E

Expected Utility Hypothesis

Mark J. Machina

AbstractThe expected utility hypothesis – that is, thehypothesis that individuals evaluate uncertainprospects according to their expected level of‘satisfaction’ or ‘utility’ – is the predominantdescriptive and normative model of choiceunder uncertainty in economics. It providesthe analytical underpinnings for the economictheory of risk-bearing, including its applica-tions to insurance and financial decisions, andhas been formally axiomatized under condi-tions of both objective (probabilistic) and sub-jective (event-based) uncertainty. In spite ofevidence that individuals may systematicallydepart from its predictions, and the develop-ment of alternative models, expected utilityremains the leading model of economic choiceunder uncertainty.

KeywordsArrow–Pratt index of absolute risk aversion;Bernoulli, D.; Bernoulli, N.; Cobb–Douglas

functions; Comparative likelihood; Consumertheory; Cramer, G.; Environmental economics;Expected utility hypothesis; First-order sto-chastic dominance preference; Increasingrisk; Independence axiom; Inequality(measurement); International trade; Lotteries;Malinvaud, E.; Marschak, J.; Menger, K.;Objective vs. subjective uncertainty; Ordinalrevolution; Preference functions; Preferenceorderings; Probability; Risk; Risk aversion; StPetersburg paradox; Stochastic dominance;Subjective probability; Sure-thing principle;Transitivity; Uncertainty; von Neumann–Mor-genstern utility function

JEL ClassificationsD8

The expected utility hypothesis is the predomi-nant descriptive and prescriptive theory of indi-vidual choice under conditions of risk oruncertainty.

The expected utility hypothesis of behaviourtowards risk is the hypothesis that the individualpossesses (or acts as if possessing) a ‘vonNeumann–Morgenstern utility function’ U(�) or‘von Neumann–Morgenstern utility index’ {Ui}defined over some set X of alternative possibleoutcomes, and when faced with alternative riskyprospects or ‘lotteries’ over these outcomes, willchoose the prospect that maximizes the expected

This chapter was originally published in The New PalgraveDictionary of Economics, 2nd edition, 2008. Edited bySteven N. Durlauf and Lawrence E. Blume

# The Author(s) 2008Palgrave Macmillan (ed.), The New Palgrave Dictionary of Economics,DOI 10.1057/978-1-349-95121-5_127-2

value ofU(�) or {Ui}. Since the outcomes could bealternative wealth levels, multidimensional com-modity bundles, time streams of consumption, oreven non-numerical consequences (such as a tripto Paris), this approach can be applied to a tre-mendous variety of situations, and most theoreti-cal research in the economics of uncertainty, aswell as virtually all applied work in the field (forexample, insurance or investment decisions) isundertaken in the expected utility framework.

As a branch of modern consumer theory (forexample, Debreu 1959, ch. 4), the expected utilitymodel proceeds by specifying a set of objects ofchoice and assuming that the individual possessesa preference ordering over these objects whichmay be represented by a real-valued maximandor ‘preference function’ V(�), in the sense that oneobject is preferred to another if and only if it isassigned a higher value by this preference func-tion. However, the expected utility model differsfrom the theory of choice over non-stochasticcommodity bundles in two important respects.The first is that, since it is a theory of choiceunder uncertainty, the objects of choice are notdeterministic outcomes but rather uncertain pros-pects. The second difference is that, unlike in thenon-stochastic case, the expected utility modelimposes a very specific restriction on the func-tional form of the preference function V(�).

The formal representation of the objects ofchoice, and hence of the expected utility preferencefunction, depends upon the set of possible out-comes. When the outcome set X ={x1,. . ., xn} isfinite, we can represent any probability distributionover this set by its vector of probabilitiesP = (p1,. . .,pn) (where pi = prob(xi)) and theexpected utility preference function takes the form

V Pð Þ ¼ V p1, . . . , pnð Þ �X

Uipi:

When the outcome set consists of the real line orsome interval subset of it, probability distributionscan be represented by their cumulative distribu-tion functions F(�) (where F xð Þ ¼ prob ~x � xð Þ),and the expected utility preference function takesthe form V(F) � Ð

U(x)dF(x) (orÐU(x)f(x)dx

when F(�) possesses a density function f(�)).

