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Subjective expected utility with non-additive probabilities on finite state spaces
JOURNAL
OF ECONOMIC
THEORY
51,
346366
(1990)
Subjective Expected Utility with Non-additive Probabilities on Finite State Spaces YUTAKA NAKAMURA University
of
Institute of Socio-Economic Planning, Tsukuba, l-l-l Tennoudai, Tsukuba, Ibaraki
305, Japan
Received July 12, 1988; revised September 5, 1989
Schmeidler and Gilboa’s representation generalizes subjective expected utility to cope with the Ellsberg paradox, so that the probability measure over states of the world need not be additive. This paper examines a similar generalization in Savage’s framework when the set of states is finite, while Savage’s states are continuously divisible. Our axiomatization requires that the set X of consequences be infinite in contrast to Savage’s arbitrary A’. Three representational forms are axiomatized to give non-additivity, complementary additivity, and additivity of probability measures, respectively. Journal of Economic Literature Classification Numbers: 022, 026. 0 1990 Academic Press, Inc.
1. INTR~O~CTI~N Theories of decision making under uncertainty provide subjective expected utility models that represent numerically the personal beliefs (subjective probabilities) and preferences of a decision maker. Under various frameworks numerous axiom systems and their numerical representations have been proposed (see the survey by Fishburn [8]). One of the most elegant and well-known axiomatizations of subjective probability and utility was given by Savage [17]. He developed an act-oriented theory which applies a preference relation over the set of acts. Acts are defined as functions from the states of the world into the consequences. There is abundant evidence that people’s carefully considered decisions often violate the assumptions of subjective expected utility theories (see Allais [ 11, Davidson, Suppes, and Siegel [6], Ellsberg [7], Slavic and Tversky [19 J, Kahneman and Tversky 1131, Grether and Plott [12], and others.) Recently, Schmeidler [ 18 J and Gilboa [lo] generalized subjective expected utility theory to accommodate Ellsberg-type violations of additive probability measures. Schmeidler adopted the idea of lottery acts (functions from the states of the world into probability distributions on the consequence space) introduced by Anscombe and Aumann [2]. Gilboa used 346 0022-0531/90
$3.00
Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
NON-ADDITIVE
PROBABILITIES
347
Savage’s basic formulation, so that the set of states is required to be continuously divisible, but required at least three consequences. Moreover, his theory provides that utility must be bounded as in Savage’s This paper axiomatizes a generalization of subjective expected utility to accommodate Ellsberg-type violation of additive probability in Savage’s framework when the set of states is finite as in Wakker [22]. Wakker assumed that the set X of consequences is a connected separable topological space, and derived a non-additive probability measure over states and a continuous utility function on X. Also, Wakker [21] extended his result to arbitrary state spaces. Our approach is different in that we adopt Savage’s framework and X is not necessarily a connected separable topological space, ‘, and it then gives rise to a more general result. The axiomatization requires that the set of consequences be infinite, while it is arbitrary in Savage’s utility theory. The axioms are shown to give a utility function over the consequence space, and a non-additive probability measure over states. The expectation of a utility function with respect to non-additive probability measures is given as Choquet integration as in Schmeidler and Gilboa’s representation, which is discussed in the next section. Section 3 states the axioms and the representation theorem that includes two special representations with complementary additivity and additivity of probability measures, respectively. Section 4 establishes a representation for all binary acts, then Section 5 completes the proof of the representation theorem in Section 3.
2. REPRESENTATIONS WITHOUT ADDITIVITY Let S be the set of states. Subsets of S are called events. For events in 2’ = {A : A c S}, denote the complement S\A of A by A”. An act is a function from S into the set X of consequences. F is the set of all acts. Let 5 on F be the binary preference relation with N and < defined in the usual way:forf,gEF,fwgiffdgandgdf;figiffdgandnot (gsf). Schmeidler and Gilboa’s representation has the following form: for all
f, g EF,
where u is a real valued function
on X, and rr is a monotonic
(not
’ Wakker [21,22] noted that the topological approach needs topological separability only for ordered partitions w of the set of states with exactly one o-essential event in his terminology.
348
YUTAKANAKAMURA
necessarily additive) measure on 2’ that satisfies n(S) = 1 and z(empty)=O. Monotonicity of n means that if A EB then rc(A) 5 n(B). Moreover, u is unique up to a positive linear transformation, and rc is unique, so that u and $A) for A E 2’ can be interpreted as a utility function on X and a subjective probability for an event A, respectively. The integration in the representation is defined in Choquet’s sense (see Choquet [S]) to account for non-additive probability measures as follows. Since we are concerned with a finite S, assume that S= {s,, .... s,} and l.416 . . . s U”, where ui= u(f(si)) for SUES and all i. Let zf denote the cumulative distribution function induced by 71through f so that
q(t)= n((s: 4fO)) 5 t>, with rcJ(t) = 0 for t < u1 and rcY(t) = 1 for t 2 u,. Then the integration given by n-1 sS
‘(f(‘))
dn(s)=
1
(“i-*i+l)
nf(“i)+
is
un,
i=l
which is referred to as the lower Choquet integral in Gilboa [ 111. Chew, Karni, and Safra [4] adopted this definition in risky situations. This means that expected utilities are calculated with respect to cumulative distributions. On the other hand, the Choquet integration, referred to as the upper Choquet integral in Gilboa [li], is defined by the expectations with respect to the decumulative distribution functions induced by R through acts. Schmeidler and Gilboa’s representation applies this definition. In risky situations, Quiggin [16] used it in the anticipated utility representation. Let rr* be the dual measure of rr such that n*(A) = 1 -n(A”) for all A E 2’. Gilboa [ 111 showed that the lower Choquet integral with respect to rt is equivalent to the upper Choquet integral with respect to K*. We say zhat q-71=71*, then rc is complementarily additive. Throughout the paper, the integrations always stand for the lower Choquet integral. 3. AXIOMS AND THEOREM Several additional notations and definitions will be useful in stating axioms. When F, and Fz are subsets of F, F1 5 F2 means f 5 g for all ffzF,andallgEFz.Wewritef~gg,ihwhenf~gandg~h;f
NON-ADDITIVE
PROBABILITtES
349
A E 2’ and x, y E X, denoted by xAy. A constant act is an act f = s x for some x E X, so every x E X is identified with a constant act. A partition of S is a sequence of non-empty events that are mutually disjoint and whose union equals S. For an n-partition P= {A,, .... A,,}, if f =A, xi for all i, then sometimes we write &(x1, .... x,) instead off: Also, xAy is sometimes written by f;l(x, y). Suppose that (x,, .... x,} = {xi ,,..., xi,}, P={A, ,..., A,}, P’=(A, ,,..., A,}, and P=P’. Then by the definition of the acts, &(x1, .... x,) = fP(xi,, .... xJ. If P” = .... x,). In {A, u AZ, A3, .... A,) and x, =x2, then &(x1, .... x,)=&.(x,, particular, xAx = x and xAy = yA”x. A null event A E 2’ is an event for which for all x, y, z E X, xAz - yAz whenever x 5 y 5 z. A universal event A E 2’ is an event for which for all x, y, z E X, xAy - xAz whenever x 5 y 5 z. An empty event is null. If x 4 y for some x, y E X, then S is not null but universal. If < is empty, then every A E 2’ is null and universal. Note that if there is an event which is neither null nor universal, then S includes at least two states, and i is not empty, so that x < y for some x, y E X. We say that 5 is bounded if, for each f E F, there are x, y E X such that x 5 f 5 y. Let N be any set of consecutive integers. Given an event A E 2’ which is neither null nor universal, we define a standard sequence as a set (a,: aie X, iE N) for which there exist a, b E X such that not(a - b), and either {a, b} 5 {ai} and aAa, for all bAai + 1 for all i, i+ 1 EN, or {a,} 5 {a, b} and a,Aa-a,+,Ab i,i+l~N. The following axioms apply to all .f E F, all x, y, z E X, and all A, B E 2’: Al. 5 on F is a bounded weak order. A2. If xAz 5 f 5 yAz then f- aAz for some a E X. A3. If A is not null and {x, y } 5 z, then x 5 y iff xAz 5 yAz; if A is not universal and z i, {x, y 1, then x 5 y iff zAx i, zAy. A4. If x 5 y and A E B, then xBy 5 xAy. A5. Every strictly bounded standard sequence is finite. A bounded weak order means by definition that 5 is bounded, complete, and transitive. A2 is a restricted solvability axiom. A3 and A4 are monotonicity axioms. A5 is an Archimedean axiom. Since 5 is bounded, for each f~ F, there are x, y E X such that x 5 f 5 y. By the definition of acts, x = XSZ and y = ySz. Therefore, by A2, f - aSz for some a E X. Thus f - a. Let m be a mapping from F into X that assigns a constant act m(f) for each f E F such that f -m(f). Let M be the set of all such mappings, so that if m, m’ EM, then by Al, m(f) - m’(f) for all f~ F. For an n-partition P = {A,, .... A,}, if f =A, xi for all i, then m(f) is denoted by m,(x,, .... x,). When P= {A, A”} and f = xAy, we denote m(f) = m,(x, y). We say that < is dense if, for each
350
YUTAKANAKAMURA
x,y~X with x Y,, E X for A, BEAM: A6.
there is an UEX such that x
for A3, .... all
If xi 5 .. . 5 x, and y1 5 . . 5 y, with xi ,$ y, for all i, then
fA(m,(x ,, .... h), m,(y,, .... yJ) -f,(m,&, A7. mA(mBk A8. .f..(mp(xl,
, ~~1,-., m,Jx,, Y,)).
Y), m,(z, w)) - m,(m,Ax, z), mh, .... 4, m,b,, ...+ y,))-fp(mA(xl,
w)). ~~1, .... m,&,,
VA).
A6 is a version under uncertainty of the condition under risk proposed in Fishburn [9, Chap. 31 that modifies the third axiom in Chew [3]. Chew’s axiom was originally motivated by Quiggin [ 161. When P = {B, B”}, A7 is similar to the isometry condition if A #B, and the bisymmetry condition if A = B in Pfanzagl [IS]. A8 is a version of the isometry condition. The purpose of the paper is to prove the following: THEOREM 1. Suppose that S is finite, and that there is an event which is neither null nor universal. If Al-A6 hold, then
(1) there are a monotonic measure TCon 2’ and a real valued function u on X such that for all f, g E F,
(2) (3) (4)
n is unique, and u is unique up to a positive linear transformation ; n is complementarily additive if A7 holds; 7t is additive $A8 holds instead of A6.
The proof of the theorem will appear in Section 5. We note that the axioms except Al, A3, A4, and A5 are not necessary for the representation, as shown in the following example. Suppose that X is the set of all rational numbers, and that u(x) = x for all x E X. If rt takes on values of irrational numbers, then A2 does not hold, so that M is empty. In this case, A2 requires that rc take on values of rational numbers. Theorem l(4) gives an expected utility representation with an additive subjective probability measure in Savage’s framework when S is finite. Wakker [20] obtained a similar representation in the additive case when X is a connected separable topological space.
NON-ADDITIVE
4. A
351
PROBABILITIES
REPRESENTATION
FOR
BINARY
ACTS
We assume throughout the rest of the paper that Al-A6 hold, and there is an event which is neither null nor universal. Note that S includes at least two states, and < is not empty but dense. This section proves the following proposition, which implies that the representation of Theorem 1( 1) holds for all binary acts. PROPOSITION 1. There are two real valued functions, n on 2’, and u on X, such that n(empty) =O, n(S) = 1, and for all x, y, z, w E X, and all A,BE~~, ifxsyandzLw then
xAy 5 zBw AEB
iff
n(A)u(x)+(l-n(A))u(y)
imphes
1rc(B)~(z)+(l n(A) 2 7c(B).
