MATHEMATICS OF OPERATIONS RESEARCH
Vol. 10. No.3. August 1985
Printed in U.S.A.
SEMIV ALVES OF POLITICAL
ECONOMIC
GAMES*
ABRAHAM NEYMAN
Hebrew University
The class of continuous
nonatomic games .
semivalues is completely characterized
for various spaces of
..
1. Introduction. Semivalues are defined in [4] where characterization of the semivalues is given for two basic spaces: the space of all finite games, and the space of
I "differentiable"
nonatomic games, i.e., pNA. In the purely economic situation, we
usually encounter games in pNA (or in pNAD); but in many political economic
situations, as in the Aumann-Kurz models of power and taxation ([1],[2]), we face
games which are the product~ of weighted majority games by games in pNA. These
games are members of other spaces which contain pNA and which we will refer to as
spaces of political economic games. In this paper we will characterize all semivalues on
spaces of political economic games. §3 presents a characterization of all continuous
semivalues on a typical class of political economic games, followed by a detailed proof.
In §4 we introduce further results without proofs. The proofs of the results in §4 are
more involved than that of §3, but actually are based on the same ideas and thus we
decided to omit them from our paper.
I
.\
t
t
I,
2. Preliminaries. Most of the definitions and notations are according to [3]. Let
(I,-~) be a measurable space isomorphic to ([0, l],.@), where.@ is the cr-field of Borel
subsets of [0,1]. A set function (or game) is a function v : ~ ~ IR with v(4)) = O. A set
function v is monotonic if for all T and S in~,
T c S ~ veT) < v(S). The space of all
set functions on (I,~)
that are the difference of two monotonic set functions is
denoted BV. The space of all bounded finitely additive set functions is denoted FA,
and its subspace of all nonatomic measures is denoted NA. If Q C BV then Q +
denotes the subset of Q of all monotonic set functions. A mapping '1': Q ~ BV is
positive if '1'( Q +) C BV+. Let G denote the group of automorphisms of (1, ~). Each 9
in G induces a linear mapping 9* of BV onto itself that is given by (9*v)(S)
= v(9S)
for all S in~.
A subset Q of BV is symmetric if for each 9 in G, 9*Q C Q.
Let Q be a ~ymmetric subspace of BV. A semiva/ue on Q is a positive linear
mapping IjI from Q into FA that satisfies:
IjI is symmetric, i.e., 1j19* = 9*1jI
if
j
I
i~
i
v E Q n FA
then
for all 9 in G.
(2.1)
(2.2)
IjIv = v.
The bounded variation norm of a set function v in BV is defined by IIvii = inf(u(1) +
w(I)) where the infinum ranges over all pairs of monotonic set functions u, w with
~ = u - w. A ~ondecreas~
sequence of sets ~ ~ o.f the .form Q : So C SIC ... C Sn
IS called a cham. The vanatlOn of v over a cham Q IS defmed by IIvll~ = 2:7= tJv(SJ v(Si-t)l. It is known [3, Proposition 4.1] that IIvii = supllvll~ where the supremum is
*Received February 5, 1979; revised December 18, 1983.
AMS 1980 subject classification. Primary 90D12, 90D13.
1AOR 1973 subject classification. Main: Games.
OR/ MS Index 1978 subject classification. Primary: 234 Games/group
Key words. Game theory, Shapley value, semivalue.
decision, cooperative.
390
0364-765X/85 /1003 /0390$01.25
Copyright © 1985, The Institute of Management Sciences/Operations Research Society of America
391
SEMIV ALVES OF POLITICAL ECONOMIC GAMES
taken over all chains Q. If Q is a subspace of BV and 0/: Q--'7BVis linear then 110/11 is
defined as sup{llo/vll : v E Q, Ilvll = l}.
The space pNA is the closed subspace of BV that is generated by powers of
non atomic measures.
Let f denote the family of all measurable functions from I to [0,1] (measurable
with respect to the a-fields -g and &?J). There is a partial order on f: f ;> g if
f(s) ;> g(s) for all s in I. A real valued function w on f with w(O) = 0 is call~d an
ideal set function; it is called monotonic if f :> g implies w(/) ;> w(g). For every ideal
set function w we denote by Ilwll the supremum of2:7=llwCD - wC.t-I)1 taken over
all increasing sequences 0 < fa < fl < ... < in < 1 measurable functions. The indicator function of a set S in -g is denoted Xs i.e., Xs(s) = 1 if s E S and equals 0 iL~·_!l_S.
We will sometimes write S for Xs, t for tXI and tS for tXs.
It is known [3, Theorem G] that there is a unique linear mapping that associates
with each set function v inpNA an ideal set function v* such that (vw)* = v*w* for all
v, w in pNA, v* is monotonic wherever v is in pNA +, IIvl[= Ilv*[I, and such that
p.*(/) = fddp. for all p. in NA and all f in f.
Denote 3v*(t, S) = (d/ dT)v*(t + TS)T=O' By Theorem H of [3] we know that for
each v in pNA and each S in -g, the derivative 3v*(t, S) exists for almost all tin [0, 1]
and is integrable over [0, 1] as a function of t.
We denote by W the set of nonnegative functions g in Loo([O,1]) with f~g(t)dt
= 1.
The characterization of the class of semivalues on pNA is given in [4, Theorem 2].
THEOREM
2.3 ([4, Theorem 2]).
given by
For each g in W the mapping o/g:pNA
--'7
FA that is
is a semivalue. Moreover, every semivalue on pNA is of this form. The map g--'7 o/g of W
onto the class of semivalues on pNA is a linear isometry.
Define DIAG to be the set of all v in BV satisfying: there exist a positive integer k, a
k-dimensional vector g of probability measures in NA, and a neighborhood U in IRk of
the diagonal [0, g(I)] such that if g(S) E v then v(S) = O. A semivalue 0/ on a
symmetric subspace Q of BV is diag~nal if \{Iv= 0 for all v E Q n DIAG.