When the outcomes are commodity bundles of theform (z1, . . . , zm), cumulative distribution functionsare multivariate, and the preference function takesthe form

Ð. . .

ÐU(z1, . . . , zm) dF(z1, . . . , zm).

The expected utility model derives its name fromthe fact that in each case the preference functionconsists of the mathematical expectation of thevon Neumann–Morgenstern utility function U(�),U(�, . . . ,�) or utility index {Ui} with respect to theprobability distribution F(�), F(�, . . . ,�) or P.

Mathematically, the hypothesis that the prefer-ence function V(�) takes the form of a statisticalexpectation is equivalent to the condition that it be‘linear in the probabilities’, that is, either aweighted sum of the components of P (i.e. �Ui pi)or else a weighted integral of the functions F(�) or f(�) (

ÐU(x)dF(x) or

ÐU(x)f(x)dx). Although this

still allows for a wide variety of attitudes towardsrisk depending upon the shape of the utility func-tionU(�) or utility index {Ui}, the restriction that V(�) be linear in the probabilities is the primaryempirical feature of the expected utility model,and provides the basis for many of its observableimplications and predictions.

It is important to distinguish between the pref-erence function V(�) and the von Neumann–Mor-genstern utility functionU(�) (or index {Ui}) of anexpected utility maximizer, in particular withregard to the prevalent though mistaken beliefthat expected utility preferences are somehow‘cardinal’ in a sense not exhibited by preferencesover non-stochastic commodity bundles. As withany real-valued representation of a preferenceordering, an expected utility preference functionV(�) is ‘ordinal’ in that it may be subject to anyincreasing transformation without affecting thevalidity of the representation – thus, the preferencefunctions

ÐU(x)dF(x) and [

ÐU(x)dF(x)]3 represent

identical risk preferences. On the other hand, thevon Neumann–Morgenstern utility function U(�) is‘cardinal’ in the sense that a different utility functionU�(�) will generate an ordinally equivalent prefer-ence function V�(F) � Ð

U�(x)dF(x) if and only ifit satisfies the cardinal relationshipU�(x) � a � U(x)+ b for some a> 0 (in which caseV�(F) � a � V(F)+ b. However, the same distinction holds in thetheory of preferences over non-stochastic commodity

2 Expected Utility Hypothesis

bundles: the Cobb–Douglas preference functiona � ln(z1) + b � ln(z2) + g � ln(z3) (written here inits additive form) can be subject to any increasingtransformation and is clearly ordinal, even though avector of parameters (a�, b�, g�) will generate anordinally equivalent additive form a� � ln (z1) + b��ln (z2) + g� � ln(z3) if and only if it satisfies thecardinal relationship (a�, b�, g�) = l � (a, b, g) forsome l > 0.

In the case of a simple outcome set of the form{x1, x2, x3}, it is possible to graphically illustrate the‘linearity in the probabilities’ property of expectedutility preferences. Since every probability distribu-tion (p1, p2, p3) over these outcomes must satisfyp1 + p2 + p3 = 1, we may represent such distribu-tions by points in the unit triangle in the (p1, p3)plane, with p2 given by p2 = 1 � p1 � p3 (Figs. 1and 2). Since they represent the loci of solutions tothe equations

U1p1 þ U2p2 þ U3p3 ¼ U2 � U2 � U1½ � � p1þ U3 � U2½ � � p3

¼ constant

for the fixed utility indices {U1, U2, U3}, the indif-ference curves of an expected utility maximizerconsist of parallel straight lines in the triangle,with slope [U2 � U1]/[U3 � U2], as illustrated bythe solid lines in Fig. 1. Indifference curves whichdo not satisfy the expected utility hypothesis (thatis, are not linear in the probabilities) are illustratedby the solid curves in Fig. 2.

