Expected utility with an interval ordered structure

JOURNAL

OF MATHEMATICAL

Expected

PSYCHOLOGY

Utility

32, 298-312 (1988)

with

an Interval

YUTAKA Department

of Precision

Ordered

Structure

NAKAMCJRA Engineering,

Osaka

University

This paper examines an interval ordered structure under risky conditions and proves that it has an expected utility representation with a threshold function. In addition to the assumption of an interval order, two independence axioms and a strong Archimedean axiom are necessary and sunicient for the representation. The threshold is given by a nonnegative linear functional. We also explore a special structure which gives a nonnegative constant threshold function. :i” 1988 Academic

Press. Inc

1. INTRODUCTION The theory of preference underlying utility theory generally assumes that the indifference relation is transitive. With regard to the vagueness of a decision maker’s preference, however, it has been argued that transitivity of the indifference relation is objectionable since utility is not perfectly discriminated (see Lute (1956), Vedder (1973), and others). Lute (1956) writes that “it is not implausible that the phenomenon of imperfect response sensitivity to small changes in utility is closely related to intransitivity of the indifference relation.” He proposed a semiorder to accommodate the intransitive indifference relation. Fishburn (1985) and others have examined an interval order as a more general order. In risky situations, intransitivity might likely occur unless a decision maker can discriminate small changes of utility difference among gambles and sure outcomes. Aumann (1962), Fishburn (1971, 1982), and others have examined partially ordered structures to derive one-way utility representations which preserve the preference relation but not the indifference relation. Lute (1956, 1973), Fishburn (1968), and Vincke (1980) have investigated semiordered structures under risky conditions and Vincke developed necessary and sufficient axioms for the existence of expected utility with a threshold function on semiordered mixture spaces. Vincke’s axioms use auxiliary relations defined from the preference relation. This paper investigates an interval ordered structure under risky conditions. We present necessary and sufficient axioms for the existence of expected utility with a threshold function. In addition to the assumption of an interval order, the axiom system consists of two independence axioms and a strong Archimedean axiom. The Reprint requests should be sent to Yutaka Nakamura, Institute of Socio-Economic Planning, University of Tsukuba, l-l-l Tennoudai, Tsukuba, Ibaraki 305, Japan.

298 0022-2496/88 $3.00 Copynght 0 1988 by Academic Press, Inc. All rights of reproduction m any form reserved.

INTERVALORDERED

STRUCTURE

299

threshold function is shown to be a nonnegative linear functional. We also present an axiom that yields a nonnegative constant threshold function. The paper is organized as follows. Section 2 states the axioms and two interval representation theorems. Section 3 proves some implications of the axioms. Section 4 examines structural properties of an interval ordered structure. Section 5 gives the proofs of two interval representation theorems. 2. AXIOMS AND REPRESENTATIONS

Let P be a convex set of probability measures on a set of consequences, so @ + (1 - i)q E P when p, q E P and 0 < %< 1. A strict preference relation on P is denoted by < (read as “is less preferred than”). We shall let - be the symmetric complement of <, so that p-q iff neither p< q nor q< p, and interpret it as an indifference or incomparability relation. Also, let 5 = < u -. As is customary in the literature, it is implicitly assumed throughout that two identical probability measures, no matter how they are structured, are substitutable for one another. In particular, for p, q, r E P, if p = q and p - r, then q - r. The following axioms apply to all p, q, r, s E P and all 0 < 3,< 1: Al.

p
A2.

< is irreflexive.

rp
or r
A3. p-q,

r-s+Ap+(l

A4. p
ris*1p+(l--,I)r
A5. p
-i)r-Rq+(l

-2)s.

and /?p+(l-P)r
for some CZ,~E(O, 1).

Axioms Al and A2 imply that i is asymmetric and transitive, and by definition that < is an interval order. Axioms A3 and A4 are independence axioms introduced by Fishburn (1968). Axiom A5 strengthens the Archimedean axiom which says that ifp
P < 4 *u(p)

+ o(p) < u(g).

If u’ and o’ > 0 satisfy the representation also, then there are two real numbers, a > 0 and b, such that u’ = au + b and CT’= ao.