-x(B))u(w);
Moreover, rc is unique, and u is unique up to a positive linear transformation, First we show three lemmas, and then give the proof of Proposition For aEX, let X*=
{xEX:b
X”=
{xEX:xsa),
X,=
{xEX:asx},
1.
x max=(xEX:x 5 for all i, i + 1 E N, or {ai} 5 {a, a’} 5 {a, a’} 5 {a,} and bAai-cAa,+, (6, c} and a,Ab-a,+, AC for all i, i + 1 EN. Let 5” be a binary relation on X” x X,, induced by 5 as follows: for ordered pairs xy, zw EXO x X,., xysAzw iff xAyszAw. We say that the triple (X”, X,,, 5”) is an additive conjoint structure (Krantz et al. [ 14, Chap. 61) if for all x, y, z E X” and all x’, y’, z’ E X,. , the following five conditions hold for a weakly ordered 5 : 1. Independence : If xAb’ 5 yAb’ for some b’ E X,,, then xAx’ 5 yAx’ ; if bAx’ 5 bAy’ for some b E X”, then xAx’ 5 xAy’.
352
YUTAKA NAKAMURA
2. Thomsen condition: If xAz’ - zAy’ and zAx’ - yAz’, then xAx’ - yAy’. 3. Restricted solvability: If xAx’ 5 yAy’ 5 PAX’, then bAx’ N yAy’ for some b E X”; if xAx’ 5 yAy’ 5 xA.z’, then xAb’ - yAy’ for some b’ E X,. . 4. Archimedean: Every strictly bounded standard sequence with respect to a and a’ is finite. 5. Essentiality: not(bAb’ - cAb’) for some 6, c E X” and some b’ E .I’,. ; not(bAb’ w bAc’) for some b E X” and some b’, c’ E X,.. LEMMA 1. If A E 2’ is neither null nor universal, then fir a, a’ E X* with a 5 a’, the triple (X”, X,., 5” ) is an additive conjoint structure. LEMMA 2. If A E 2’ is neither null nor universal, then there are two real valuedfunctions, 4 and $, on X such that for all x, y, z, w E X with x 5 y and z 5 w,
(1) .xAySzAw iff~(x)+Il/(.v)~~(z)+~(u’); (2) if 4’ and t+Ysatisfy (1) instead of 4 and t,!t, respectively, then there exist constants a > 0, PI, and f12 such that for all
x E X\X,,,,
for all
x 6 X\X,,,.
LEMMA 3. If A E 2’ is neither null nor universal, then there is a real valued function u on X such that for some 0 < GIc 1, and for all x, y, z, w E X with x 5 y and z 5 w.
xAy 5 zAw
ifs
ffu(x)+(l
-(W)u(y)~cYu(z)+(1
-cY)u(w).
Moreover, a is unique, and u is unique up to a positive linear transformation. Proof of Lemma 1. Suppose that A is neither null nor universal. By A 1, 5 is a weak order. It easily follows from A2, A3, and A5 that the independence, the restricted solvability, and Archimedean conditions of the additive conjoint structure hold. The essentiality condition follows from a, a’ E X* and the assumption that A is neither null nor universal. In what follows, we show the Thomsen condition. Suppose that xAz’- zAy’ and zAx’ - yAz’. Then by Al, m,(x, z’) m,(z, y’) and mA(z, x’)-m,(y, z’). We are to show that m,(x, x’) m,(y, y’), so xAx’- yAy’. The following three cases cover all possibilities: (x’, y’} 5 z’; {x’, z’ } 5 y’; { y’, 2’) 5 x’.
NON-ADDITIVE
353
PROBABILITIES
Suppose first that (x’, y’> 5 z’. By A3, m,Jx, z’) 5 mA(x’, z’) and m,(y, z’) 5 mA(y’, z’). Thus A3 and A6 imply
.Lh,(x,
xl m,4(z’,z’)) -jxm,(x, -fa(m,(z,
so by AL frc(m.&, ~‘1, m/h’,
z’), m&‘, 2’)) Y’), mAx’, z’))
(by A3)
-f/i(m,(z,
-q, m,(y’,
z’))
(by A6)
-fA(@~,(Y,
z’), m/h’,
;‘)I
(by A3)
-f,(m,(y,
Y’), mAz’, z’)),
z’)f-M=,(Y,
~‘1, ~/AZ’,
z’)).
(by A61
(by ‘46) BY
Al
and A3, {m,(x, x’), m,(y, y’)} 5 mA(z’, z’). Hence by A3, m,(x, x’) m,(y, Y’). Suppose next that (x’, z’} 5 y’. Then similarly we obtain fA(m,(x, x’), m,kz’, Y’)) --f,(mAy, y’h m,&‘, ~‘1). BY Al and A3, (m,dx, x’), m,(y, y’)} 5 m,(z’, y’). Hence by A3, nz,(x, x’) - m,(y, y’). Suppose last that { y’, z’} 5 x’. Then similarly, f,Jm,(x, x’), m,(z’, x’)) -f,(m,(y, y’), mA(z’, x’)). By Al and A3, {m,(x, x’), m,(y, y’)} 5 mA(z’, x’). Hence by Q.E.D. A3, m,(x, x’) - m,(y, ~‘1. Proof of Lemma 2. Suppose that A is neither null nor universal. Lemma 1 implies that for a E X*, the triple (X0, X,, 5”) is an additive conjoint structure. Thus by Theorem 2 of Chapter 6 in Krantz et al. [ 141, there exist two functions, 4, on X” and II/, on X,, such that for all x, y E X” and all z, WAX,, XAZ 5 YAW
iff
40(x) + ll/a(z) 5 4,(Y) + Ic/Aw).
If a 5 b for a, b E X*, then by Lemma 1, the triple (X0, Xb, 5”) is an additive conjoint structure. Since X” x X, c X” x X, and X” x X, _CXb x X,, for all x, y E X” and all z, w E X,, we have xAz 5 yAw
iff
4J-9 + k(4
5 UY)
+ Icl,(w)
iff
4bcX)
=< db(y)
+
+
$btZ)
@btw).
By the uniqueness of the additive representations for the additive conjoint structure, there are real valued functions k(a, b) > 0, k,(a, b), and ,&(a, b) for all a, b E X* with a 5 b such that #a(X)
=
k(a,
b)
$bcX)
+
k,(a,
6)
Ic/&) = Ma, b) @t,(x) + &(a, b)
for all
x E X”,
for all
x E X,.
354
YUTAKA
NAKAMURA
Given 4, and $a for a fixed a E X*, scale bb and $b for b E X,\X,,,,,, and 4, and II/, for CE X’\X,,, such that k(a, b)=k(c, a)= 1 and kj(a, b) = kj(c, a) = 0 for i = 1, 2. Then it easily follows that if b 5 c and b, c E X*, then db = 4, on Xb, and tjb = $, on XC. Therefore, for all ?cE X, define if
i(x) = h(x)
xEX*,
for some
= 4&7(x)
if
$(x1 = h(x)
if
x E Xmin,
aEX*
if
XEX,,,,
XE X*,
forsome
= *a(x)
a E X*
so 4 on X\X,,, and $ on X\X,,
are uniquely specified. If x, y E X\X,,,, then (x, y> 5 z for some ZE X\X,,,, since < is dense. Thus for all if {x, y} sz, then x, y9 z E x\x,,,, xsy
iff
XAZ 5 yAz
iff
#z(x) + $A)
iff
4x(x) 5 4,(Y)
(by scaling)
iff
4(x) 5 d(Y).
(by definition)
Similarly, for all x, y E X\X,i,, obtained, let
(by A3)
(by Lemma 1)
5 d,(Y) + 4+,(z)
x 5 y iff $(x) 5 $(y). With 4 and rj thus
~(x*)~suP{&):-=X\X,,,~
if
x* E X,,, ;
It/(x,) Sinf(ti(x):
if
x* E X,,,.
xEX\Xminl
Since < is dense, we obtain that 4(x) < 4(x*) for all XE X\X,,, and all < tj(x) for all XE X\X,i, and all x.+ E Xmin. x* E x,,,; +(x,) We are to show that 4 and II/ satisfy (1) and (2). Suppose that x 5 y and z 5 w. First, assume that z 5 y and x 5 w. Then by Al, A2, and A3, {x,z}~a~{y,w} for some aEX. IfaEX*, then by Lemma1 and the constructions of 4 and $, xAy 5 zAw
Note
iff
4&J
iff
4.x(x) + Ii/,(Y) 5 &7(z) + Iclw(w)
iff
d(x) + t4Y) s 4(z) + b+(w).
+ @a(Y) I4a(z)
+ $a(w)
that 4(x.+.) for x, E Xmin is finite, otherwise, x,Ax< {x, z}, so we obtain
a, XE X*. If ac Xmin, then a-
a for all
NON-ADDITIVE
xAy <, zAw
355
PROBABILITIES
iff aAy<_aAw
(by Al and A3)
iff
ysw
(by A3)
iff
$(Y) <=Ii/(w)
iff
4(a)+$(~)54(a)+ti(w)
iff
4(x)+W)54(z)+$(w).
If aEXim,x, then (1) similarly follows. Next assume that either y < z or w < x. Then (1) easily follows from Al, A3, and the constructions of 4 and rj. The constructions of 4 and J/ immediately give (2). Q.E.D. Proof of Lemma 3. Suppose that A is neither null nor universal. Let # and $ satisfy Lemma 2(l) with 4(x*) zsup{d(x): x E X*} if x* l Xmaxr and IC/(x,)sinf{lC/(x): x6X*} if x* EX,,,~“. Then we show that choosing the appropriate 4(x*) and y?(x,) if x* l Xmax and x.+ EX,.,.,~,,, there are constants, /? > 0 and y, such that $(x) = @j(x) + y for all x E X. Define
ha(x) = d(mA4
x)1,
$dx)
= $(mAa, xl)
for
+xE X,
if
a $ X,,,;
L(x)
a)),
$2Jx) = IL(m,(x, a))
for
XE x1
if
a$ Xmin.
= 4(m,(x,
For x, y, z, w E X, and a $ X,,,, if x 5 y and z 5 w, then by Al and A3, mAa, x) 5 mAa, y) and m,(a, z) 5 mAta, w), so #l&f iff
+ CltdYf 5 #la(Z) + $1,(w) &mAa,
xl) + Nm,(a,
y)) 5 #(mAa, ~1) + $(mAa, w)) iff .LAm,Ja, xl, mAa, Y)) 5f.A m A a, z 1, m,(a, w)) (by Lemma 2(l)) iff .fa(mA(a, a), mAx, Y)) 5fA iff
m,dx, v) 5 m,&, w)
iff
xAy 5 zAw.
mA a , a 1, m,(z, w))
(by Al andA6)
(by A31
If x 5 y and z 5 w for x, y, z, w E X” and a $ X,,, , then similarly xAyszAw
iff
h(x)
+ WY)
S dd4
we get
+ Il/zh4.
Thus di, and t,ki, satisfy Lemma 2( 1) on X, for i = 1, and on X” for i = 2. Therefore, there are real valued functions, ki>O and kli, on X\X,,, for i = 1 and X\X,i” for i = 2 such that for a 4 X,,, and b 4 Xmin, 4(mAa, xl) = k,(a) d(x) + k,,(a)
for
x~X,\X,,,,
ti(m.h
for
X E Xb \Xmin.