When the outcomes consist of different wealthlevels x1 < x2 < x3, this diagram can be used toillustrate other possible features of an expectedutility maximizer’s attitudes towards risk. On theprinciple that more wealth is better, it is typicallypostulated that any change in a distribution(p1, p2, p3) which increases p3 at the expense ofp2, increases p2 at the expense of p1, or both, willbe preferred: this property is known as ‘first-orderstochastic dominance preference’. Since suchshifts of probability mass are represented bynorth, west, or north-west movements in the dia-gram, first-order stochastic dominance preferenceis equivalent to the condition that indifferencecurves are upward sloping, with more preferred

indifference curves lying to the north-west. Alge-braically, this is equivalent to the conditionU1 < U2 < U3.

Another widely (though not universally)hypothesized aspect of attitudes towards risk isthat of ‘risk aversion’ (for example, Arrow 1974,ch. 3; Pratt 1964). To illustrate this property, con-sider the dashed lines in Fig. 1, which representloci of solutions to the equations

x1p1 þ x2p2 þ x3p3 ¼ x2 � x2 � x1½ � � p1þ x3 � x2½ � � p3

¼ constant

and hence may be termed ‘iso-expected value loci’.Since north-eastmovements along any of these lociconsist of increasing the tail probabilities p1 and p3at the expense of the middle probability p2 in amanner which preserves the mean of the distribu-tion, they correspond to what are termed ‘mean-preserving increases in risk’ (Rothschild andStiglitz 1970, 1971). An individual is said to be‘risk averse’ if such increases in risk always lead toless preferred indifference curves, which is equiv-alent to the graphical condition that the indiffer-ence curves be steeper than the iso-expected valueloci. Since the slope of the latter is given by[x2 � x1]/[x3 � x2], this is equivalent to the alge-braic condition that [U2 � U1]/[x2 � x1] > [U3

� U2]/[x3 � x2]. Conversely, individuals who pre-fer mean-preserving increases in risk are termed‘risk loving’: such individuals’ indifference curveswill be flatter than the iso-expected value loci, andtheir utility indices will satisfy [U2 � U1]/[x2 �x1] < [U3 � U2]/[x3 � x2].

Note finally that the indifference map in Fig. 1indicates that the lottery P is indifferent to theorigin, which represents the degenerate lotteryyielding x2 with certainty. In such a case theamount x2 is said to be the ‘certainty equivalent’of the lottery P. The fact that the origin lies on alower iso-expected value locus than P reflects ageneral property of risk-averse preferences,namely, that the certainty equivalent of any lotterywill always be less than its mean. (For risk lovers,the opposite is the case.)

Expected Utility Hypothesis 3

When the outcomes are elements of the realline, it is possible to represent the above (as wellas other) aspects of preferences in terms of theshape of the von Neumann–Morgenstern utilityfunction U(�), as seen in Figs. 3 and 4. In eachfigure, consider the lottery which assigns the prob-abilities 2/3:1/3 to the outcome levels x0: x00. The

expected value of this lottery, x ¼ 2=3 � x0 þ 1=3

�x00, lies between these two values, two-thirds of theway towards x0. The expected utility of this lottery,u ¼ 2=3 � U x0ð Þ þ 1=3 � U x00ð Þ lies between U(x0)and U(x00) on the vertical axis, two-thirds of theway towards U(x0). The point x,uð Þ thus lies on theline segment connecting the points (x0, U(x0)) and(x00, U(x00)), two-thirds of the way towards theformer. In each figure, the certainty equivalent ofthis lottery is given by the sure outcome c that alsoyields a utility level of u.

The property of first-order stochastic domi-nance preference can be extended to the case ofdistributions over the real line (Quirk andSaposnick 1962), and it is equivalent to the con-dition that U(x) be an increasing function of x, asin Figs. 3 and 4. It is also possible to generalize thenotion of a mean-preserving increase in risk todensity functions or cumulative distribution func-tions (Rothschild and Stiglitz 1970, 1971), and theearlier algebraic condition for risk aversion gen-eralizes to the condition that U00(x) < 0 for all x,that is, that the von Neumann–Morgenstern utilityfunction U(�) be concave, as in Fig. 3. As before,risk aversion implies that the certainty equivalentof any lottery will lie below its mean, as seen inFig. 3; the opposite is true for the convex utilityfunction of a risk lover, as in Fig. 4. Two of theearliest and most important analyses of risk atti-tudes in terms of the shape of the von

1

p3 P

p1 10

Increasing preferences

Expected Utility Hypothesis, Fig. 1 Expected utilityindifference curves

1

p3

p1 10

Increasing preferences

Expected Utility Hypothesis, Fig. 2 Non-expected util-ity indifference curves

U (x′)

xc ″x′ –x

U(x″)

–u

U (.)

