The core of a repeated n-person cooperative game

European Journal of Operational Research 127 (2000) 519±524

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Theory and Methodology

The core of a repeated n-person cooperative game Jorge Oviedo Instituto de Matem atica Aplicada San Luis, Universidad Nacional de San Luis, Ejercito de los Andes 950, 5700 San Luis, Argentina Received 15 June 1998; accepted 4 May 1999

Abstract We study the core of a repeated cooperative game. We de®ne the repeated cooperative game as a repeated game where in each round the agents play a cooperative game. We introduce an imputation sequence and a dominated imputation sequence. We de®ne the core of the repeated game as the set of all undominated imputation sequences. We show that the core of a repeated cooperative game contains the core of the original cooperative game. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Cooperative game; Repeated cooperative game; Core

1. Introduction In this note we repeat a cooperative game and study the core. We begin with an n-person game with transferable utility in characteristic function in which cooperation is permitted or simply a cooperative game and repeat this game m times (m may be 1). Consider the following cooperative game 1 in characteristic function form with transferable utility, N ˆ f1; 2g and v…f1g† ˆ v…f2g† ˆ 0; v…f1; 2g† ˆ 1. We repeat this game two times. If we consider this repeated game as a cooperative game: we have four players N2 ˆ f10 ; 20 ; 11 ; 21 g, where it denotes player i ˆ 1; 2 in round t ˆ 0; 1: The grand

1

E-mail address: [email protected] (J. Oviedo). The de®nition is in Section 2.

coalition f10 ; 20 ; 11 ; 21 g means that both players in each round are able to make a binding agreement. The coalition f10 ; 11 g means that player 10 and 11 make a binding agreement. It is equivalent to player 1 not making a binding agreement with player 2 in any round of the game. The coalition f10 ; 21 g makes a binding agreement is not possible, because they are in di€erent rounds. In this sense, we cannot see this repeated cooperative game as a cooperative game. We consider the repeated cooperative game as a repeated game where in each round the agents play a cooperative game. An agent of the repeated cooperative game decides in that round t whether to take part in a coalition (of the stage cooperative game) or not. We de®ne the set of a coalition sequence of the repeated cooperative game as a sequence of coalitions formed in each stage of the repeated cooperative game.

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 3 3 5 - 5

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J. Oviedo / European Journal of Operational Research 127 (2000) 519±524

We de®ne the repeated cooperative game by a set of a coalition sequence (where a coalition sequence is a sequence of stage game's coalition) and a repeated characteristic function. This de®nition has the same structure (characteristic function form) as the stage cooperative game. We use a ddiscounted payo€ to de®ne the repeated characteristic function. We de®ne an imputation sequence or payo€ allocation sequence as a sequence of imputations or payo€ allocations for each stage game. Our domination de®nition compares the total payo€ of the agents that form the coalition sequence. We show that if we choose a sequence of payo€ allocations that is in the core of the cooperative game, this sequence is in the core of the repeated cooperative game. Moreover, the core of the repeated game admits a payo€ allocation sequence such that the payo€ allocation for each period are not in the core of the corresponding cooperative game (see Example 1). This result is analogous to the repeated non-cooperative game. Repeated games for the non-cooperative case have been studied extensively (Kalai (1990) is an excellent survey). One such game is the prisoner's dilemma. Consider the following payo€ matrix (see Table 1). If the game is played once, …N ; N † is the unique Nash equilibrium. If the game is repeated in®nite number of times, we have that any payo€ vector …v1 ; v2 † such that v1 ; v2 P 0 may be supported (in particular …0; 0†) as an Nash equilibrium strategy of the repeated game. This means that there exists a Nash equilibrium strategy such that the vector payo€ is …v1 ; v2 †. In non-cooperative games, the set of Nash equilibria of the one shot game is contained in the set of Nash equilibria of the repeated game. To the best of our knowledge, repeated cooperative games have not been formally studied in game theory literature.

