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All Rights
rrom JOURNAL OF ECONOMIC THEORY
Reserved
by Academic
Press, New York
Vo1.27,No.I,June1982
Printed in Belgium
and London
The Number of Commodities Required to
Represent a Market Game*
SERGIU HART
Department 0/ Statistics, Tel-Aviv University,
. Tel-Aviv, Israel
Received June 25, 1980; revised October 27, 1980
An n-person game with side payments that can arise from an economic market is
called a market game. It is proved here that any n-person market game can be
obtained from a market with at most n - I commodities. Moreover, no smaller
number will suffice in some cases (e.g., the unanimity game). This result is then
applied to settle a question regarding market games with a continuum of traders.
Journal a/Economic Literature Classification Numbers: 021, 022.
The following problem is studied here: What is the minimal number of
commodities required to represent a totally balanced n-player game (with
side payments)? I Shapley and Shubik [4] showed that every such game is a
market game-i.e., it can be derived from a "market"; their construction (of
the "direct market") requires n commodities, It is proved here that n - 1
suffice (Proposition 2); moreover, this bound is exact-there
are n-player
market games for which n - 1 goods are necessary (Proposition 1). This last
result is then applied to non-atomic games (where the original
question-setlled
here as Proposition 3-was raised).
The n-player unanimity game is defined as follows: v(S) = 0 for all S. N
and v(N) = 1, where N = {I, 2,..., n}. It is clearly a totally balanced game; a
market which generates it, is for example,2 (N, 1R~-l,A, U), where, for ii= n,
* This research was supported by the Institute for Advanced Studies, the Hebrew
University, Jerusalem, Israel, and by National Science Fout;ldation Grant SOC75-21820-AOI
at the Institute for Mathematical Studies in the Social Sciences, Stanford University. The
author wishes to thank R. J. Aumann, J. F. Mertens and L. S. Shapley for very helpful
discussions.
I For the non-side-payments case, where the problems are not yet completely settled, see
16], which includes an overview and references.
2 The notations of Shapley and Shubik [4] are followed here. N is the set of agents, IR~ the
commodity space, A = jallieN the initial endowments and U= {ui},eN continuous concave
utility functions (not necessarily non-decreasing). The market game v is defined by v(S) =
max/LieS Ui(XI) I LieS xt = LieS ai, all Xi E IR~} for all SeN.
163
0022-0531/82/030163-07$02.00/0
Copyright @ 1982 by Academic Press, Inc.
All rights or reproduction in any rorm reserved.
.
164
SERGIU
HART
al is3 the ith unit vector in IRn-1 and ui ==-0; for i = n, an = 0 and
Un(Xl' x2'..., Xn-I) = min{xl' X2,"" Xn-J I.
PROPOSITION1. Let (N, IR~ , A, U) be
unanimity game v on N. Then m ~ n - 1.
a
market
generating
the
Since v({il) = 0, we have ul(al) = 0; since v(N\UI) = 0, we have
Proof
!~
0 = max
~ xl = ~
ul(xl)
L"d
I
al, all xl E IR~ /,
!
1*1
1*1
hence the maximum is attained
= al
at xl
for all j
Ll"'iO=O).
Now v(N) = 1, hence there is an allocation (/)leN
,-' Y I \' al
=
'
leN
L Ul(/)
leN
leN
*-
= tui(al)
= 1.
+ 00)
as follows:
tal + (1 - t)/,
for x =c=
+ (1 - t) Ul(/),
with 0 ~ t
~ 1,
otherwise.
=-00,
Then al are polyhedral concave functions
and al ~ Ul. Let {J be the market game
(J~ v; however, since ul(al) = ul(al) and
(J= v.
Now consider the functions us' defined
us(x) = sup!
=
such that
Let us define now new concave functions al: IR~ -+ [-00,
al(x)
i (L1*1 ul(al)
/L
for S eN,
L Xl
UI(XI)
leS
(cf. Rockafellar [3, pp. 172-173]),
generated by (N, IR~,A, 0), then
al(yl) = Ul(/), we actually have
I
as follows:
= x, all Xl E IR~ /.
!
leS
Then Us are polyhedral concave functions (cf. Rockafellar
19.3.4]). In particular, for each i E N, the function
at I2JeN\Ii) al (cf. Rockafellar
uNIJ/)
[3, Corollary
is subdifferentiable
[3, Theorem 23.10]); i.e., there is pi E IRm
such that
pi E auN\(/)
( L ).
al
leN\(/)
J Subscripts will be used for coordinates, and superscripts for players.
NUMBER
165
OF COMMODITIES
But this clearly implies pi E 8uk(ak) for all k 1=i; indeed,
uAk (x ) = '"
jf:1.k
j
Aj
U (a )
<UN\(i)
+ u (x )
Ak
A
/'
uN\(i)
"'"
(?:.aj)+pi. (.2;:
J*'
= uk(ak) + pi . (x - ak).
