PGN-value for dynamic games with changing partial cooperation Текст научной статьи по специальности «Математика»

PGN-Value for Dynamic Games with Changing Partial Cooperation1

Hong-Wei Gao2, Ye-Ming Dai2, Qian Wang2

2 College of Mathematics, Qingdao University,

Qingdao,266071, P. R. China

Abstract. The game with partial cooperation with perfect information in extensive form is considered. The optimal solution PMS-vector in such a game has been proposed in [Petrosjan, 2000].

In our paper the characteristic functions are defined for each coalition S (S C N) according to some unified principle (for example, the best response to Nash equilibrium), but they are not necessarily supper additive.

A new principle of optimal behavior in such a game is established, based on the nucleolus as optimality principle for the allocation of coalitional payoff. On the first part of this paper, we have made an assumption that once the player announced that he would take cooperative behavior and never change this announcement, namely, he could not leave the coalition.

Based on this assumption, we construct algorithm for the solution of the game. And in the second part in this paper, we try to eliminate this limitation and, so, we construct a new method to achieve the goal. Algorithm of PGN-value of this kind of a game is offered and the optimal trajectory is found. The existence and uniqueness of nucleolus leads to the existence and uniqueness of the new solution.

Keywords: Game with changing partial cooperation, nucleolus, Nash equilibrium,

perfect information, PGN-value.

1 Project 70571040 supported by National Natural Science Foundation of China. Corresponding author: Hong-Wei Gao, Professor, College of Mathematics, Qingdao University, Qingdao, P. R. China, 266071. Fax: +86-532-85953532. E-mail: gaosai@public.qd.sd.cn

1. Dynamic games with partial cooperation and the monotonous increase of player’s coalition

1.1. Introduction

In recent years, the study on cooperative game has gone through the development from complete cooperation to partial cooperation. People have been already used to the so-called cooperative assumption in the research of the static or dynamic complete cooperative game, i.e. game rule agrees that players act to maximize payoff of the affiliated coalition. The player participates in the affiliated coalition throughout the game process, and it’s decided that a coalition of players should not change after being formed in the whole game process. In the game defined in literature [Petrosjan, 1998] and [Ayoshin, 1998] any player may proceed with cooperative activity from a given stage instead of in the whole game process. To be specific, before the game starts each player must, independent of other players, point out a particular stage, to cooperate with other players and participate in the coalition of players who are ready to cooperate, but keep individual rational behavior and don’t cooperate with any other player before this stage. Players are not permitted to alter the declared options in game process. Following the above rule, one player begins cooperation from the stage he has chosen in any case, no matter which concrete path is taken.

Literature [Petrosjan, 2000] weakens the conditions mentioned above. Different from the game defined in [Petrosjan, 1998] and [Ayoshin, 1998], the identical player might choose cooperation or non-cooperation respectively in different paths on the same stage according to [Petrosjan, 2000]. However, there’s an obvious limitation to the definition of partial cooperation, i.e. player can not still change the declared choice in the game process, which is displayed in the demand of the monotonous increase of players’ coalition. The existing optimal solution defined in the complete cooperative game is unable to get in the study of partial cooperative game based on non-cooperative games in finite extensive form. The limitation is obvious for [Petrosjan, 2000] to establish the optimal solution of game through defining PMS-vector, because the computation of PMS’-vector bases the value of characteristic function of complete cooperative game on the value of 2-person zero sum game in special coalition, then assign payoff of the coalition with the distribution principle of Shap-ley value. The limitation of the above-mentioned PMS-vector can only be overcome through changing the structure principle of characteristic function in the coalition, but the change of structure principle of characteristic function will influence the choice of the optimal rule of behavior directly, namely, if the characteristic function established fails to satisfy super additivity, the application of Shapley vector will be nonsensical (because it is not individually reasonable at this moment), so PMS’-vector loses the foundation of existence.

1.2. Notations and definition

Let r be a n-person non-cooperative game in finite extensive form with perfect information and without chance moves. Denote the set of players by N = {1,...,n}.

Let K(xo) be the game tree with the origin xo. According to the definition of a game in extensive form, on K(x0) there exists a partition P1,...,Pn, Pn+1 of the set of game tree nodes, where Pi(i G N) is the set of decision points of player i, and Pn+i is the set of endpoints. The payoff of player i is specified by terminal real-valued functions hi : Pn+i ^ R+, i G N.

Definition 1. fi : Pi ^ {0, 1} ,i G N, is called a cooperative function of player i, if for a given path {xo, ...,x', x", ..., x}, where x' G Pi and X G Pn+i, from fi(x") = 1 it follows that fi(y) = 1 for each y G Pi fl {x'',...,x}. Player i keeps cooperative behavior when fi(x) = 1 and plays individually when fi(x) = 0.

Definition 2. f = (fi, ..., fn) is called a cooperative function of the game.

Definition 3. According to cooperative function f = (fi, ..., fn), players can cooperate or play individually in the switched game process, identical player may choose coopemtion or non-cooperation in diffident paths of one stage of game process. The switched game is called a partial cooperative game rf (xo). Function f = (fi, . ..,fn) defines a special coalition structure on every node of the ga,me tree K(xo).

