One-way flow two-stage network games Текст научной статьи по специальности «Математика»

UDC 519.83 Vestnik of St. Petersburg University. Serie 10. 2014. Issue 4

L. A. Petrosyan, A. A. Sedakov

ONE-WAY FLOW TWO-STAGE NETWORK GAMES*)

St. Petersburg State University, 7/9, Universitetskaya embankment, St. Petersburg, 199034, Russian Federation

In the paper two-stage network games are considered in which at the first stage players form a directed network, whereas at the second stage they choose feasible controls. It is assumed that payoffs of players depend on both the network and controls chosen by their "neighbors". In a cooperative case we find cooperative behavior of players and consider the Shapley value as a solution of the game. It is proved that the Shapley value is time-inconsistent, therefore the dynamic Shapley value is defined with the use of imputation distribution procedure. Bibliogr. 9. Fig. 1. Table 1.

Keywords: network, cooperation, Shapley value, time-consistency.

Л. А. Петросян, А. А. Седаков

ДВУХСТУПЕНЧАТЫЕ СЕТЕВЫЕ ИГРЫ НА ОРИЕНТИРОВАННЫЕ СЕТЯХ

Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7/9

В работе строится двухэтапная сетевая игра, в которой на первом этапе игроки формируют ориентированную сеть, а на втором — выбирают некоторые допустимые управления. Предполагается, что выигрыши игроков зависят от самой топологии сети, выбранного управления и от выбранного управления «соседей» в сформированной сети. При рассмотрении кооперации игроков находится кооперативное поведение, а в качестве решения используется вектор Шепли. Показывается, что вектор Шепли является динамически неустойчивым решением, в связи с чем строится динамически устойчивая процедура распределения вектора Шепли. Библиогр. 9 назв. Ил. 1. Табл. 1.

Ключевые слова: сеть, кооперация, вектор Шепли, динамическая устойчивость.

Introduction. In the theory of dynamic cooperative games, time-consistency of a solution is the key problem. Namely, having agreed on the particular solution before the game starts, players have to get the payoff prescribed by this solution at the end of the game. Such a problem is quite common for dynamic cooperative games, since during the game in the case of time-inconsistency, players may break initial agreement by their actions. Time-consistency of the cooperative solution based on a special payment scheme [1] stimulates players to follow agreed upon cooperative behavior.

In [2-4] two-stage network games were considered, in which players form a network [5] at the first stage, and then at the second stage they choose admissible controls. In particular, it was proved that in the cooperative setting, the cooperative solution - the

Petrosyan Leon Aga,ne.sovich — dean of faculty, doctor of physical and mathematical sciences, professor; e-mail: [email protected]

Sedakov Artem Aleksandrovich — candidate of physical and mathematical sciences, senior lecturer; e-mail: [email protected]

Петросян Леон Аганесович — декан факультета, доктор физико-математических наук, профессор; e-mail: [email protected]

Седаков Артем Александрович — кандидат физико-математических наук, старший преподаватель; e-mail: [email protected]

*) The reported study was supported by Russian Foundation for Basic Research (Research project N 13-01-91160-GFEN_a) and Saint Petersburg State University (Research project N 9.38.245.2014).

Shapley value [6] - is time-inconsistent. Time-inconsistency of other cooperative solutions, e.g. the core or the r-value, was shown in [7]. In contrast to the mentioned above papers, in which two-stage models were developed for undirected networks (models with two-way flow), in the present research we focus on two-stage models for directed networks (one-way flow models).

1. The model. Let N = {1,...,n} be a finite set of players, and g be a given network - the set of pairs (i,j) G N x N where (i,j) G g means that there is a direct link connecting players i and j and such a link generates communication of player i with player j. As it was mentioned before, the network under consideration is one-way flow network, that is any link (i, j) is direct link, i.e. (i, j) = (j, i).

Consider a two-stage model. At the first stage, players choose their partners - players with whom they want to form links. Once the partners are chosen, a communication structure, i.e. a network is formed. At the second stage, players choose admissible control variables, which together with the formed network, affect their payoffs. Consider the model in details.