6)) = h(b) d4x) + kdb)
Suppose that x 5 ( y, z) 5 w for y, z E X*. Then x & X,,,,, and w $ Xmin.
356
YUTAKA
NAKAMURA
By Al and A3, m,(x, y) 5 m,(z, w) and m,(x, z) 5 m,(y, w). It follows from A6, Lemma 2( 1 ), and the preceding paragraph that
iff
$(m&,
Y)) + Il/(m.&,
iff
k,(x) 4(.v) + M-4
~1) = 4(mAx, =)I + 4WA(~, w))
+ MN
Nz) + Mw)
= k,(x) d(z) + k,,(x) + b(w) Ii/(Y) + k,,(w) iff
k,(xNti(.v)
-d(z))
= Mw)($(Y)
- W)).
Since the above equations are satisfied for all x, v, z, w E X with y, z E X* and x 5 {y, z} 5 w, there are constants, 13.and 6, such that for all x 4 X,,, and all w 4 Xmin, and
k,(x)=,?>0 Thus 4Kv)
-d(z))
= @KY)
- W)),
k2(w) = 6 > 0. so W(u)
- MY)
= W(z)
- U(z)
for
all y, z E X*. Therefore, there are constants, /I > 0 and y, such that $(x) = /Q~(x) +y for ail xEX*. Suppose that X,,, is not empty. If sup 4(x)= + cc, then by the preceding paragraph, Ii/(x*) = + cc for x* l Xmax. By Lemma 2(l), this implies that a < xAx* for all a, x E X *. Therefore, sup d(x) must be finite. The preceding paragraph implies that Ii/(x*) >=sup Ii/(x) = /I sup 4(x) + y for x* E x,,,. Thus define 4(x*) = ($(x*)-7)/j?, so that $(X)=&~(X) + y for XEXIXI,,. Suppose next that Xmin is not empty. Then similarly, we can define @(x.+)=&~(x*)+Y for x.+EX,,,. Hence Il/(x)=/@(x)+y for all XE X. With /I> 0, 4(x*) for x* E X,,,,,, and tj(x,) for x* E Xmin thus obtained, let CY= l/(1 + 8). Then 0 -CO!< 1. Define U(X) = $(x)/a for all x E X. Then the representation of the lemma easily follows. To show the uniqueness of U, suppose that U’ also satisfies the representation. By the uniqueness of 4 and +, it easily follows that there are constants, /I > 0 and y, such that u’(x) = flu(x) + y for all x E X*. Suppose that x* Ay -z for x* E X,,, and y E X*. Then z E X*. We obtain au(x,)+(l -a)u(y)=u(z), au/(x,)
+ (1 -a) u’(y) = u’(z),
so u’(x*) = flu(x,) + y. Similarly, u’(x*) = flu(x*) + y for x* E X,,,. Hence Q.E.D. the uniqueness part of the lemma obtains. Proof of Proposition 1. If A is neither null nor universal, then Lemma 3 implies that there is a real valued function uA on X such that for some O
xAy 5 ZAW
iff
aAuA(x)+(l--clA)~A(y)~aAuA(z)+(l-aA)~~(w).
NON-ADDITIVE
357
PROBABILITIES
Thus we define rc on 2’ as follows: n(A)=cc,
if
A is neither null nor universal;
= 1
if
A is universal;
=o
if
A is null.
Suppose that A and B are neither null nor universal. Then we show that uA = clus + fi for some constants, CI> 0 and /?. Given aeX*, a binary relation 5” on X” x X, is defined as follows: for xy, zw E X” x X,,
By Al and A6,
Since by Al and A3, m&x, a) 5 m,(a, C = A or B, Lemma 3 implies that xy5”zw
iff
44)
uAm&, 5 @I
iff
n(B) do&, 5 n(B)
y) and m,(z, a) 5 m,(u, w) for
~1) + (1 - W)) u,4(mIk
a)) + (1 -Ml)
4) + (1-V)) ~&AZ,
~,&,(a,
Y)) u,(m,(a,
+OK&~
~1) + (1 - 4B))
w))
Y)) u&Au,
~1).
By the uniqueness of additive representations, there are real valued functions, w >O and c, on X* such that for all UEX* and all XEXO,
WI U.&&G ~1) = 4~) 4B) u,(m,(x, ~1) + 4~). Since m,(x, y) E X” for x, y E X”, it follows from Al, A6, Lemma 3, and the above equation that for x, y E X” with x 5 y,
W) uAm,(m,(x, Y)?m,4(4 a))) = xc(A)uAm,(m,(x> ah m,(Yv = x(A)2 =da)
a)))
U,4(~Lf(X, a)) + (1 - W)) dB){W)
~&Ax,
n(A)
a)) + Cl-
u,(m,(y, 44))
u,b,(y,
a)) aI)> + 4~).
On the other hand, by the preceding paragraph, Lemma 3, Al, A3, and A6,
358
YUTAKA
NAKAMURA
Therefore, the above two equations give
u,(m&,(x, a), ?.4(YYa))) =71(A) u,(m,(x, a)) + (1 - NA)) u,(m,(y3 a)). We show that u,(m,(x, y)) = x(A) uLI(x) + (1 - n(A)) uB(y) for all x, y E X with x 5 y. Thus by Al and Lemma 3, we must have uA = clus + /I for some constants c1> 0 and /I. Suppose first that x 5 y and x, y E X*. If zAy <, x for some z E X, then by A2, wAy w x for some w E X. Therefore, the desired result easily follows from the preceding paragraph. Thus assume that x< zAy for all z with z
~B(mB(m,b~ Y), m,(y, Y))) = WQ u/Am& y)) + (1 - n(B)) UB(Y), and
u,(mAw Y)) = 44) UAW)+ (1 - 44)) U,(Y) = n(A) @) UB(X)+ (1 -n(A) n(B)) UB(Y). By Al, A3, and A6,
m,4(w,Y) -m,(m,bG YL mLi(y, Y)) - m*(mAx, Y), m,(y, Y))?
sou~(~~(w,Y))=u~(~~(~~(x,Y),~,(Y,Y))). Therefore, W)4B)u&)+ (1- 64) n(B))U,(Y)= 49 u,(m,(x,Y))+ (1-n(B)) DAY). This is rearranged to give the desired result. Hence u,(m,(x, y)) = x(A) uB(x) + (1 -z(A)) us(y) for all x, y E X* with x 5 y. If either x, y E Xmin or x, y E X,,,, then the desired result easily follows, so we show that it also holds when XE X,,, or y E X,,,,,. Suppose that x E Xmin and y E X*. When x E X* and y E X,,, , the proof is similar. Then
NON-ADDITIVE
PROBABILITIES
359
y 5 z for some ZE X*. Since m,(x, y) and m,(y, z) are in X* for C = A or B, Lemma 3 and the preceding paragraph imply that
u&,(m/i(x, Y), m.4bs z))) = z(B) ~B(~/l(& Y)) + (1 -n(B)) u,(m,(.Y, z)) =71(B)Us(mA(X,y))+(f--(B)){~n(A)u,(y)+(l-n(A))u,(z)}, u&&,(x,
VI, m,(y, z)))
= n(A) u,(m,(x, Y)) + (1 -n(A)) uB(m,(Y>z)) = ~W{W) UB(X)+ (1 - 4B)) Us(Y)) +(l-n(A)){71(A)ug(y)+(l--(B))u,(z)}. BY A6 mAmAx, Y), m,h, ~1) ~m,(m&, Y), mB(y, ;)I, so by Lemma 3, u,b&Ax~ ~),m,(y,z)))=u,(m,(m,(x, Y)TAY,~))). Thus u,(mAx, Y)) = 64) u&J + (1-7-U)) Us(Y). Suppose last that XE Xmin and ye X,,,. Applying the results in the obtains. Hence preceding paragraphs, the desired result similarly u,(m,(x, JJ)) = x(A) uB(x) + (1 - n(A)) uB(y) for all x, y E X with x <, y. Under appropriate positive linear transformations, there is a real valued function u on X such that u = uA for all A E 2’ which are neither null nor universal. Noting that if x < y then xAy - y when A is null, and xAy -x when A is universal, it follows from the preceding paragraphs that for all A E 2’, and all x, y E X with x 5 y,
u(mAx, Y)) = n(A) u(x)+ (1 -n(A)) U(Y). Thus for all A, BE 2’ and all x, y, z, w E X with x 5 y and z 5 w, xAy 5 zBw
iff
m,(x, y) 5 m,(z, w)
iff
u(m,k
iff
~A)~~)+(~-~(A))u(~)~~(B)u(z)+(~-Tc(B))u(w),
Y)) 2 uh,(z,
w))
so the representation of the proposition holds. It easily follows from A4 that A G B implies n(A) 5 n(B). The uniqueness of rc and u follows from the construction of u, and Lemma 3. Q.E.D.
642/51/2-9
360
YUTAKANAKAMURA
5. PROOF OF THE THEOREM
Throughout the section, we shall assume that S is finite, and let u on X, and rr on 2’ be real valued functions obtained in Proposition 1. We extend 2.4on X as follows:
u(f) = 4x)
when
u(f) = u(m(f ))
for all
f=sx; f EF.
Thus by Al, for all f, ge F, f 5 g iff u(f) s u(g). We note by Al and A6 that if xi - yi for i = 1, .... n, then fp(x, , .... x,) N f,(y,, .... y,) for all n-partitions. Proof of Theorem 1. (1) We are to show that u(f) = Js u(f(s)) LIZ(S), so of (1) obtains. Let P= (A,, .... A,} be an n-partition,
that the conclusion and
fn=A,xi
for
i=l,...,n,
where x1 5 ... 5 x, and xi E X for all i. Also let BO be empty, and Bi = Cj= r Aj, so B, = S. For all fi E F, Proposition 1 gives the desired result. Thus we assume n > 2. Suppose that the conclusion is true for all n-partitions with n
since this easily leads to the definition of the lower Choquet integral. If xi - xi for some 1 5 i < j 5 k, then the desired result easily follows from Al, A6, and the hypothesis of the induction. Thus we shall assume that x1-c ...
xAxzwxl
and x,_,Ay*x,
for some x, y E X and some
AEON.
CASE 2. either xAx, - x1 for some XE X and some nonuniversal A E 2’ or xk ~ I By - xk for some y E X and some nonnull BE 2’. CASE 3. not(xAx, -x,) and not (xk- r By -xk) nonuniversal A E 2’. and all nonnull BE 2’.
for all x, .YEX, all
Let P’ = {A,, .... Ak _ 2, A, _, v Ak} and PI’ = {A, u A,, A,, .... Ak}, so they are (k - 1 )-partitions.
NON-ADDITIVE
361
PROBABILITIES
Case 1. Since xAx, - x, and xk _ i Ay - xk for some x, y E x and some A E 2’, A3 implies that x < x1 and xk < y It follows from Al and A6 that fk
=
fPtxl
> ...Y xk)
-fp(m,(x,
4,
‘-f.4(m,(x,
x2,
‘-f,=,(m&,
m/&z,
-4,
. . . . xkp2,
x2,
. . . . mA(Xk--I,
xk&1,
. . . . XL-l),
xk-l)>
xk-I),
mA(Xk-I,
mP(x2~
m,“b2,
x2,
...> xkp1,
x39
v)) ..., xkLl,
y))
v)),
SO by Al, fk - Wl,(m,.(X, X2, .... +-I), m&X,, .... x&l, J’)). Since XAX, - x1 and xk- lAy - xk, Proposition 1 implies that u(xi) = and u(xk) = x(A) u(x& 1) + (1 -n(A)) U(y). 71(A) U(X) + (1 -x(A)) U(X2) Thus by the definition of u(fk), and the hypothesis of the induction, we get u(fk) = ~(m,.,(mP’b, =
n(A)
x2,
u(m,.(x,
. . . . xk-
x2,
I),
m,“(X,,
. . . . xk-
. ..) Xkpl))+(l
-n(A))
1, y))) u(mP”(x2~~~9
xk-1,
y))
k-2 =
WI
4Bl)
4x)
+
1 i==2
i
+
t1 -dBk-2))
(44)
-@-I))
4Xi)
dxk--1) I k-1
+
t1
-71(A))
n(B2)
u(x2)
+
1
i
+
t1
(71(Bi)-n(Bj-l))
u(xi)
i=3
-nn(Bk-l))
dy) I
=
ii,
bCBi)
-
71(Bi-
1))
u(xi).