Definition 4. Let Ki = {K(xi),...,K(xq)} be a combination of non-intersecting subtrees of K(xo), with their origins x1 ,...,xq being in Pi. The combination Ki is called the cooperative region of player i, i.e., player i pledges himself to proceed his cooperative behavior on the decision points in Ki fl Pi. On the nodes in Pi\Ki player i plays individually. Suppose that f has been defined and after several moves the

game party came to a decision point x of player i. Assume that cooperative function of player i satisfies fi (x) = 1. Consider the set

Sj(x) = {j G N\1y G Pj f {xo, ...,x} : fj (y) = 1} (1)

Sj (x) consists of players who are ready to cooperate on x and the players who have

made a move to cooperate before x. According to the definition of the cooperative function, players in Si (x) will continue to cooperate on every node of the subtree K (x) with the initial node x.

Definition 5. A subtree K(x) is the trustiness region (TR) of player j if for every y G Pj f K(x),j G N, the cooperative function fj(y) = 1.

Hence,

S 2(x) = {j G N\Sj \K (x) is TR of player j} . (2)

S'2 (x) consists of the players who haven’t taken cooperative action in the path

{xo,... ,x} but are going to cooperate on K(x). Saying that player i proceeds the cooperative behavior on a node x G K(xo), we mean that on x player i acts in the interests of the coalition

Sf (x) = Sj U Sj. (3)

Players in Sf(x) are not permitted to leave the coalition once the coalition is determined. The rest of the players in N\Sf (x) are considered as individual ones on x.

Since Sf (x) is defined by the cooperative function f, the whole coalition structure

Sf (x), {j1} , {j2} ,...^j\N\S f (x)| } (4)

is specified by f as well. The player set Nf in r f (xo) consists of the subsets of the set N and are formed according to cooperative function f. The player set Nf is defined as follows. Take an arbitrary decision point x. Suppose that x G Pi. Introduce

i Sf(x), if/,(x) = 1 f (x)^ , if/,(x) = 0 . (5)

Consider the set

I(S) = {x G K(xo) \if (x) = S} , (6)

where S C N is independent of x, i.e., for all x G I(S) the sets if (x) coincide. The set (coalition) S C N will be considered as a player in game rf (xo) making decisions on the nodes x G I(S). The payoff of player S of rf (xo) is defined as the sum of payoffs of player i G S on the endpoints of K(xo):

hs(x) = ^2 hi(x), x G Pn+i, hi(x) > 0, i G N. (7)

iES

Obviously, player S in game rf (xo) may only consist of one player in game r, i.e., we could find decision point x G Pi, such that if (x) = {i}. It may also happen that the game rf (xo) will be a one-player game (the set Nf consists of only one player

N). This occurs in the case when if (x) = N for all x G Pi,i G N. In the most

complicated case the set Nf may consist of all subsets of the set N.

During the explanation we will often use the following notations. Assume that x is an arbitrary node. Let the set of immediate successors of x be Z(x). Denote the decision making player on x, x G Pi,i G N by i(x) G N. The decision of player i(x) on x leads to the node x G Z(x). Finally, the rule cf is determined by a cooperative function f = (fi,..., fn) if x is a decision point of player i, where

( ) i 1, if fi(x) = 1

cf(x) = 1 n -fff ^ n . (8)

[ 0, if fi (x) = 0

Suppose that the longest path of the tree K(xo) passes through T decision points. Introduce a partition of all nodes on T +1 sets Xo, Xi,...,Xt,..., XT and XT = {xo}, where Xt is composed of nodes which are reachable from xo after T—t sequential moves. Denote decision points belonging to Xt by xt,t = 1,..., T.

1.3. The construction and algorithm of the optimal path

Consider the game rf (xo). In this section we shall try to construct the solution concept for rf (xo) which will lead to the construction of the corresponding optimal path. The optimal path is determined by means of backward induction, moving from

the final nodes toward the initial one. The procedure is similar to the one used in the scheme of sub game-perfect Nash equilibrium construction, but the difference as follows is essential as well.

Let K(x) belong to a cooperation region of player i. Then, on the endpoints of K(x) we have to consider the payoffs of a coalition which includes player i instead of the payoffs of player i. By the Nash scheme the decisions of player i maximizing the payoff of coalition (in which he is included) can be easily determined with respect to K(x). However, since the player i’s payoff is not picked out from the coalition payoff, there occur difficulties on the decision points of player i between x and the root xo, where player i plays individually. Therefore, the definition of players’ payoffs corresponding to nodes where the individual behavior is replaced by the cooperative one is the main problem considered in the algorithm.

The initial stage. Consider the set Pn+i of endpoints. Since no player makes any move on Pn+i, the coalition structure on x and that on its immediate predecessor xi,x G Z(xi) are the same. On the node xi the given f specifies coalition structure Sf (xi), {ji} ,...,{j\N\sf (xi) }. The terminal payoffs hi(x),..., hri(x) in r specify the new payoff structure in rf (xo) in correspondence with the coalition structure on xi. Hence, the coalition Sf (xi) gets ^ hi(x) on x and an individual player

ieSf (xi)

jk ,k = 1,..., \N\Sf (xi)\, gets hjk (x) on x.

