M,tthematical Social Sciences 11 (1986) 83-87

N,,l lh-Itolland

83

A P R O O F T H A T T H E C O R E OF A N O R D I N A L C O N V E X

G A M E IS A V O N N E U M A N N - M O R G E N S T E R N S O L U T I O N

Bezalel P E L E G

Department o f Mathematics, The Hebrew University, Jerusalem 91904, Israel

Communicated by K.H. Kim

Received 5 April 1985

It is proved that the core of an ordinal convex game is a yon Neumann-Morgenstern solution.

The proof makes a strong use of reduced games of cooperative games without side payments.

Key words: Core; von Neumann-Morgenstern solution; convex games without side payments.

I. Introduction

In this note it is proved that the core of an ordinal convex game (without side

payments) is a v o n Neumann-Morgens t e rn solution. Our result generalizes earlier results of Vilkov (1977) and Sharkey (1981) for restricted families of ordinal convex

games. Our proof is based on an investigation of the structure of reduced games (of

ordinal convex games). Reduced games of cooperat ive games with side payments

were in t roduced in Maschler and Peleg (1967). Maschler, Peleg and Shapley (1972)

investigated the structure of reduced games of convex games with side payments.

Reduced games of ordinal convex games were also used in Greenberg (1982).

Demange (1985) has found impor tant applications of our theorem to social choice (see Remark 2.13).

2. Cores of ordinal convex games

Let N = { 1 . . . . , n} be a set of players and let R+ be the set of all non-negative real numbers . If S is a coalition (i.e. S C N ) then we denote by ~s+ the set of all func-

tions f rom S to ~+. I f x e ~+x and S is a coalit ion, then we denote b y x s the restric- tion o f x to S. Let S be a coalition and let xS, ySe ~s. We write xS>_y s ifxi>_y i for

all i e S, and xS>>y s if xi>} 'i for all i e S. Finally, ~s+ is considered as a (topological)

subspace of the Eucl idean space Ns (of all real functions on S).

Following Greenberg (1982) we use the following definition.

Definition 2.1. An n-person cooperative game without side payments is a pair

0165-4896/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-ttolland)

84 B. Peleg / Core o f an ordinal convex game

(N, o), where N is a set of players and v is a function that assigns to every subset S of N a subset o(S) of R+x such that

v (0) = 0.

If S ~ 0 , then o(S)#:O.

o(N) is closed.

If x e o(S), y e ~N+, and xS>__y s, then y e o(S).

up(S)= {xS lxe o(S)} is bounded.

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

Definition 2.2. Let (N, o) be an n-person cooperative game without side payments. (N, o) is an ordinal convex game if for all S, T C N

o(S) (*I o(T)C o(S f'I T) U o(SU T). (2.6)

Let (N, o) be an ordinal convex game.

Definition 2.3. Let TCN, T~O. The subgame (T, Or) is defined by the following rules:

or(O) = O.

OT(S)={Xz[xeo(S)} , if OvLS~e T, SCT.

Or(T) = closure ( {xTIxe v(T)}).

(2.7)

(2.8)

(2.9)

We remark that (7, of) satisfies (2.1)-(2.6).

Definition 2.4. Let TCN, 0=/= T=/=N, let z r e Or(T), and let M = N - T. The reduced game (M, o~4) is defined by the following rules:

o~(0) = 0. (2.10)

v~t(S) : U { x M I x e u ( S U R ) a n d x T ~ > z r } , i f O ~ S : / : M , S C M . (2.11) RC T

oh(M) = {xM l (x M, z T) ~ o(N)}. (2.12)

We remark that (M, v~) satisfies (2.1)-(2.5).

Definition 2.5. Let (N, v) be an n-person cooperative game without side payments and let x, y e o(N); x dominates y via a coalition S, written x Dom (S)y, if xS~> y s and x e v(S); x dominates y, written x Dom y, if there exists a coalition T such that x Dom (T)y. The core of (N, v) is the set of undominated vectors in o(N) and it is denoted by C(N, o).