Case 2. Suppose that xk- i By - xk for some y E X and some nonnull BE 2? A similar proof applies when xAx, -x1 for some x E X and some nonuniversal A E 2S. By the assumption and A3, B is neither null nor universal. Then let z N x1 Bx2 for some z E X, so xi < z < x2. Thus by Al and A6, fP(Z, x2,
since
fk
-
mp(x,,
4mp(z, x2,
.*.,
xk)
-~s-(~B(~,
9 x2)~
mB(x2~
-,fLh&l,
***, xk),
m,(x,,
‘-fB(mP(Xlr
..*, xk),
m,f’(X2,
. . ..
xk), Proposition
I implies
. .. .
xk))
+
=
44
u(fk)
(1
-n(B))
x2)~
...Y m,(xk, x2,
x3,
xk))
. . . . xk))
..*, xk)).
+.+2,
. . . . xk)).
362
YUTAKA
NAKAMURA
Noting that u(mp(z, x2, .... X/C))=~(~I) Utz)+Cf=2 (n(Bi)-n(BiL1)) by Case 1, and u(z) = z(B) u(x,) + (1 -z(8)) u(xz) by Proposition above equation is rearranged to give n(B)uUk)=n(Bl)u(z)+
i i=2
u(xi)
1, the
(nc(Bi)-n(Bi-l))u(xi)
=n(B) ‘lBl) ‘Cx,)+ f: (n(Bi)-71(Bi-l)) i
U(Xi)
i=2
. I
Hence the desired result obtains. Case 3. Let z N xi Ax, for some A E 2’ which is neither null nor universal. Then x1
xl, m,(.Y, z))) = 4m,(m,(.Y,
Y), m,4(x, z))).
Since by Al and A2, m,(y, x) < m,(y, z), (1) implies that the left hand side of the above equation is given as follows: 4mAmA.K
x)3 m,(x
z)))
= n(A) 4m,(.Y, xl) + (l-7@))
4mA.Y9 z))
= ~W{~W)
NY))
u(x)+ (1 -WC))
+(1-71(A)){~(A)U(y)+(l--n(A))u(z)} = n(A) 7c(AC)u(x) + n(A)(2 - 7c(A) - n(K))
u(y) + (1 - z(A))2 U(Z).
We have two cases to examine: m,(y, y) 5 mA(x, z); mA(x, z) < m,(y, y). Case 1 (m,(y, 4mAmA.h
y) 5 mA(x, z)).
It follows from (1) that
Yh m,(x, z)))
= 4-4) 4mA.K
Y)) + (1 -n(A))
4m.4(4
z))
=n(A)u(y)+(l-71(A)){7L(A)U(X)+(l--(A))U(Z)} = 7&4)( 1 - n(A)) U(X) + n(A) u(y) + (1 - n(A))2 U(Z).
NON-ADDITIVE
363
PROBABILITIES
Since u(m,(m,(y, xl, m,(y, 2))) = u(m,(m,dy,
Y), m,&,
n(A) n(AC) u(x) + n(A)(2 - n(A) - +q)
z))),
we obtain
u(y)
=n(A)(1-71(A))U(X)+~(A)u(Y),
which is rearranged to give rc(A)(l - x(A) - x(A”))(u(x) - u(y)) = 0. Since U(X) < u(y) and A is not null, we must have n(A) + rc(A”) = 1. Case 2 (m,(x, z)
y)). It follows from (1) that
Yh m,b,
z)))
= 4A”) u(~.&> =n(A”){n(A)
z)) + (1 - n(A”)) u(m,Ay, Y)) u(x)+ (1 -n(A))
u(z)} + (1 -n(AC))
u(y)
= n(A) n(A”) u(x) + (1 - n(A”)) u(y) + n(A”)( 1 -n(A))
Since u(m,.&,(y,
xl,
m,(y,
~1))
=
u(m.Jm.d~,
Y),
m,h,
z))),
we
u(z). obtain
7c(A)(2 - n(A) - n(A”)) u(y) + (1 - x(A))~ u(z) = (1 - n(A=)) u(y) + x(A’)( 1 - n(A)) u(z), which is rearranged to give (1 - n(A))(l -n(A) - n(AC))(u(z) - u(y)) = 0. Since u(y) < u(z) and A is not universal, we must have n(A) + n(A”) = 1. Next we assume that A is null. When A is universal, the proof is similar. We are to show that A’ is universal, so that n(A) + n(A”) = 1. If A” is neither null nor universal, then the preceding paragraphs imply that A is neither null nor universal. This is a contradiction, so that A’ is either null or universal. Assume that A” is null. Also assume that x < y
x)9 m‘4(y, 2))) = u(m.(Y, z)).
On the other hand, (1) implies that u(m,(m,(y,
Y)? m&3
z))) = u(mdx, z)) = WB(Y,
Y))
if
mdy, Y) 5 m,(x, z);
if
m&,
z)
y),
Since B is neither null nor universal, Al and A3 imply that hdx, z), mh Y,} im,(y, z). Thus by (I), m,(m,(~, Y), mdx> z))< m,(m,(y, x), m,((y, z)). This contradicts A7. Hence AC must be universal.
364
YUTAKANAKAMURA
(4) Suppose that A8 holds. Note that A8 implies A6 and A7. To show additivity of rc, it suffices to prove that if A c B then n(B) = n(A) + n(B\A). By the hypothesis of the theorem, suppose that C E 2’ is neither null nor universal, and .x< y for some x, y EX. Let z = m,(~, y) and w=m,(z,x), so by A3, x
w, z) - f&,(x,
x), m,(z, xl, m,(x,
- fc(mi& -fc(m
Y))
z, xl, mP(x, x3 u)) B&,
Ay)ym&,
~1).
Since x < w < z, (1) implies 4fp(x,
w, z)) = 4‘4) u(x)+ (n(B) - @\A))
u(w) + Cl- @)I
Note by (1) and (3) that U(Z) = X(C) U(X) + (1 -n(C)) rc(C) u(z) + (1 - rc(C)) u(x). Thus (1) and (3) imply
u(z).
u(y) and U(W) =
u(fc(m,,,(z~ XL mB(xvY)) = n(C) 4m,,,k xl) + Cl- 4C)) 4m&, Y)) =~~(C)(~~(B\A)U(Z)+(~--(B\A))U(X)} + (1- 4C))(4B) u(x)+ (1 -n(B)) U(Y)} = n(B\A){W) 4~) + (1 - n(C)) u(x)> + (4B) - NB\A)) 4x1 + (I- 4B))i4C) u(x)+ (1 - 40) U(Y)) = x(B\A) Since u(fAx,
u(w) + (n(B) - n(B\A)) u(x) + (1 -n(B))
w, z)) = 4fc(m,,,(z,
u(z).
XL m,(x, y))), we obtain
44) u(x)+ (n(B) - n(B\A)) u(w) = n(B\A) u(w) + (n(B) - nn(B\A)) 4x1,
which is rearranged to give (n(A) - n(B) + n(B\A))(u(x) Hence, n(B) = r(A) + n(B\A).
- u(w)) = 0.
Q.E.D.
6. CONCLUSIONS
The purpose of this paper has been to axiomatize a generalization of the subjective expected utility in Savage’s framework when the set of states is finite. Our axiomatization required that the set X of consequences be infinite, in contrast to Savage’s arbitrary X. However, X is not necessarily a connected topological space. A generalized representational form yielded
NON-ADDITIVE
PROBABILITIES
365
a utility function u on X and a monotonic probability measure over states, and adopted Choquet integration as in Schmeidler and Gilboa’s representation to account for non-additive probability measures. The six axioms were shown to be sufficient for the representation, where the sixth axiom was crucial in the finite state formulation. Also, we examined two additional axioms that gave complementarily additive or additive probability measures over states, respectively.
ACKNOWLEDGMENTS The author gratefully acknowledges two anonymous referees for pointing out errors in an earlier version of the paper and informing him of Wakker’s related works, and Irving H. LaValle for informing him of Gilboa’s recent work.
REFERENCES 1. M. ALLAIS, Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’ecole americaine, Econometrica 21 (1953), 5033546. 2. F. J. ANSCOMBEAND R. J. AUMANN, A definition of subjective probability, Annals Math. Statist. 34 (1963), 199-205. 3. S. H. CHEW, An axiomatization of the rank dependent quasilinear mean generalizing the Gini mean and the quasilinear mean, preprint, Department of Political Economy, Johns Hopkins University, Baltimore, 1984. 4. S. H. CHEW, E. KARNI, AND Z. SAFRA, Risk aversion in the theory of expected utility with rank dependent probabilities, J. Econ. Theory 42 (1987), 370-381. 5. G. CHOQUET,Theory of capacities, Ann. Inst. Fourier 5 (1955), 131-295. 6. D. DAVIDSON, P. SUPPES,AND S. SIEGEL,“Decision Making: An Experimental Approach,” Stanford Univ. Press, Stanford, CA, 1957. 7. D. ELLSBERG,Risk, ambiguity and the Savage axioms, Quart. J. Econ. 75 (1961), 643-669. 8. P. C. FISHBURN,Subjective expected utility: A review of normative theories, Theory and Decision 13 (1981), 139-199. 9. P. C. FISHBURN,“Nonlinear Preference and Utility Theory,” Johns Hopkins Univ. Press, 1988. 10. I. GILBOA, Expected utility with purely subjective non-additive probabilities, J. Math. Econ. 16 (1987), 65-88. 11. I. GILBOA, Duality in non-additive expected utility theory, Ann. Oper. Rex, in press. 12. D. M. GRETHERAND C. R. PLO~T, Economic theory of choice and the preference reversal phenomenon, Amer. Econ. Rev. 69 (1979), 6233638. 13. D. KAHNEMAN AND A. TVERSKY, Prospect theory: An analysis of decision under risk, Econometrica 47 (1979), 263-291. 14. D. H. KRANTZ, R. D. LUCE, P. SUPPES,AND A. TVERSKY, “Foundations of Measurement,” Vol. 1, Academic Press, New York, 1971. 15. J. PFANZAGL, “Theory of Measurement,” Wiley, New York, 1968. 16. J. QUIGGIN, A theory of anticipated utility, J. Econ. Behau. Organ. 3 (1982), 323-343. 17. L. J. SAVAGE, “The Foundations of Statistics,” Wiley, New York, 1954; Second revised ed., Dover, New York, 1972.