1.1. First stage: network formation. A network is formed as a result of simultaneous choices of all players. Let Mi C N \ {i} be the set of players to whom player i G N can offer a link, and ai G {0,...,n-1} be the maximal number of links which player i can offer. At the first stage, behavior of player i G N is a profile gi = (gii,..., gin), where

From (1) we get gii = 0, i G N, which excludes loops from the network, whereas the condition (2) shows that the number of "offers" is limited. Note that if Mi = N \ {i}, player i can offer a link to any player, whereas if ai = n - 1, he can offer any number of links.

Denote the set of all possible behaviors of player i G N at the first stage satisfying (1), (2) by Gi. The set ni£w Gi is the set of behavior profiles at the first stage. Supposing that players choose their behaviors gi G Gi, i G N, simultaneously and independently of each other, the behavior profile (g1,...,gn) is formed. A resulting network g consists of directed links (i,j) s.t. gij = 1.

Define the closure of network g as an undirected network g, where gij = max{gij, gji}. Denote neighbors of player i in the network g by Ni (g) = {j G N\{i} : (i,j) G g}, whereas neighbors of player i in the closure g are denoted by Ni(g) = {j G N \ {i} : (i,j) G g}.

Example 1. Consider a four player case. Let N = {1,2, 3,4} and players choose the following behaviors: gi = (0, 0,0,1), g2 = (0,0,1,0), g3 = (0,1, 0,1), gA = (1,0, 0,0). The network g consists of five links g = {(1, 4), (2, 3), (3, 2), (3, 4), (4, 1)}, whereas its closure gg consists of three undirected links g = {(1,4), (2, 3), (3,4)} (see figure, a and b). Note that, for instance, N4(g) = {1} while N4(g) = {1, 3}.

1.2. Second stage: controls. To allow players to break formed links at the first stage (we introduce this possibility to punish neighbors from the complementary coalition in the case of zero-sum game which can appear at coalition formation stage), we define an n-dimensional profile di (g) as follows:

1, if player i offers a link to player j G Mi: 0, otherwise,

(1)

subject to the constraint:

(2)

jeN

L, if player i keeps the link formed at the first stage dij (g) = { with player j e Ni (g) in network g, (3)

0, otherwise.

Networks for examples 1, 2 a - network g; b - network g; c - network gd; d - network gd.

Denote all profiles di (g) satisfying (3) by Di (g), i e N .At the second stage players simultaneously and independently choose di(g), i e N, thus the profile (di(g),...,dn(g)) changes network g forming a new network which is denoted by gd.

Example 2. Suppose that players choose their profiles gi, i e N, as in example 1 forming the network g = {(1,4), (2, 3), (3, 2), (3,4), (4,1)}. Let di(g) = (0,0,0,1), d2(g) = (0, 0, 0, 0), d3(g) = (0, 0, 0,1), d4(g) = (1, 0, 0, 0), i.e. Player 1 keeps the link with Player 4, Player 2 breaks the link with Player 3, Player 3 breaks the link with Player 2, and Player 4 keeps the link with Player 1. Then we have a new network gd = {(1, 4), (3, 4), (4,1)}. The closure gd consists of two undirected links gd = {(1, 4), (3, 4)} (see figure, c and d).

Also at the second stage player i e N chooses control ui from a given set Ui. Then behavior of player i e N at the second stage is a pair (di (g), ui).

Payoff function Ki of player i depends on network gd, his control ui and controls Uj, j e Ni (gd) of his neighbors in the closure gd. More formally,

Ki (Ui ,uN. (gd)) : Ui x

jENi (gd)

Uj ^ R, i e N,

is a real-valued function where notation uNi (gd) means the profile of chosen controls uj of all player j e N (gd) in network gd.

2. Cooperation in one-way flow two-stage network games. In this section we study the cooperative case: we allow players to coordinate their actions and choose behaviors jointly. Players being rational, choose their behaviors gi e Gi, (di(gd),ui) e

Di(g) x Ui, i e N, to maximize the joint payoff, the value:

Ki(ui ,uNi(3d)). (4)

ieN

It can be easily seen that to maximize the total sum (4) of players' payoffs (supposing that maximum in (4) exists), it is sufficient to form the network at the first stage without changing it at the second stage, i.e. di(g) = gi, for all i e N and

max y^ Ki(ui,UNi(gd)) = max "S^ Ki(ui,UNi(q)).

