DYNAMIC NASH BARGAINING SOLUTION FOR TWO-STAGE NETWORK GAMES Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, XI, 66-72

Dynamic Nash Bargaining Solution for Two-stage Network

Games

St. Petersburg State University 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: [email protected]

Abstract In this paper, two-stage network games are studied. At first stage of the game players form a network, while at second stage they choose strategies according to the network realized at the first stage. However, there are two kinds of two-stage networks. The first is a special class of two-stage network games when players have the opportunity to revised their network which they formed before. And the second is classical two-stage network. Cooperative setting is considered. In the cooperative case, we use Nash Bargaining Solution as a solution concept. It is demonstrated that the Nash Bargaining Solution satisfies the time consistency property for the special class of two-stage network game. But its not true for a classical two-stage network game.

Keywords: network, time-consistency, Nash Bargaining solution.

1. Model

Consider the model in details. Let N = {1 ,...,n} be a finite set of players who can interact with each other. The interaction between two players means the existence of a link connecting them and,therefore,communication between them. On the contrary, the absence of a link connecting players means the absence of any communication between them. Under these assumptions cooperation of players is said to be restricted by a communication structure (or a network). A pair (N, g) is called a network, where N is a set of nodes (it coincides with the set of players), and g € N x N is a set of links. If a pair (i, j) e g, there is a link connecting players md j, and, therefore, generating communication of players in network. Below to simplify notations, the network will be identified with a set of its links and denoted by g, and a link (i, j) in the network will be denoted by ij. All links are non-directed, so ij = ji. The two stage network game under consideration we denote as G.

2. First stage: network formation

Having the player set N given, define the link formation rule in a standard way: links are formed as a result of players' simultaneous choices. Let Mj C N/ {i} be the set of players whom player i e N can offer a mutual link, and a e {0,..., n — 1}

i

offer). Behavior of player i e N at the first stage is an n-dimensional profile g» = (gii,..., gjn) whose entries are defined as:

Jie Junnan

1, if player i offer a link to j G M¿

0, otherwise

subject to the constraint:

J2gij - ai • (2)

jew

The condition gij = 0, i G N excludes loops from the network, whereas (2) shows that the number of possible links is limited. If Mi = N/ {i}, player i can offer a link to any player, whereas if ai = n — 1, he can maintain any number of links. A set of all possible behaviors of player i G N at the first stage satisfying (1), (2) is denoted by Gj. The Cartesian product nieN Gi is the set of behavior profiles at the first stage. It is supposed that players choose their behaviors at the first stage simultaneously and independently of each other. In particular, player i G N choose gi G G.j, and as a result the behavior profile gi = (gii,..., gin) is formed. Under the above assumptions, an undirected link ij = ji is established in network g if and only if gij = g^, g consists of mutual links which were offered only by both players.

3. Second stage: choosing controls

Denote the game on the second stage over the network g, as r (g). Having formed the g

player i in the network g as elements of set Ni(g) = {j G N \ {i} : ij G g}. Players are allowed to reconsider their decisions made at the first stage by giving them

n

dimensional profile di(g) as follows:

di 11, player i doesn't break the link formed with player j G N (g). ij l0, otherwise

Elements di(g) satisfying (2), (3) are denoted by Di(g), i G N. It is obvious that profile (di(g)... dn(g)) applied to the net work g changes its structure and forms a new network,denoted by gd. Network gd is obtained from g by removing links ij such that either dij (g) = 0 or dji (g) = 0.

Moreover, at the second stage player i G N looses control ui form a finite set U^ Then, behavior of player i G N at the second stage is a pair (di (g), ui): it defines, on the one hand, links to be removed (di (g), and, on the other hand, control ui.