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NAKAMURA
Subjective probability and expected utility without additivity, preprint 84, Institute for Mathematics and Its Applications, Univ. of Minnesota, Minneapolis, 1984. 19. P. SLOVIC AND A. TVERSKY, Who accepts Savage’s axiom? Behau. Sci. 19 (1974) 368-373. 20. P. P. WAKKER, Concave additively decomposable representing functions and risk aversion, in “Recent Developments in the Foundations of Utility and Risk Theory” (L. Daboni, P. Montesano, and M. Lines, Eds.), pp. 249-262, Kluwer, 1986. 21. P. P. WAKKER, Nonadditive probabilities and derived strengths of preferences, Internal report 87 MA 03, University of Nijmegen, Department of Mathematical Psychology, Nijmegen, The Netherlands. 22. P. P. WAKKER, Continuous subjective expected utility with nonadditive probabilities. J. Math. Econ. 18 (1989) l-27. 18. D. SCHMEIDLER,
OF ECONOMIC
THEORY
51,
346366
(1990)
Subjective Expected Utility with Non-additive Probabilities on Finite State Spaces YUTAKA NAKAMURA University
of
Institute of Socio-Economic Planning, Tsukuba, l-l-l Tennoudai, Tsukuba, Ibaraki
305, Japan
Received July 12, 1988; revised September 5, 1989
Schmeidler and Gilboa’s representation generalizes subjective expected utility to cope with the Ellsberg paradox, so that the probability measure over states of the world need not be additive. This paper examines a similar generalization in Savage’s framework when the set of states is finite, while Savage’s states are continuously divisible. Our axiomatization requires that the set X of consequences be infinite in contrast to Savage’s arbitrary A’. Three representational forms are axiomatized to give non-additivity, complementary additivity, and additivity of probability measures, respectively. Journal of Economic Literature Classification Numbers: 022, 026. 0 1990 Academic Press, Inc.
1. INTR~O~CTI~N Theories of decision making under uncertainty provide subjective expected utility models that represent numerically the personal beliefs (subjective probabilities) and preferences of a decision maker. Under various frameworks numerous axiom systems and their numerical representations have been proposed (see the survey by Fishburn [8]). One of the most elegant and well-known axiomatizations of subjective probability and utility was given by Savage [17]. He developed an act-oriented theory which applies a preference relation over the set of acts. Acts are defined as functions from the states of the world into the consequences. There is abundant evidence that people’s carefully considered decisions often violate the assumptions of subjective expected utility theories (see Allais [ 11, Davidson, Suppes, and Siegel [6], Ellsberg [7], Slavic and Tversky [19 J, Kahneman and Tversky 1131, Grether and Plott [12], and others.) Recently, Schmeidler [ 18 J and Gilboa [lo] generalized subjective expected utility theory to accommodate Ellsberg-type violations of additive probability measures. Schmeidler adopted the idea of lottery acts (functions from the states of the world into probability distributions on the consequence space) introduced by Anscombe and Aumann [2]. Gilboa used 346 0022-0531/90
$3.00
Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
NON-ADDITIVE
PROBABILITIES
347
Savage’s basic formulation, so that the set of states is required to be continuously divisible, but required at least three consequences. Moreover, his theory provides that utility must be bounded as in Savage’s This paper axiomatizes a generalization of subjective expected utility to accommodate Ellsberg-type violation of additive probability in Savage’s framework when the set of states is finite as in Wakker [22]. Wakker assumed that the set X of consequences is a connected separable topological space, and derived a non-additive probability measure over states and a continuous utility function on X. Also, Wakker [21] extended his result to arbitrary state spaces. Our approach is different in that we adopt Savage’s framework and X is not necessarily a connected separable topological space, ‘, and it then gives rise to a more general result. The axiomatization requires that the set of consequences be infinite, while it is arbitrary in Savage’s utility theory. The axioms are shown to give a utility function over the consequence space, and a non-additive probability measure over states. The expectation of a utility function with respect to non-additive probability measures is given as Choquet integration as in Schmeidler and Gilboa’s representation, which is discussed in the next section. Section 3 states the axioms and the representation theorem that includes two special representations with complementary additivity and additivity of probability measures, respectively. Section 4 establishes a representation for all binary acts, then Section 5 completes the proof of the representation theorem in Section 3.
2. REPRESENTATIONS WITHOUT ADDITIVITY Let S be the set of states. Subsets of S are called events. For events in 2’ = {A : A c S}, denote the complement S\A of A by A”. An act is a function from S into the set X of consequences. F is the set of all acts. Let 5 on F be the binary preference relation with N and < defined in the usual way:forf,gEF,fwgiffdgandgdf;figiffdgandnot (gsf). Schmeidler and Gilboa’s representation has the following form: for all
f, g EF,
where u is a real valued function
on X, and rr is a monotonic
(not
’ Wakker [21,22] noted that the topological approach needs topological separability only for ordered partitions w of the set of states with exactly one o-essential event in his terminology.
348
YUTAKANAKAMURA
necessarily additive) measure on 2’ that satisfies n(S) = 1 and z(empty)=O. Monotonicity of n means that if A EB then rc(A) 5 n(B). Moreover, u is unique up to a positive linear transformation, and rc is unique, so that u and $A) for A E 2’ can be interpreted as a utility function on X and a subjective probability for an event A, respectively. The integration in the representation is defined in Choquet’s sense (see Choquet [S]) to account for non-additive probability measures as follows. Since we are concerned with a finite S, assume that S= {s,, .... s,} and l.416 . . . s U”, where ui= u(f(si)) for SUES and all i. Let zf denote the cumulative distribution function induced by 71through f so that
q(t)= n((s: 4fO)) 5 t>, with rcJ(t) = 0 for t < u1 and rcY(t) = 1 for t 2 u,. Then the integration given by n-1 sS
‘(f(‘))
dn(s)=
1
(“i-*i+l)
nf(“i)+
is
un,
i=l
which is referred to as the lower Choquet integral in Gilboa [ 111. Chew, Karni, and Safra [4] adopted this definition in risky situations. This means that expected utilities are calculated with respect to cumulative distributions. On the other hand, the Choquet integration, referred to as the upper Choquet integral in Gilboa [li], is defined by the expectations with respect to the decumulative distribution functions induced by R through acts. Schmeidler and Gilboa’s representation applies this definition. In risky situations, Quiggin [16] used it in the anticipated utility representation. Let rr* be the dual measure of rr such that n*(A) = 1 -n(A”) for all A E 2’. Gilboa [ 111 showed that the lower Choquet integral with respect to rt is equivalent to the upper Choquet integral with respect to K*. We say zhat q-71=71*, then rc is complementarily additive. Throughout the paper, the integrations always stand for the lower Choquet integral. 3. AXIOMS AND THEOREM Several additional notations and definitions will be useful in stating axioms. When F, and Fz are subsets of F, F1 5 F2 means f 5 g for all ffzF,andallgEFz.Wewritef~gg,ihwhenf~gandg~h;f
NON-ADDITIVE
PROBABILITtES
349
A E 2’ and x, y E X, denoted by xAy. A constant act is an act f = s x for some x E X, so every x E X is identified with a constant act. A partition of S is a sequence of non-empty events that are mutually disjoint and whose union equals S. For an n-partition P= {A,, .... A,,}, if f =A, xi for all i, then sometimes we write &(x1, .... x,) instead off: Also, xAy is sometimes written by f;l(x, y). Suppose that (x,, .... x,} = {xi ,,..., xi,}, P={A, ,..., A,}, P’=(A, ,,..., A,}, and P=P’. Then by the definition of the acts, &(x1, .... x,) = fP(xi,, .... xJ. If P” = .... x,). In {A, u AZ, A3, .... A,) and x, =x2, then &(x1, .... x,)=&.(x,, particular, xAx = x and xAy = yA”x. A null event A E 2’ is an event for which for all x, y, z E X, xAz - yAz whenever x 5 y 5 z. A universal event A E 2’ is an event for which for all x, y, z E X, xAy - xAz whenever x 5 y 5 z. An empty event is null. If x 4 y for some x, y E X, then S is not null but universal. If < is empty, then every A E 2’ is null and universal. Note that if there is an event which is neither null nor universal, then S includes at least two states, and i is not empty, so that x < y for some x, y E X. We say that 5 is bounded if, for each f E F, there are x, y E X such that x 5 f 5 y. Let N be any set of consecutive integers. Given an event A E 2’ which is neither null nor universal, we define a standard sequence as a set (a,: aie X, iE N) for which there exist a, b E X such that not(a - b), and either {a, b} 5 {ai} and aAa, for all bAai + 1 for all i, i+ 1 EN, or {a,} 5 {a, b} and a,Aa-a,+,Ab i,i+l~N. The following axioms apply to all .f E F, all x, y, z E X, and all A, B E 2’: Al. 5 on F is a bounded weak order. A2. If xAz 5 f 5 yAz then f- aAz for some a E X. A3. If A is not null and {x, y } 5 z, then x 5 y iff xAz 5 yAz; if A is not universal and z i, {x, y 1, then x 5 y iff zAx i, zAy. A4. If x 5 y and A E B, then xBy 5 xAy. A5. Every strictly bounded standard sequence is finite. A bounded weak order means by definition that 5 is bounded, complete, and transitive. A2 is a restricted solvability axiom. A3 and A4 are monotonicity axioms. A5 is an Archimedean axiom. Since 5 is bounded, for each f~ F, there are x, y E X such that x 5 f 5 y. By the definition of acts, x = XSZ and y = ySz. Therefore, by A2, f - aSz for some a E X. Thus f - a. Let m be a mapping from F into X that assigns a constant act m(f) for each f E F such that f -m(f). Let M be the set of all such mappings, so that if m, m’ EM, then by Al, m(f) - m’(f) for all f~ F. For an n-partition P = {A,, .... A,}, if f =A, xi for all i, then m(f) is denoted by m,(x,, .... x,). When P= {A, A”} and f = xAy, we denote m(f) = m,(x, y). We say that < is dense if, for each
350
YUTAKANAKAMURA
x,y~X with x
there is an UEX such that x
for A3, .... all
If xi 5 .. . 5 x, and y1 5 . . 5 y, with xi ,$ y, for all i, then
fA(m,(x ,, .... h), m,(y,, .... yJ) -f,(m,&, A7. mA(mBk A8. .f..(mp(xl,
, ~~1,-., m,Jx,, Y,)).
Y), m,(z, w)) - m,(m,Ax, z), mh, .... 4, m,b,, ...+ y,))-fp(mA(xl,
w)). ~~1, .... m,&,,
VA).
A6 is a version under uncertainty of the condition under risk proposed in Fishburn [9, Chap. 31 that modifies the third axiom in Chew [3]. Chew’s axiom was originally motivated by Quiggin [ 161. When P = {B, B”}, A7 is similar to the isometry condition if A #B, and the bisymmetry condition if A = B in Pfanzagl [IS]. A8 is a version of the isometry condition. The purpose of the paper is to prove the following: THEOREM 1. Suppose that S is finite, and that there is an event which is neither null nor universal. If Al-A6 hold, then
(1) there are a monotonic measure TCon 2’ and a real valued function u on X such that for all f, g E F,
(2) (3) (4)
n is unique, and u is unique up to a positive linear transformation ; n is complementarily additive if A7 holds; 7t is additive $A8 holds instead of A6.
The proof of the theorem will appear in Section 5. We note that the axioms except Al, A3, A4, and A5 are not necessary for the representation, as shown in the following example. Suppose that X is the set of all rational numbers, and that u(x) = x for all x E X. If rt takes on values of irrational numbers, then A2 does not hold, so that M is empty. In this case, A2 requires that rc take on values of rational numbers. Theorem l(4) gives an expected utility representation with an additive subjective probability measure in Savage’s framework when S is finite. Wakker [20] obtained a similar representation in the additive case when X is a connected separable topological space.