In the theorem u is a utility function, and c is a threshold function. Linearity means that u(Lp + (1 - A)q) = k(p) + (1 - A) u(q) for all p, q E P and all 0 -CII < 1. Let u = u + (T. Then the representation of the theorem is modified to p < go u(p) <

300

YUTAKANAKAMURA

u(q) - a(q). Alternatively, we have p < q o u(p) < u(q) with u(p) Q u(p) for all p E P. We say that (P, <) is an interval ordered structure if < satisfies Al, A2, A3, A4,

and A5. The proof of the theorem appears in Section 5. We introduce the following axiom, which applies to all p, q E P: A6. pmaq+(l-a)p This axiom Theorem 1:

for some Oq-ccp+(l-a)q.

yields the following

representation

theorem

as a special case of

THEOREM 2. Suppose (P, <) is an interval ordered structure with < nonempty. Then A6 holds tf and only if there is a linear functional u on P such that for some a>O,

If MI and a’ > 0 satisfy the representation also, there are two real numbers, b > 0 and c, such that u’ = bu + c and a’ = ba.

The proof appears in Section 5. Throughout

the paper, we shall assume that

(P, < ) is an interval ordered structure.

3. IMPLICATIONS

OF THE AXIOMS

This section introduces some notations and proves implications of an interval ordered structure (P, i ). When P, and P, are subsets of P, P,
(2)

(3)

1. (1)

Zfp
thenq-ap+(l-a)rforsomeO
Ifp 4 q and q - r, then for some 0
Zfp-q

Ap+(l-l)r-q

for all 0 d A < a,

%pps(l -I)r
foralla
and q
then for some O
1.

1,

q - E,r+ (1 - A)p

for all 0 < 1”,< a,

q
foralla
(4)

Zfp < q, then p 5 Ap + (1 - E.)q 5 q for all 0 < A< 1.

(5)

Zfp
thenAp+(l-A)rsAq+(l-i)rforallrEPandallO
Proof (1) Suppose that p
INTERVAL

ORDERED

STRUCTURE

301

(g.1.b.) of S, so by A5, 66~~1. By A5, q<&+(l -/I)r for some O
q-ap+(l

-a)r.

(2) Suppose that p < q and q mr. Let S=(a:ccp+(l-sr)r
Similar

to (2).

(4)

This easily follows from (2) and (3).

(5) Suppose thatp
302

YUTAKANAKAMURA

S,={a:q-crr+(l-cr)p,Odadl}, S,={cr:q-ccp+(l-cr)r,Ol, Lemma 1, q-crr+(l -a)p for 1 -B
-a)p -a)r
for

A-ccc< 1 whenq
for

B
1 whenp
Suppose that p < q. When p - Aq + (1 - A)p, A is the maximum indifference value a that satisfies p waq+(l--a)p for O
either piq (2) Up-r, Bq+(l-B)r,Cq+(l-C)(Iwp+(l-l)r)},

or q
or {q,r}
andfor

A,BE(O,~), pw then C*=LA*+

andfor O
Proof (1) Suppose that the hypotheses of (1) hold with p < {q, r}. When {q,r}
For convenience we define a subset [p, q] of P as [p,q]={1p+(l-1)q:p,qEPandallO
By A3 and A5, if r<[p,q], LEMMA

(2)

3. (1)

rfr<

If (s, t} -KY, r-

then r
r
if [p,q]
thenpir

or q
{s, t}, rw p and r< [p, s], then r< [p, t]. p and [p, s] < r, then [p, t] < r.

Proof. We show (1). A similar proof applies to obtain (2). Suppose that the hypotheses of (1) hold. Let rw{A1s+(l-AI)r, A,t+(l-A,)r, Bt+(l-B)p, Cls+(1-C,)x,C,t+(1-C,)x},wherex=ar+(1-a)pforO
303

INTERVALORDEREDSTRUCTURE

First we shall assume A,=O. Then by A5, as+(l -cr)r
We note

whereb=(1+~)/2andc=41/(1+~)2.Sincer-p,A3impliesas+(1-u)r-us+ (1 --a)~. Thus by A3 and Lemma 1, us+(l-u)r
-A,)r)+(l

-b)(kt+(l

-k)x),

where a=A,(l-A,)/d, b=C,(l-A,)/d, k=C,A2(1-A,)/e, d=A,(l-A,)+ C,(l -A,), and e= C1(A2 - A,) + A,(1 -AZ), Therefore, by A3 and A4, we must have k=C,. Thus C:=k*=C:A:/A:=A:/cl. Therefore, B=O, so r<[p,t]. Q.E.D. 4. STRUCTURAL

PROPERTIES

This section shows structural properties of the interval ordered structure. According to Fishburn (1985, Chap. 2), we define two binary relations, < - and < +, on P as p <- q ifp-riqfor some rEP, p < + q ifp
some rEP.

