Measuring Nonmonetary Utilities in Uncertain Choices: The Ellsberg Urn Author(s): Vernon L. Smith Source: The Quarterly Journal of Economics, Vol. 83, No. 2 (May, 1969), pp. 324-329 Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1883087 Accessed: 17/04/2009 11:28 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=mitpress. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact
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MEASURING NONMONETARY UTILITIES IN UNCERTAIN CHOICES: THE ELLSBERG URN VERNON L. SMITH * Introduction, 324.-I. The Ellsberg problem, 326.-II. Probabilistic interpretation, 326.-III. Utility interpretation, 327. IV. Measuring nonmonetary utility, 328.
Several years ago EllsbergI raised some very fundamentaldecision theoretic questions regardingthe probabilisticinterpretation of experimentalchoice behaviorin contingencysituations not characterizedby von Mises type public probability measures. Fellner2 has since elaboratedwhat he calls a semiprobabilisticinterpretation of the kind of behavior (revealed in various experimentalsettings) which violates the Savage axioms of personalistic expected utility theory. My purpose here is to explicate the utility interpretationof these violations by means of the expected utility calculus, and to suggest a schemefor measuringthe utility of money rewardsin such contingencysituations. Fellner speaksof the utility interpretationof such violations as "Version2," but prefers to discuss the problem in terms of the "Version1," probabilisticinterpretation. Let me make my own position clear at the outset: I stand with those, like Savage, Raiffa, and Schlaifer, who say they would not want to violate the axioms consciously. The axioms and their implications are a guide to patterns of behavior under uncertainty which I find desirablefor me. If I violate the axioms I have made a mistake. Certainly,I think this would be my position in most ordinary decision situations under uncertainty, although I am not preparedto rule out the possibility that undersome circumstancesI would find the decisionimplicationsof the axiom unacceptable. However,having stated this, I am not preparedto assertthat he who seriously and consciouslyviolates the axioms, and in my judgment "knowswhat he is doing," is thereby simply making a mistake," and should be given a little more conditioningand "educa* I wish to acknowledge support by the National Science Foundation. 1. D. Ellsberg, W. Fellner, and H. Raiffa "Symposium: Decisions under Uncertainty," this Journal, LXXV (Nov. 19615, 643-94. See also H. Roberts, "Risk, Ambiguity, and the Savage Axioms: Comment," this Journal, LXXVII (May 1963). 2. W. Fellner, Probability and Profit (Homewood, Ill.: Irwin, 1965).
MEASURING NONMONETARY UTILITIES
325-
tion." To me, his thoughtful choices are inviolate, and Fellner is right in exploringthe implications of such violations. Even if the axioms are,to be regarded as basically a normative theory, that theory can also do valuable service in helping us to understand actual behavior. But I do not care for the probabilisticinterpretation of the violations. To me probabilitiesare probabilitiesin the sense of nonnegativity, additivity and the property of the unit measureover the whole event space. I grant the right of a man to have systematicand deliberatepreferencesfor rewardsbased on dice game contingencies over the same rewards based on Dow-Jones stock price contingencies. But if he insists also that he is less than certain that the Dow-Jones average will either rise or not rise by five points or more tomorrow,then so far as I am concernedhe is now makinga "mistake."He does not understandwhat is (or should be) meant by probability. He is entitledto his tastes, but not to any new definitionsof probability. Fortunately,this is not what subject violators do. They merely violate the axioms,without the necessity for the probabilisticinterpretation. As I see it, it is much more plausible to say that violators in "nonstandardprocess" contingencies,such as the stock price example,sufferutility losses (or gains) relative to what is experienced in less controversial"standardprocess"contingencies,such as dice games. In nonstandardprocess collectives, characterizedby subjective probabilitieswhichvary widely amongindividuals,theremay be real or imaginedelementsof skill which increaseor reducethe subjective value of the outcomes "lose" or "win." Alternatively, an individual may have a low psychologicaltolerance for the "ambiguity"associatedwith nonstandardprocessevents,whichwe canvery reasonablyand naturally describein terms of "utility losses." Thus, if a man loses a dice game bet he and his associates might consider that he was merely the victim of bad luck, and the poor showing may be socially excusable. But if he loses from incorrectlypredicting a rise in the Dow-Jones, he may perceive that his colleagues feel that he should have known better, that he is not so smart after all, that they are glad to see his "ignorance"revealed,and so on. Or, if he knows nothing about the stock market,then the mysteries and ambiguitiesin the Dow-Jones may generate special uncomfortand anxietieswhen he gambleson such contingencies.In all such cases, we are simply saying that the utility of money or other rewardsis not independentof the circumstancesunder which it is obtained. The utilities in the payoff matrix may have argumentsother than what appearsto be the "objective"reward.
326
QUARTERLY JOURNAL OF ECONOMICS
I. THEELLSBERGPROBLEM In the Ellsberg counterexamplewe imagine confrontinga subject with two urns. Urn I contains100 red or black balls. The number of red and the numberof black are not specified, nor is there specified a procedurewhereby the compositionof balls in the urn is determined.Urn II contains50 red and 50 black balls. Define the following four "random"variables or contingencies based on a randomdrawfromUrns I or II: Urn I N red, 100-N black
Urn II 50 red,50 black
N unknown
Rl-=r x, if red (p1unknown) l,
Bi=
r
if black (q1unknown)
0, if red (pi unknown) x, if black (q1unknown)
(P2=A) r l, x, ifif red black (q2=A)
B2=
f 0, if red (P2=
x, if black (q2=2)
The results of Ellsberg's interrogationsand Fellner's similar experimentswith monetaryrewardsare that many subjectsshowthe followingpattern of tastes: 1. Rl.Bl 3. R2PR1
2. R2~B2 4. B2PB1
where _ means "indifferentto" and P means "preferredto." That is, the subjects are indifferentbetween the contingencies"win $x on red, $0 on black," and "win $0 on red, $x on black" in Urn I, and similarly for Urn II. But as between Urn I and Urn II, "win $x on red, $0 on black" in Urn II is preferredover the same contingency in Urn I, and similarly "win $0 on red, $x on black" in Urn II is preferredover the same contingencyin Urn I. There is a consistentpreferencefor Urn II contingenciesover Urn I contingencies.