~8aln, lel (N, v)

B. Peleg / Core o f an ordinal convex game 85

be an ordinal convex game. We now prove the following

~ m m ~ 2.6. Let T c N , O4:T~N, let z r e v r ( T ) , and let M = N - T . I f | F e ('1 1', ~Jr) and for RC T, R ~ T, zr~ closure (or(R)), then (M, v,~t) satisfies (2.6) [L¢, it is a convex game).

P ~ o f . I.et SiCM, i=1 ,2 . We have to show that

o ~ ( S l ) n O h ( S 2 ) c * S OM( 1 n $2) U o~4(S 1 u 32). (2.13)

Thus, we may a s s u m e O = / : S i ~ M , i = 1, 2. Let xXt• @t(Si), i= 1,2. Then there exist ~,~ 1' and xir,>z r, i= 1, 2, such that

(x~4, x [ ) • v ( S i U R i ) , i= 1,2.

I el .r~ = min(xl, x~), i • T, and let x = (x M, x,r). Then

x • o(St U R]) n o(S 2 U R2).

Because (N, v) is convex,

X • u((S 1 N $2) O (R 1 n R2) ) O o ( S 1 O S 2 U R 1 U R2).

We distinguish the following possibilities:

x • t)((S1 n $2) U (R 1 n Rz)). (2.14)

Because xr.,> z r and z r • C( T, Vr), Si nS2=/:0. Hence, by (2.11), xM • v~4(Sj AS2).

X • o ( S 1 U S 2 U R I U R2). (2.15)

it S] U $2 e: M, then, again by (2.11), xM• v~t(S1 U $2). Thus, it remains to consider the case SI US2=M. We further distinguish these subcases:

R 1 U R 2 = T. (2.16)

lhen x • v(N). Hence, by (2.4), (x M, z ~) • v(N), and xM• v~(M):

R l U R 2 ~ T . (2.17)

l,et R = R ~ U R 2. By our assumption zr¢closure (vr(R)). However, z r •c losure IrJv(T)). Hence, there exists a sequence (z[), k = l, 2, ..., such that z [ ~ z r, I.v M, z[) • o(T), (x M, zD ¢ v(R), and (x M, z[) • v (MU R), k = 1,2 . . . . . By (2.6), I.v M, z[) • v(N). Hence, by (2.3), (x M, z r) • v(N). Thus, by (2.12), x M • o~(M).

I.emma 2.7. Under the assumptions o f Lemma 2.6, if C(M, v~t) ~eO, then there exists x • C(N, v) such that x r = z r (in particular, C(N, v)SO).

Proof. Let yM • C(M, v~t). By (2.12), x o = (yM, z r) • v(N). Choose x • o(N) such

86 B. Peleg / Core o f an ordinal convex game

that: (1) xT=z f, (2) xM>>-y M, and (3) if wev(N) , wT=z T, and wM>---y M, then

w i. We claim that x e C(N, v). Indeed, assume on the contrary thai there exist w~v(N) and a coalition S such that wDom(S)x. If S c T , then w T Dora (S)z T in the game (T, OT). Because zTe C(T, OT), this is impossible. Hence, Q=Sf]M:/:O. If Q:/:M, then wM Ev~4(Q). Since wQ>>x(2>_y Q, wMDom(Q)y M in the game (M, v~t). Because y~4e C(M, v~t), this is impossible. Hence, Q=M. Let R = S f ] T. If R = T, then S=N. By (2.4) (wM, z r )ev (N) . Because wM>>x M, this is impossible. Hence, R:/:T. By our assumption zr~closure (or(R)). However, zreclosure(vp(T)). Hence, there exists a sequence (zT), k= 1,2, . . . , such that z [ ~ s T, (w M,zf.)ev(S), (w M,z [)EO(R), and (w M,z T) ev(T) , k = 1,2, . . . . By (2.6), (w it, z [) ~ v(N), k = 1,2, . . . . Hence, by (2.3), (w M, z T) ~ v(N), which is impossible.

