This research started at the Center for Operations. Research and .... if v(P) = / v [. In order to justify the name of this section we will call 5, for a given game v,.
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
Cores of Exact Games, DAVID
Center
for Research
40,
I*
SCHMEIDLER+
in Management Science, University Berkeley, California 94720 Submitted
1.
214-225 (1972)
of California,
by Ky Fan
INTRODUCTION
A game v is a bounded, real nonnegative valued function, defined on a field .Z of subsetsof a set P, which mapsthe empty set to zero (i.e., v(+) = 0). The purposeof this paper is to characterize those gamesfor which any solution to the following problem is countably additive: minimize A(P) where X is an additive real valued function on Z such that X(S) > w(S) for all S in 2’. The conditions for the existence of minimum in this problem and its value are well known. (Referencesare discussedin Section 5.) To every game v we associatean extended real number 1a 1,called the norm of v: j 21/ = sup{Ziaiv(Si) j (ai , St) is a finite sequencein ai 3 0 and &aJ&) < 1 for all p in P>,
R x
Z,
(1)
where R is the set of real numbers and Is is the indicator function of S in 2, i.e., 1&) = 1 for p in S and Is(p) = 0 for p in the complement of S. Then the minimum exists and is equal to ) v 1iff 1v ) is finite. This problem arose in the theory of cooperative games, where it was stated somewhatdifferently. An outcome of a gamev is an additive real valued function, say h, on Z such that h(P) = v(P). If an outcome h fulfills also h(S) > o(S) for all S in 2 it belongs,by definition, to the core of the game. Our main result (Theorem 3.2) statesthe condition on v which is equivalent to the property that every elementin the core of v is countably additive. This condition is sequential continuity of v at P where a sequence(S,) in Z convergent to P if (S,) is monotone increasing and P = u S, . (S, r P). If Z is a u-field, this condition implies also(Theorem 3.10) the existenceof a measureon 2 such that every element in the core is obsolutely continuous with respect to it. We can, then, and via the Radon-Nikodym theorem, * This research started at the Center for Operations Research and Econometrics at the Catholic University of Louvain, and was completed at the University of California at Berkeley under the support of NSF Grant GS-3274. + Present address: Departments of Economics and Statistics, Tel Aviv University, Tel Aviv, Israel.
214 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
CORES
OF EXACT
GAMES,
215
I
represent every element in the core by a real-valued function on P, i.e., to indicate the personal outcome or payoff to every player, as is called an element of P. (Whereas set functions indicate the payoffs of coalitions, i.e., elements of 2, only.) However, Theorems 3.2 and 3.10 are stated only for the class of exact games. This class is introduced in Section 2. Theorem 2.2 describes the mapping that associates with every game v of finite norm an exact game v’ so that if v has nonempty core then v” has the same core. Hence the results of Section 3 can be and are restated for any game of finite norm (Corollaries 3.3 and 3.12). (The corollaries in this paper are not followed by proofs because of their simplicity.) Two additional results, 4.2 and 4.4, are proved in Section 4. Theorem 4.4 connects our work with that of Kannai [4, Theorem 11. This connection is discussed in Section 5. It should be remarked that the main results of Kannai [4] are for games defined on a topological space, which is outside the scope of this paper.
2. EXACT ENVELOPE OF A GAME 2.1. Notations and conventions. For S in Z we denote by S the elementP\S in L!Yand by S* the indicator function of S. Similarly .Z* denotesthe set {S* / S E Z) and B is the Banach spacespannedby Zc* with the sup norm. The spaceof bounded additive set functions on .E, ba is isometrically isomorphic to the norm-dual of B; for h in ba we denote by h* the correspondingcontinuous linear functional on B. Inequalities between functions are assumedto hold pointwise. The number 0 denotesthe origin of any linear spacementioned in the paper. IfL is a spaceof functions L, = (X EL / x 3 0}, in particular R, is the set of nonnegative real numbers. A functional f on a linear spaceis superlinearif it is superadditive (i.e.,f (x + Y) 3 f(x) + f (y)) and posi‘t’ive homogenous(i.e., f (Xx) = Af(x) for h E R, . It is sublinearif -f is superlinear. To avoid uninteresting complications it is assumedthroughout this paper that the sets P and C are nonempty and v is not identically zero. THEOREM 2.2. Let v be a game with $nite norm on 2 and considerthe following three real valued functions on .Z:
6(S) = inf{A(S) / h Eba, h > v and h(P) = j v I}
SE.z
(2)
V(S) = sup(Zaiv(S,) / (ai , &) is a$nite sequence in R, x .Z and Z’aiSi* 6 S*}
SE 2
t3)
V=(S)= sup(Zaiv(SJ - a 1v j 1(ai , Si) is a jnite sequence in R, x Z, a E R, and ZaiSi* - aP* < S*> SE2
(4)
216
SCHMEIDLER
Tken G(P) = E(P) = v(P)
and
6=v’>v.