PROPOSITION
2.4.
Continuous semivalues are diagonal.
PROOF. The proof in [7] that continuous values are diagonal does not make use of
the efficiency axiom and therefore the same proof works here.
Another result which will be used in the proofs of the present paper is:
PROPOSITION
2.5. Let Q be a symmetric subspace of BV, and let 0/ be a semivalue on
+ and f is defined on the range of p. with fop. E Q, then o/(f 0 p.) = ap.for
some constant a in IR.
Q. If p. E NA
PROOF. Follows from the proof of Proposition 6.1 in [3].
3. Characterization of the semivalues on a class of political economic games. In the
purely economic situation, we usually encounter games in pNA (or in pNAD-· the
closed linear space generated by pNA and DIAG) but in many political economic
situations we face games of the form .v = uq where q is in pNA and u is a jump
function with respect to a given NA probability measure p., i.e.,
u(8)= {~
S) ;> a,
if
J..L (
if
J..L(S)<a.
Such games arose for instance in models for taxation (see Aumann-Kurz
[1] and [2]).
392
ABRAHAM
NEYMAN
We denote by u * pNA the minimal linear symmetric space containing
games of the form uq where q E pNA and a is a fixed number in (0,1).
pNA
THEOREM A. For any pair (a, g), a E IR+, g E W, there is a semivalue
u * pNA such that Jor any q E pNA
=
(o/(a,g)q)(S)
Moreover,
g(t)aq*(t,S)dt
+
semivalue on u * pNA
Ilo/(a,g)II = max(a, IlqIL"J·
any continuous
o/(a,g) is 1-1 and
o/(a,g) on
1
= aq*(a)p.(S)
(o/(a,g)(uq))(S)
~
L
and all
(3.1 )
and
L
1
(3.2)
g(t)aq*(t,S)dt.
is oj that Jorm.
The mapping
(a, g)
The proof of the theorem is accomplished
in several stages. First we shall state and
prove a result on the range of vector of members of pNA. This is a generalization
of a
result of Dvoretzky, Wald and Wolfowitz [5, p. 66, Theorem 4].
LEMMA3.3. Let v be a Jinite dimensional vector oj measures in NA, and let m be a
positive integer. Then Jor each m-tuple JI' ... , Jm of ideal sets such that fl
+ fm
= 1 = XI' and each k-tuple ql' ... , qk of members of pNA, and each E
0 there is a
partition (TI' ... , Tm) oj I, Ti in ~ such that for all A C {I, ... , m} and aliI < j < k
+ ...
>
V
(
UTi)
J(
=
iEA
"£
iEA
fi)dv
and
CJj( UTi)
- q/(
is replaced
by pNA'
/
REMARK. The same result holds if pNA
iEA
)1 <
j;
"£
iEA
E.
(replace
in the proof
II II by II II')·
PROOF. From the definition of pNA it follows that for each 1 < j < k there exists a
polynomial vj of NA-measures;
vj = P/p.(, ... , p.~) with Ilqj - v)1 < E. By Theorem 4
of [5] there is a partition (TI' ... , Tm) of I, Ti in ~ such that for all 1 < i < m,
1 < j < k, and 1 < I < nj
v (Ti)
=
J ./;
dv
and
=
p.1(Ti )
J
j; diLl·
From the finite additivity
of members
of NA, we deduce that for each A C
{1, : .. , m}, I < j < k and I < 1< "i' v(T(A»
= J(A)dv and p./(T(A» = J(A)
dp.1 where T(A) = UiEA Ti and J(A) = 2:iEA j;. From the last equalities and the
properties of the mapping V~
v*, it follows that also v/T(A»
= v/(j(A».
Thus
J
ICJj(T(A)) - ql(J(A))1
<
Iqj(T(A))
- vj(T(A))1
+ Iv/ (J(A))
and as Ilv*11 = Ilvll for each v E pNA,
proof of Lemma 3.3 .•
J
+ Ivj(T(A))
- q/ (J(A))I
<
IIqj - v)1
ICJj(T(A» - q/(j(A»1
We will use in our proof the following
immediat~
- v/(J(A))I
corollary
< 2E.
+ 0 + Ilv/ -
q/II
This completes
of Lemma
the
3.3.
COROLLARY3.4. Let v be a Jinite dimensional vector of measures in NA, and let m be
a positive integer. Then Jor each m-tuple JI' ... , Jm oIideal sets such that JI < J2 < ...
< Jm' and each k-tuple ql, ... , qk oj set Junctions in pNA, and each E > 0 there is an
m-tuple T1, T2, ... , Tm oj sets in ~ such that TI C ... C Tm and for aliI < j < k and
all 1 < i < i' < m
v ( Ti )
Iqj(Ti) - CJj*(j;)1 <
E
and
=
J./;
dv
and
Iqj(Tr - Ti) - ql(./;' -
,/;)1 <
E.
393
SEMIV ALVES OF POLITICAL ECONOMIC GAMES
PROOF.
fm-I,1
Follows by applying Lemma 3.3 to the m + I-tuple fl' f2 - fl' ...
,fm-
- fm'
We will proceed in order to show that (3.1) and (3.2) define a unique linear
symmetric operator from u * pNA into FA. For this we shall need the following lemma.
LEMMA
3.5.
0i E G, i = 1, ...
If w = v + 2:,7= ](Oi*U)qi is monotonic, where v E pNA,
, n, then for any S E ~ and gEL 00' g :> 0,
S) dt:> 0,
la g(t)av*(t,
(]
J
a
and
(3.6)
(I
n
g(t)av*(t,S)dt+
qi E pNA
i~IJa
g(t)aqj*(t,S)dt:>
0
(3.7)
and
n
2:
j=
P.(OiS)qj*(a)
:>
(3.8)
Q.