By Al and A2, it is easy to see that < - and < + are asymmetric and transitive. We define two subsets of P by P-=(p:r
304

YUTAKANAKAMURA

The interval ordered structure implies that P-, P+, P\P-, and P\P’ are convex sets. We note by Theorem 2 in Fishburn (1985, Chap. 2) that < - on P- and < + on P+ are weak orders. The following lemma shows relations among < -, < +, and maximum indifference values. LEMMA

O
4. (1)

If

thenp <-q

(2) If {p,q}
r<{p,q} ijjf B
and r-{Ap+(l-A)r, {Ap+(l-A)r,

Bq+(l-B)r}

Bq+(l-B)r}

for O
for then

We the symmetric First we shall p c-q. Then

show (1). A similar proof applies to obtain (2). Let - - on P- be complement of < ~ on P-. Suppose that the hypotheses of (1) hold. assume A = B and show p - - q. Suppose on the contrary that p-sA, we obtain r
Applying

Lemma 4, we obtain the following structural property:

PROPOSITION 1. Suppose that P- and P t are not empty. Then there are two linear functionals, u and v, on P such that

P < - q=u(p)
for

p, 4EP-;

P <- q*u(p)
for

p,qEP;

P -= + 4 * O(P) < p(q)

for

p, qEP+;

P <+ q=v(p)
for

P, qE P.

If 1.4’and VI satisfy the representationsalso, there are four real numbers,a, > 0, a, > 0, b,, and b,, such that u’=a,u+b, and v’=a2v+b,.

INTERVAL

ORDERED

305

STRUCTURE

Proof Suppose that P- and Pf are not empty. Note that < - on Pm- and < + on P+ are weak orders. First, we show that < ~ on P- satisfies for all p, q, rE Pand all 0<1<1,

Bl.

p <-q=%r+(l-I)p

B2. p<-q, a, B 6 (0, 1).

<-ir+(l-l)q.

q<-r=t-ar+(l-a)p<-q,

q<-jr+(l-P)p

for

some

The weakly ordered < - on P- with Bl and B2 implies the first representation in the proposition (see Fishburn (1982)). To show Bl, we shall suppose that p < - q for p, qE Pp. Then p- t < q for some t. By Al, s< {p, q, r) for p, q, rE P- and some SEP. Suppose first that s< [s, p] forallssuchthatsi(p,q,r).ByA3,~r+(1-I)p-/Zr+(1-A)rforallO~~~1. Wearetoshowthat~r+(1-~)tB. Let s- {A,(;lr+(l -,I)p)+(l -A,)s, B,(ir+(l--1)q)+(lLB,)s). Thus by Lemma2(1), A:=IC*+(l-1)A* and B:=K’*+(l-A)B*, so A,>B,. Therefore, by Lemma4(1), l.r+(l-l)p
for some 0 < /3 < 1. Hence q < ~ Br + (1 - p) p. Therefore, B2 holds. Similarly we obtain that < + on P+ satisfies Bl and B2, where < - is replaced by <+. The preceding analyses show that there are two linear functionals, u on Pand v on Pi, that satisfy the first and the third representations, which are unique up to positive linear transformations. In what follows, we show that the second holds by linearly extending U. A similar proof applies to get the fourth. We first show that if ~EP\P~ and rEP-, then ap+(l-a)rEPand ap+(l-a)r<-rforsomeO
306

