Semiorders and risky choices

JOURNAL

OF MATHEMATICAL

5, 358-361

PSYCHOLOGY:

Semiorders PETER Research

Analysis

(1968)

and Risky Choices C.

FISHBURN

Corporation,

McLean,

Virginia

22101

If a preference relation on a set is a semiorder then the indifference relation is not necessarily transitive. However, if a preference relation on a set of probability distributions is a semiorder and satisfies a simple sure-thing axiom, then indifference is transitive. Modifications of preference theory under risk that preserve intransitive indifference are suggested.

THEOREM

Let 9 be a set of probability distributions on a set of consequences, with QP + BQ in 9 when P and Q are in 9, where in general olP + (1 - a)Q is the direct linear combination of P and Q. The purpose of this note is to prove and discuss the

THEOREM. If, for all P, Q, R, and S in 8, a binary relation Al.

< on 9 satisjies

Not P < P,

A2. IfP
RthenP<

SorS<

R,and

A3. IfP
and < on 9 dejined by

P-QifandonlyifnotP
P,

D2. P
P,

- on .9 is an equivalence (transitive, rejexive, symmetric) and < on 9 is a weak order (connected OYcomplete, and transitive). Axioms Al and A2 are two of the three Scott-Suppes (1958) semiorder axioms, the third of which is: if P < Q and R < S then P < S or R < Q. Under Dl and D2, their definition is equivalent to the earlier definition of semiorder stated by Lute (1956). Axiom A3 is a simple version of an independence (Samuelson, 1952) or sure-thing (Savage, 1954) axiom for risky choices. Lute (1956) introduced the concept of a semiorder as a generalization of weak order in which indifference - is not assumed to be transitive. For example, (P -Q, Q - R, P < R) satisfies the three Scott-Suppes semiorder axioms but - is not transitive. 358

SEMIORDERS

AND

RISKY

CHOICES

359

Since < on P is a binary relation, either P < Q or not P < Q, so that, by D2, either P < Q or Q < P is true. Hence < is complete or connected. Moreover, Al and A2 imply that < is transitive (set S = R in A2, then use Al). To complete the proof of the theorem it will suffice to show that, in the presence of Al and A2, A3 destroys the possibility of intransitive indifference. Assume P N Q and Q~R.IfP
P < Q if and only if E(u, P) < E(u, Q) is obtained, where u is a real-valued function on the set of consequences is the expected value of u with respect to P.

and E(u, P)

DISCUSSION

From the examples for semiorders presented by Lute and others in the riskless, or nonprobabilistic, context, it is readily apparent that in some cases it makes good sense not to assume that indifference is transitive. In the context of risky choices the same thing seems to be true. For a simple example, let the consequences be amourits of money considered as potential increments to one’s present wealth, and let

P($lOO) = 1 Q($lOl)

= I

S($ 40) = S($200) = .5. Surely P < Q. However, it seems entirely possible that the individual will have no clear preference between P and S, or between Q and S, so that (in the spirit of D2) we obtain P M S and Q N S, along with P < Q. If, in fact, we wish to preserve the possibility of intransitive indifference among risky choices and proceed from < as the primitive binary relation, then (granting Al) either A2 or A3 must be weakened. An example of weakening A2 occurs by assuming only that ( is transitive along with Al. Indeed, there seems to be some sense in not retaining the full strength of A2 in the

360

FISHBURN

risky setting. For example, suppose R($102) = 1 is adjoined to P, Q, and S in the preceding example. Then surely P < Q < R. A2 then implies that P < S or S < R. But, reasoning as before, it would not seem surprising if P N S and S N R, in which case A2 does not hold. Turning our attention to A3, we can readily see that this axiom would not be expected to hold in many riskless situations. For example, A3 along with transitivity for < implies that if P < Q then P < *P + $Q < Q. Suppose that P, Q,... are tones, $P + fQ is a tone whose pitch is produced mechanically so as to be midway in pitch between the pitches of tones P and Q, and P < Q means that a listener believes that tone P has a lower pitch than tone Q. As a bisection axiom “P < Q implies P < -$P + +Q < Q” would not be expected to hold since its repeated application implies that the listener can distinguish between any two tones whose pitches are different though arbitrarily close together. In the risky-choice context A3 does not always agree with casual judgmental or choice behavior, but it is often defended as a rational preference criterion. The same is true of the following sure-thing, axiom, which implies A3: A4. ~~O<<
-ol)R.