1
Assume that w is monotonic. For proving (3.6) it is enough. to show that
for any t with I 0 < t < a for which av*(t, S) is defined, av*(t, S) :> O. Let 0 < t < a
and let 0 < h be such that t + h < a. For such t and h, t + hS < t + h and therefore
PROOF.
(OtJ.L)*(t
+
hS)
< (Wp.)*(t + h)
= t
+h <a
for each
i = 1,
,n.
For any E > 0 we could apply Corollary 3.4 to the'vector v = (OiJ.L,
, O:p.) of
nonatomic measures, the 2-tuple t, t + hS and the set function v in pNA to show the
existence of two sets TI' T2 in ~ with TI C T2 and such that (Oi*J.L)(TI) = (OJ*J.L
)(t) - t
< a, (OJ*J.L)(T~ = (OtJ.L)(t + hS) < a for all i = 1, ... , n and such that Iv*(t) - v(TI)1
< E and Iv*(t + hS) - veT ~I < E. Therefore, on the one hand, (Wu)(TI) = (Otu)(T ~
= 0 which implies that weT ~ - w(Tl) = veT ~ - v(TI)'
and on the other hand,
veT ~ - v(TI) < v*(t + hS) - v*(t) + 2E. Altogether v*(t + hS) -v*(t)
:> weT ~ w(TI) - 2E. As w is monotonic we deduce that v*(t + hS) - v*(t) :> -2E, and as this
holds for any E> 0 we conClude that v*(t + hS) - v*(t) :> 0 and therefore av*(t, S)
:> 0 for any 0 < t < a for which av*(t"S) is defined. This completes the proof of (3.6).
For proving (3.7) it is enough to prove that for any t with a < t < I for which all the
derivatives av*(t, S) and aqt(t, S) exist,
n
av*(t,
S). +
2: aqj* (t, S) :> O.
i= 1
Applying Corollary 3.4 to the vector (OiJ.L,... , O:J.L) of nonatomic measures, the
2-tuple t, t + hS (where 0 < h is such that t + h < 1) and the members v, q], ... , qn in
pNA we have for every E> 0 two sets TI C T2 in ~ such that for every 1 < i ...;n,
a < t = (OtJ.L)*(t) = (OtJ.L)(Tl) < (OJ*J.L)(T~ and Iqj(Tl) - qi*(t) I < E, Iqj(T~
- qt(t +
hS)1 < E, and Iv*(t) - v(T])1 < E, Iv*(t + hS) - v*(T ~I < E. Therefore,
n
w(T2) - weT])
=
v(T2)
-
v(TI)
+
2: qj(Tz)
j=
- qj(T])
1
n
...;v*(t
+
hS) - v*(t)
+
2: qt(t
+
hS)
i=1
- qj*(t)
+ 2(n + I)E.
.
Again as this holds for all E > 0 and as w is monotonic, it follows that av*(t, S) +
2,7= ]aqt(t, S) :> 0, which completes the proof of (3.7).
The proof of (3.8) will make use of
LEMMA
functions
3.9. Let J.LI' ••. , J.Lnbe nonatomic probability
in pN A. Then for every 0 < a < 1 and every E
measures and ql' ... , qm set
and every 1 ...;k < 1) there
>0
1 Along the proof a and J1. stand for the fixed scalar and the probability measure, respectively, that are
used in the definition of the set function u.
394
ABRAHAM NEYMAN
are two sets TJ, T2 in ~,
TJ
Iqj(Tt)
C
T2 such that for all 1
< E,
- q/(O')I
:< i :<
n and for all
<E
Iqj(T2) - q/(O')I
Iqj(T2\Tt)1
< E,
fLi( T2 \ Tt)
< E,
fLi(T2) ;;;. a> fLi(Tt)
iff
1 :< j :< m
fLi = fLk'
PROOF.
Let K = {I :< i :< n : fLi = fLk}' By Lyapunov's theorem there is T in ~ such
that fLi(T) = fLk(T) iff fLi = fLk' Let b = fLk(T). Observe that for sufficiently small y > 0,
a + y(T - b) is an ideal set and that fLt(O' + y(T - b)) = a iff i E k (i.e., iff fLj = fLk)'
If (J,)~= I is a sequence in f that converges uniformly to fin f then for every q in
pNA, q*(j,.) converges to q*(j). (All that is needed for that conclusion is that fL*(jr)
converges to fL(j) for every nonatomic measure fL.) Therefore there is y > 0 sufficiently
small so that 0'+ y(T - b) is an ideal set and such that for all 1 :< j :< m, Iq/(O' +
y(T - b)) - q/(O')I < E/3. Fix such a y > 0, and observe that fJ(O' + y(T - b))~
0'+
y(T - b) as fJ < I converges to 1. Therefore for sufficiently large fJ < 1, for all
l:<j:<m
+ y(T-
Iq/(fJ(O'
b»)1 < E/3,
b»)) - q/(O' + y(T-
Iq/(fJ(O'
As fJ-;*(O' + y(T - b))
have
b»))1 < E/3,
fJ)(O' + y(T-
Iq/«I-
+ y(T-
and thus
< 2E/3.
b»)) - q/(O')I
iff i E K, it follows that for sufficiently large fJ
= a
iff
fLt ( a + y ( T - b» ;;;.a > fLt ( fJ (a + y ( T - b)))
<
1 we also
i E K.