( '\'
j
a + x
j~k
)
aj+x-?:.aj
J*"k
J*'
)
Hence {8uk(ak)}Z=1 is a collection of n convex sets in IRm, such that the
intersection of every n - 1 of them is non-empty; if m n - 2, Helly's
theorem (Rockafellar [3, -Theorem 21.6]) would imply the existence of a
vector p with p E8uk(ak) for all kEN. Then
<
n
1=
L
n
Uk(yk)
k= I
< k=L 1 !uk(ak) + p . (yk -
ak)] = 0,
since uk(ak) = 0 and Lk yk = Lk ak. This contradiction shows that m;;;::
n - 1.
Remark. Note that each Ui could be just a concave function with values
in [-00, +00) (i.e., not necessarily finite on all IR~).
PROPOSITION2. Let (N, v) be a totally balanced game. Then v can be
derived from a market with n - I goods. Moreover, the set of competitive
payoff vectors of this market and the core of v coincide.
Remark. In Shapley and Shubik [5], it is shown that the competitive
vectors of the direct market-with
n goods--enjoy the same property.
Proof. Consider the restriction of v to N' = {I, 2,..., n - I}. It is a totally
balanced game of n - 1 players; let (N', IRn- I, A " U') be the corresponding
direct market. Define an =0 and un: IR~-l~ IR by
un(x)
= max
\L
Ysv(S)
ISC:N
.
I
LYseS
Sc:N
= (x, 1), all Ys;;;:: Of,
where (x, I) = (x P X2 ,..., xn I> 1) E IRn, and es E IR~ is the characteristic
vector of S (i.e., ef = 1 or 0 according to i E S or i E S). It is easy to check
that un is a continuous concave function. Denote by (J the market game
generated by (N, 1R~-I,A, U).
We show now that (J= v. For S eN', it is clear that 6(S) = v(S). When
n E S, we have
6(S);;;:: un
L ai ) + L Ui(O),
( ieS
ieS'
l
166
SERGIU
HART
= S\{n}. Now Ui(O) = 0 and (Lies ai, 1) = es; therefore, choosing
Ys= I and YT= 0 for T -1=S in the definition of Un(Lies at), we obtain
v(S) ~ v(S). Conversely, let (xi)ies be an S-allocation (Le., Lies Xi =
such that
LieS at). Then, for each i E S', we obtain Y~~ 0 for all TeN'
where S'
Ui(Xi)
=
L
y~v(T),
TeN'
y~eT = (Xi, 0)
L
TeN'
(see Shapley and Shubik [4, Eqs. (4-2) and (4-3)1). For i= n, we have
0 for all TeN such that
Y~ ~
un(xn)
L
TeN
=
L
Y~V(T),
TeN
y~eT = (xn, 1).
Now let YT= LieS Y~for TeN',
and YT= y~ for T3 n; then
2:YTeT = (2: 1)= (2:ai, I )= eS,
Xl,
TeN
ieS
ieS
which implies
2:YTV(T)~ V(S),
TeN
since v is totally balanced. Hence
L
Ui(Xi)
ieS
=
L
YTV(T) ~ V(S),
TeN
~ v(S). This4 completes the proof that the
above market indeed generates v.
We come now to the second part. We have to show that if fJ =
(fl1,fJ\...,fJn) belongs to the core of v, then it is a competitive payoff vector
from which it follows that v(S)
of
our
market
(the
other
direction
is
well
known).
Denote
n
=
(fl1,fJ2,...,fJn-l)ElRn-l, xl=(O,O,...,O)ElRn-1
for i-l=n and xn=
(1, 1,..., 1) E IRn-l. We will show that (n, x) is a competitive equilibrium,
and that the corresponding payoff vector is precisely fl.
4
The reason the above construction need not work if we do not use the direct market for
N', is that the vectors LIeS a' for S c N' may be linearly dependent and the function u"
cannot be defined.
I
NUMBER
167
OF COMMODITIES
Indeed, Lr:/ 7ri = LiEN' pi ~ v(N'); hence, for all i EN', we have Ui(X) ~
7r' x for all x E IR~-l (the same proof as Shapley and Shubik [5, Eq. (16)]).
Therefore.
pi
= Ui(O) -
7r . (0 - ai)
~
- 7r . (x
Ui(X)
- ai).
As for i = n, note that
un(xn) = v(N)
(see the proof above that v(S)
un(xn)-7r'
= O(S),
for S = N); hence
I
(xn-an)=v(N)-
pi=pn.
iEN'
Let x E IR~-l, then
LYseS
ScN
implies, since pEcore
v, that
,-' Ysv(S)~
ScN
= (x, 1)
I
ys(eS 'P)=(x,
1),p=n.x+pn;
ScN
therefore
un(x) - 7r . (x - an) ~ pn = un(xn) - 7r . (xn - an),
completing the proof.
Remark. The idea behind the above representation of v is that the nth
player does not "bring" to the market any goods-he contributes, instead,
his "good" utility function (or, production function--depending on the interpretation). This is possible because the utilities are transferable.