Corollary 2.8. C(N, o):/:0.

Proof. The proof is by induction on n, the number of players. The case n = 1 follows from (2.2)-(2.5). Now let n_>2, let T={n} , and let M = N - { n } . Denote

"=sup{xnlx a{n})}.

Then T and z n satisfy the assumptions of Lemma 2.6. Hence, the game (M, v~t) is convex. Because M contains n - 1 players, C(M, v~4):/:O. Hence, by Lemma 2.7, C(N, v) :/: 0.

Remark 2.9. Corollary 2.8 is due to Greenberg (1982).

Corollary 2.10. Under the assumptions of Lemma 2.6 there exists x e C(N, v) such that x r= z r.

Proof. By Lemma 2.6, (M, v~) is convex. By Corollary 2.8, C(M, v~t):/:O. Thus, by Lemma 2.7, there exists x6 C(N, v) such that xT=z T.

We recall that C(N, v) is a von Neumann-Morgenstern solution if for every y e o(N), y~C(N, v), there exists x~C(N, v) such that x D o m y .

Definition 2.11. x e v(N) is weakly Pareto-optimal if there is no y6 v(N) such that y ~ x .

The following property of (N, v) is required in the sequel:

o(S) is closed for every S c N . (2.18)

Theorem 2.12. I f (N, v) satisfies (2.18) then C(N, v) is a yon Neumann-Morgenstern solution.

B. Peleg / Core o f an ordinal convex game 87

Proof. Let y e u ( N ) - C(N, t)). We distinguish the following possibilities"

y is weakly Pareto-optimal. (2.19)

let T be a minimal coalition with the following property: there exists z e v ( T ) ,ttch that zr>>y r. Because of (2.19), T:#N. Also, by (2.1), T#:0. If z e u ( T ) and :~> v r, then z c u ( R ) for all RC T, RV:T. Hence, there exists z. re C(T, ur) such ~t~a~ zr>>y r and zrCur(R) for R C T , R=/:T. By Corollary 2.10, there exists ~cC(N,t~) such that x r = z I By (2.4), x e u ( T ) . Thus, xDom(T).v.

y is not weakly Pareto-optimal. (2.20)

l 'here exists y , e t)(N) such that (1)y,>>y, and (2 )y , is weakly Pareto-optimal. c lcarly, y , Dora (N)y. Thus, if y , ¢ C(N, t)), then there exists x e C(N, ~)) such that v l )omy, . Because y,>>y, x D o m y .

Remark 2.13. Demange (1985) has obtained important applications of Theorem 2.12 to social choice theory. In particular, she has shown that it implies that the core correspondence of a convex effectivity function is nonmanipulable. (The reader is ~cferred to Chapter 6 of Peleg, 1984, for a study of cores of effectivity functions.)

References

(;. Demange, Non manipulable social choice correspondences, Laboratoire d'Econometrie de l'Ecole Polytechnique, Paris, France, 1985.

I. Greenberg, Cores of convex games without side payments, University of Haifa, Israel, 1982. M. Maschler and B. Peleg, The structure of the kernel of a cooperative game, SIAM Journal of Applied

Mathematics 15 (1967) 569-604. .',1. Maschler, B. Peleg and L.S. Shapley, The kernel and bargaining set for convex games, International

Journal of Game Theory 1 (1972) 73-93. P,. Peleg, Game Theoretic Analysis of Voting in Committees (Cambridge University Press, Cambridge,

1984). W.W. Sharkey, Convex games without side payments, International Journal of Game Theory 11 (1981)

101-106. V.B. Vilkov, Convex games without side payments, Vestnik Leningradskiva Universitata 7 (1977) 21-24

(in Russian).