(9i,di (g),ui )£Gi XDi(g)xUi, "i(il ) (giUi)eGi xUi, i(g)

ieN ieN ieN ieN

The profile (g*,u*), i e N, maximizing (4) we call the cooperative profile. Behavior profile (g*,..., gn) forms the network g* and

y2Ki(ui,UNi(r)) = ( mpax U Y^ Ki(ui,UNi(g)).

(gi ,ui) eGi X Ui , ieN ieN ieN

To allocate the maximal sum of players' payoffs according to some solution concept, one needs to construct a cooperative TU-game (N, V). The characteristic function V is defined in the sense of von Neumann and Morgenstern as:

V(N) = £ Ki(u*,u*Nig)),

ie N

V (S) = max min y^ Ki(ui,uN. (gd)),

(gi,di(g),ui )eGi-Di (g)-Ui, (gj,dj (g),uj )eGj -Dj (g)xUj, i(g >'

ieS jeN\s ieS

V(0) = 0,

where the network g is formed by profile (g*,..., gn) and the network gd is formed by profile (d*(g), ...,dri(g)).

Consider a non-empty coalition S c N. Denote a network, formed by profiles gi, i e N, s.t. gj = (0,...,0) for all j e N \ S, by gS. Let gS be the closure of gS. For any controls ui, i e S let controls Uj(uS), j e N \ S, where uS = {ui}, i e S, solve the following optimization problem:

y^Ki (ui,uNi(gs )nS, u(N\S)nNi(gs )(uS ))

ieS

mi^^^ 1 J2Ki (ui,uNi(gs)nS, u(N\S)nNi(gs)) .

uj ,je(N\S)nNi(gs

ie S

Here uNi(gS )nS is the profile of controls chosen by all neighbors of player i from coalition S in the network gS, and u(N\S)nNi(gS)(uS) is a profile of controls chosen by all players from coalition N \ S who are neighbors of player i in the network gS.

Proposition 1. Suppose that functions Ki, i e N, are non-negative and satisfy the following property: for any two networks g and g' s.t. g' C g, controls (ui,uNi(g)) e Ui x rijeNi(g) Uj and player i, the inequality Ki(ui,uNi(g)) > Ki(ui,uNi(g,)) holds. Then for all S c N we have

V(S) = (gij„maGxxUij Ki (ui,uNi(gs)nS,u(N\S)nNi(gs)(uS^ . iiesi ieS

Proof. Consider the maxmin value for coalition S C N:

V (S ) = max min > Ki(ui,uN. (gd)).

(9i,di(g),ui )tGiXDi(g)xUi, (gj:dj (g),uj )eGj X Dj (g)XUj, i(g> >'

ies \s ieS

Since the presence of a link (j, i) e g, i e S, j e N \ S, increases payoff of coalition S according to the property formulated in the statement of Proposition 1, therefore, player j e N \ S, as a neighbor of i, changes his component dji(g) from 1 to 0 in the profile dj(g), i.e. removes link (j, i) to minimize the payoff of coalition S. Thus, to minimize the value ieS Ki(ui, uNig)) players from N\S remove all links with players from S and use controls uj(uS), j e N \ S. Note that it is not important for coalition S how players from its complement N \ S are connected to each other. Therefore, without loss of generality assuming that dj (g) = (0,..., 0) for all j e N \ S, we obtain

V (S) = £ Ki (ui,uNi(gs )nS ,u(N\S)nNi(gs )(uS ^ .