Expected Utility Hypothesis, Fig. 3 Von Neumann-Morgenstern utility function of a risk averse individual

4 Expected Utility Hypothesis

Neumann–Morgenstern utility function are thoseof Friedman and Savage (1948) andMarkowitz (1952).

Analytics

The tremendous analytic capabilities of theexpected utility model derive largely from thework of Arrow (1974) and Pratt (1964), whoshowed that the ‘degree’ of concavity of the utilityfunction provides a measure of an expected utilitymaximizer’s ‘degree’ of risk aversion. Formally,the Arrow–Pratt characterization of comparativerisk aversion is the result that the following con-ditions on a pair of (increasing, twice differentia-ble) von Neumann–Morgenstern utility functionsUa(�) and Ub(�) are equivalent:

• Ua(�) is a concave transformation of Ub(�) (thatis, Ua(x) � r(Ub(x)) for some increasing con-cave function r(�)),

• �U00a xð Þ=U0

a xð Þ � �U00b xð Þ=U0

b xð Þ for each x,• if ca and cb solve Ua(ca) =

ÐUa(x)dF(x) and

Ub(cb) =ÐUb(x)dF(x) for some distribution

F(�), then ca � cb,

and if Ua(�) and Ub(�) are both concave, theseconditions are in turn equivalent to:

• if r> 0, E ~z½ � > r, prob ~z < rð Þ > 0, and aa andab maximize

ÐUa((I – a) � r + a � z)dF(z) andÐ

Ub((I – a) � r + a � z)dF(z) respectively, thenaa � ab.

The first two of these conditions provide equiv-alent formulations of the notion that Ua(�) is amore concave function than Ub(�). The curvaturemeasure R(x) � � U00(x)/U0(x) is known as the‘Arrow–Pratt index of (absolute) risk aversion’,and plays a key role in the analytics of theexpected utility model. The third condition statesthat the more risk averse utility functionUa(�) willnever assign a higher certainty equivalent to anylottery F(�) than will Ub(�). The final conditionpertains to the individuals’ respective demandsfor risky assets. Specifically, assume that eachmust allocate $I between two assets, one yieldinga riskless (gross) return of r per dollar, and theother yielding a risky return ~z with a higherexpected value but with some chance of doingworse than r. This condition says that the lessrisk-averse utility function Ub(�) will generate atleast as great a demand for the risky asset as themore risk-averse utility functionUa(�). It is impor-tant to note that it is the equivalence of the aboveconcavity, certainty equivalent and asset demandconditions which makes the Arrow-Pratt charac-terization such an important result in expectedutility theory. (Ross 1981, provides an alternative,stronger, characterization of comparative riskaversion.)

Although the applications of the expected utilitymodel extend to virtually all branches of economictheory (for example, Hey 1979), much of the flavourof these analyses can be gleaned from Arrow’s(1974, ch. 3) analysis of the portfolio problem ofthe previous paragraph. If we rewrite(I � a) � r + a � z as I � r + a � (z � r), the first-order condition for this problem can be expressed as:

ðz � U0 I � r þ a � z� rð Þð ÞdF zð Þ

�ðr � U0 I � r þ a � z� rð Þð ÞdF zð Þ ¼ 0,

U(.)U(x″)

U (x ′)

–u

c x″x′ –x

Expected Utility Hypothesis, Fig. 4 vonNeumann–Morgenstern utility function of a risk lovingindividual

Expected Utility Hypothesis 5

that is, the marginal expected utility of the lastdollar allocated to each asset is the same. Thesecond-order condition can be written as:

ðz� rð Þ2 � U000

I � r þ a � z� rð Þð Þ dF zð Þ < 0

and is ensured by the property of risk aversion(i.e. U00(�) < 0).