NON-ADDITIVE
4. A
351
PROBABILITIES
REPRESENTATION
FOR
BINARY
ACTS
We assume throughout the rest of the paper that Al-A6 hold, and there is an event which is neither null nor universal. Note that S includes at least two states, and < is not empty but dense. This section proves the following proposition, which implies that the representation of Theorem 1( 1) holds for all binary acts. PROPOSITION 1. There are two real valued functions, n on 2’, and u on X, such that n(empty) =O, n(S) = 1, and for all x, y, z, w E X, and all A,BE~~, ifxsyandzLw then
xAy 5 zBw AEB
iff
n(A)u(x)+(l-n(A))u(y)
imphes
1rc(B)~(z)+(l n(A) 2 7c(B).
-x(B))u(w);
Moreover, rc is unique, and u is unique up to a positive linear transformation, First we show three lemmas, and then give the proof of Proposition For aEX, let X*=
{xEX:b
X”=
{xEX:xsa),
X,=
{xEX:asx},
1.
x max=(xEX:x
352
YUTAKA NAKAMURA
2. Thomsen condition: If xAz’ - zAy’ and zAx’ - yAz’, then xAx’ - yAy’. 3. Restricted solvability: If xAx’ 5 yAy’ 5 PAX’, then bAx’ N yAy’ for some b E X”; if xAx’ 5 yAy’ 5 xA.z’, then xAb’ - yAy’ for some b’ E X,. . 4. Archimedean: Every strictly bounded standard sequence with respect to a and a’ is finite. 5. Essentiality: not(bAb’ - cAb’) for some 6, c E X” and some b’ E .I’,. ; not(bAb’ w bAc’) for some b E X” and some b’, c’ E X,.. LEMMA 1. If A E 2’ is neither null nor universal, then fir a, a’ E X* with a 5 a’, the triple (X”, X,., 5” ) is an additive conjoint structure. LEMMA 2. If A E 2’ is neither null nor universal, then there are two real valuedfunctions, 4 and $, on X such that for all x, y, z, w E X with x 5 y and z 5 w,
(1) .xAySzAw iff~(x)+Il/(.v)~~(z)+~(u’); (2) if 4’ and t+Ysatisfy (1) instead of 4 and t,!t, respectively, then there exist constants a > 0, PI, and f12 such that for all
x E X\X,,,,
for all
x 6 X\X,,,.
LEMMA 3. If A E 2’ is neither null nor universal, then there is a real valued function u on X such that for some 0 < GIc 1, and for all x, y, z, w E X with x 5 y and z 5 w.
xAy 5 zAw
ifs
ffu(x)+(l
-(W)u(y)~cYu(z)+(1
-cY)u(w).
Moreover, a is unique, and u is unique up to a positive linear transformation. Proof of Lemma 1. Suppose that A is neither null nor universal. By A 1, 5 is a weak order. It easily follows from A2, A3, and A5 that the independence, the restricted solvability, and Archimedean conditions of the additive conjoint structure hold. The essentiality condition follows from a, a’ E X* and the assumption that A is neither null nor universal. In what follows, we show the Thomsen condition. Suppose that xAz’- zAy’ and zAx’ - yAz’. Then by Al, m,(x, z’) m,(z, y’) and mA(z, x’)-m,(y, z’). We are to show that m,(x, x’) m,(y, y’), so xAx’- yAy’. The following three cases cover all possibilities: (x’, y’} 5 z’; {x’, z’ } 5 y’; { y’, 2’) 5 x’.
NON-ADDITIVE
353
PROBABILITIES
Suppose first that (x’, y’> 5 z’. By A3, m,Jx, z’) 5 mA(x’, z’) and m,(y, z’) 5 mA(y’, z’). Thus A3 and A6 imply
.Lh,(x,
xl m,4(z’,z’)) -jxm,(x, -fa(m,(z,
so by AL frc(m.&, ~‘1, m/h’,
z’), m&‘, 2’)) Y’), mAx’, z’))
(by A3)
-f/i(m,(z,
-q, m,(y’,
z’))
(by A6)
-fA(@~,(Y,
z’), m/h’,
;‘)I
(by A3)
-f,(m,(y,
Y’), mAz’, z’)),
z’)f-M=,(Y,
~‘1, ~/AZ’,
z’)).
(by A61
(by ‘46) BY
Al
and A3, {m,(x, x’), m,(y, y’)} 5 mA(z’, z’). Hence by A3, m,(x, x’) m,(y, Y’). Suppose next that (x’, z’} 5 y’. Then similarly we obtain fA(m,(x, x’), m,kz’, Y’)) --f,(mAy, y’h m,&‘, ~‘1). BY Al and A3, (m,dx, x’), m,(y, y’)} 5 m,(z’, y’). Hence by A3, nz,(x, x’) - m,(y, y’). Suppose last that { y’, z’} 5 x’. Then similarly, f,Jm,(x, x’), m,(z’, x’)) -f,(m,(y, y’), mA(z’, x’)). By Al and A3, {m,(x, x’), m,(y, y’)} 5 mA(z’, x’). Hence by Q.E.D. A3, m,(x, x’) - m,(y, ~‘1. Proof of Lemma 2. Suppose that A is neither null nor universal. Lemma 1 implies that for a E X*, the triple (X0, X,, 5”) is an additive conjoint structure. Thus by Theorem 2 of Chapter 6 in Krantz et al. [ 141, there exist two functions, 4, on X” and II/, on X,, such that for all x, y E X” and all z, WAX,, XAZ 5 YAW
iff
40(x) + ll/a(z) 5 4,(Y) + Ic/Aw).
If a 5 b for a, b E X*, then by Lemma 1, the triple (X0, Xb, 5”) is an additive conjoint structure. Since X” x X, c X” x X, and X” x X, _CXb x X,, for all x, y E X” and all z, w E X,, we have xAz 5 yAw
iff
4J-9 + k(4
5 UY)
+ Icl,(w)
iff
4bcX)
=< db(y)
+
+
$btZ)
@btw).
By the uniqueness of the additive representations for the additive conjoint structure, there are real valued functions k(a, b) > 0, k,(a, b), and ,&(a, b) for all a, b E X* with a 5 b such that #a(X)
=
k(a,
b)
$bcX)
+
k,(a,
6)
Ic/&) = Ma, b) @t,(x) + &(a, b)
for all
x E X”,
for all
x E X,.
354
YUTAKA
NAKAMURA
Given 4, and $a for a fixed a E X*, scale bb and $b for b E X,\X,,,,,, and 4, and II/, for CE X’\X,,, such that k(a, b)=k(c, a)= 1 and kj(a, b) = kj(c, a) = 0 for i = 1, 2. Then it easily follows that if b 5 c and b, c E X*, then db = 4, on Xb, and tjb = $, on XC. Therefore, for all ?cE X, define if
i(x) = h(x)
xEX*,
for some
= 4&7(x)
if
$(x1 = h(x)
if
x E Xmin,
aEX*
if
XEX,,,,
XE X*,
forsome
= *a(x)
a E X*
so 4 on X\X,,, and $ on X\X,,
are uniquely specified. If x, y E X\X,,,, then (x, y> 5 z for some ZE X\X,,,, since < is dense. Thus for all if {x, y} sz, then x, y9 z E x\x,,,, xsy
iff
XAZ 5 yAz
iff
#z(x) + $A)
iff
4x(x) 5 4,(Y)
(by scaling)
iff
4(x) 5 d(Y).
(by definition)
Similarly, for all x, y E X\X,i,, obtained, let
(by A3)
(by Lemma 1)
5 d,(Y) + 4+,(z)
x 5 y iff $(x) 5 $(y). With 4 and rj thus
~(x*)~suP{&):-=X\X,,,~
if
x* E X,,, ;
It/(x,) Sinf(ti(x):
if
x* E X,,,.
xEX\Xminl
Since < is dense, we obtain that 4(x) < 4(x*) for all XE X\X,,, and all < tj(x) for all XE X\X,i, and all x.+ E Xmin. x* E x,,,; +(x,) We are to show that 4 and II/ satisfy (1) and (2). Suppose that x 5 y and z 5 w. First, assume that z 5 y and x 5 w. Then by Al, A2, and A3, {x,z}~a~{y,w} for some aEX. IfaEX*, then by Lemma1 and the constructions of 4 and $, xAy 5 zAw
Note
iff
4&J
iff
4.x(x) + Ii/,(Y) 5 &7(z) + Iclw(w)
iff
d(x) + t4Y) s 4(z) + b+(w).
+ @a(Y) I4a(z)
+ $a(w)
that 4(x.+.) for x, E Xmin is finite, otherwise, x,Ax< {x, z}, so we obtain
a, XE X*. If ac Xmin, then a-
a for all
NON-ADDITIVE
xAy <, zAw
355
PROBABILITIES
iff aAy<_aAw
(by Al and A3)
iff
ysw
(by A3)
iff
$(Y) <=Ii/(w)
iff
4(a)+$(~)54(a)+ti(w)
iff
4(x)+W)54(z)+$(w).
If aEXim,x, then (1) similarly follows. Next assume that either y < z or w < x. Then (1) easily follows from Al, A3, and the constructions of 4 and rj. The constructions of 4 and J/ immediately give (2). Q.E.D. Proof of Lemma 3. Suppose that A is neither null nor universal. Let # and $ satisfy Lemma 2(l) with 4(x*) zsup{d(x): x E X*} if x* l Xmaxr and IC/(x,)sinf{lC/(x): x6X*} if x* EX,,,~“. Then we show that choosing the appropriate 4(x*) and y?(x,) if x* l Xmax and x.+ EX,.,.,~,,, there are constants, /? > 0 and y, such that $(x) = @j(x) + y for all x E X. Define
ha(x) = d(mA4
x)1,
$dx)
= $(mAa, xl)
for
+xE X,
if
a $ X,,,;
L(x)
a)),
$2Jx) = IL(m,(x, a))
for
XE x1
if
a$ Xmin.
= 4(m,(x,
For x, y, z, w E X, and a $ X,,,, if x 5 y and z 5 w, then by Al and A3, mAa, x) 5 mAa, y) and m,(a, z) 5 mAta, w), so #l&f iff
+ CltdYf 5 #la(Z) + $1,(w) &mAa,
xl) + Nm,(a,
y)) 5 #(mAa, ~1) + $(mAa, w)) iff .LAm,Ja, xl, mAa, Y)) 5f.A m A a, z 1, m,(a, w)) (by Lemma 2(l)) iff .fa(mA(a, a), mAx, Y)) 5fA iff
m,dx, v) 5 m,&, w)
iff
xAy 5 zAw.
mA a , a 1, m,(z, w))
(by Al andA6)
(by A31
If x 5 y and z 5 w for x, y, z, w E X” and a $ X,,, , then similarly xAyszAw
iff
h(x)
+ WY)
S dd4
we get
+ Il/zh4.
Thus di, and t,ki, satisfy Lemma 2( 1) on X, for i = 1, and on X” for i = 2. Therefore, there are real valued functions, ki>O and kli, on X\X,,, for i = 1 and X\X,i” for i = 2 such that for a 4 X,,, and b 4 Xmin, 4(mAa, xl) = k,(a) d(x) + k,,(a)
for
x~X,\X,,,,
ti(m.h
for
X E Xb \Xmin.
6)) = h(b) d4x) + kdb)
Suppose that x 5 ( y, z) 5 w for y, z E X*. Then x & X,,,,, and w $ Xmin.
356
YUTAKA
NAKAMURA
By Al and A3, m,(x, y) 5 m,(z, w) and m,(x, z) 5 m,(y, w). It follows from A6, Lemma 2( 1 ), and the preceding paragraph that
iff
$(m&,
Y)) + Il/(m.&,
iff
k,(x) 4(.v) + M-4
~1) = 4(mAx, =)I + 4WA(~, w))
+ MN
Nz) + Mw)
= k,(x) d(z) + k,,(x) + b(w) Ii/(Y) + k,,(w) iff
k,(xNti(.v)
-d(z))
= Mw)($(Y)
- W)).