YUTAKA

NAKAMURA

andpr+(l-P)si[r,pr+(l-P)s]f or some 0 < fi < 1. Then by A3, [s, r] < r, so cts+(l -a)r
where a =A/(1 -a(1 -A)). By A3, ~-ax + (1 -a)~ Let s- Bx+ (1 - f?)s, so B>u. We have B>A since u>A. By Lemma4(1), x <- r. Let u on P- satisfy the first representation. Let r, s E P- and p E P\P-. By the preceding paragraph, let x=ap+(l-a)r and y=j?p+(l--fi)s for some a,pE(O,l)suchthatx,yEP~,x<~r,andy<~s.SinceP~isconvex,wehave upx+(l-up)s=uay+(l -ua)rEP-, where u= l/(a+P-afi). Thus we obtain &(x) + (1 - up) U(S) = uau( y) + (1 - ua) u(r). Rearrangement gives u(x)/a a*u(r) = u(y)//? - j?*u(s). Hence for p E P\P-, we define u(p) = u(x)/a - a*u(r) as a linear extension of u on P-, where rEP-, x=ap+(l-a)rEP-, and xc-r. Since x < - r, U(X) < u(r). Thus we have u(r) -u(p) = (u(r) - u(x))/a >O. Therefore, u(p)
=k(p)+(l

u(r)

-A)u(q).

Hence u on P\P- is linear. It is easily verified that u is unique up to a positive linear transformation. In what follows, let u and v be two linear functionals on P in Proposition define -C - and < + on P as

Q.E.D. 1. We

p < - q-=u(p)
Let N ~ and N + on P be the symmetric respectively. For r, s EP, we define

complements

of < ~ and < + on P,

INTERVAL

ORDERED

STRUCTURE

LetZ,=~if~itfornorEP,andJ,=~iftisfornotEP.ForrEP+andsEP-, Z, and J, are uniquely specified by Lemma 3. Before stating our last structural property for an interval (P, <), we need to prove the following lemma: LEMMA

5.

qEJ,,, qEJ,ap

(1)

(2)

qEZp, qEZ,=>p

(3)

pcJ,,pEP+oqdp,

(4) (5)

P>qEz,=+p

Iv

307

ordered structure

IV- r.

k + r. qEP-.

9.

P,qEJr*p-+q.

Proof (1) Suppose that qe J, and qE J,. Then p, re P-, so by Al, t< (p, r> forsomet.ByA3,~~+(l-i)t-;Iq+(1-I)tforallO
(5)

Similar to (4).

Q.E.D.

According to Fishburn (1985), let P- and P, be two disjoint copies of P defined by P- = {(P, -1: PEP}, P, = ((p, +): pE P}.

We write (p, -) as Ap-+(l-A)q-, P-UP, by

480/32/3-7

asp-, and (p, +) asp+. Moreover, we write (Ap + (1 - A)q, - ) and (1p+(l-I)q, +) as Lp++(l--A)q+. Define co on

308

YUTAKANAKAMURA

1. p-
q, q,

3. P+
orp-qfor

q$J, andp$I,.

Then we have the following structural property: PROPOSITION

2.

co is an asymmetric weak order.

Proof. We show first asymmetry of -C0, so x co y implies that not( y <,, x). If then asymmetry easily follows. Suppose (x, y) = either x, YEP~~ or x, YEP,, and q + co p -. Then by definition, p 5 q and q < p, a con(p-3 q+)> P-
INTERVALORDEREDSTRUCTURE

Case 3. (x, y, 2) = (p’, q-, r-).

not(q- co r-). Thus by definition, contradiction.

309

Then p+ co r-, not(p+ co 9 1, and p < r; q 5 p; not(q < - r). Therefore, q < - I, a

Case 4.

(x, y, z)= (p-, q+, r+). Similar

to Case 1.

Case 5.

(x, y,z)=(p+,q-,r+).

to Case2.

Case 6.

(x, y, z) = (p’, q+, r-). Similar to Case 3.

Similar

Q.E.D.