2. Basic de®nitions Assume that the stage game …N ; v† is an nperson cooperative game (with transferable utility) in characteristic function form, where v, is a realvalued function, de®ned on the subsets, of the player or agent set N ˆ f1; . . . ; ng v satis®es, v…;† ˆ 0 and for all coalitions, S; T  N ; v…S [ T † P v…S† ‡ v…T † if S \ T ˆ ;: We repeat m times (m may be 1) this cooperative game. We denote by …N ; vt ; t† the stage cooperative game …N ; v† that the agents play in period t: We assume that the characteristic function of the game …N ; vt ; t† has a discount factor dt , 0 < d < 1. The characteristic function of the game …N ; vt ; t† is de®ned for all S  N by vt …S† ˆ …1 ÿ d†dt v…S†: The normalization factor …1 ÿ d† allows the stage-game and repeated-game characteristic function to be measured in the same unities. An imputation vector or payo€ allocation vector of the cooperative game …N ; vt ; t†; xt ˆ …xt1 ; xt2 ; . . . ; xtn † is n X iˆ1

xti ˆ vt …N † ˆ …1 ÿ d†dt v…N †

and for all i ˆ 1; . . . ; n xti P vt …fig† ˆ …1 ÿ d†dt v…fig†: Let xt ; y t be two imputation vectors of the game …N ; vt ; t† and let S be a coalition. We say xt dominates y t through S …xt S y t † if

Table 1 Prisoner's dilemma C N

The remainder of this paper is organized as follows. In Section 2 we de®ne the repeated cooperative game. In Section 3, we study domination and the core.

C

N

3,3 4; ÿ1

ÿ1; 4 0,0

xti > yit and

for all i 2 S

J. Oviedo / European Journal of Operational Research 127 (2000) 519±524

X i2S

xti 6 vt …S† ˆ …1 ÿ d†dt v…S†:

We say xt dominates y t (xt  y t † if there exists a coalition S such that xt S y t : The core of …N ; vt ; t† is the set of all undominated imputation vectors for cooperative game …N ; vt ; t† and is denoted by C…vt † or C……1 ÿ d†dt v†. We de®ne the repeated cooperative game, so that it has the same structure as the cooperative game, we must specify the coalitions set and the characteristic function. We denote by S t  N a coalition formed at time t: Suppose that the game begins in time t ˆ 0. For t P 0, let …S 0 ; S 1 ; . . . ; S t † be the formed coalition sequence through period t, t and let H t ˆ …2N † be the space of all possible period-t coalition sequences, where 2N is the set of subset of N. We denote by H the set of all coalition sequences. Let H ˆ …S 0 ; S 1 ; . . .† ˆ …S t †mtˆ0 2 H be a coalition sequence. We de®ne the repeated characteristic function w as w…H† ˆ …1 ÿ d†

m X

dt v…S t †:

…1†

tˆ0

In the case m ˆ 1 w…H† exists because v…S t † is bounded and 0 < d < 1. w…H† is the utility that the members of H (i.e., the members of each coalition of the sequence) can get from the repeated cooperative game. Note that if the coalition sequence is the empty b ˆ …;; ;; . . .†, then set in each period, that is H b ˆ 0: w… H†

De®nition 1. Let …N ; v† be an n-person cooperative game in characteristic function form with transferable utility. We de®ne the repeated n-person cooperative game in characteristic function form that is the repeated cooperative game of …N ; v† by …H ; w†, where H is the set of coalition sequences and w is the repeated characteristic function de®ned by (1). The new game …H ; w† is de®ned as a cooperative game. The players of a coalition sequence H ˆ …S k †mkˆ0 sign a contract that says that in period t he/she is playing as member of the coalition S t in the stage game …N ; vt ; t†. We de®ne the imputation sequence or payo€ allocation sequence for the repeated cooperative game as: De®nition 2. An imputation sequence or payo€ allocation sequence x ˆ …x0 ; x1 ; . . .† for the repeated cooperative game is a sequence of the stage game's imputations or payo€ allocations. 3. Domination or improving an imputation sequence Given two di€erent imputation sequences x; y there are some players that prefer Pm that Pxm to y, those x gives a bigger total payo€ … tˆ0 xti > tˆ0 yit †. Then to de®ne domination, we will compare the total payo€ of the players that are in a coalition sequence of the repeated cooperative game. We have the following de®nition. De®nition 3. Let x; y be two imputation sequences and let H be a coalition sequence. We say x dominates y through H …x H y† if m X

m …S t †tˆ0

2 H, W ˆ Let H ˆ coalition sequences such 0 6 t 6 m; S t \ T t ˆ ; then

m …T t †tˆ0

2 H be two that for all

w…H [ W† P w…H† ‡ w…W†; m

where H [ W ˆ …S t [ T t †tˆ0 : The repeated cooperative game is formally de®ned below.