Since the above equations are satisfied for all x, v, z, w E X with y, z E X* and x 5 {y, z} 5 w, there are constants, 13.and 6, such that for all x 4 X,,, and all w 4 Xmin, and
k,(x)=,?>0 Thus 4Kv)
-d(z))
= @KY)
- W)),
k2(w) = 6 > 0. so W(u)
- MY)
= W(z)
- U(z)
for
all y, z E X*. Therefore, there are constants, /I > 0 and y, such that $(x) = /Q~(x) +y for ail xEX*. Suppose that X,,, is not empty. If sup 4(x)= + cc, then by the preceding paragraph, Ii/(x*) = + cc for x* l Xmax. By Lemma 2(l), this implies that a < xAx* for all a, x E X *. Therefore, sup d(x) must be finite. The preceding paragraph implies that Ii/(x*) >=sup Ii/(x) = /I sup 4(x) + y for x* E x,,,. Thus define 4(x*) = ($(x*)-7)/j?, so that $(X)=&~(X) + y for XEXIXI,,. Suppose next that Xmin is not empty. Then similarly, we can define @(x.+)=&~(x*)+Y for x.+EX,,,. Hence Il/(x)=/@(x)+y for all XE X. With /I> 0, 4(x*) for x* E X,,,,,, and tj(x,) for x* E Xmin thus obtained, let CY= l/(1 + 8). Then 0 -CO!< 1. Define U(X) = $(x)/a for all x E X. Then the representation of the lemma easily follows. To show the uniqueness of U, suppose that U’ also satisfies the representation. By the uniqueness of 4 and +, it easily follows that there are constants, /I > 0 and y, such that u’(x) = flu(x) + y for all x E X*. Suppose that x* Ay -z for x* E X,,, and y E X*. Then z E X*. We obtain au(x,)+(l -a)u(y)=u(z), au/(x,)
+ (1 -a) u’(y) = u’(z),
so u’(x*) = flu(x,) + y. Similarly, u’(x*) = flu(x*) + y for x* E X,,,. Hence Q.E.D. the uniqueness part of the lemma obtains. Proof of Proposition 1. If A is neither null nor universal, then Lemma 3 implies that there is a real valued function uA on X such that for some O
xAy 5 ZAW
iff
aAuA(x)+(l--clA)~A(y)~aAuA(z)+(l-aA)~~(w).
NON-ADDITIVE
357
PROBABILITIES
Thus we define rc on 2’ as follows: n(A)=cc,
if
A is neither null nor universal;
= 1
if
A is universal;
=o
if
A is null.
Suppose that A and B are neither null nor universal. Then we show that uA = clus + fi for some constants, CI> 0 and /?. Given aeX*, a binary relation 5” on X” x X, is defined as follows: for xy, zw E X” x X,,
By Al and A6,
Since by Al and A3, m&x, a) 5 m,(a, C = A or B, Lemma 3 implies that xy5”zw
iff
44)
uAm&, 5 @I
iff
n(B) do&, 5 n(B)
y) and m,(z, a) 5 m,(u, w) for
~1) + (1 - W)) u,4(mIk
a)) + (1 -Ml)
4) + (1-V)) ~&AZ,
~,&,(a,
Y)) u,(m,(a,
+OK&~
~1) + (1 - 4B))
w))
Y)) u&Au,
~1).
By the uniqueness of additive representations, there are real valued functions, w >O and c, on X* such that for all UEX* and all XEXO,
WI U.&&G ~1) = 4~) 4B) u,(m,(x, ~1) + 4~). Since m,(x, y) E X” for x, y E X”, it follows from Al, A6, Lemma 3, and the above equation that for x, y E X” with x 5 y,
W) uAm,(m,(x, Y)?m,4(4 a))) = xc(A)uAm,(m,(x> ah m,(Yv = x(A)2 =da)
a)))
U,4(~Lf(X, a)) + (1 - W)) dB){W)
~&Ax,
n(A)
a)) + Cl-
u,(m,(y, 44))
u,b,(y,
a)) aI)> + 4~).
On the other hand, by the preceding paragraph, Lemma 3, Al, A3, and A6,
358
YUTAKA
NAKAMURA
Therefore, the above two equations give
u,(m&,(x, a), ?.4(YYa))) =71(A) u,(m,(x, a)) + (1 - NA)) u,(m,(y3 a)). We show that u,(m,(x, y)) = x(A) uLI(x) + (1 - n(A)) uB(y) for all x, y E X with x 5 y. Thus by Al and Lemma 3, we must have uA = clus + /I for some constants c1> 0 and /I. Suppose first that x 5 y and x, y E X*. If zAy <, x for some z E X, then by A2, wAy w x for some w E X. Therefore, the desired result easily follows from the preceding paragraph. Thus assume that x< zAy for all z with z
~B(mB(m,b~ Y), m,(y, Y))) = WQ u/Am& y)) + (1 - n(B)) UB(Y), and
u,(mAw Y)) = 44) UAW)+ (1 - 44)) U,(Y) = n(A) @) UB(X)+ (1 -n(A) n(B)) UB(Y). By Al, A3, and A6,
m,4(w,Y) -m,(m,bG YL mLi(y, Y)) - m*(mAx, Y), m,(y, Y))?
sou~(~~(w,Y))=u~(~~(~~(x,Y),~,(Y,Y))). Therefore, W)4B)u&)+ (1- 64) n(B))U,(Y)= 49 u,(m,(x,Y))+ (1-n(B)) DAY). This is rearranged to give the desired result. Hence u,(m,(x, y)) = x(A) uB(x) + (1 -z(A)) us(y) for all x, y E X* with x 5 y. If either x, y E Xmin or x, y E X,,,, then the desired result easily follows, so we show that it also holds when XE X,,, or y E X,,,,,. Suppose that x E Xmin and y E X*. When x E X* and y E X,,, , the proof is similar. Then
NON-ADDITIVE
PROBABILITIES
359
y 5 z for some ZE X*. Since m,(x, y) and m,(y, z) are in X* for C = A or B, Lemma 3 and the preceding paragraph imply that
u&,(m/i(x, Y), m.4bs z))) = z(B) ~B(~/l(& Y)) + (1 -n(B)) u,(m,(.Y, z)) =71(B)Us(mA(X,y))+(f--(B)){~n(A)u,(y)+(l-n(A))u,(z)}, u&&,(x,
VI, m,(y, z)))
= n(A) u,(m,(x, Y)) + (1 -n(A)) uB(m,(Y>z)) = ~W{W) UB(X)+ (1 - 4B)) Us(Y)) +(l-n(A)){71(A)ug(y)+(l--(B))u,(z)}. BY A6 mAmAx, Y), m,h, ~1) ~m,(m&, Y), mB(y, ;)I, so by Lemma 3, u,b&Ax~ ~),m,(y,z)))=u,(m,(m,(x, Y)TAY,~))). Thus u,(mAx, Y)) = 64) u&J + (1-7-U)) Us(Y). Suppose last that XE Xmin and ye X,,,. Applying the results in the obtains. Hence preceding paragraphs, the desired result similarly u,(m,(x, JJ)) = x(A) uB(x) + (1 - n(A)) uB(y) for all x, y E X with x <, y. Under appropriate positive linear transformations, there is a real valued function u on X such that u = uA for all A E 2’ which are neither null nor universal. Noting that if x < y then xAy - y when A is null, and xAy -x when A is universal, it follows from the preceding paragraphs that for all A E 2’, and all x, y E X with x 5 y,
u(mAx, Y)) = n(A) u(x)+ (1 -n(A)) U(Y). Thus for all A, BE 2’ and all x, y, z, w E X with x 5 y and z 5 w, xAy 5 zBw
iff
m,(x, y) 5 m,(z, w)
iff
u(m,k
iff
~A)~~)+(~-~(A))u(~)~~(B)u(z)+(~-Tc(B))u(w),
Y)) 2 uh,(z,
w))
so the representation of the proposition holds. It easily follows from A4 that A G B implies n(A) 5 n(B). The uniqueness of rc and u follows from the construction of u, and Lemma 3. Q.E.D.
642/51/2-9
360
YUTAKANAKAMURA
5. PROOF OF THE THEOREM
Throughout the section, we shall assume that S is finite, and let u on X, and rr on 2’ be real valued functions obtained in Proposition 1. We extend 2.4on X as follows:
u(f) = 4x)
when
u(f) = u(m(f ))
for all
f=sx; f EF.
Thus by Al, for all f, ge F, f 5 g iff u(f) s u(g). We note by Al and A6 that if xi - yi for i = 1, .... n, then fp(x, , .... x,) N f,(y,, .... y,) for all n-partitions. Proof of Theorem 1. (1) We are to show that u(f) = Js u(f(s)) LIZ(S), so of (1) obtains. Let P= (A,, .... A,} be an n-partition,
that the conclusion and
fn=A,xi
for
i=l,...,n,
where x1 5 ... 5 x, and xi E X for all i. Also let BO be empty, and Bi = Cj= r Aj, so B, = S. For all fi E F, Proposition 1 gives the desired result. Thus we assume n > 2. Suppose that the conclusion is true for all n-partitions with n
since this easily leads to the definition of the lower Choquet integral. If xi - xi for some 1 5 i < j 5 k, then the desired result easily follows from Al, A6, and the hypothesis of the induction. Thus we shall assume that x1-c ...
xAxzwxl
and x,_,Ay*x,
for some x, y E X and some
AEON.
CASE 2. either xAx, - x1 for some XE X and some nonuniversal A E 2’ or xk ~ I By - xk for some y E X and some nonnull BE 2’. CASE 3. not(xAx, -x,) and not (xk- r By -xk) nonuniversal A E 2’. and all nonnull BE 2’.
for all x, .YEX, all
Let P’ = {A,, .... Ak _ 2, A, _, v Ak} and PI’ = {A, u A,, A,, .... Ak}, so they are (k - 1 )-partitions.
NON-ADDITIVE
361
PROBABILITIES
Case 1. Since xAx, - x, and xk _ i Ay - xk for some x, y E x and some A E 2’, A3 implies that x < x1 and xk < y It follows from Al and A6 that fk
=
fPtxl
> ...Y xk)
-fp(m,(x,
4,
‘-f.4(m,(x,
x2,
‘-f,=,(m&,
m/&z,
-4,
. . . . xkp2,
x2,
. . . . mA(Xk--I,
xk&1,
. . . . XL-l),
xk-l)>
xk-I),
mA(Xk-I,
mP(x2~
m,“b2,
x2,
...> xkp1,
x39
v)) ..., xkLl,
y))
v)),
SO by Al, fk - Wl,(m,.(X, X2, .... +-I), m&X,, .... x&l, J’)). Since XAX, - x1 and xk- lAy - xk, Proposition 1 implies that u(xi) = and u(xk) = x(A) u(x& 1) + (1 -n(A)) U(y). 71(A) U(X) + (1 -x(A)) U(X2) Thus by the definition of u(fk), and the hypothesis of the induction, we get u(fk) = ~(m,.,(mP’b, =
n(A)
x2,
u(m,.(x,
. . . . xk-
x2,
I),
m,“(X,,
. . . . xk-
. ..) Xkpl))+(l
-n(A))
1, y))) u(mP”(x2~~~9
xk-1,
y))
k-2 =
WI
4Bl)
4x)
+
1 i==2
i
+
t1 -dBk-2))
(44)
-@-I))
4Xi)
dxk--1) I k-1
+
t1
-71(A))
n(B2)
u(x2)
+
1
i
+
t1
(71(Bi)-n(Bj-l))
u(xi)
i=3
-nn(Bk-l))
dy) I
=
ii,
bCBi)
-
71(Bi-
1))
u(xi).