INTERPRETATION II. PROBABILISTIC
Assume now that we can represent an individual's choices in regardto R2 and B2, in terms of an expectedutility calculationusing the "objective" probabilities P2 = q2= J. Assume further that his choices involving R1 and B1 can be represented by an expected util-
ity calculationin which he behaves as if he imputessubjectiveprob-
MEASURINGNONMONETARYUTILITIES
327
ability weights, or degreesof belief, Pi to red and q, to black. Then, if u(m) is the utility of m units of money: (2.1)
R1iB1 = >E [U (R1)] =plu(x) +qlu(O) = E[U(B1)] =plu(0) +qlu(x),
and, therefore,pi= ql, if u (x) =/=u(0). Indifferencebetweenred and black in Urn I implies that red and black are equally probable (but not necessarily = -1). Also,
(2.2)
R2,B2=
>E[U(R2)] *E[U(B2)
=P2u(x) +q2u(0)
=
+q2u(x),
]p2u(0)
which is consistent with the hypothesis that the individual acts as on an expectedutility calculation,with his subjective probabilities, P2
and q2, equal to the "objective" value
1.
P2=q2=
Behaviorally
we can only say that P2 = q2, but we assume for the individualthat he takes P2 = q2= , as in the probability textbooks. We also have
(2.3)
R2PR1=>E[U(R2)]
=p2U
(X)
+q2u
>
(0)
E[U(R1)] =plu(x) +qlu(0) and
(2.4)
B2PB1=>E[U(B2)]
=p2U
(0) + q2u(X)
>
E [U (B1)] =plu (0) +qiu (x). if Hence, adding the inequations (2.3) and (2.4), p2+q2>pl+ql, u(O)+u(x) >O; therefore pl+ql E [U(R1)]=p1u1 (x) +q1ui (0) = (3.1) E [ U(B1)] =piu (0) +q1u1(x), and p1= q1. Since pi+q =1, pi q1=. (3.2)
R2~B2= >E[U(R2)] =p2U2() +q2u2 (0) = E[U(B2)] =P22 (0) + q2u2(x) As before,this is consistentwith P2= q2 = (3.3) R2PR1= >EtU(R2)] =p2u2(x)+q2u2 (0) >
E [ U (R1)] = p1u1(x) + q1u1(0). But from (3.1) and (3.2), pl=ql=P2=q2=J, and, the: U2 (X) +U2
(0) > U1 (X) +U1
(0)
(3.4)
B2PB1= >E[U(B2)] =p22 (0) +q2u2 (X) > E [ U (B1) ] = p1u1(0) +q1u1 (x), and, from (3.1) and (3.2), u2(x) +u2 (0) > u1(x) +u1 (O preferences are consistent provided that u2 (m) > ul (mn) IV. MEASURING NONMONETARY UTILITY From the above analysis, our subject is behaving as if he suffered utility losses at all levels of money reward m for Urn I contingencies as compared with Urn II contingencies. Suppose we assume that such losses are additive with monetary utilities. Then (4.1) U2() = u1(m) + (m) where x(m) is utility loss, due to ambiguity, in situation 1. u1(m) and X(n) can be measured, given an individual's von Neumann-Morgenstern utility function u2(m) for standard process collectives (in this case, Urn II), by finding the premium ir (x) >0, for each x>0, such that R'1,-R2, where X+7r (X),
R'0= l X,
R2 =
P1-
N 0
(since R1iB1),
q, = 100N =j (since R1~B1); p2 = i
MEASURINGNONMONETARYUTILITIES
329
Given x, and r(x), such that R'1-R2, we can write E[U(R'1)] =iul[x+7r(x) ] +Jul (O)= E[U(R2)] =iU2 (x) +iU2 (0)i
If we assumeu (0) =u2(0), then (4.2)
Ul [X+7(x)]
= u2 (x);
or, from (4.1) (4.3)
A[XI+
(X)] =U2 [X+
(X) ]-Ul
[X+
=U2 [X+r
(X) ]-U2
(X).
(X)]
Note that ul(0) =u2(0) impliesir(0) =0. A graphic illustration of the measured functions u1 (m), A(m),
given u2(m), is shownbelow: U1,u
x +
uz(m)
I-
U,(m) UPM)-(m)
Trrx)J
X
m
X+Tr(X) FIGURE
I:
In closing, I would conjecturethat the kind of choice behavior revealedby the Ellsberg counterexampleis not just a characteristic of "known"versus "unknown"probabilitysituations. I suspectthat the preferencefor Urn II gamblesover Urn I gambleswould be little changedfor many, if not most, subjects,if they were guaranteedthat the numberof red balls in Urn I had been determinedby a random draw from the integers0-100. Urn I is then a 50-50 gamble,except that it is a compoundgamble, rather than a simple gamble as in Urn II. The question then is not "Are there uncertaintiesthat are not risks?",as posed by Ellsberg, but "Arethere risks that are not risks?" OF MASSACHUSETTS UNIVERSITY