Stage 1. Shift back from the endpoints x to their predecessors xi. Consider an arbitrary taken xi. If cf (xi) = 1, player i(xi) cooperates on xi, from which it follows that i(xi) maximizes the payoff of the coalition Sf (xi) (thus he is playing in rf (xo) as player if (x) = Sf (xi), if (x) G Nf). We purpose him to select xi G Z (xi) from the condition

max \ hi(x) = } hi(xi). (9)

xEz(xi)

i*ESf (xi) i^Sf (xi)

In case cf (xi) = 0, player i(xi) maximizes the payoff of the coalition if (xi) consisting of himself only:

m&X hi(xi)(x) hi(xi )(x^i). (10)

xEZ(xi)

In the same way, we can construct trajectories starting from the arbitrary nodes on Xi. Therefore, instead of considering the terminal payoff function hi,i G N, on Pn+i, we may deal with payoff function rj : Xi ^ R+, i G N, on Xi such that

= < h«<^ i>- i GPn+>'. (11)

hi(xi), if xi G Pn+ i.

Stage t. Continue moving toward the tree root. Since the procedures on the further stages are the same, omitting explanation of every stage we deal with a stage t as an example of the general approach. Hence, suppose that we have reached a set of nodes Xt by continuing the moving on the game tree toward the origin xo. Let

r--payoffs obtained on the stage t — 1 for Xt-i. We don’t deal with the endpoints belonging to Xt f Pn+i. Find the decisions of players on the set of non-terminal nodes Xt\Pn+i. Let Y(xt) = Yi(xt) U Y2(xt), where

Yi(xt) = {x G Z(xt)\cf (xt) = 0 and i(xt) G Sf (x)}

Y2(xt) = {x G Z(xt)\cf (xt) = 1 and Sf (x)\Sf (xt) = 0}

If individually playing player i(xt) enters into multi-player coalition Sf (yt-i) on yt-i G Z(xt), or coalition Sf (xt) gets a new member i(xt) on yt-i, the coalition structure will be changed on yt-i. For each node xt G Xt we deal with two main cases.

1) Assume that Y(xt) = 0 for all xt G Xt\Pn+i. In this case, the functions rt-i

specify the payoff obtained at the end of the game for each player i(xt), i.e., if the decision of player i(xt) leads to node xt G Z(xt), then at the end of the game the

coalition Sf (xt) will get ^ rt—i(xt), and the payoff of individual player jk will

ieSf (xt)

be rtj-i(xt). Therefore, we can easily determine the nodes xt, where xt G Z(xt) and

xt G Xt\Pn+i.

If cf (xt) = 0, then xt has to satisfy

max rt-is(x) = rt-\(xt) (13)

xez(xt) i(x)y ’ i(xt)y tJ K >

Now assume that cf (xt) = 1. By the definition of the cooperative function, the coalition Sf (xt) is included in coalition Sf (xt-i) for each xt-i G Z(xt). Therefore, the coalitions Sf (xt) and Sf (xt-i) coincide since Y(xt) = 0. Then since player i(xt) belongs to the coalition Sf (xt) on xt, the node xt has to satisfy

max rt-\(x) = rt-\(xt). (14)

xeZ(xt) ^ i(xt)K ' ^ i(xt)K J

ieSf (xt) ieSf (xt)

2) Now, suppose that there exists xt such that the subset Y(xt) of nodes where the payoff of the coalition including player i(xt) is not defined by the functions rt-i, is not empty. When cf (xt) = 0 our procedure did not define the payoff of the player

i(xt). On the other hand, in the case of cf (xt) = 1, we have Sf (xt-i)\Sf (xt) = 0. Once Sf (xt) C Sf (xt-i), the payoff of the coalition Sf (xt) is included into the payoff of the coalition Sf (xt-i) and, thus, is not defined either.

To construct a path on K(xt), it is necessary to define some imputation of payoff of coalition Sf (yt-i) for each yt-i G Y(xt). We do it by considering an auxiliary cooperative game Gf (yt-i, Sf (yt-i)) on the subtree K(yt-i) with the set of players Sf (yt-i) and the characteristic function vf (yt-i,R), R C Sf (yt-i), for each yt-i G

Y(xt). The payoff of the grand coalition in Gf (yt-i, Sf (yt-i)) is defined as

vf (Vt-i,Sf (yt-i)) = ri 1 )•

i^Sf (yt-i)

The explanation of the cooperative function construction will be provided in section 4. Consider the nucleolus of the cooperative game

where PGNi(yt-i) = Nuf (yt-i), i G Sf (yt-i). Hence, the changed payoffs on Xt-i are specified by functions f*-i : Xt-i R+, i G N, such that for xt-i G Z(xt)

Suppose that cf (xt) = 0. Then player i(xt) chooses xt G Z(xt) from the condition

If cf (xt) = 1, then player i(xt) cooperates on xt with the coalition Sf (xt). Hence, xt has to satisfy

Finally, we know the evolution of game on any subtree K(xt) since the decisions of players have been determined for every node xt G Xt. Hence, to construct the path on a subtree K (xt+i) we have to consider just the decisions of player i(xt+i). When

Y(xt) = 0, the payoffs of players are different from those in the case of Y(xt) = 0. Define the payoffs on Xt by functions r\: Xt ^ R+, i G N, such that for xt G Xt and i G N

Definition 6. We construct a path which is realized using the above algorithm if the

it optimal path of the partial cooperative game rf (xo).