Fix such a fJ < 1 and apply Corollary 3.4 to the 2'-tuple fJ(O' + y(T - b)) «a +
y(T - b)) with E/3, the vector (fLJ, ... , fLn) of nonatomic measures and the members
qt, ... , qm of pNA to show the existence of Tt, T2 E ~ with Tt C T2,
fLi(Tt) = fLt (fJ(
y(T - b»)),
0'+
fLi(T2) = fLt(O' + y(TIq/ (a
+ y(T
- b») - qj(T2)1
< E/3,
<
Iqj(TJ) - q/(O')I
Iqj(T2 \ Tt)1
fLi(T2)
>
a
< E/3,
i :<n, 1 :< j
Iqj(T2) - q/ (0')1
n,
1 :< j :< m,
1 :< j
b»)) - qj(T2\Tt)1
fJ)(O' + y(T-
Altogether, we conclude that for all I
:< i :<
1 :< i:< n,
b»),
!q/( fJ( 0'+ y(T - b»)) - £]j(T1)1
Iq/«l-
1
<
:<
m,
< E/3.
m
< f./3 + f./3 < f.,
< f./3 + f.!3 + E/3 < E,
< f./3 + f./3 < E,
> fLj(Tt)
iff
and
fLi = fLk'
which completes the proof of Lemma 3.9.
We return now to proof of (3.8) of Lemma 3.5. Observe that
that for every 1 :< k :< n if K(k) denotes the set of all 1 :<
then 2-iEK(k)qt(O');;;' O. Apply Lemma 3.9 to the .nonatomic
Or fL.. , 0: fL and the set functions v, q), ... , qn in pNA to
it is sufficient to prove
i :< n with 0t fL = 0: fL
probability measures
show the existence of
395
SEMIV ALVES OF POLITICAL ECONOMIC GAMES
Tj, T2 in ~
with Tj C T2 and such that for all I
i <; n,
<:;;;
- v*(a)1
< E,
Iv(T2) - v*(a)1
< E,
JqJTj) - qj*(a)I
< E,
Iqj(T2) - qj*(a)1
<
Iv(TI)
OJ*p.(T2) ;> a>
O;*p.(Tj)
iff
E,
i E K(k).
Therefore
2:
<:;;;2E+
q;*(a)+2€ll
jEK(k)
2:
<
j
q;*(a) + 2(n + I)E.
E K(k)
As this holds for every E > 0 the assumption that W is monotonic implies that
2:jEK(k) qi*(a) ;> 0 which completes the proof of Lemma 3.5.
3.10. Let g be in Wand
semivalue o/(a,g) on u * pNA.
a in R +. Then (3.1) and (3.2) defines (uniquely) a
LEMMA
PROOF. Any element W in u * pNA is of the form W = v + 2:7= lOj*u)qj,
v, qj E pNA. By linearity and symmetry, it follows from (3.1) and (3.2) that
o/(a,/)w( S) =
10
n
I
S) dt+
g(t)3v*(t,
j~1
i
Oi E G,
I
g(t)3q;* (t, S) dt
n
+
2: aqj* (a)(
i= I
OJ* J.L)(S).
(3.9)
.
We have to show that o/(a,g) is well defined, i.e., that it is independent of the
representation of w. Because of the linearity it is enough to show that if W = 0 then
o/(a,g)w = O. If W = 0 then by Lemma 3.5 we conclude that o/(a,g)w(S) ;> 0, and that
o/(a,g)(- w)(S) = - o/(a,g)w(S) ;> 0 which means that o/(a,g)w = O. Linearity and symmetry of o/(a,g) follow from the definition. The finite additivity of aq*(t, S) (q E pNA) as
well as that of 0;*p. implies that o/(a,g)w is finitely additive. Positivity of o/(a,g) follows
now from Lemma 3.5 and the finite additivity of o/(a,g)w. Obviously u * pNA is
reproducing; hence that positivity of o/(a,g) and the finite additivity of o/(a,g)w implies
that o/(a,g)w is in FA whenever W is in p. * pNA. Now let W E (u * pNA) n FA. We have
to show that o/(a,g)w =w. Without loss of generality we may assume that W = v +
2:7= j(O;* p.)qj where v E pNA and qj E pNA and 0;*p. = 0/ p. iff i = j. First we shall show
that q:(a) = 0 for each k, 1 <:;;; k <:;;; n. Let 1 <:;;; k <:;;; n be given. Applying Lemma 3.9 to
the nonatomic probability measures 0;*J.L,... , 0: J.L,the set functions v, ql' ... , qn in
pNA we have for every 0 < E two sets Tj, T2 E~,
TI C T2 and such that for all
1 <:;;; i <:;;; nand
OJ*p.(T2) ;> a>
O;*J.L(T1)
JV(T2) - v(TI)1
< E,
Iqj(T2) - qj(TI)1
Assuming
E
<a
<E
we find that Iw(T2\T1)1
iff
i = k,
JV(T2 \ Tj)1
and
< E,
Oi*p.(T2\Tj)
= Iv(T2\T1)1
< E.
< E.
On the other hand,
n
Iw(T2) - w(Tj)1
>
qk(T2) -lv(T2)
- v(TI)I-
2: Iqi(T
2)
-
i= I
;> Iq:(a)l-
E- E~
nE = Iq:(a)l-
(n + 2)E.