521

tˆ0

xti >

m X tˆ0

yit

for all i 2

[

and for all t X i2

S k

Sk

xti

Sk ;

…2†

:

…3†

k

t

6 …1 ÿ d†d v

[

! S

k

k

Condition (2) states that the players that are at least in a coalition of H in some period t get strictly

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J. Oviedo / European Journal of Operational Research 127 (2000) 519±524

higher total (note that the sum is over all periods) payo€ in x than y. Condition (3) states that they are capable of obtaining for each period-t what xt gives them in the corresponding stage game. De®nition 4. We say x dominates y …x  y† if there exists a coalition sequence H such that x H y. It is easy to see that the relation H is a partial order relation. The set of all undominated imputation sequences for a repeated cooperative game w we call the core and denote by C…w†: The next proposition shows that the core, contains the core in each period. Proposition 1. Let y be an imputation sequence such that for all t, y t 2 C……1 ÿ d†dt v†. Then m C……1 ÿ d†dt v†  C…w†. y 2 C…w†, i.e., Xtˆ0 Proof. Assume that y 62 C…w†: Then there exists an imputation sequence x and a coalition sequence H satisfying (2) and (3). By condition (3) and the hypothesis, we have that if we sum over all t ! m m X [ X X t t k x 6 …1 ÿ d†d v S S k i k tˆ0 tˆ0 i2

k

S

6

m X

X

tˆ0 i2

S k

yit ;

i2

S k

S k tˆ0

xti 6

m X X i2

S k

S k tˆ0

0 6 xt1 6 3…1 ÿ d†dt ; 1…1 ÿ d†dt 6 xt2 ÿ 6 4…1 ÿ d†dt ; 2 1 ÿ d†dt 6 xt3 6 5…1 ÿ d†dt : We repeat this cooperative game three times. Given x0 ˆ …1 ÿ d†…3 ‡ d; 3 ÿ d; 3†; x1 ˆ …1 ÿ d†d…1; 5; 3†; x2 ˆ …1 ÿ d†d2 …1; 2; 6†; we have that xt 62 C……1 ÿ d†dt v† for t ˆ 0; 1; 2. But x ˆ …x0 ; x1 ; x2 † 2 C…w†. Assume that there exists an imputation sequence y and a coalition sek 3 quence S H ˆ …S †kˆ0 such that y H x. If 2 kˆ0 S k ˆ 3 …j
j is the cardinality S of …
†† then y 2 is not an imputation. If kˆ0 S k ˆ 2, and S2 k kˆ0 S ˆ f2; 3g, then v…f2; 3g† ˆ 6. Condition (3) implies y20 ‡ y21 ‡ y22 ‡ y30 ‡ y31 ‡ y32 ‡ 6 …1 ÿ d†  …6 ‡ 6d ‡ 6d2 †:

…4†

By condition (2), we have that

Sk

this implies that m X X

C……1 ÿ d†dt v†  ˆ xt 2 R3 : xt1 ‡ xt2 ‡ xt3 ˆ 9…1 ÿ d†dt ;

yit :

y20 ‡ y21 ‡ y22 > …1 ÿ d†…3 ÿ d ‡ 5d ‡ 2d2 †;

…5†

y30 ‡ y31 ‡ y32 > …1 ÿ d†…3 ‡ 3d ‡ 6d2 †:

…6†

The converse of Proposition 1 is not true. That is, there exists x belonging to the core of the repeated cooperative game and for each t ˆ 0; . . . ; m; xt is not in the core of the cooperative stage game …N ; vt ; t†.

The sum of (5) and (6) yields an inequality that contradicts (4). The other cases are similar. Thus, x ˆ …x0 ; x1 ; x2 † 2 C…w†. The following proposition is a characterization of the core. Some authors (see Myerson, 1991) give a version of this Proposition for a stage game as the de®nition of the core.