A payoff function Ki of player i G N depends on both new network gd and controls ui;i G N. Specifically, it depends on player i's behavior at the second stage as well as behavior of his neighbors in the network gd. i.e., Ki (ui;uNi(gd)) is a nonnegative real-valued function defined on Ui x rijeNi(gd) j Here uNi(gd) denotes a profile of controls u3- chosen by all neighbors j G Ni(gd) of players i in the network gd. Assume that fun étions Ki; i G N, satisfy the following property:

(P): For any two networks ^d g' s.t. g' Ç g, controls (ui,uNi(gd)) G Ui x rijeNi(gd) Uj, and players i, the inequality |Ni (g) | > |Ni (g') | implies the inequality Ki (ui,uN.(g^ > Ki (ui,uN.(g')). Also we suppose that the payoff of an isolated player is equal to 0.

3.1. Introduce the definition of Nash Bargaining solution

Let K be the set of all possible payoffs in the game. Denote vi the lower value of the zero-sum game between player i and player N/i, with the payoff of player i equal

to Kj. Consider the following expression:

max (K - vi )... (K„ - v„) =

= (Ki - vi)(i?2 - v2) ... (K„ - v„)

Vector K = (ifi, K2,..., K„) is called Nash Ba^aming solution. Suppose, we use Nash Bargaining solution K = (ifi, K2,..., K„) in two stage game, this leads us to network g, which is formed on the first stage, and subgame r (g) on the second stage. Pair (g, r (g)) we shall call Nash Bargaining trajectory.

Proposition 1. Nash Bargaining solution is time-consistent in G (two-stage game), if Nash Bargaining solution computed for game G coincides with Nash Bargaining solution computed for subgame r (g). Nash Bargaining solution is time-consistent G

Proof. Consider the Nash Bargaining solution in two-stage game.Because players from the set N \ i have the possibility not to form links with player i, the lower value of zero sum game vj will be equal to 0, since player i can be isolated. K = (Ki, K2,..., K„) is the Nash Bargaining solution in two-stage game, and we have

max (Ki - vi )... (Kn - v«

Ki>vi,Ki>Vi,...,Kn>vn ;(Ki,K2,...,K„)eK.

= (KTi - vi) (K - v2) ... (Kn - v„) =H Ki

K

Consider the Nash Bargaining solution on the second stage.

max (Ki' - vi')... (K«' - v«

Ki'>vi',...,K„'>v„';(Ki',K2',...,K„')eK'.

= (KTi' - vi') (KÎ2' - v2')... (Kg«' - v„') ^ KK

'

Where K' c K is the set of all possible payoffs on the second stage. Where K' c K is the set of all possible payoffs on the second stage. The value of zero sum game v/ still will be equal to v = 0 since player i can be isolated (because

players can break links on the second stage). We have that K = (Kgi, K2,..., K„)

()

trajectory of the cooperative game, K will remain in the subs et K ' (K G K '). Hence, Nash Bargaining (olution on the second stage K' = (KV, K2',..., Kn') will always be equal to K = (Kgi, Kg2,..., Kn).

Therefore, Nash Bargaining solution is time-consistent in this model.

4. Example

In this section, we consider a three-person game as an illustration, i.e., the set of players N = {1, 2,3}. Assume that player 1 can maintain 2 links and players 2,

3 can only maintain 1 link. Moreover player 3 can offer a link only to player 1. Under these restrictions, we have: subsets of players to whom each player can offer links are M1 = {2, 3} , M2 = {1, 3} , M3 = {1}, a number of links each player can maintain: a 1 = 2, a2 = a3 = 1. Therefore, at the first stage sets of players' behaviors are: G i = {(0, 0,0); (0,0,1); (0,1, 0); (0,1,1)} G2 = {(0, 0,0); (1,0, 0); (0,0,1)}, G3 = {(0,0,0); (1,0,0)}, and only four networks can be formed at the first stage of the game: the empty network (the network without links, g = 0), g = {1, 2}, g = {1, 3} g = {1, 2,3}. Suppose that sets of controls Ui at the second stage for any network g realized at the first stage are the same U = U2 = U3 = {A, B}, and payoff functions are defined as: Ki(ui) K^A^O, Ki(B)=0; i = 1, 2,3

k12

(u 1,

U2)

K1 '2(A, A) = 2; kJ'2(A, B ) = 4; KÍ'2(B,A) = 1;