^ i i ies i ies

To maximize the sum ^ieS Ki(ui,uNi(gs)nS,-U(N\s)nNi(gS)(us)), it is sufficient for players from coalition S to form the network at the first stage without changing it at the second stage, i.e. di(g) = gi, for all i e S. Then we get

V (S) = (g.„ImaXxU. Ki(ui,uNi(gs )ns ,u(N\S)nNi(gs )(uS )) iiesi ieS

which proves the statement. □

An imputation in the cooperative two-stage network game is a profile £(V) = (£i(V),..., £n (V)) s.t. Zi£N & (V) = V (N) and & (V) > V ({i}) for all i e N .We denote the set of all imputations in the game (N,V) by I(V). A solution concept (or simply solution) of TU-game (N, V) is a rule that uniquely assigns a subset of I(V) to the game (N,V). For example, if the solution concept is the Shapley value $(V) = (^i(V),..., $n(V)), its components can be calculated as

SCN,ieS 1

3. Time-consistency problem. In this section we study time-consistency of the Shapley value $(V). We already found behavior profiles (g*,u*), i e N, of players which maximize the sum (4) allowing players to get the value V(N). Allocating V(N) according to the Shapley value, we obtain the solution $(V) = (V),..., $n(V)). In other words, in the cooperative two-stage network game player i e N should receive the amount of (V) as his payoff.

After the first stage (after forming network g*) players may recalculate the solution according to the same solution concept. To find the new, recalculated solution, one needs to consider the subgame (one-stage game) starting from the second stage, provided that players chose behavior profile (g*,..., gn) at the first stage, and therefore formed network g*. Consider this subgame. The characteristic function for the subgame will depend on a parameter - the network g* - formed at the first stage, and we denote this function as v(g*,S) to stress the dependence on the network. The characteristic function v(g*,S) in the sense of von Neumann and Morgenstern is defined as follows:

v(g*,N) = £ Ki(u*,uNi(g.)) = V(N),

ieN

v(g ,S) = max mm > Ki(uí,un.(gd*}),

(di(g* ),ui)£Di (g*)XUi: (dj (g* )€Dj (g* )xUj, i(g >'

ies \s iES

v(g* , 0) = 0.

The following proposition can be proved similarly to Proposition 1.

Proposition 2. If functions Ki, i G N, are non-negative and satisfy the property stated in Proposition 1, the value v(g*, S) can be calculated by formula

v(g*,S)= max Ki(ui,uNi(3* )ns, ü{N\s)nNi{g* )(us)),

^es"' ies

where Uj(us), j G N \ S, solve the following optimization problem:

J2Ki (Ui,UNi(g* )nS,U(N\S)nNi(g* )(uS^ =

ies

AsSvi N-(-* } ^ Ki (Ui, UNÁrs )nS, U(N\S)nNi(g **})

Uj ,je(N\S)nNi(g*s ) ,es

and gS is the closure of network gS, formed by profiles g*, i G N, s.t. g* = (0,..., 0) for all j gN \ S.

In the subgame, an imputation £(g*,v) = (£i(g*,v),...,£n(g*,v)) satisfies two conditions: J2ieN &(g*,v) = v(g*,N) and &(g*,v) > v(g*, {i}), i G N. Recalculate players' payoffs in the subgame using the same solution concept - the Shapley value &(g*,v) = (&1(g*,v),..., &n(g*,v)), where its components can be computed as

9i{g ,V)= -HyTj-H9 ,S) -v{g ,S\ {«})J

scN,ies 1

for all i G N .

Definition 1. The Shapley value &(V) is time-consistent if:

h(V) = &(g*,v), i G N. (5)

The equality (5) means that if we use the imputation £(V) = &(V) at the first stage, and then at the second stage recalculate players' payoffs according to the same solution concept £(g*,v) = &(g*,v), i.e. calculate a new imputation £(g*,v) = &(g*,v), subject to formed network g , players' payoffs prescribed by this imputation will not change. Since in most games the condition (5) is not satisfied (in our setting characteristic functions V(S) and v(g*,S) are different), the time-consistency problem arises: player i G N, who initially expected his payoff to be equal to (V), can receive different payoff (g*,v). To avoid such situation in the game, we propose a stage payments mechanism-imputation distribution procedure [1] for the Shapley value &(V).

Definition 2. Imputation distribution procedure for &(V) in the cooperative two-stage network game is a matrix

( /311 3i2

3 =

\ Pul Pn2

s.t. i(V) = Pil + Pi2 for all i G N.