Definition 7. With the construction of the optimal path x(f) we get the final payoffs

Nuf (y

\Sf (yt-i)l .

where Nufk.(yt-i) = vf (yt-i,Sf (yt-i)) is taken as an optimal imputation

j=i 3

of coalition Sf (yt-1) payoff.

Note

PGN(yt-i) = (PGNi(yt-i),.PGNn(yt-i)),

(17)

PGNi(xt-i), if xt-i G Y(xt) and i G Sf (xt-i); rt-i(xt-i), otherwise.

(18)

max r

xeZ(xt)

(19)

max

xeZ(xt)

(20)

ieSf (xt)

ieSf (xt )

r- (Xt), if Y (xt) = 0; ftt-i (xt), if Y (xt) = 0;

hi(xt), if xt G Pn+i.

(21)

cooperative function f = (fi, ..., fn) is given in r. Denote this path by x(f) and call

rT(xo) of players, we shall call it PGN-value of partial cooperative game Yf (xo).

1.4. The concept of the best response to Nash equilibrium in cooperative subgames and construction of characteristic functions

In this section we establish the characteristic function vf (x,R), R C Sf (x), of Gf (x, Sf (x)), x G Y(xt). When constructing the optimal path in section 3 we define the behavior of players in each of its decision points. Such a fixed behavior regarded as a function of decision point is called a strategy. Denote n-tuple of strategies defined in section 3 by (■) = №(■),.. .,^*n(■)).

The cooperative game is constructed with the help of these strategies.

Consider the trace ^x(') = (^*x('),... ,^nx(')) of the n-tuple in K(x). For

i G Sf (x) fix the strategies ^*x(-) and consider the subgame rf (x) of rf (x0) in which the choice of player i G Sf (x) is fixed according to ^*x(■) in his corresponding decision points. Thus, the subgame rf (x) is a game among the players in the coalition Sf (x).

Introduce the concept of the best response to Nash equilibrium. It is easy to know that deviation of the single player to Nash equilibrium will not cause the increase of individual payoff, however, the deviation of part of the coalition (including individual), or the whole coalition, will probably lead to the increase of coalition payoff.

Definition 7. Suppose that the set of all players is M in subgame. The behavior that the deviation of part of the coalition R or the whole coalition R, maximizes the realized payoff of the coalition R when the players of M\R act on Nash equilibrium for any given coalition R C M, is called the best response to Nash equilibrium of coalition R.

We will get the characteristic function of cooperative subgame G(x, Sf (x)) using the concept of the best response to Nash equilibrium. Firstly, we must find the Nash equilibrium of subgame Tf (x), then consider the best response to Nash equilibrium for every subcoalition R C Sf (x), and take the realized maximum of payoff of coalition R which is obtained by the best response as the value v(x, R) of characteristic function of R. We can decrease the number of possible equilibrium through assuming every player serves as a well-meaned restriction to the other players since the Nash equilibrium can be more than one. It can be shown that the payoff of every coalition R defined in such a way can not exceed v(x,Sf (x)) = ^ r\(x). Using

ieSf (x)

the above v(x,R), one can construct the nucleolus Nuf (x) of cooperative subgame Gf (x,Sf (x)) and define PGN-value.

1.5 The example of PGN-value

Example 1. Consider a non-cooperative game ri with the tree K (x0) given in Figure 1 (broken line is excluded). The set of players is N = {1, 2, 3, 4, 5}. The player 1’s decision points are x0,x9,x2i, player 2’s -xi,xi2,x23, player 3’s - x4,xi3,x25, player 4’s-x5, xi_6, x26, player 5’s -x8, xi7, x29. Terminal payoffs are written vertically, with in every column the payoff of player 1 being the upper number, and so on.

Suppose that the cooperative function fi has the following form: f/(x0) = fi(x2i) = 0, ftx) = 1, f2(xi) = 0, f2(xi 2) = f2 (x23) = 1, f3i(x4) = f3(xi 3) =

f3(x25) = 0, f4 (x5) = f4i(xi6) = fi(x26) = 0, f5 (xi7) = 1, f5(xs) = f5 (x29) = 0. Thus, the player set in the game rf i (x0) is Nf i = {1, 2, 3, 4, 5, {1, 2} , {1, 2, 5}}. In rf i (x0) player 1 makes decision on x0, x2i, player 2 - on xi, player 3 - on x4, xi3, x25, player 4 - on x5,xi6,x26, player 5 - on x8,x29, player {1, 2} - on x9,x23, player {1, 2, 5} - on xi2, xi7. The payoff of player {1, 2} G Nf in Yf i (x0) is defined as the sum of corresponding terminal payoffs of players 1 and 2 from N in r. The payoff of player {1, 2, 5} G Nf in Yf i (x0) is defined as the sum of corresponding terminal payoffs of players 1, 2 and 5 from N in r.