KÍ'2(B,B) = 3;

KjL'3(M1, u3) : K1'3(A, A) = 3; K1'3(A,B) = 5; K1'3(B, A) = 1;

K13(B,B)

3

K21,2(W2,W1) :

K2'2(A,A) = 2;

k12(A,B ) = 4;

k1'2(B,A) = 1;

K1'2(B,B)

3;

K3'3

(u3

U1)

K3'3(A,A) = 2; K3'3(A,B ) = 5; K¿'3(B, A) = 1;

K13(B,B)=4;

KÍ'2;1'3(u1,U2;u1,U3) : KÍ'2;1'3(A, A, A) = 6; KÍ'2;1'3(A, A, B) = 7;

K1'2;1'3(A,B,A) = 5; K1'2;1'3(A,B,B ) = 2; K1'2;1'3(B, A, A) = 4; K1'2;1'3(B,A,B ) = 6; K1'2;1'3(B,B, A) = 1; k1'2;1'3(B, B, B )=9 Consider the network ^{1, 2}:

(2, 2,0) (4,1,0) \ & ( (2, 2, 0) (4,1, 0) (1, 4,0) (3, 3,0W& I (1,4, 0) (3, 3, 0)

(4)

Here player 1 chooses the rows of the matrix (the first row corresponds to the choice of the strategy A and the second of B), player 2 choose the columns of the matrix (the first column corresponds to the choice of the strategy A and the second to B), and player 3 chooses one of the matrices (the first matrix corresponds to the choice of the strategy A and the second to B). In the described game,the Nash bargaining solution gives the payoffs K1(B, B) = K2(B, B) = 3, K3(A) = K3(B) = 0

Strategy profiles are (d1(g),u[) = ((0,1,0),B), (d2(g),«2) = ((1,0,0),B)

Consider the network g={1,3}:

(3,0, 2) (3,0, 2) \ „ /(5,0,1) (5, 0,1)

(1,0, 5) (1,0, 5),/~ V(3,0, 4)(3, 0,4); (5)

In the described game, the Nash bargaining solution gives the payoffs K (B, B) = 3,Ks(B,B) =4,K2(A) = K2(B) =0,

Strategy profiles are (di (g),«2) = ((0,0,1),B), (d£(g),u|) = ((1,0,0),B). Consider the network g={1,2, 3}:

(6, 2, 2) (5, 2, 2)N& f(7, 2,1) (2,1,1) (4,4, 5) (1, 3, 5) y 1(6,4, 4) (9, 3,4)

(6)

In the described game, the Nash bargaining solution gives the payoffs Ki(B, B, B) = 9, K2(B, B, B) = 3, Ks(B, B, B) = 4,

Strategy profiles are (di (g), u|) = ((0,1,1), B), (d2(g), «2) = ((1,0,0), B) (d3(g),u|) = ((0,0,1), B).

Now, consider two stage game, the Nash bargaining solution gives the payoffs K1(B, B, B) = 9, K2(B, B, B) = 3, K3(B, B, B) = 4, and strategy profiles are (g 2, d 2 (g),u 2) = ((0,1,1), (0,1,1), B), (g2 , d2 (g),« 2 ) = ((1,0,0), (1,0,0), B),

(g2, (g),«3) = ((0,0,1), (0,0,1),B). In the subgame starting from the second stage, after realized Nash Bargaining solution computed for two stage game on the

(9, 3, 4)

(9, 3, 4)

solution in the game starting from the first stage. We see that the Nash bargaining solution is a time-consistent solution concept.

5. The classical two-stage network game

The first stage, is as in previous case. However, at the second stage, we do not give the players opportunity to revise the network. So they just choose control at the second stage. Player i e N chooses only controls « form a finite set Uj.

A payoff function K of player i e N depends on both network g and controls «i, i e N. Specifically, it depends on player i's behavior at the second stage as well as behavior of his neighbors in the network g. i.e., K (ui,uNi(s)) is a nonnegative real-valued function defined on U x njeNi(s) Uj. Here denotes a profile of controls «j chosen by all neighbors j e Nj(g) of players i in the network g.

