))nS

Substitute calculated above values in the formula of the Shapley value (15)

^i[v]= £ (|S | — ^ — |S l)! E (hi(xi (z),xj (z)) + hj (xj (z),xi(z))), i e N.

SCN,ieS ! jeNi(g(z)) ns

(66)

This formula does not require the calculating the values of characteristic function ScN

only the structure of network g(z).

4.3. IDP-Core and Strongly Time-Consistency

As previously MV is set of all imputations G(z0).

Introduce IDP ftj = (ft0,..., ft1), i G N, f°r imputation £ G I(V), which satisfy

i

= £ ft[, i G N. (67)

r=0

Consider as optimality principle subset of the core for game G(zk) — IDP-core (Wolf, Zakharov and Petrosian, 2017; Petrosian, Gromova and Pogozhev, 2018) C(V(S; zk)), i. e. set of IDP, which satisfied following conditions:

i

]Tfti ^ V(S; Zk) = ^^ hi (Xi(zr ),Xj (zr)), S C N, S = N,

¿es r=k ies jewi(9(zr))ns

(68)

N i

]Tfti = V (N; = hi(Xi (zr ),Xj (zr)), (69)

¿=1 r=fci£NjeNi (g(zr ))nN

where fti = (ftl,..., ftk,..., ftl) satisfy following:

Eftk ^ v(S, Zfc ) = ^ ]T hi(Xi (zfc ),Xj (zfc)), S C N, (70) ¿es ¿es jeNi(j(zfc))ns

N

]Tftifc = v(N, Zk) = £ ]T hi(Xi(zfc),Xj(zfc)). (71)

i=1 ¿eN jeNi(g(zt ))nN

Suppose, that all C(V(S; zk)) = 0.

Definition 8. Optimality principle C(V(S; z0)) = 0 is strongly time-consistent in game G(z0), if

1) C(V(S; Zfc)) = 0, k = M;

2) For every imputation x G C(V(S; z0)) there exists such IDP ft = (ft0,..., fti),

£ = ^r=0 ftj' such that k

]Tftr © C(V(S; Zk+i)) C C(V(S; Z0)), k = 0,1.

r=0

Here symbol © means, that if a G Rn, B C Rn,then a © B = {a + b : b G B}. Proposition 6. Optimality principle C(V(S; z0)) is strongly time-consistent.

4.4. Example

This example is similar to one from the previous section. Consider the case with N = 3, I = 3, i. e. game consist four stages and starts in state z0. In this state sets Mj of players, whom player i can offer a connection, are given.

Mi = {2, 3}, M2 = {1, 3}, M3 = {1, 2}.

Restrictions for number of connections for every player are also given.

ai = 1, = 1, = 2.

In state zo players choose ve ctors b^zo), formed the net work g(zo) and pass to the state z^ In every state zk, k > 1 players choose controls y'(zk) = (b'(zk),x'(zk)), where bi(zk) — player's regulation connections vector (with components 1 and 0), and xi(zk) is equal to

xi(zfc) = xi(z) G Xi = {x^z),xi(z)},

X2(zfc) = x2(z) G X2 = {x^z^x^z)}, x3(zfc) = x3(z) G X3{x^(z),x|(z)}

i. e. every player i has the same set of control components Xi in every state zk.

For all acceptable states zk, k > 1, and all possible strategies payoffs hi(xi ( zr), x j (zr)) are given in following way: h'(xj ( zr), x j (zr)) and hi(xi (zr),x

j(zr))-

In state z1 game takes place with payoffs h(xi(z1),xj-(z1)^. In state z1 every player i G N choose his component of control xi(z1), and if all xi(z1) = x1 (z1),i G N, then players pass to the state z2, in game, where payoffs are similar hi(xi(z2),xj(z2)). If at least one of components xi(z1) = x2(z1),i G N, then in state z2 players play game with payoffs hi(xi(z2),xj-(z2)). Similarly, the pass to the state z3: if all xi(z1) = x1(z1),i G N, then in state z3 players use payoffs hi(xi(z3),xj(z3)), if at least one of the components xi(z1) = x2(z1 ),i G N, then payoffs are h'i(xi(z3),xj(z3)).