Construct the optimal path of the partial cooperative game Yf i (x0). The procedure of optimal path construction starts on the endpoints xi9,x20, x30,x3i. The coalition structure on xi9, x20 is the same as on xi7, i.e., Sf (xi7) = {1, 2, 5} , {3} , {4}. Hence, for players from Nf i, the payoffs on xi9 and x20 are given by triples (8,4, 5) and (6,0, 0) respectively, where the first component is the payoff of player {1, 2, 5}, the second one - player 3’s and the third - player 4’s. On xi7, player {1, 2, 5} goes left to get 1+2+5=8. Hence, ri(xi7) = (1, 2, 4, 5, 5)*(notation* denotes transposition) and ri(xi8) = (1,1,1, 4,1)*. Since r4(xi7) > r^x^), on xi6 for player 4 it is optimal to go left to get 5. Hence, r2(xi6) = (1, 2,4, 5, 5)*, r2(xi5) = (1,1, 3,1,1)*.

Since r3(xi6) > r3(xi5), on xi3 for player 3 it is optimal to go right to get 4. Hence, r3(xi3) = (1, 2,4, 5, 5)*, r3(xi4) = (1, 3,1,1,1)*. On xi2 player 2 maximizes the payoff of the coalition {1, 2, 5}, and should go left to get r3(xi3) + r3(xi3) + rf(xi3) = 8. Hence, r4(xi2) = (1, 2, 4, 5, 5)*. We get r4(x23) = (2,1, 3, 4, 5)* similarly.

On x9 player {1, 2} makes decision. But his share in the proposed payoff of {1, 2, 5} is not known. Construct the cooperative subgame Gf i (xi2, Sf i (xi2)), Sf i (xi2) = {1, 2, 5}, on the subtree K(xi2) with the initial node xi2. Fixing the above determined decisions of player 3 on xi3 and 4 on xi6, we can define the characteristic function vf i (xi2, R), R C Sf i (xi2) with the concept of the best response to Nash equilibrium. The values of vf i (xi2, R) are as the following: vf i (xi2, {1, 2, 5}) = 8, vf i (xi2, {1}) = 1, vf i (xi2, {2}) = 3, vf i (xi2, {5}) = 1,vf i (xi2, {1, 2}) = 4, vf i (xi2, {1, 5}) = 2, vf i (xi2, {2, 5}) = 7. Thus, the nucleolus

Nuf (x 12) = (Nu{ (x\2),Nu2 (X12), Nu[ (xi2)) equals (1, |, |)*.

Hence, PGN(x 12) = (1, §, §)*, f4(x 12) = (1, |, 4, 5, |)*. It isn’t necessary to construct cooperative subgame on subtree K(x23) because node x23 G Y(x9) = 0. Thus, player {1, 2} should choose xi2 to get payoff f\(xi2) + ff(xi2) = 1 + | = for maximizing his own payoff. Hence, r5(x9) = (1, |, 4, 5, |)*, r5(x22) = (1,1, 2,1,1)*. On x2i player 1 does not belong to coalition {1, 2} and plays individually. Since his share in the proposed payoff of {1, 2} is not known, construct the cooperative subgame Gf i (x9,Sf i (x9)), Sf i (x9) = {1, 2}, on the subtree K(x9) with the initial node x9. Fixing the above determined decisions of player 3 on xi3, x25, 4 on xi6, x26 and 5 on xi7, x29, we can define the characteristic function vf i (x9, R),R C Sf i (x9) The values of Vf 1 (x9, R) are as the following: Vf 1 (x9, {1, 2}) = 4r, Vfi (x9, {1}) = 2

Vfi(xg, {2}) = 1. Thus, the nucleolus Nuf (x9) = (Nu{ (xg),Nuf (x9)) = (^, |)*

Hence, PGN(x9) = (f,| )*, f5(x9) = (f, |,4, 5, | )*, r6(x21) = (f,f,4,5,§)* r6(xi0) = (2,1,1,1,1)*. According to the given fi, comparing rf(x2i) and rf(xi0)

player 5 does not cooperate on xg and goes left to get | for maximizing his own payoff. Hence, rr(xg) = |, 4, 5, !)*, rr(xr) = (1,1,1, 3,1)*. Comparing r\{xi) and

r\(xg), it is optimal for player 4 to go right to get 5. Hence, r8(x5) = (^, |, 4, 5, |)*, r8(x6) = (1,1, 3,1,1)*. On x4 it is optimal for player 3 to go left to get 4. Hence, r9(x4) = (^p, |, 4, 5, |)*, r9(x3) = (1,1,1,1,1)*. According to the given J1, player 2 does not cooperate on x\ and go right to get | for maximizing his own payoff. Hence, r10(xi) = (^, |, 4, 5, |)*, r10(x2) = (2,1,1,1,1)*. Player 1 does not cooperate on xo and should go left to get Hence, rn(x0) = (^, |,4, 5, |)*. Thus, for the game rf i (x0) the optimal path is

x(f ) {x0, x1, x4, x5, x8, x2i, x9, x12, xi3, x16, xi7, xi9 }, (22)

and PGN -value of Tf i (x0) is

r1(xo) = (^,^4,5,^r. (23)

Consider a complete cooperative game G(x0) constructed on the same tree K(x0) in Figure 1. The path x* = {x0,... .,xi9} giving the maximal payoff to the grand coalition N in G(x0), coincides with the optimal path x(fi) of the game Tfi (x0). The Shapley value of G(x0) is Sh^x0) = 4^, 4^, 3, 3}. The optimal path using

the algorithm of [Petrosjan, 2000] coincides with x(fi) of the game rf i (x0), but the PMS'-value is (|, |, 4, 5, |)*.