Proposition 2. Nash Bargaining solution is time-inconsistent in G (two-stage game)

Proof. Consider the Nash Bargaining solution in two-stage game.

Because players from the set N \ i have the possibility not to form links with player i. The lower value of zero sum game vi will be equal to 0, since player i can be isolated. K = (if1, K2,..., K„) is the Nash Bargaining solution in two-stage game, and we have

max (Ki - vi)... (Kn - v„) =

Ki,Ki,...,Kn ;Ki>vi,Ki>Vi,...,Kn>Vn ;(Ki,K2,...,K„)eK.

rK?i - viC rK?2 - V2C... rK - v„c = n Ki.

Here K is the set of all possible payoffs in two stage game. Consider the Nash Bargaining solution on the second stage.

max (Ki' — vi ')•

n

• (Ki' — Vi')... (Kn' — vn' )= (Kfi' — viO (K2' — V2 0 ...(Kin' — VnO = £[ K',

i=i

Where K' c K is the set of all possible payoffs on the second stage. The value of zero sum game v/ will be v| = maxminKi'(ui; U{Ni(gd)}), We have that K =

(ifi, if• • •, Kn) is the Nash Bargaining solution in two-stage game where vi will be equal to 0. Also along the Nash Bargaining trajectory of the cooperative game, K will remain in the oubset K' (K € K'}. So, obviously, K = (K^, K2,..., Kn) is not coincides with (Ky, K2',..., KT^') Therefore, Nash Bargaining solution is time-inconsistent in this model.

5.1. Example

By using the same example, we can get the following: Consider the network {1, 2}:

(2, 2,0) (4,1,0) ' & ((2, 2, 0) (4,1, 0) ' (1, 4,0) (3, 3,0)j&^(1,4, 0) (3, 3, 0)J (i)

Here player 1 chooses the rows of the matrix (the first row corresponds to the choice of the strategy A and the second of B), player 2 choose the columns of the matrix (the first column corresponds to the choice of the strategy A and the second of B), and player 3 chooses one of the matrices (the first matrix corresponds to the choice of the strategy A and the second of B). In the described game,the Nash bargaining solution gives the payoffs Ki(B, B) = K2(B, B) = 3, K3(A) = K3(B) = 0.

Consider the network {1, 3}:

' (3, 0, 2) (3, 0, 2) ' „ ( (5,0,1) (5,0,1)'

(1, 0, 5) (1, 0, 5);& ^(3,0, 4) (3,0, 4)/ (8)

In the described game, the Nash bargaining solution gives the payoffs Ki(B, B) 3, K3(B, B) = 4, K2(A) = K2(B) = 0. Consider the network {1, 2,3}:

' (6, 2, 2) (5, 2, 2) ' „ ((7, 2,1) (2,1,1)'

(4, 4, 5) (1, 3, ^(6,4, 4) (9, 3, 4)/ (9)

In the described game, the Nash bargaining solution gives the payoffs Ki(B, A, B) = 6, K2(B, A, B) = 4, K3(B, A, B ) = 4.

Now, consider two stage game, the Nash bargaining solution gives the payoffs Ki(B, B, B) = 9, K2(B, B, B) = 3, K3(B, B, B) = 4.

In the subgame starting from the second stage, after realized Nash Bargaining solution computed for two stage game on the first stage, we obtain the Nash bar-

(6, 4, 4)

(9, 3, 4)

We see that the Nash bargaining solution is not a time-consistent solution concept.

6. Conclusion

In the special class of two-stage network where players have the opportunity to revise their network which they formed before, Nash Bargaining solution is time-consistent. We also consider in the classical two-stage network game where players are just choosing controls at the second stage. Cooperative setting is considered. In the cooperative case, we use Nash Bargaining Solution as a solution concept. It is demonstrated that the Nash Bargaining Solution satisfies the time consistency property for this special class of two-stage network game. But it does not satisfy the time consistency property for the classical two-stage network game.

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