Payoffs hi (x' (zr), xj (zr)):

h1(x1 x2) = 4 h 1 (x 1 x3) = 5, h2(x1 x3) =5

h1(x1! x2) = 3 h1(x2 x3) = 3 h2(x2 x3) =1

h1(x1 x2) = 5, h^xj; x3) = 1 h2 (x2 x33) =4

h1(x1! x2) = 5 h1(x2 x3) = 2, h2(x2 x33) =1

h2(x2 x1) = 4 h3(x1 x311) = 5 h3(x1 x2) =5

h2(x2 x 1) = 3 h3(xg x21) = 3 h3(x1 x2) =1

h2(x2 x 1) = 5, h3(x2 x111) = 1 h3(x2 x2) =4

h2(x2 x 1) = 5 h3(x2 x21) = 2 h3(x2 x2) =1

payoffs hj (x' (zr), xj (zr)):

h 1 (x 1 x12) = 8 h 1(x1 x3) = 6 h2 (x2 7 x3) = 12

h21(x211 x12) = 3 h 1 ( x2 x3) = 5 h2 (x2 7 x3) = 10

h 1(x11 x222) = 7 h 1 (x 1 x3) = 4 h2(x2,x§) = 5

h 11(x211 x222) = 4 h 1 (x2 x3) = 3 h2(x2, x3) = 4

h2(x2 x1) = 8 h.3^3 x311) = 6 h3(x3> x2) = 12

h2(x1 x21) = 3 h3(x3 x21) = 5 h3(x3i x2) = 10

h2 (x2 x111) = 7 h3(x2 x111) = 4 h3(x3,x1) = 5

h2 (x2 x21) = 4 h3(x2 x21) = 3 h3(x3J x2) = 4

zo

players:

b1(zo) = (0, 0,1), b2(zo) = (0,0,1), b3(zo) = (1,1,0). As result we have the following network:

Fig. 5. Network ou t.lie first stage

Players have an ability to delete connections on all stages, except the first, but to maximize the joint payoff it is beneficial for players to be in touch with all neighbors throughout the game, i. e. 6j(z0) = bj(zi) = bj(z2) = &j(z3), for all i G N. Components of controls yi(z) for players:

xi (zi) = xl, X2(zi) = x^ X3(zi) = x1, Xi (z2 ) = xi, x2(z2) = x2, x3(z2) = x^ xi (Z3) = xi, x2(z3) = x2, x3(z3) = x;j. Calculate the values of characteristic function v(S; z) in all states on cooperative zo

not get any payoffs.

S {1} {2} {3} {12} {13} {23} {123}

; Zi) 0 0 0 6 6 10 16

v(S; Z2) 0 0 0 14 12 20 32

v(S; Z3) 0 0 0 14 12 20 32

z1

ponds from this choice. In every state every player has only two alternatives: choose controls and pass in state where play game r(zk) with payoffs ^¿^(zk),Xj(zk)), or pass to the state with game, where payoffs are h^^(zk),Xj(zk)).

Fig. 6. Tree of all possible states in game

Numbers 1 and 2 above arrows (fig. 6) indicate which payoffs will be used by players in the next state: 1 means ^¿^(zk),Xj(zk)), 2 — h!i^¿(z/t),Xj(zk)).

Cooperative path in game G(z0): z = (z0, z1, z22, z|) = (z0, z1, z2, z3). Calculate the values of characteristic function in multistage game G(z0):

S {1} {2} {3} {12} {13} {23} {123}

V(S; Z3) 0 0 0 14 12 20 32

V(S; Z2) 0 0 0 28 24 40 64

V(S; Zi) 0 0 0 34 30 50 80

Consider an imputation q G C(V(S; z1)):

x1 > 0, x2 > 0, x3 > 0,

x1 + x2 > 34, (72)

x1 + x3 > 30, x2 + x3 > 50, x1 + x2 + x3 = 80.

As ¡3k in IDP P = (po, p1, p2, p3) we will take imputations ak G C(v(S; )), which satisfy (70), (71), ¡3k = a, k =1, 2,3; po set equal to zero, and satisfies the system

a1 > 0, a2 > 0, a3 > 0,

a1 + «1 > 6, (73)

a1 + a1. > 6,

«2 + «3 > 10,

a1 + «2 + a1 = 16, P2 and P3 satisfy the inequalities, where k = 2,3:

ak > 0,

ak > 0,

a3k > 0,

ak + ak > 14, (74)

ak + ak > 12, ak + ak > 20, ^ ak + ak + ak = 32.

Summing the left and right sides of the inequalities of the last two systems and we get

91 > 0,

92 > 0,

93 > 0,

< 91 + 92 > 34, (75)

q1 + 93 > 30, 92 + 93 > 50, „ 91 + 92 + 93 = 80.

From last inequality follows, that 9 G C(V(S; zo)). In other words, imputation from set C(V(S; zo)), can be decomposed on the sum of imputations from the sets C(v(S; zk)),k = 1, 2, 3,. This proves strongly time consistency of optimality principle C(V(S; zo)).

5. Conclusion

The paper discusses the results in the field of cooperative network games with pairwise interaction. Two-stage and multistage games and various approaches to

the determination of the characteristic function in each of them are considered. The cooperative solutions, such as the Shapley value and the core, are examined and the properties of their time-consistency as well as strongly time-consistency are investigated and the conditions for it are found. Special types of networks are considered, such as a complete network, a star-network, and the possibilities for solutions that give such geometric structures are investigated. All results are illustrated by examples.

Acknowledgements. We acknowledge for the support Russian science foundation (project No.17-11-01079).

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