P($50) = 1, Q($O) = .2, Q($lOO) he might

also have no clear preference

between

= .8,

R and S, where

R($50) = .lO, R($SO) = .45, R($lOO)

= .45,

S($O) = .02, S($SO) = .45, S($lOO)

= .53,

and Savage so that R - S. However, using an argument like that given by Friedman (1952, pp. 468-469) or Savage (1954, p. 103), we might claim that if the person is shown that

R = .lP + .9T, S = .lQ + .9T, where T($80) = T($lOO) = .5, then he might decide after all that R < S. According to Savage and others the main value of A3 or A4 is to enable a person to cross-analyze preference judgments in this way. If the person is in fact persuaded that he ought to conform to A4 and if he is certain that P < Q, then he will decide that R < S if he realizes how R and S are related to P and Q even though he might feel that R and S are “very close together.” Despite the fact that A3 and A4 destroy the possibility of intransitive indifference in the presence of Al and A2, some other sure-thing axioms do not have this effect.

SEMIORDERS

AND

RISKY

CHOICES

361

To illustrate, we recall from Scott and Suppes (1958) that if < on .Y is a semiorder and if 9 is a finite set then there is a real-valued function u on 9 such that, for all P and Q in .Y, P < Q if and only if u(P) + 1 < u(Q). Proofs of this are given by Scott (1964), Scott and Suppes (1958), Zinnes (1963). Its obvious counterpart in the risky-choice setting is

and Suppes

and

P < Q if and only if E(u, P) + 1 < E(u, Q),

(1)

which generally violates A3 and A4. However, each of the following axioms, the first three of which are independence axioms, with A8 a typical Archimedean condition, is implied by (1). A5. 1f P < Q and 0 < a: < 1 then not aQ + (1 - a)R < o~P + (I - a)R. A6.IfP
IfPmQ,RN

S, and 0 < 01 < 1 then CXP+ (I -

AX. IfP
ol)R N EQ + (1 - ol)S.

-a)R
-,B)R

for some 01,/3 strictly between 0 and 1. Note also that (1) implies Al, A2, and the third Scott-Suppes semiorder condition” Thus, we have identified two main routes for the preservation of intransitive indifference with risky choices: first, retain A3-A4 and weaken A2; second, retain A2 and use independence axioms like A5-Al but not A33A4. Both routes await further exploration.

REFERENCES FISHBURN,

P. C. Bounded

expected

utility.

Annals

of Mathematical

Statistics,

1967,

38,

1054-

1060. M. AND SAVAGE, L. J. The expected-utility hypothesis and the measurability of Journal of Political Economy, 1952, 60, 463-474. JENSEN, N. E. An introduction to Bernoullian utility theory. I. utility functions. Swedish lownai of Economics, 1967, 69, 163-183. LUCE, R. D. Semiorders and a theory of utility discrimination, Econometrica, 1956, 24, 178-191. SAMUELSON, P. A. Probability, utility, and the independence axiom. Eco?tometrica, 1952, 20. 670-678. SAVAGE, L. J. The foundations of statistics. New York: Wiley, 1954. SCOTT, D. Measurement structures and linear inequalities. Journal of Mathematical Psychology, 1964, 1, 233-247. SCOTT, D. AND SUPPES, P. Foundational aspects of theories of measurement. ~oournal of Symbolic Logic, 1958, 23, 113-128. SUPPES, P. AND ZINNES, J. L. Basic measurement theory. In R. D. Lute, R. R. Bush, and E. Galanter (Eds.), Handbook of mathematical psychology, Vol. 1. New York: Wiley, 1963. FRIEDMAN,

utility.

RECEIVED:

September

13, 1967