An n-player game v can be regarded as a (real) function defined on the
vertices of the unit n-cube (by identifying S c N with es E {O,I} n). Then v is
totally balanced if and only if it can be (uniquely) extended to a continuous,
concave, and positively homogeneous of degree one function on the unit ncubes [0, lY (or even, on all IR~); see Shapley and Shubik ([4, Sect. 4; 5,
Footnote 5D.
This suggests a natural candidate for the space of non-atomic market
games. With the additional requirement of differentiable utilities (rather than
just continuous), Aumann and Shapley [1, p. 240] consider the space H of
all games in pNA (the space of "differentiable" non-atomic games) that are
, This
is the so-called set of "fuzzy" or "ideal" subsets of N.
Il
168
SERGIU HART
concave6 and positively homogeneous of degree one. H + is then the subset of
all monotone games in H.
It is known that H + contains the differentiable non-atomic market
games-see Aumann and Shapley [1, proof of Proposition 40.26 D. We show
here that the converse is false.
PROPOSITION
3. There are non-atomic games in H + that are not market
games (i.e., cannot be generated by markets).
We prove first a preliminary result which is a direct consequence of
Proposition 1. Let (/,~) be a measure space, assumed to be isomorphic to
the unit interval [0, 1] with its Borel a-field. Let 11.,112"'" Iln be n mutually
singular non-atomic probability measures on (/, ~), and define
v= \111.'112""
'Iln.
Then v is a market game. For example, it can be generated using n
commodities as follows: let Tp T2,..., Tn be a partition of / such that Ti is a
support of Ill; the initial endowment aCt) of each t E TI is the ith unit vector
el E IR~; the utility function of each t E / is u(x) = n . \Yx. . X2 . ... . Xn ,
and the population measure is 11= (p 1 + 112+ ... + Iln)/n. Actually, n - 1
commodities suffice, as in Proposition 2.
PROPOSITION4. A non-atomic market generating the above v has at
least n - 1 commodities.
Proof Let «/,~, v), IR~, {a(t)}tEP {UtIlE/) be a market generating v. Let
T., T2,..., Tn be as above, and define, for i = 1,2,...,n,
al= fT,adv,
UI(X)
= sup!
Ut(x(t» dv(t)
IfT,
fT,
x dv
I
= x, x(t) E IR~ for all tl.
!
Then UI is a concave function (with possible values -00).
replace each Tj by a player i, we obtain the n-player
moreover, (N, IR~, {al}iEN' {UI}IEN)is an n-player market
Proposition 1 .(see the remark following its proof), m ~ n -
Note that, if we
unanimity game;
generating it. By
1.
Proof7 of Proposition 3. Let {llntln=I,2,...;I=I,2,...,n be a (countably)
infinite triangular array of mutually singular non-atomic probability
6 Their definition
actually
requires super additivity
instead
presence of homogeneity,
the two concepts are equivalent.
7
The idea of this proof is due to R. J. Aumann.
of concavity;
however,
in the
NUMBER
.
169
OF COMMODITIES
measures on (1, 'if) (for example, take the normalized restrictions of a nonatomic measure
to a countable
partition
of 1 into non-null
sets). Define
CD
v=
I
2-n
<!llnl'lln2""
'Ilnn,
n=l
and let vN be the partial sum to N. Then vNE pNA (by Theorem C in
Aumann and Shapley [1, p. 25 j, vr;.;:;;E pNA, which is an algebra), UNis
positively homogeneous of degree 1 (see [1, Eq. (22.18)]), it is superadditive
and monotone. Hence vN E H +. Moreover, v ---:vN is also a monotone game,
hence
.
CD
IIv -
vN11
= (v -
un)(I)
=
1
L 2n =
n=N+ 1
1
2N
~
0,
.
which, since H + is closed, implies v E H + .
If v is generated by a market, then, by considering the restriction to the
support of jllni}7= it must have at least n - 1 goods (Proposition 3). This is
I'
true for all n, so no finite number of commodities suffice.
The importance of this last result lies in showing that, in order to study
(differentiable) market games, one need not consider the whole space H +;
the finite dimensional "character" of the markets may be a useful property.
A similar situation occurs in the non-differentiable case-cr. Hart [2, Open
Problem A]. See also the forthcoming paper of Dubey and Neyman.
REFERENCES
l. R. J. AUMANNAND L. S. SHAPLEY, "Values of Non-Atomic Games," Princeton Univ.
Press, Princeton, N.J., 1974.
2. S. HART, Measure-based values of market games, Math. Operations Res. 5 (1980),
197-228.
3. T. R. ROCKAFELLAR,"Convex Analysis," Princeton Univ. Press, Princeton, N.J., 1970.
.
4. L. S. SHAPLEYAND M. SHUBIK,On market games, J. Econ. Theory I (1969),9-25.
5. L. S. SHAPLEY AND M. SHUBIK, Competitive outcomes in the cores of market games,
Internat. J. Game Theory 4 (1975),229-237.
6. R. J. WEBER, Attainable sets and markets: An overview, in "Generalized Concavity in
Optimization and Economics" (M; Avriel, S. Schaible and W. T. Ziemba, Eds.), Academic
Press, New York, 1981.
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