qj(TI)1
396
ABRAHAM
NEYMAN
The assumption that w is finitely additive will imply that E > Iw(Tz \ T))I = Iw(Tz) w(T1)1 ;> Iqt(a)l- (n + 2)E, i.e., that Iqt(a)\ -< (n + 3)E. As this is true for every
0< E < a we conclude that qt(a) = O. Let S be in ~, with p,(O;S) < a. In that case
w(S) = v(S), and by using the finite additivity of wand Lemma 3.3, we see that
v*(hS) = h(v(S)) for any rational 0 -< h ..;;1 and then by continuity of v* we deduce
that v*(hS) = hv(S) for any realh, 0 -< h -< 1. Therefore av*(O, S) = v(S). Now, let
0< t < a, and let S E ~ with p,(O;S) < a. Again using Lemma 3.3 to the vector
measure 0;*p" 1 ..;;i -< n; and the game v E pNA and the 3-tuple hS, t, 1 - t - hS,
h < a - t we have for any E > 0 a partition (T), Tz, T3) of I with Iv(T)) - v*(hS)1 < E,
Iv(Tz) - v*(t)1 < E and \v(TI U Tz) - v*(t + hS)1 < E and 0fp,(T) U Tz) < a. Hence
w(TI U Tz) = v(TI U Tz), w(TI) = veT)) and w(Tz) = v(Tz)· Therefore, using the
finite additivity of w we have veT) U Tz) - v(Tz) = veT)) - v(0) = v(T)), and as
Iv(TI U Tz) - v*(t + hS)1 < E, Iv(Tz) - v*(t)1 < E and Iv(T)) - v*(hS)1 < E, I[v*(t +
hS) - v*(t)) - v*(hS)1 < 3E and as this holds for any E > 0, v*(t + hS) - v*(t)
= v*(hS) = hv(S) and therefore av*(t, S) exists and equals v(S). In a similar way, by
using Lemma 3.3 to the vector measure 0;*p,; 1 ..;;i..;; n, and the games v E pNA, q;,
1 ..;;i ..;;n and the 3-tuple hS, t, 1 - t - hS, h < 1 - t we can prove that for a < t <-1,
a(2:7= I qJ*(t, S) + av*(t, S) = v(S). Therefore as fb get) dt = 1 we conclude that
tV(a,g)w(S) = v(S) = w(S) whenever S is in ~ with p,(O;S) < a. For S in ~ there exists
always a partition S = S) U ... U Sk with Sp i = 1, ... , k in ~ and p,(O;S} < a,
I ..;;
i ..;;n, 1 ..;;j ..;;k. Therefore by the finite additivity ofw as well as that of tf;w we
have tV(a,g)w(S) =
I tf;(a,g)w(SJ =
w(SJ = w(S) which completes the proof of
Lemma 3.10.
2:7=
2:7=)
LEMMA 3.11.
Let g be in Wand a in R +. Then the semivalue tV(a,g)on u * pNA
defined by (3.1) and (3.2) is continuous and 1Itf;(a,g)
II = max{a, II gilL..)'
PROOF. Let w be in u * pNA. Without loss of generality we may assume that
w = v + 2:7=1(9;*u)q where v is in pNA, q; EpNA for 1 -< i -< nand O;*p,= O/p, iff
i = j (1 ..;;i ..;;j ..;;n).
Therefore we have to prove that the right-hand side is at most max{a,
II glkJllwll·
n
l(tV(a,g)w)(S)1
=
a ~1
+
+ ~aav*(t,S)g(t)dt
q;*(a)(O;*p,)(S)
.fa(
v+
;~1
q;)*(t,S)g(t)dt
n
..;;a
2: Iq;*(a)I(O;*p,)(S)
;=1
,
+ IIgIlL_( flaV'(t, S)I
<; maxi a, IIgilL.}
dt + .c a( v + It,q )'(t,
[flav'(t,
+
S)I dt
+
,t,lq" (a)I(O,' ~)(
S)
dt)
.c a( + It, )'(t,
v
S) ]
q
S) dt
As
397
SEMIVALUES OF POLITICAL ECONOMIC GAMES
it is sufficient to prove that
n
IIWIl > i~1Iqt(a)1
L (Ia(
1
+
+ ~a(lav*(t,S)1
+ lav*(t,
S)I +
v + itl qi)*(t,
1 - S)I)dt
la( v + it1
(3.12)
qi)*(t, 1 -
S)I) dt.
First assume that v and qi' 1 < i < n are polynomials)n
nonatomic probability
measures. For every integer k > 2 we will construct a chain'" Qk so that IlwllQkwill
converge as k ~ 00 to the right-hand side of (3.12).
Observe that there is f in J with (Oi*JJ.)*(f) = (O/JJ.)*(j) iff i = j. We may assume
that 1/2 < f < 1. (Otherwise replace f by (1 + j)/2.) For every k> 1 let I be the
largest integer with 1< ak. Without loss of generality we may assume that for
1 < i,j < n (OtJJ.)*(j) > (O/J-L)*(j) iff i <j. Therefore for each 1 < i < n there is a
(unique) /3i = /3Jk) with 0 </3i :< 2/k and (Oi*JJ.)*(l/k + /3j) = a. Obviously all the
/3/s are different and 0 < /3i < /3j < 2/k whenever 1 < i <:j < n. Define (gJ7=1 by
gi = 1/ k + /3J and define (jJ7~tn-3
by
I
2k
i-I
S
-2-k- + k
1=
1
k + gi-21
i+3-n
2k
i+2-n+S
2k
k
if
i
< 21 is an
even integer,
if
i
< 21 is an
odd integer,
if
21
if
21 + n
<
if
21 + n
< i and
< i < 21 + n,
i and i - n is an odd integer,
i - n is an even integer.
Apply Corollary 3.4 to the vector (OiJJ., ... , O:JJ.) of nonatomic measures, the members v, q1'
, qn of pNA and € = 1/ k(2k + n) to construct a chain Qk : (I:)7~tn-3
(To ~ T ~
~ T2k+n-3) such that for all 1 < j < n and for all 0 < i < 2k + n - 3,
(O/JJ.)(Ti) = ((}/JJ.)(1),'
< k(2k1+
1'(j(I:) - q/(1)1
Iv(Ti). - v*(1)1
Denote by Q~ the subchain (I:)7~o'
(T)i2k+n-3
i=21+n+l' Th en
I
< k(2k + n)
flt the sub chain
n) ,
.
(I:)7~'i
and Qi the sub chain
IlwllQk> IlwllQL+ IIwlb~ + IlwllQk.