The value ¡3ik is a payment to player i at stage k = 1,2. Therefore, the following payment scheme is applied: player i e N at the first stage of the game receives the payment ¡¡i1, at the second stage of the game he receives the payment ¡¡i2, thus his total payment received at both stages 3i1 + 3i2, is equal to the component of the Shapley value &i (V), which he initially wanted to get in the game as the payoff.

Definition 3. Imputation distribution procedure 3 for the Shapley value &(V) is time-consistent if for all i e N

MV) - 3ii = Mg*,v).

It is obvious that time-consistent imputation distribution procedure for &(V) in the cooperative two-stage network game can be defined as follows:

3ii = MV) - Mg*,v), 3i2 = (g*,v), i e N. (6)

4. Numerical example. To illustrate the theoretical results obtained in the previous sections, consider a three-person game as an example. Let N = {1,2, 3} be the set of players. We suppose that Player 1 can establish a link with Player 3, Player 2 can establish links with Player 1 and Player 3, and, finally, Player 3 can establish links with Player 1 and Player 2. Therefore, we have: M1 = {3}, M2 = {1, 3}, M3 = {1, 2}. Moreover, we suppose that each player can offer a limited number of links: Player 1 and Player 3 can offer only one link, while Player 2 can offer two links that is a1 = a3 = 1, a2 = 2. Thus, at the first stage the sets of behaviors of players are: G1 = {(0,0,0); (0,0,1)}, C2 = {(0,0, 0); (1, 0,0); (0,0,1);(1, 0,1)}; G3 = {(0, 0,0); (1,0, 0);(0,1,0)}.

Let ui be behavior of player i e N at the second stage, and Ui = [0, <x>) for all i e N. The payoff function of player i depends on players connected with player i as well as on players with whom player i established links and have the following form [8, 9] (the expression of the payoff function is justified in above mentioned papers and is used in network models of public goods):

Ki(g,u) =ln 11 +Wj +

- CjUj - k\Nj(g)\,

jeNi(g)

u

3

where parameters c1 = 0.2, c2 = 0.25, c3 = 0.4, k = 0.75, and network g is formed by the profile (g1,g2,g3), and u = (u1,u2,u3). To find cooperative behavior, one needs to maximize the total payoff, i.e. to solve the optimization problem:

max

(gi,ui)eOiXUi i=l ,2,3

Y,Kj(g,

u) = max

(gi,ui)eOiXUi,

i=1,2,3 j=1

E

ln I 1 + Uj + Y^

- Cj Uj - k\Nj (g)\

jeNi(g)

From table we conclude that V(N) = 3.8242 which is attained at two different profiles:

U

3

g* = (0,0,1), g2 = (1,0,0), g3 = (0,0,0),

14, 0, 0,

and

g*

g2 gt

(0, 0,0), u*

(1,0,0), u2

(1, 0,0), ut

14, 0, 0,

which form networks {(1, 3), (2,1)} and {(2,1), (3,1)} at the first stage respectively.

By the definition of characteristic function V(S), find its values for all S C N. For i G N we have

max min Ki(g,u)

(gi ,di(g),ui )eGiXDi(g)xUi (a j ,dj (g),u j)eG j xD j (g)xU j ,

j = i

max TT Ki(g,u)\g,=0,uj=0, j=i =

(gi ,ui)eGixUi

max [ln (1 + ui) — ciui].

uieUi

Total payoffs of players

Network g

3

E Ki

i= 1

Network g E Ki

_i=1

0 1.7620 {(1,3)} 2.6915 {(1, 3), (2, l)} 3.8242 {(1, 3), (2, l), (2, 3)} 3.0742 {(1, 3), (2, l), (2, 3), (3, l)} 2.3242 {(l, 3), (2, 1), (2, 3), (3, 2)} 2.3242 {(1, 3), (2, l), (3, l)} 3.0742 {(l, 3), (2, l), (3, 2)} 3.0742 {(1, 3), (2, 3)} 2.2870 {(1, 3), (2, 3), (3, l)} 1.5370 {(l, 3), (2, 3), (3, 2)} 1.5370 _{(1,3), (3, 1)} 1.9415