As usual, we may differentiate the first-ordercondition to obtain the effect of a change in someparameter, say initial wealth I, on the optimal levelof investment in the risky asset (the optimal valueof a). Differentiating the first-order condition(including a) with respect to I, solving for da/dI,and invoking the second-order condition and thepositivity of r yields that this derivative possessesthe same sign as:

ðz� rð Þ � U00 I � r þ a � z� rð Þð Þ dF zð Þ:

Substituting U00(x) � � R(x) � U0(x) and sub-tracting R(I � r) times the first-order conditionyields that this expression is equal to:

�ð

z� rð Þ � R I � r þ a � z� rð Þð Þ � R I � rð Þ½ �

� U0 I � r þ a � z� rð Þð ÞdF zð Þ:

On the assumption that a is positive and R(�) ismonotonic, the expression (z � r) � [R(I � r + a �(z� r)) � R(I � r)] will possess the same sign as R0

(�). This implies that the derivative da/dI will bepositive (negative) whenever the Arrow–Pratt indexR(x) is a decreasing (increasing) function of theindividual’s wealth level x. In other words, anincrease in initial wealth will always increase(decrease) demand for the risky asset if and only ifU(�) exhibits decreasing (increasing) absolute riskaversion in wealth. Further examples of the analyticsof the expected utility model may be found in theabove references, as well as the surveys ofHirshleifer and Riley (1979), Lippman and McCall(1981), Machina (1983) and Karni andSchmeidler (1991).

Axiomatic Development

Although there exist dozens of formal axiomat-izations of the expected utility model, most pro-ceed by specifying an outcome space andpostulating that the individual’s preferences overprobability distributions on this outcome spacesatisfy the following four axioms: completeness,transitivity, continuity and the IndependenceAxiom. Although it is beyond the scope of thisentry to provide a rigorous derivation of theexpected utility model in its most general setting,it is possible to illustrate the meaning of theaxioms and sketch a proof of the expected utilityrepresentation theorem in the simple case of afinite outcome set {x1, . . . , xn}.

Recall that in such a case the objects of choiceconsist of probability distributions P =(p1, . . . , pn) over {x1, . . . , xn}, so that the fol-lowing axioms refer to the individuals’weak pref-erence relation � over these prospects, whereP� � P is read ‘P* is weakly preferred (that is,preferred or indifferent) to P’ (the associated strictpreference relation and indifference relation ~are defined in the usual manner):

• Completeness: For any two distributions P andP*, either P� � P, P � P�, or both.

• Transitivity: If P�� � P� and P� � P, thenP�� � P.

• Mixture continuity: If P�� � P� � P, thenthere exists some l � [0, 1] such thatP� ~ l � P�� + (1 � l) � P.

• Independence: For any two distributions P andP�,P� � P if and only ifl � P� + (1 � l) � P��� l � P + (1 � l) � P�� for all l � [0, 1]and all P��

where l � P + (1 � l) � P�� denotes the l :(1 � l) ‘probability mixture’ of P and P��, that is,the lottery with probabilities (l � p1 þ 1� lð Þ � p��1, . . . , l � pn þ 1� lð Þ � p��n ).

The completeness and transitivity axioms areanalogous to their counterparts in standard con-sumer theory. Mixture continuity states that if thelottery P** is weakly preferred to P* and P* isweakly preferred to P, then exists some probability

6 Expected Utility Hypothesis

mixture of the most and least preferred lotterieswhich is indifferent to the intermediate one.