As (O/JJ.)(T21) = ((}/JJ.)(AI) = 1/ k
<
a
2/
IlwllaL= i~llv(I:)
for all 1< j
2/
- v(Ti_I)1
> i~llv*(.t:)
<
n,
21
__
- v*(.t:-1)1-
k(2k + n) .
As v is a polynomial in nonatomic measures, L:7~dv*(j,) - V*(1-1)1 converges as
k~oo
to jg(lav*(t,S)1 + lav*(t, 1- S)l)dt (see for instance pp. 45, 46 of [3] or
observe that for 1 < i < 2/Iv*(1)
- V*(1-1)1 = lav*(i/2k,S)I/k
+ o(l/k) where S
= S if i is odd and S = 1 - S if i is even). Thus
liminfllwIlQ'>
k~oo
k
+ lav*(t,
J(aC1av*(t,S)1
o
1 - S)l)dt.
398
ABRAHAM
NEYMAN
Similarly
liminfllwll~l>
J=1
~ ( 1) +2/- I) I
J=1
*(t,I-S)I)dt.
For each fixed 1 < j
We turn now to the estimation of Ilwll~.
> (Ot",,)(1}+Z/-I) iff i = j. Thus
I w ( 1) +Z/) -
tqj)
±lJJ)*(t,S)I+la(v+
(1(la(v+
Ja
k~oo
;;;. 19/
< n,
(Ott.t)1}+z/) > a
1) +2/ ) I - Iv ( 1) + Z/) - v ( 1) + Z/- I) I
n
- ~
Iqi(1}+Z/) - qj(1}+Z/-I)I·
;=1
Thus Ilwllui > "'2/)=,q/1}+z/) -llvllfJi - 2:j=,lIqjll~i'
For each fixed 1 < j < n, 9/1}+z/)~
q/(a) as k~
and also Ilvll~2i ~o as k~
00 and thus liminfk~oollwlll~
conclude that
IIwll > liminfllwlluk
;;;.liminfllwll~!
00
~o
and IIqjll~
> 2:j=t!q/(a)l·
+ liminfllwll~
and k~
Altogether
00,
we
+ liminfllwllul
n
;;;.;~Ilqt
+
S)I + lav*( t, 1 - S )1) dt
(a)1 + ~\Iav*(t,
f([v( V + It, q')'(I,
S)[ +
1"(V + It, q')'(I,
1 - S) ) dt
which proves (3.12) in the case that v and qi are polynomials in, nonatomic measures.
For the general case let £ > 0 and approximate
v and q; by polynomials of NAmeasures v and q; respectively with lit) - vii < E:, !q; - lJ;1< E:, and let w = v +
2:7=I(O/U)q;. As II000ull= 1 and IIvdlllv211 < IIvdlllv211 for all V!,V2 in BV, Ilw -wll
< Ilv - vII + 2:7= ,II(O/u)(q; - lJJ11 < (n + 1)£. Using Lemma 23.1 of [3] we have for all
SElf,
la1av*(t,S)
- av*(t,S)ldt
..c[v( v + It, ijl)'(I,
S) -
<
<
Also Iq/(a) - q/(a)1
Ilq/ - q;*11
E:.
v( V +
<
it
Ilv - v*11 <
E:
q')'(t,S)! dt.;
and
(n
+ I).,
Altogether,
n
(n + 1)£ + IIwll > Ilwll > i~Ilqt(a)l-
n£ + ~alav*(t,S)ldt-
+ ~alav*(t,
1 - S)ldt-
E:
E:
+ .£I[a( v + j~1
qJ*(t,
+ iI[a(
q;)*(t, 1 - S)[ dt- (n + 1)E:.
v + ;~I
S) dt- (n + 1)£
As this is true for all £ > 0, (3.12) is proved which completes
the proof of Lemma 3.11.
PROOF OF THEOREM A. We have already seen that for a in R + and g in W, (3.1)
and (3.2) define (uniquely) a (continuous) semivalue tP(a,g) on u *pNA. Now we have
to show that any continuous semivalue on u *pNA is of that form. Let tP be a
continuous semi value on u *pNA. In particular, tP induces a semivalue on pNA and
399
SEMIVALUES OF POLITICAL ECONOMIC GAMES
therefore by Theorem 2.3 there is g in W with
11g(t)dv*(t,S)dt
1/;v(S)=
for each
vinpNA.
(3.13)
Let v be a probability measure in NA, and k a positive integer. For any 8
8 < (l/2)min{ a, 1 - a} define Es : [0, 1]~ R + by
ro
Es(x)=
if
~ 1
II -
1/8(lx - al-
if
if8
8)
> 0,
Ix - al ;;. 28,
Ix -:-"al ~. 8,
<Ix - al < 28,
and define Vs by:
(3.14)
First we shall show that
Iluvsll <
32k8.
(3.15)
Define U = {S E -G': 0 < [.1.(S) - a < 20, Iv(S) - al < 28}. Then for S in U, [.1.(S)
;;. a and therefore u(S) = 1 and also for S in u, -lp(S) - [.1. (S)I < 48 and thus for S in
U, Ivk(S) - [.1.k(S)1.,;;48k, and IVs(S)1 .,;;48k. For every S in -G'\U either [.1.(S) < a
and thus u(S) = 0 or Ip(S) - al > 28 and thus vs(S) = 0, or [.L(S) - a ;;. 28 and thus
vs(S) = O. In any case for S El U, (uvs)(S) = O. Let Q: So C Sl C ... C SL be a
chain. Let io be the first index for which Sjo E U and let jo be the last index for which
~o
E U. Then from the definition of U it follows that Sj E U iff io .,;; i < jo. Therefore,
as (uvs)(S) = 0 whenever S El U, and l(uvs)(S)1 .,;; 48k whenever S E U we deduce
that
L
Iluvsll~
= i=l
2: l(uvs)(Sj)
- uV.s(Sj_1)1
.