{(1, 3),(3,2 {(2, 1 {(2, 1),(2,3

{(2, 1), (2, 3),(3, 1 {(2, l), (2, 3),(3,2 {(2, 1),(3, 1 {(2, 1),(3,2 {(2, 3 {(2, 3),(3, 1 {(2, 3),(3,2 {(3, 1

_{(3,2

2.2870 2.3715 3.2047 3.0742 2.4547 3.8242 3.2047 2.4683 2.2870 1.7183 2.6915 2.4683

For all i, j e {1, 2, 3}, such that either i e Mj or j e Mi, m = N \ {i,j}

V ({i,j })

max min \Ki(g,u) + Kj (g, u)]

(gi,di(g),ui )EGj XDi(g)xUi (gm ,dm(g) ,Um)EGm X Dm(g) X Um gj,dj (g),uj )EGj XDj (g)x.Uj

max [Ki(g, u) + Kj(g,u)]

(gi,di(g)ui )EGj x Dj(g) x Ui ; 'U

(gj,dj (g),uj )EGj X Dj (g)xUj

gm = 0,um = 0

= max{ max [ln (1 + ui ) — ciui + ln (1 + Uj ) — Cj Uj ]

uj EU.

max [2 ln (1 + ui + Uj ) — ciui — Cj Uj — fc]}

UiEU, u jEu.

= max{V ({«}) + V ({j }); max [2ln(1 + ui + uj ) — ciui — Cj uj — fc]}.

Thus after solving the corresponding maximization problems, we obtain values of characteristic function V(S):

S {1,2,3} {1,2} {1,3} {2,3} {1} {2} {3} V(S) 3.8242 2.0552 2.0552 1.6589 0.8094 0.6363 0.3163

The Shapley value $(V) = ($1 (V),$2(V(V)), calculated for characteristic function V(S), is

$(V) = (1.5179, 1.2331, 1.0731), (7)

i.e. choosing behaviors jointly at both stages, players get the total payoff of 3.8242 and allocating the amount according to the Shapley value at the end of the game, each player gets $i (V), i = 1,2, 3, as his payoff in the game.

To show that the Shapley value $(V) is time-inconsistent, consider the subgame of the two-stage game, starting from the second stage, provided that players chose the cooperative behaviors at the first stage. Select the cooperative profile g\ = (0,0,1), g2 = (1,0,0),

ujEUj

g3 = (0,0,0), and u* = (u1 ,u2,u3) = (14,0,0). The cooperative behaviors at the first stage (g*,g\,g3) form the network g* = {(1, 3), (2,1)}. To prove that the Shapley value ) is time-inconsistent it is sufficient to compute the Shapley value $(g*,v) and show that $(V) = $(g*,v). For this purpose calculate characteristic function v(g*,S) in the subgame for all S c N. Note that v(g*,S) = V(S) = 3.8242. For i e {1,2, 3}, we have

v(q*,\i\) = max min KAq* ,u) =

(di(g* ),ui)eDi(g*)xUi (dj (g* )uj )£Dj (g* )xUj, = (di(g* )^(g*),U, ^^ (g* )=0'Uj =

= max [ln (1 + ui) — CiUi] = V({i}).

Ui^Ui

For all i, j e {1, 2, 3}, m = N \ {i,j} we get: v(g*Ai,^) = max min [Ki (g*,u) + Kn (g*,u)] =

Kg,X,jIJ (di(g*)Ui )€Di (g* )xUi (dm(g*),um)eDm(g* )xUml A q, j q, J'

(dj (g* ) ) eDj (g*)xUj

max [K,-(g*,u) + K^(g *,u)], , ^ „ „ =

H)eni(g* )X.Uii iy ' 3V >nm(g*) = 0,um = 0

j (g*)-Uj

max [ln (1 + ui) — ciui + ln (1 + uj) — Cjuj)], {i,j} = {2, 3},

Uj

max{ max [ln (1 + ui) — ciui + ln (1 + uj) — Cj uj ]

" uieU,

Uj

= Uj

max [2 ln (1 + ui + uj) Ci i~u i Cj j~u j — fc]}, otherwise.