5. PROOFS OF THE THEOREMS

In this section we give the proofs of Theorems 1 and 2 by applying the results in the preceding section. Let wO be the symmetric complement of co. The proof of Theorem 1 needs the following lemma: LEMMA 6. Ifr--,p+ands--,q+, for all O
thenIre+(1-i)s--,Ilp++(l-i)q+

Proof Suppose that the hypotheses of the lemma hold. By definition, we note that r- mOp+ opEJ,orrElp,ands--,q+oqEJ,orsEly.First,weshallsuppose that p E J, and qE J,. When r E Ip and SEZ~, the proof is similar. Since r,sEP-, Al implies t < {r, s> for some t G P. It follows from A4 that [@ + (I- R)q, t] < Rr + (1 - 2)s for all 0 < A < 1, since [p, t] < r and [q, t]
310

YUTAKA

NAKAMURA

Proof of Theorem 1. Suppose that < is nonempty. The necessity of the axioms easily follows, so we show their sufficiency. We construct a real-valued functionalf on P- u P, such that for all x, y E P- u P,,

and f is linear on P_ and P + , respectively. Since p- -co q- o p < - q, we define f(p- ) = u(p) for all p E P, where u is a linear functional obtained by Proposition 1. Let Q= {p: p+ Nor- for some rEP} and Q, = {p’:p~Q}. It easily follows from Lemma 6 that Q is a convex set. Since co is a weak order, we define f(p+)=u(r) for all PEQ when P+-~Y~. Suppose that ~+-~r~ and q+wos-. Then by Lemma 6, we obtain f(Ap++(l-I)q+)=u(Ar-+(l-A)s-) =224(r-)+(l-A)u(s-) =Y(P+)+(1--)f(q+).

Hence f on Q + is linear. It is easy to see that x co y-f(x)


for all x, y E

P-uQ+.

Since P+ E Q, and < o on P + agrees with < + on P, there are real numbers a > 0 and b such that f(p+ ) = au(p) + b for all p E Q, where u is a linear functional obtained by Proposition 1. Without loss of generality, we shall assume a = 1 and b = 0. Definef(p+) = u(p) for all p E P\Q. Thus p+ co q+ of(p+)
Note that xc0 y for all XEP- and y~p+\Q+ since P’rQ. To prove that it suffices to show thatf(p-)
By the preceding analyses, f = u on P- and f = v on P, for some linear functionals u and v imply that ~<~y~f(x)
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311

STRUCTURE

Therefore, letting a(p) = u(p) - u(p) for all p E P, the representation of the theorem is obtained. The uniqueness parts of the theorem easily follow from Proposition 1. Q.E.D. Proof of Theorem 2. Suppose that (P, i) is an interval ordered structure with < nonempty. First we show the necessity of A6. Suppose that p w aq + (1 - CI)p for some O
By A5, orr+(l-cr)piq for some O
6. CONCLUSION

This paper examined an interval ordered structure in risky decision making. We showed that the interval ordered structure, which consists of the interval ordered preference, two independence axioms, and the strong Archimedean axiom, has linear utility with a linear threshold functional. The strong Archimedean axiom was first introduced in the paper. This axiom plays an important role in deriving the interval ordered representation. However, some weaker axiom system for the representation might exist.

ACKNOWLEDGMENTS The author gratefully for pointing out errors

acknowledges in the paper.

the helpful

comments

of R. D. Lute

and two anonymous

referees

REFERENCES AUMANN, R. J. (1962). Utility theory without FISHBURN, P. C. (1968). Semiorders and risky

the completeness choices. Journal

axiom. Econometrica, 30, 445462. of Mathematical Psychology, 5, 358-361.

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with finite consequence spaces. Annals of 42, 572-577. FISHBURN, P. C. (1982). Foundations of expected utility. Dordrecht, Holland: Reidel. FISHBURN, P. C. (1985). Interval orders and interval graphs. New York: Wiley. LUCE, R. D. (1956). Semiorders and a theory of utility discrimination. Econometrica, 24, 178-191. LUCE, R. D. (1973). Three axiom systems for additive semiordered structures. SIAM Journal on Applied Mathematics, 25, 41-53. FISHBURN,

P. C. (1971). One-way expected utility

Mathematical

Statistics,

J. N. (1973). Multiattribute decision making under uncertainty using bounded intervals. In J. L. Cochrane and M. Zeleny (Eds.), Multiple criteria decision making (pp. 93-107). University of South Carolina, Columbia, SC: Univ. South Carolina Press. VINCKE, P. (1980). Linear utility functions on semiordered mixture spaces. Econometrica, 48, 771-775. VEDDER,

RECEIVED:

July 20, 1987