Example 2. Consider game r0 with the tree K (x0) given in Figure 1 composed of broken line part and bold part of Figure 1. The values of cooperative function f0 are the same as those of fi of the game r i. We can get the game tree K(x0) of example

2 in [Petrosjan, 2000] through changing the payoff on x3 to (1, 2,1,1,1)*. We can get the same optimal path as x(fi) in rf 0 (x0) when the payoff on x3 is (1,1,1,1,1)* in game To according to [Petrosjan, 2000], the PMS'-value is (|, |, 4, 5, 2)*. Consider a complete cooperative game G(x0) in r0. The path giving the maximal payoff to the grand coalition N in G(x0) still coincides with x(fi), and the Shapley value of G(x0) is

Sh(Xo) = {if; if; if; 3; ¿5} • The optimal path of game T^o(xo) coincides with x(fi ) when characteristic function is constructed using the concept of the best response to Nash equilibrium, PGN-vslue is r°(x0) = (^, |, 4, 5, ^)*.

Example 3. Denote the new game which the value of cooperative function on xi for player 2 is changed into 1 in game r0 by r2, denote cooperative function by f2. We can get the same optimal path as x(fi) in rf2 (x0) according to [Petrosjan, 2000], the PMS'-value is (|, |, 4, 5, 2)*. Consider a complete cooperative game G(x0) in r2. The path giving the maximal payoff to the grand coalition N in G(x0) still coincides with x(fr), and the Shapley value of G(x 0) is Sh(x 0) = 4^, 4^,3, 3}.

The optimal path of gamerf 2 (x0)coincides with x(f i)when characteristic function is constructed using the concept of the best response to Nash equilibrium, PGN-value is r2(X0) = (^^4)5)nr_

1.6. Conclusion

An algorithm of the optimal path and PGN-value for partial cooperative game in finite extensive form with perfect information are proposed to overcome the limitation of PMS-value in this paper. To be specific, the PGN-value proves that some players would make necessary sacrifice for the sake of constructing coalition under the supposition of giving up the extremely opposing behavior to other players.

Comparing the component of PGN-value, PMS-value and Shapley value for player

2 in the above three examples, we find that the payoff of player 2 according to PGN -value is less than those according to PMS-value and Shapley value, which shows that player 2 makes more sacrifice for the sake of urging other players to cooperate. At the same time we can avoid the limitation of the superadditivity which the characteristic function should satisfy using the nucleolus concept.

Besides, according to the above three examples and the solution, we know that all optimal paths are showing no difference although the PGN-value got by the algorithm in this paper is different from the PMS-value and Shapley value. We find that the corresponding components of PGN-value, PMS-value for player 3 and 4 outnumbering the Shapley value Sh(x0) in complete cooperative game. In this way, PGN-value and PMS-value prove that the cooperation between player 3 and 4 is loose under the supposition of complete cooperation. The algorithm of the optimal path built in this article has introduced a more stable optimal solution in this paper, and the research will promote the construction of the system of partial cooperative game.

Changing the cooperative function f we get a class rF (x0) of all partial cooperative game rf (x0) which can be defined on K (x0). We can get more stable optimal solution if the optimal payoff of complete cooperative game is established with the help of the PGN-value of partial cooperative games Tf (x0) G rF (x0). Summing up the above algorithm and basing on the existence and uniqueness of nucleolus of complete cooperative game we get:

Theorem. The PGN-value of partial cooperative game infinite extensive form with perfect information exists and keeps unique.

2. Dynamic games with partial cooperation and the free-changing structure of coalition

2.1. Introduction

In the game defined in [Petrosjan, 1998] and [Ayoshin, 1998], one player begins cooperation from the stage he has chosen in any case, no matter which concrete path is taken. [Petrosjan, 2000] weakens the conditions mentioned above. The identical player might choose cooperation or non-cooperation respectively in different paths on the same stage.

However, players are not permitted to alter the declared options in game process of [Petrosjan, 1998], [Ayoshin, 1998], [Petrosjan, 2000] which is displayed in the demand of the monotonous increase of players’ coalition. The paper tries to cancel the limit through introduction of pivotal definition 10.

2.2. The basic model

Let r be a n-person non-cooperative game in finite extensive form with perfect information and without chance moves. Denote the set of players by N = {1,...,n}. Let K (x0) be the game tree with the origin x0. On K (x0) there exists a partition Pi,..., Pn, Pn+i of the set of game tree nodes, where Pi(i G N) is the set of decision points of player i, and Pn+i is the set of endpoints. The payoff of player i is specified by terminal real-valued functions hi : Pn+i ^ R+,iG N.