Case 2. Suppose that xk- i By - xk for some y E X and some nonnull BE 2? A similar proof applies when xAx, -x1 for some x E X and some nonuniversal A E 2S. By the assumption and A3, B is neither null nor universal. Then let z N x1 Bx2 for some z E X, so xi < z < x2. Thus by Al and A6, fP(Z, x2,
since
fk
-
mp(x,,
4mp(z, x2,
.*.,
xk)
-~s-(~B(~,
9 x2)~
mB(x2~
-,fLh&l,
***, xk),
m,(x,,
‘-fB(mP(Xlr
..*, xk),
m,f’(X2,
. . ..
xk), Proposition
I implies
. .. .
xk))
+
=
44
u(fk)
(1
-n(B))
x2)~
...Y m,(xk, x2,
x3,
xk))
. . . . xk))
..*, xk)).
+.+2,
. . . . xk)).
362
YUTAKA
NAKAMURA
Noting that u(mp(z, x2, .... X/C))=~(~I) Utz)+Cf=2 (n(Bi)-n(BiL1)) by Case 1, and u(z) = z(B) u(x,) + (1 -z(8)) u(xz) by Proposition above equation is rearranged to give n(B)uUk)=n(Bl)u(z)+
i i=2
u(xi)
1, the
(nc(Bi)-n(Bi-l))u(xi)
=n(B) ‘lBl) ‘Cx,)+ f: (n(Bi)-71(Bi-l)) i
U(Xi)
i=2
. I
Hence the desired result obtains. Case 3. Let z N xi Ax, for some A E 2’ which is neither null nor universal. Then x1
xl, m,(.Y, z))) = 4m,(m,(.Y,
Y), m,4(x, z))).
Since by Al and A2, m,(y, x) < m,(y, z), (1) implies that the left hand side of the above equation is given as follows: 4mAmA.K
x)3 m,(x
z)))
= n(A) 4m,(.Y, xl) + (l-7@))
4mA.Y9 z))
= ~W{~W)
NY))
u(x)+ (1 -WC))
+(1-71(A)){~(A)U(y)+(l--n(A))u(z)} = n(A) 7c(AC)u(x) + n(A)(2 - 7c(A) - n(K))
u(y) + (1 - z(A))2 U(Z).
We have two cases to examine: m,(y, y) 5 mA(x, z); mA(x, z) < m,(y, y). Case 1 (m,(y, 4mAmA.h
y) 5 mA(x, z)).
It follows from (1) that
Yh m,(x, z)))
= 4-4) 4mA.K
Y)) + (1 -n(A))
4m.4(4
z))
=n(A)u(y)+(l-71(A)){7L(A)U(X)+(l--(A))U(Z)} = 7&4)( 1 - n(A)) U(X) + n(A) u(y) + (1 - n(A))2 U(Z).
NON-ADDITIVE
363
PROBABILITIES
Since u(m,(m,(y, xl, m,(y, 2))) = u(m,(m,dy,
Y), m,&,
n(A) n(AC) u(x) + n(A)(2 - n(A) - +q)
z))),
we obtain
u(y)
=n(A)(1-71(A))U(X)+~(A)u(Y),
which is rearranged to give rc(A)(l - x(A) - x(A”))(u(x) - u(y)) = 0. Since U(X) < u(y) and A is not null, we must have n(A) + rc(A”) = 1. Case 2 (m,(x, z)
y)). It follows from (1) that
Yh m,b,
z)))
= 4A”) u(~.&> =n(A”){n(A)
z)) + (1 - n(A”)) u(m,Ay, Y)) u(x)+ (1 -n(A))
u(z)} + (1 -n(AC))
u(y)
= n(A) n(A”) u(x) + (1 - n(A”)) u(y) + n(A”)( 1 -n(A))
Since u(m,.&,(y,
xl,
m,(y,
~1))
=
u(m.Jm.d~,
Y),
m,h,
z))),
we
u(z). obtain
7c(A)(2 - n(A) - n(A”)) u(y) + (1 - x(A))~ u(z) = (1 - n(A=)) u(y) + x(A’)( 1 - n(A)) u(z), which is rearranged to give (1 - n(A))(l -n(A) - n(AC))(u(z) - u(y)) = 0. Since u(y) < u(z) and A is not universal, we must have n(A) + n(A”) = 1. Next we assume that A is null. When A is universal, the proof is similar. We are to show that A’ is universal, so that n(A) + n(A”) = 1. If A” is neither null nor universal, then the preceding paragraphs imply that A is neither null nor universal. This is a contradiction, so that A’ is either null or universal. Assume that A” is null. Also assume that x < y
x)9 m‘4(y, 2))) = u(m.(Y, z)).
On the other hand, (1) implies that u(m,(m,(y,
Y)? m&3
z))) = u(mdx, z)) = WB(Y,
Y))
if
mdy, Y) 5 m,(x, z);
if
m&,
z)
y),
Since B is neither null nor universal, Al and A3 imply that hdx, z), mh Y,} im,(y, z). Thus by (I), m,(m,(~, Y), mdx> z))< m,(m,(y, x), m,((y, z)). This contradicts A7. Hence AC must be universal.
364
YUTAKANAKAMURA
(4) Suppose that A8 holds. Note that A8 implies A6 and A7. To show additivity of rc, it suffices to prove that if A c B then n(B) = n(A) + n(B\A). By the hypothesis of the theorem, suppose that C E 2’ is neither null nor universal, and .x< y for some x, y EX. Let z = m,(~, y) and w=m,(z,x), so by A3, x
w, z) - f&,(x,
x), m,(z, xl, m,(x,
- fc(mi& -fc(m
Y))
z, xl, mP(x, x3 u)) B&,
Ay)ym&,
~1).
Since x < w < z, (1) implies 4fp(x,
w, z)) = 4‘4) u(x)+ (n(B) - @\A))
u(w) + Cl- @)I
Note by (1) and (3) that U(Z) = X(C) U(X) + (1 -n(C)) rc(C) u(z) + (1 - rc(C)) u(x). Thus (1) and (3) imply
u(z).
u(y) and U(W) =
u(fc(m,,,(z~ XL mB(xvY)) = n(C) 4m,,,k xl) + Cl- 4C)) 4m&, Y)) =~~(C)(~~(B\A)U(Z)+(~--(B\A))U(X)} + (1- 4C))(4B) u(x)+ (1 -n(B)) U(Y)} = n(B\A){W) 4~) + (1 - n(C)) u(x)> + (4B) - NB\A)) 4x1 + (I- 4B))i4C) u(x)+ (1 - 40) U(Y)) = x(B\A) Since u(fAx,
u(w) + (n(B) - n(B\A)) u(x) + (1 -n(B))
w, z)) = 4fc(m,,,(z,
u(z).
XL m,(x, y))), we obtain
44) u(x)+ (n(B) - n(B\A)) u(w) = n(B\A) u(w) + (n(B) - nn(B\A)) 4x1,
which is rearranged to give (n(A) - n(B) + n(B\A))(u(x) Hence, n(B) = r(A) + n(B\A).
- u(w)) = 0.
Q.E.D.
6. CONCLUSIONS
The purpose of this paper has been to axiomatize a generalization of the subjective expected utility in Savage’s framework when the set of states is finite. Our axiomatization required that the set X of consequences be infinite, in contrast to Savage’s arbitrary X. However, X is not necessarily a connected topological space. A generalized representational form yielded
NON-ADDITIVE
PROBABILITIES
365
a utility function u on X and a monotonic probability measure over states, and adopted Choquet integration as in Schmeidler and Gilboa’s representation to account for non-additive probability measures. The six axioms were shown to be sufficient for the representation, where the sixth axiom was crucial in the finite state formulation. Also, we examined two additional axioms that gave complementarily additive or additive probability measures over states, respectively.
ACKNOWLEDGMENTS The author gratefully acknowledges two anonymous referees for pointing out errors in an earlier version of the paper and informing him of Wakker’s related works, and Irving H. LaValle for informing him of Gilboa’s recent work.
REFERENCES 1. M. ALLAIS, Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’ecole americaine, Econometrica 21 (1953), 5033546. 2. F. J. ANSCOMBEAND R. J. AUMANN, A definition of subjective probability, Annals Math. Statist. 34 (1963), 199-205. 3. S. H. CHEW, An axiomatization of the rank dependent quasilinear mean generalizing the Gini mean and the quasilinear mean, preprint, Department of Political Economy, Johns Hopkins University, Baltimore, 1984. 4. S. H. CHEW, E. KARNI, AND Z. SAFRA, Risk aversion in the theory of expected utility with rank dependent probabilities, J. Econ. Theory 42 (1987), 370-381. 5. G. CHOQUET,Theory of capacities, Ann. Inst. Fourier 5 (1955), 131-295. 6. D. DAVIDSON, P. SUPPES,AND S. SIEGEL,“Decision Making: An Experimental Approach,” Stanford Univ. Press, Stanford, CA, 1957. 7. D. ELLSBERG,Risk, ambiguity and the Savage axioms, Quart. J. Econ. 75 (1961), 643-669. 8. P. C. FISHBURN,Subjective expected utility: A review of normative theories, Theory and Decision 13 (1981), 139-199. 9. P. C. FISHBURN,“Nonlinear Preference and Utility Theory,” Johns Hopkins Univ. Press, 1988. 10. I. GILBOA, Expected utility with purely subjective non-additive probabilities, J. Math. Econ. 16 (1987), 65-88. 11. I. GILBOA, Duality in non-additive expected utility theory, Ann. Oper. Rex, in press. 12. D. M. GRETHERAND C. R. PLO~T, Economic theory of choice and the preference reversal phenomenon, Amer. Econ. Rev. 69 (1979), 6233638. 13. D. KAHNEMAN AND A. TVERSKY, Prospect theory: An analysis of decision under risk, Econometrica 47 (1979), 263-291. 14. D. H. KRANTZ, R. D. LUCE, P. SUPPES,AND A. TVERSKY, “Foundations of Measurement,” Vol. 1, Academic Press, New York, 1971. 15. J. PFANZAGL, “Theory of Measurement,” Wiley, New York, 1968. 16. J. QUIGGIN, A theory of anticipated utility, J. Econ. Behau. Organ. 3 (1982), 323-343. 17. L. J. SAVAGE, “The Foundations of Statistics,” Wiley, New York, 1954; Second revised ed., Dover, New York, 1972.
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Subjective probability and expected utility without additivity, preprint 84, Institute for Mathematics and Its Applications, Univ. of Minnesota, Minneapolis, 1984. 19. P. SLOVIC AND A. TVERSKY, Who accepts Savage’s axiom? Behau. Sci. 19 (1974) 368-373. 20. P. P. WAKKER, Concave additively decomposable representing functions and risk aversion, in “Recent Developments in the Foundations of Utility and Risk Theory” (L. Daboni, P. Montesano, and M. Lines, Eds.), pp. 249-262, Kluwer, 1986. 21. P. P. WAKKER, Nonadditive probabilities and derived strengths of preferences, Internal report 87 MA 03, University of Nijmegen, Department of Mathematical Psychology, Nijmegen, The Netherlands. 22. P. P. WAKKER, Continuous subjective expected utility with nonadditive probabilities. J. Math. Econ. 18 (1989) l-27. 18. D. SCHMEIDLER,