fo+1
=
2:
l(uvs)(Sj)
- (uvs)(Sj_l)1
i= ;0
fo
= luvs (Sjo) + luv.s(~0)1 +
1
2:
luvs (Sj) - uVs(Sj-1)1
;= io+ 1
fo
.,;;88k
2:+
+
; = io
I( uvs)(
Sj) - (uv.s)( Sj-1)1·
1
fo
2:
;= ;0+ 1
Iv.s (S;)
- V.s (Sj-1)1
fo
=
2:
I( Es
0
p)( Es
0
[.1.)( Sj )(pk - [.1.k)( Sj)
;=io+1
- (E.s 0 v)(E.s
0
[.1.)(Si)(pk - [.1.k)(Sj-c1)
+ (Es
0
[.1.)(Sj)(pk - [.1.k)(Si_1)
0
[.1.)(Sj_1)(Vk - [.1.k)(S;_I)1
0
p)(Es
- (E.s 0 v)(E.s
400
ABRAHAM NEYMAN
But maxl(F/J
fo
2:
0
v)(F/J
0
J-L)(S)I< 1 and
I(vk - J-Lk)(S;) - (vk - J-Lk)(S;_I)1
< (vk + J-Lk)(8.io)
- (vk
+ J-Lk)(S;o)
;=;0+ 1
< 88k
maxl(vk
SEU
< 4k8
- J-Lk)(S)1
and
II(F8o V)(F8
and
J-L)II< IIF8 0 vllllF8
0
0
Therefore 2:1~;o+ dV8(S) - v8(S;_I)1 < 88k + (48k)4 = 24k8, hence lIuv/Jllf!
this holds for any chainH, (3.15) is proved. Define G8: [0, 1]~ R + by
o
G8(x)=
if
if
if
I
{ 1-!(x-a-8)
x
x
a
> a + 28,
< a + 8,
+8<x <
a
p,11
< 4.
<
32k8. As
+ 28,
and define 138 by
(3.16)
First observe that 138EpNA (although (G o. v)(G 0 J-L) f£pNA). Define!!fl to be the
diagonal neighborhood defined by !!fl = {S: I p,(S) - v(S)1 < 8}. Let S E!!fl and
denote v = vk - J-Lk; then u(v - v8)(S) = (v - V8)(S), because if J-L(S) < a and S E !!fl
then v(S) < a + 8 and therefore V8(S) = v(S) and of course then (u(v - V8))(S) = 0
= (v - v8)(S),
and if p,(S) > a then u(v - V8)(S) = (v - V8)(S), and v(S) > a - 8.
But for x > a - 8, G8(x) = F8(x) which yield that (v - v8)(S) = (v - V8)(S), whenever S E !!fl with J-L(S) > a. Thus we have seen that
u(v - v/J) coincides with v ~ 138 ona diagonal neighborhood, v - 138 E pNA and u(v - 138) E U * pNA.
(3.17)
As t/J is continuous Proposition 2.4 and (3.17) implies that
t/J(u(v
- 138» = t/J(v - 138),
(3.18)
Now we claim that
To prove (3.19) observe that if t
and if 0
< t < a + 8 and
h
<
[(G8o v)(G8
> a + 28
0 with t
0
and h
> a + 28,
< a + 8.
if
t
if
t
>0
then
(3.19)
+ h > 0 then
p,)(vk - p,k) J*(t
+ hS)
=
v*(t
+ hS).
As v - v/Jis in pN A, (3.13) and (3.18) imply that :
t/J( v - 138)(S) - (I dV*(t, S )g(t) dt
Ja+28
l
I
<
(a+28 g(l)
IJa+8
I d( v
- 138)*(1, S) dt
O.
~
l
8~O
(3.20)
If we let 8~0,
(3.20), (3.18) and (3.15) imply that
t/J(U(pk
- J-Lk»)(S)
=
i
1
g(t)d(Vk
- J-Lk)*(t,S)dt.
(3.21)
401
SEMIVALUES OF POLITICAL ECONOMIC GAMES
Observe that u E u * pNA. By Proposition 2.5 1/;u= aIL, and by the positivity of 1/;,
+ . Now let B be the subset of pN A of all games q for which 1/; (uq) =
(a,g)(uq).
k
k
By (3.21) pk - ILk E B. Observe that UlL - aku is inpNA and hence 1/;(UlL - aku)(S)
= J~g(t)a(lLk)*(t,S)dt
and 1/;(aku) = akalL. Therefore it is easily verified that ILk E B.
But B is obviously a linear subspace of pNA and therefore as it contains ILk and
pk _ ILk it contains pk for any probability measure in NA and hence any polynomial in
NA + measures. As both "" and 1/;(a,g) are continuous and IIuqll .;;;IIulill qll it follows
that B is closed, thus B = pNA. Now as both 1/; and 1/;(a,g)~e ~ear
and continuous we
deduce that they coincide on u * pNA, which completes the proof of Theorem A .•
r
a E R
.4. Further results and remarks. We are able to characterize the set of all continuous semivalues on many other important spaces, like bv' NA and bv' NA * pIYA. As the
proof uses similar methods to those presented in the former sections we will just give a
sample of results.
Notations.