Definition 9. According to cooperative function f = (fi, ..., fn), players can cooperate or play individually in the switched game process, identical player may choose cooperation or non-cooperation in different paths of one stage of game process. The switched game is called a partial cooperative game rf (x0). Function f = (fi, . ..,fn) defines a special coalition structure on every node of the game tree K(x0).

Definition 10. Suppose that f has been defined and after several moves the game party came to a decision point x of player i. Consider the set

made a move to cooperate before x and don't leave the coalition.

Definition 10 indicates that the players in coalition Hf (x) still can leave the coalition after coalition Hf (x) is formed on node x. The players from the set N\Hf (x)

is specified by g as well on node x.

Now let’s define a kind of partial-cooperative game rf (xo) with perfect information in an extensive form. The game rf (x0) is generated by the game r and the cooperative function f. The game tree of r f (x0) coincides with the game tree of r. Take an arbitrary decision point x. Suppose that x G Pi. Introduce

The player set Nf in rf (x0) consists of the subsets of the set N (player set in r). The player set Nf is denoted as

The payoff of player S of r f (x0) is defined as the sum of payoffs of player i G S on the endpoints of K(x0)

fj (y) = 1,y is player j’s nearest personal decision | point to x on path {x0,. ..,x} ,y G Pj \

Hf ( x) consists of players who are ready to cooperate on x and the players who have

on x are considered as individual players. Since Hf (x) is defined by the cooperative function f, the whole coalition structure

Hf(x), {ji} , {h} \H f (æ)|} •

Hf (x), if fi(x) = 1

{i}, if fi(x) = 0

Nf = {if (x) \x G K(xo) } •

hs(x) = ^2 hi(x), x G Pn+i, hi(x) > 0,i G N.

ies

The rule cf is determined by a cooperative function f = (fi,..., fn) if x is a decision point of player i, where

, , \ 1, if fi(x) = 1

Cf (x) = < .

\ 0, if fi(x) = 0

The algorithm of construction of the optimal trajectory

The optimal path is determined by means of backward induction. The procedure is similar to the one used in the section 1.3.

The initial stage. The coalition structure on x and that on its immediate predecessor xi, x G Z(xi), are the same. On xi the given f specifies coalition structure

Hf (x), {ji} , {j2 } ,...,{j\N\Hf (x)\}. Hence, the coalition Hf (xi) gets £ hi(x)

i€Hf (xi)

on x and an individual player jk ,k = 1,..., \N\Hf (xi)|, gets hjk (x) on x.

Stage 1. Shift back from the endpoints x to their predecessors. Consider an arbitrarily taken xi .

If Cf (xi) = 1, player i(xi) selects xi G Z (xi) from the condition

max \ hi(x) = } hi(xi).

xEz(xi)

i*EHf (xi) i^Hf (xi)

In case Cf (xi) = 0, player i(xi) maximizes the payoff of the coalition if (xi) consisting of himself only

max hi(xi) (x) hi(xi ).(xi)

xEZ(xi)

In the same way, we can construct trajectories starting from the arbitrary nodes on Xi. Therefore, instead of considering the terminal payoff function hi,i G N, on Pn+i, we may deal with payoff function r* : Xi ^ R+, i G N, on Xi such that

( hi(xi), if xi G Pn+i

(x1) =

I hi(xi), if xi G Pn+i

r

Stage t. Continue moving toward the tree root. For nodes xt G Xt\Pn+i, consider set Y(xt) = {x G Z(xt)\Hf (x) = Hf (xt)}.

For each node xt Xt we deal with two main cases.

1) Assume that Y(xt) = 0 for all xt G Xt\Pn+i. In this case, the functions rt-i specify the payoff obtained at the end of the game for each player i(xt), i.e. if the decision of player i(xt) leads to node xt G Z(xt), then at the end of the game the coalition Hf (xt) will get ^ rt-i(xt), and the payoff of individual player jk will

ieHf (xt)

be rj-i(xt). Therefore, we can easily determine the nodes xt, where xt G Z(xt) and

xt G Xt\Pn+i. If Cf (xt) = 0, then xt has to satisfy max rt(x|)(x) = rt(x|)(xt).

xeZ (xt )

Now assume that Cf (xt) = 1. The coalitions Hf (xt-1) and Hf(xt) coincide on xt-1 G Z(xt) since Y(xt) = 0. Then since player i(xt) belongs to the coalition Hf (xt) on xt, the node xt has to satisfy max ^ r*(x1)(x) = r*(x1)(xt).

x£Z(xt) iEHf (xt) ieHf (xt)

2) Now, suppose Y(xt) = 0. To construct a path on K(xt), it is necessary to define some imputation of payoff of coalition Hf (yt-1) for each yt-i G Y(xt). We do it by considering an auxiliary \Hf (yt-1)\-person cooperative game Gf (yt-1,Hf (yt-1)) on the subtree K(yt-1) with the set of players Hf (yt-1) and the characteristic function vf (yt-i,R),R C Hf (yt-1), for each yt-i G Y (xt). The explanation of the cooperative function vf (yt-i,R) is provided in [Kuhn, 1953]. The payoff of the grand coalition Hf (yt-i) in Gf (yt-i, Hf (yt-i)) is defined as vf (y—i,Hf (yt-i)) =