Example 1. v…f1; 2; 3g† ˆ 9; v…f1; 2g† ˆ 4, v…f1; 3g†ˆ 5; v…f2; 3g† ˆ 6; v…fig† ˆ 0, …i ˆ 1; 2; 3†. For all t the core of …1 ÿ d†dt v is

Proposition 2. Let x ˆ …xt †tˆ0 be an imputation sequence, x 2 C…w† if and only if for all coalition sequences H ˆ …S k †mkˆ0

This contradicts condition (2).



m

J. Oviedo / European Journal of Operational Research 127 (2000) 519±524 m X X i2

S k

S k tˆ0

m X

xti P

[

…1 ÿ d†dt v

[

Example 1 (continued). The core of the repeated cooperative game C…w† for this example is

Sk

k

tˆ0

ˆ …1 ÿ d†v

!

! Sk

k

m X

dt :

…7†

C…w†

tˆ0

ˆ

( z ˆ …z0 ; z1 ; z2 † :

2 X 3 X tˆ0

Proof. Assume that x 62 C…w†: Then there exists an imputation sequence y and a coalition sequence H satisfying conditions S (2) and (3). Summing condition (2) over all i 2 k S k and condition (3) over all periods t yields a contradiction to condition (7). Hence x 2 C…w†:

06

i2

k

Sk

…1 ÿ d†

Pm

tˆ0

Pm

tˆ0

xti

d

t

ˆv

[

! Sk

…1 ÿ d†

a ˆ v…N † ÿ v

k

! S

X

ÿ

S

i2N n

k

Pm t x tˆ0 i P m

‡ Se S k j k j tˆ0 zi ˆ a > : v…fig† ‡ nÿ S j k Sk j …1ÿd†

dt

dt 6

tˆ0

2…1 ÿ d†

2 X

2 X tˆ0

t

d 6

tˆ0

tˆ0

tˆ0

dt ;

tˆ0

zt2 6 4…1 ÿ d†

2 X

dt ;

zt3

2 X

dt ;

tˆ0

6 5…1 ÿ d†

2 X

) d

t

:

tˆ0

It is clear that x ˆ …x0 ; x1 ; x2 † 2 C…w†:

Pm

v…fig† P 0: yi ˆ

Sk

Then the last inequality follows by superadditivity. We de®ne z by 8 > <

2 X

2 X

Proof. If C…v† 6ˆ ; then for all t ˆ 0; . . . ; m, C…vt † ˆ C……1 ÿ d†dt v† 6ˆ ;, and this implies that C…w† 6ˆ ;. If C…w† 6ˆ ;. Let x ˆ …x0 ; . . . ; xm † 2 C…w†. We de®ne y ˆ …y1 ; . . . ; yn †, by

Let

k

zt1 6 3…1 ÿ d†

iˆ1 2 X

zti ˆ 9…1 ÿ d†

Corollary 1. C…w† 6ˆ ; if and only if C…v† 6ˆ ;

ÿ e:

k

[

2 X tˆ0

Conversely, assume that a coalition sequence H exists such that condition (7) is false, that is there exists e > 0 such that P S

523

if if

i2 i 62

S S

k k

Sk k

S :

It is clear that z is an imputation of the stage game m v. We de®ne the imputation sequence y ˆ …y t †tˆ0 by y t ˆ …1 ÿ d†dt z: It is easily seen that y is an imputation sequence m and, S moreover, y H x where H ˆ …T t †tˆ0 , with m t k  T ˆ kˆ0 S for all t. Hence x 62 C…w†.

xt Pmi

tˆ0

…1 ÿ d†

tˆ0

dt

;

we have y is an imputation vector of the game …N ; v† and that for all S  N X

yi P v…S†

i2S

(this is true because if we choose H ˆ …S†mkˆ0 in 7 in Proposition 2). This says that y 2 C…v†.  Acknowledgements I am grateful to Carolyn Berry, Alejandro Manelli and Ricard Torres for useful comments. Thanks are also due to the referees for their valuable suggestions.

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J. Oviedo / European Journal of Operational Research 127 (2000) 519±524

References Kalai, E., 1990. Bounded rationality and strategic complexity in repeated games. In: Ichiishi, Neyman, Tauman (Eds.),

Game Theory and Applications. Academic Press, San Diego, CA, pp. 131±157. Myerson, R., 1991. Game Theory: Analysis of Con¯ict. Harvard University Press, Cambridge, MA.