Proof. The functions B and v are well defined because 1v j is finite and G(P) = 1v 1 = u(P) < E(P). We will show that V=(P) < j v / . Otherwise there are a finite sequence (ai , SJ and a 3 0 so that ,Zaiv(Si) - a / ZJ1 > 1v / and Za,S,* - aP* < P*. Then 2 -s.a+1
v(SJ > I v 1
Z -f3a+1
and
si* < p*,
a contradiction to the definition of ) v j in (1) and the proof of the first part of the theorem is completed. The inequality E > v is obvious. Also the proof of 6 > V=is simple. Given A in ba so that h 3 v and A(P) = / v /, we get S* >, ZuiSi* - aP* implies h(S) = h*(S*)
3 h*(ZaiSi*
- aP*)
= ZaiX(Si) - ah(P) 3 Zuiv(Si)
- a 1v (.
Hence, h(S) 3 Z(S) an d as h was arbitrary 6(S) 3 E(S). The passage to the limit is unnecessary because of the following remark. Remark 2.3. The infimum in the definition of d in (2) can be changed to minimum because the set {h E ba / h 3 v and X(P) = 1v I} is compact in the B topology of ba. To complete the proof of 2.2 the inequality V=3 8 is proved via the separating theorem. To this end we define a function f on B:
f(x)
= sup(zhiv(Si) - a j v I J (ai , SJ is a finite sequence in x E B. R, x Z, a E R, and .ZaiSi* - aP* < x}
The functional f is the extension of B from Z* to B with ,Z*). f is superlinear on B. Homogenity of f superlinearity of f let X, y E B and E > 0 be given. sequences (ai , Si) and (bj , Tj) in R, x 2 and zh,S,* - aP* < x, ZbjTj* - bP* < y,
(5)
(where we identify Z is obvious. To prove Then there are finite a, b in R, so that
ZU,V(Si)- a / v I > f (x) - $ and Zb,v(T,)
- b 1v j 3 f(y)
- +.
The two inequalities in B imply Za,S,* + Zb3Tj* - (a + b) P* < x + y and this together with the other two inequalities imply f(x +Y)
>f(x)
+f(Y)
- 6.
CORES
OF EXACT
GAMES,
217
I
Given S in Z let A = {x E B I f(x)
B={xEB~x,
and C=
stB~x+S* 1
if
f(S*)
# 0.
Let D be the convex hull of B union C. The set A is convex because of the superlinearity of f. We will show later that int A n D = $3. Then by the separation theorem (See Dunford and Schwartz [2, V. 2.81) there is a nonzero linear functional F on B such that
F(x)< F(P*)= F (&
S*) 0, hence we assume that F(P*) = 1v j. For every T in Z, nT*+P*eA hence F(T*) 30. If f(T*)=O then F(T*)>f(T*). If f(T*) > 0 let r > 0 be such that f (rT*) = 1v 1. Then F(rT*) 3 I v 1 or F(T*) >f(T*). W e h ave, of course, that F(S*) = f (S*). So the set function induced by F is in the core of v’ and obtains v’(S) = f (S*) in S. We have to prove that f (x) ,< / v I f or x in D. Because of the monotonicity off it is sufficient to show that f(rP*+sf#S*)$Iuj
forO