= rt-i(yt-i). Compute the nucleolus of game Gf (yt-i,Hf (yt-i)),

ieHf (yt-i)

\Hf (yt-i )\

Nuf (yt-i) = (Nufki (yt-i),...,Nuf (yt-i)), where Nufkj (yt-i) =

\Hf (»t-i)\ j=i j

= vf (yt-i, Hf (yt-i)) is taken as an optimal imputation of Gf (yt-i, Hf (yt-i)). Note PGN(yt-i) = (PGNi (yt-i),..., PGNn(yt-i)), where PGNi(yt-i) = Nuf(y-), i G Hf (yt-i). Hence, the changed payoffs on Xt-1 are specified by functions ft-i : Xt-1 ^ R+, i G N, such that for xt-1 G Z(xt),

t-u ^ j PGNi (xt-i), if xt-i GY (xt) and i G Hf (xt-i)

rit-1(xt-1), otherwise

rT (xt-i) =

if Cf (xt) = 0, player i(xt) chooses xt G Z(xt) from the condition

max

xeZ(xt)

max. ftt(xl)(x) = ^(x!) (xt).

If Cf (xt) = 1, xt has to satisfy max ^ r* 1(x) = ^ r^ 1(xt).

xeZ(xt) ieHf (xt) ieHf (xt)

When Y(xt) = 0, the payoffs of players are different from those in the case of

Y(xt) = 0. Define the payoffs on Xt by functions rfXt ^ R+, i G N, such that for xt Xt and i N,

f rt- 1 (xt), if Y(xt) = 0

ri(xt) = < rt- 1 (xt), if Y(xt) = 0 .

{ hi(xt), if xt G Pn+i

By continuing moving backward on K(x0) toward the origin x0, players’ decisions on remaining sets XT,t = t + 1, ...,T are determined sequentially.

1

2.3 The example of PGN-value of the game with partial cooperation and the free-changing structure of coalition

Example. Consider a non-cooperative game r with the tree K(x0). We can get the game tree K(x0) of through changing the payoff on x3 to (1, 2,1,1,1)* and x2 to (1,1,1,1,1)* in broken line part and bold part of picture. The set of players is N = {1, 2, 3,4, 5}.

The player 1’s decision points are x0, x9, player 2’s - xi, xi2, player 3’s - x4, xi3, player 4’s - x5, xi6, player 5’s - x8, xi7. Terminal payoffs are written vertically, with in every column the payoff of player 1 being the upper number, and so on.

Suppose that the cooperative function g has the following form: f1(x0) = 1, fi(xg) = 0, f2(xi) = /2(xi2) = 1, /3(x4) = f3(xi3) = 0,

/4^5) = /4(xi6) = 0, f5(xg) = /5 (xi7) = 1.

Construct the optimal path of partial cooperative game rf (x0) using above algorithm. Thus, we know x(f) = {x0,xi,x4,x5,x8,xg,xi2,xi3,xi6, xi7, xig}, PGN-value of rf(xn) is ?’10(xo) = (f )*•

2.4 Note

The changing process of coalition Hf (■) is as follows:{1} ^ {1, 2} ^ {1, 2, 5} ^ {2, 5}. On xi players 1, 2 construct coalition Hf (xi) = {1, 2} because player 2 obtains ip from coalition that is more than that when selecting X3. On the other hand, player

1 can’t select X2 since he forecasts that he obtains j after constructing coalition with player 2. Similarly we know coalition {1, 2} changes into {1, 2, 5} = Hf (x8) on x8 because player 5 can obtain | after joining coalition {1, 2}, which is good for player 5. But on x9 the coalition changes into Hf (x9) = {2, 5} from {1, 2, 5}, because player 1 finds his share from coalition is less than that when leaving coalition. The case which coalition decreases occurs.

References

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Ayoshin D.,Tanaka T. 1998. The core and the dominance core in multichoice multistage games with coalitions in a matrix form. In: Proceedings of NACA98. International Coference on Nonlinear Analysis and Convex Analysis.

Petrosyan L. A., Ayoshin D. A. 2000. The value of dynamic games with partial cooperation. Proceedings of the Institute of Mathematics and Mechanics. Ekaterinburg, 6 (1-2): 160-172.

Kohlberg E. 1971. On the nucleolus of a characteristic function game. SIAMJ. Appl. Math., 20: 173-177.

Kuhn H. W., Tucker A. W. 1953. Theory of Games II. Princeton: Princeton University Press; 307-317.

Petrosjan L. A., Mamkina S. I. 2004. New Value for Dynamic Games with Perfect Information and Changing Coalitional Structure. In: Proceedings of the XI International Symposium on Dynamic Games and Applications.Tucson, Arizona: 799-813.

Nash J. F. 1950. Equilibrium Points in n-Person Games. Proc. Nat. Acad. Sci., 36: 48-49.

Nash J. F. 1951. Non-cooperative Games. Ann. of Math 54: 286-295.