Thus after solving the corresponding maximization problems, we obtain values of characteristic function v(g*,S), g* = {(1, 3), (2,1)}:

S {1,2,3} {1,2} {1,3} {2,3} {1} {2} {3}

v(g*,S) 3.8242 2.0552 2.0552 0.9526 0.8094 0.6363 0.3163

Using the values v(g*,S), the Shapley value ^(g*,v) = (^i(g *,v), $2(g*,v), ^3(g*,v)) is computed:

4>(g*,v) = (1.7533, 1.1154, 0.9554),

and from (7) we conclude that $(V) = g*,v). This shows time-inconsistency of the Shapley value ^>(V). Time-consistent imputation distribution procedure 3 of the Shapley value $(V) can be computed by formulas (6):

3*2 \ ( —0.2354 1.7533 3 = I 321 322 I = ( 0.1177 1.1154 | . (8)

331 332 J V 0.1177 0.9554

Similarly, it can be seen that the Shapley value $(V) is time-inconsistent also for the second cooperative behavior profile: g* = (0,0, 0), g\ = (1,0, 0), g3 = (1, 0,0), and u* = (u*,u2,u3) = (14,0,0). The cooperative behaviors at the first stage (g*,g\,g3) form the network g* = {(2,1), (3,1)}. One can show that the characteristic function v(g*,S) in the subgame, calculated for the given network g = {(2, 1), (3, 1)}, coincides with the characteristic function v(g, S) calculated for network g = {(1, 3), (2,1)}. Therefore, given

Uj eUj

the network g* = {(2,1), (3,1)}, we have time-inconsistency of the Shapley value ф(У), and time-consistent imputation procedure в for the Shapley value ф(У) will be the same as in (8).

5. Conclusion. Two-stage games with network formation at the first stage are considered. One of the main assumptions is that the payoff of any player depends only on his behavior and behavior of his neighbors in the network. In contrast to our previous results, the present research deals with directed network that influenced the construction of characteristic function of the game. It is shown that the Shapley value - the proposed cooperative solution of the two-stage game - is time-inconsistent, but with the use of a newly introduced payment mechanism - imputation distribution procedure - one can guarantee the realization of such solution in the game.

References

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4. Petrosyan L. A., Sedakov A. A., Bochkarev A. O. Dvukhstupenchatye setevye igry (Two-stage network games). Matematicheskaya teoriya igr i ee prilozheniya, 2013, vol. 5, no. 4, pp. 84-104.

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Литература

1. Петросян Л. А., Данилов Н. Н. Устойчивость решений неантагонистическикх дифференциальных игр с трансферабельными выигрышами // Вестн. Ленингр. ун-та. Сер. 1: Математика, механика, астрономия. 1979. № 1. С. 52-59.

2. Goyal S., Vega-Redondo F. Network-formation and Social Coordination // Games and Economic Behavior. 2005. Vol. 50. P. 178-207.

3. Jackson M., Watts A. On the formation of interaction networks in social coordination games // Games and Economic Behavior. 2002. Vol. 41, N 2. P. 265-291.

4. Петросян Л. А., Седаков А. А., Бочкарев А. О. Двухступенчатые сетевые игры // Математическая теория игр и ее приложения. 2013. Т. 5, № 4. С. 84-104.

5. Bala V., Goyal S. A. Noncooperative Model of Network Formation. Econometrica,, 2000, vol. 68, no. 5. P. 1181-1229.

6. Shapley L. S. A Value for га-Person Games. Contributions to the Theory of Games II. Princeton: Princeton University Press, 1953. P. 307-317.

7. Gao H., Petrosyan L., Sedakov A. Strongly Time-consistent solutions for two-stage network games // Procedia Computer Science. 2014. Vol. 31. P. 255-264.

8. Bramoullé Y., Kranton R. Public goods in networks // Journal of Economic Theory. 2007. Vol. 135, N 1. P. 478-494.

9. Galeotti A., Goyal S. The Law of the Few // American Economic Review. 2010. Vol. 100, N 4. P. 1468-1492.

The article is received by the editorial office on June 26, 2014.

Статья поступила в редакцию 26 июня 2014 г.