Let X be a linear subspace (not necessarily closed) of the Banach space
bv' (the space of all functions f: [0, 1] ~ IR with f(O) = 0 such that f is of bounded
variation continuous at 0 and 1, endowed with the total variation norm). We denote by
W(X) the subset of the dual X* (of the closure X of X) of all elements x* satisfying:
(1) For each monotonic nondecreasingfin
X, x*(f) > 0; (2) If X contains the function
h defined by hex) = x, then x*(h) =1. The subspace of all absolutely continuous
elements in bv' is denoted ac'. For each 0 < x < 1 define fx : [0, 1] ~ IR by fAy) = 0 iff
Y < x and fx(Y) = 1 iff Y > x and Ix: [0, 1] ~ IR by Jx(Y) =_0 iff Y .;;;x and Jx(Y) = 1 iff
y> x. The subspace of bv' generated by the functions fAfJ is denoted by rj' (lj'), and
that generated by all jump functions (i.e., by rj' and lj') is denoted by j'. If X c bv' we
denote by XNA the linear symmetric space generated by games of the form fOIL,
f E X and p. is a probability measure in NA.
THEOREM4.1.
Let X be a subspace of bv'. There is a 1-1 linear isometry from W(X)
onto the continuous semivalues on XNA; for each x* E W(X) the semivalue 1/;x*on XNA
is given by 1/;x*(f 0 p.) = x*(f)p..
REMARKS. (a) W(ac') = Wand
therefore Theorem 4.1 can be considered a
generatlization of Theorem 2.3 (ac' NA is dense in pNA).
(b) W(lj') is identified with all bQunded functions a:(O,l)~R+;
for 0< x < 1
x*(a)(jx) = a(x) and Ilx*(a)11 = sUPO<x<Ia(X). Each of the continuous semivalues on
rj' NA can be extended to a semiva1ue on its closure: However, there are discontinuous
semiva1ues on lj' NA; they can be obtained by omitting the boundness condition on a.
(c) WU') is identified with all pairs of bounded functions a,b: (0, 1)~ R + where for
0< x < 1 x*(a,b)(fx)
= a(x)
and x*(a,b)(jx)
= b. We have Ilx*(a,b)1I =
sUPO<x<I{a (x), b(x)}.
Notations.
If QI and Q2 are linear symmetric subspaces of BV we denote by
QIG) Q2 the linear symmetric space generated by games of the form V1V2 where Vi E Qi
(i = 1,2), and the space QI * Q2 is defined as the linear symmetric space generated by
QI® Q2' QI and Q2'
THEOREM4.2.
For each pair (a, g), a: (0, 1)~
semiva/ue 1/;(a,g)on lj' NA * pNA given by:
whenever
1/;(a,g/(fx
whenever v E pNA,
0
0
p.)v)(S)
<x <
=
a(x)v*(x)p.(S)
1 and p. is a probability
R + and g E W = W(ac')
V
+
EpNA,
i
r
g(t)av*(t,S)dt,
there is a
(4.3)
(4.4)
measure in NA. The semiva/ue 1/;(a,g)
ABRAHAM
NEYMAN
402
is continuous iff a is bounded. Moreover, any continuous semivalue on rj' NA * pNA is of
that form. tf;(a,g)can be extended to a semivalue on rj' NA * pNA iff a is bounded and then
II tf;(a,g)II = max(suPO<x<la(x), II gll LJ·
REMARK. Similar results hold for the spaces Ij' NA * pNA and j' NA * pNA. (In the
second case the semivalues are associated with triples (a, b, g).)
THEOREM4.5. For each pair (a, g), a : (0, 1)~1R+
and g E L';;(O, 1) there is a
semivalue tf; (a,g) on rj' N A (!) pN A given by (4.4). This semivalue is continuous if and only
if a is bounded, Moreover, any continuous semivalue is of that form.
REMARKS. (a) The semivalues on rj' NA * pNA differ from those on rj' NA@pNA
sinceNA
rj' NA(!)pNA while NA C rj' NA * pNA.
(b) The proof of Theorems 4.2 and 4.5 are similar to that of Theorem A.
(c) The fact that (a,O) is a semivalue on rj' NAG)pNA is easy to prove (see Lemma
3.5(3.8)) and actually makes use only of the property of pNA of having a continuous
extension to ideal sets satisfying Lemma 3.3. Thus it follows that the existence of such
semivalues is valid for any space of the form rj' NAG) Q where -Q has such an
extension. If Q is such a space satisfying: there exist a: (0, l)~
R + \{O} s.t. for each
v E Q and 0 < x < 1, v*(x) = a(x)v*(I)
then by setting a(x) = l/a(x),
tf;(a,O)is a
value on rj' NAG Q. However, these values are discontinuous, whenever a is not
bounded away from O.
(d) For every g in W which is continuous, there is a semivalue tf;g on DIFF (for
definitional see [6]) that is given by: for any v in DIFF and S E ~,
ct
_.
tf;gv(S) - hm
h~O
IV*U+hS)-V*U)gU)dt
h )
(l
0
.
The proof is essentially the same as in Merten's proof in [6] of the existence of a value
on DIFF.
Acknowledgement. I would like to thank Professor Aumann for helpful discussions
and both referees for thoughtful reports.
References
[I] Aumann, R. J. and Kurz, M. (1977). Power and Taxes. Econometrica 45 1137-1161.
[2] -and --.
(1977). Power and Taxes in a Multi-Commodity Economy. Israel J. Math. 27
185-234.
[3] -and Shapley, L. S. (1974). Values of Non-Atomic Games. Princeton University Press, Princeton,
N.J.
[4] Dubey, P., Neyman, A and Weber, R. J. (1981). Value Theory Without Efficiency. Math. Oper. Res. 6
122-128.
[5] Dvoretsky, A, Wald, A. and Wo1fowitz, J. (1951). Relations among Certain Ranges of Vector
Measures. Pacific J. Math. 1 59-74.
[6] Mertens, J. F. (1980). Values and Derivatives. Math. Oper. Res. 5 523-552.
[7] Neyman, A (1977). Continuous Values Are Diagonal. Math. Oper. Res. 2338-342.
MATHEMATICS
DEPARTMENT,
HEBREW UNIVERSITY,
91904 JERUSALEM,
ISRAEL