As in standard consumer theory, completeness,transitivity and continuity serve to establish theexistence of a real-valued preference functionV(p1, . . . , pn) which represents the relation � ,in the sense that P� � P if and only ifV p�1, . . . , p

�n

� � � V p1, . . . , pnð Þ. It is the Indepen-dence Axiom which gives the theory its primaryempirical content by implying that � can berepresented by a linear preference function of theform V(p1, . . . , pn) � � Uipi. To see the mean-ing of this axiom, assume that individuals arealways indifferent between a two-stage compoundlottery and its probabilistically equivalent single-stage lottery, and that P* happens to be weaklypreferred to P. In that case, the choice betweenthe mixtures l � P� + (1 � l) � P�� and l � P +(1� l) � P�� is equivalent to being presented witha coin that has a (1 � l) chance of landing tails(in which case the prize will be P**) and beingasked before the flip whether one would ratherwin P* or P in the event of a head. The normativeargument for the Independence Axiom is thateither the coin will land tails, in which case thechoicewon’t havemattered, or it will land heads, inwhich case one is ‘in effect’ facing a choicebetween P* and P and one ‘ought’ to have thesame preferences as before. Note finally that theabove statement of the axiom in terms of the weakpreference relation � also implies its counterpartsin terms of strict preference and indifference.

In the following sketch of the expected utilityrepresentation theorem, expressions such as ‘xi �xj’ should be read as saying that the individualweakly prefers the degenerate lottery yielding xiwith certainty to that yielding xj with certainty,and ‘l � xi + (1� l) � xj’will be used to denote thel : (1 � l) probability mixture of these twodegenerate lotteries.

The first step in the proof is to define the vonNeumann–Morgenstern utility index {Ui} and theexpected utility preference function V(�). Withoutloss of generality, we may order the outcomes sothat xn � xn – 1 � . . . � x2 � x1. Since xn� xi � x1 for each outcome xi, mixture continu-ity implies that there exist scalars {Ui} [0, 1]

such that xi ~ Ui � xn + (1 � Ui) � x1 for each i(which implies U1 = 0 and Un = 1). Given this,define V(P) = � Uipi for each P.

The second step is to show that each lotteryP = (p1,. . ., pn) is indifferent to the mixture l � xn+ (1� l) � x1 where l=�Ui pi. Since (p1,. . ., pn)can be written as the n-component probabilitymixture p1 � x1 + p2 � x2 + . . . + pn � xn, andeach outcome xi is indifferent to the mixture Ui �xn + (1 � Ui) � x1, an n-fold application of theIndependence Axiom yields that P = (p1,. . ., pn)is indifferent to the mixture

p1 � U1 � xn þ 1� U1ð Þ � x1½ � þ p2� U2 � xn þ 1� U2ð Þ � x1½ � þ . . .þ pn� Un � xn þ 1� Unð Þ � x1½ �,

which is equivalent toPn

i¼1 Uipi� � � xnþ

1�Pni¼1 Uipi

� � � x1.The third step is to demonstrate that a mixture

l� � xn + (1 � l�) � x1 is weakly preferred to amixturel � xn + (1 � l) � x1 if and only ifl� � l.This follows immediately from the IndependenceAxiom and the fact that l� � l implies that thesetwo lotteries may be expressed as the respectivemixtures (l� � l) � xn + (1 � l� + l) � Q and(l� � l) � x1 + (1 � l� + l) � Q, where Q isdefined as the lottery (l/(1 � l� + l)) � xn + ((1� l�)/(1 � l� + l)) � x1.The completion of the proof is now simple. For

any two distributionsP� ¼ p�1, . . . , p�n

� �and P =

(p1, . . . , pn), transitivity and the second stepimply that P� � P if and only if

Xn

i¼1Uip

�i

� �� xn þ 1�

Xn

i¼1Uip

�i

� �� x1 �Xn

i¼1Uipi

� �� xn þ 1�

Xn

i¼1Uipi

� �� x1,

which by the third step is equivalent to the condi-tion

PUip

�i �

PUipi , or in other words, that

V(P�) � V(P).As mentioned, the expected utility model has

been axiomatized many times and in many con-texts. The most comprehensive accounts of theaxiomatics of the model are undoubtedly Fishburn(1982) and Kreps (1988).

Expected Utility Hypothesis 7

Subjective Expected Utility

In addition to the above setting of ‘objective’ (thatis, probabilistic) uncertainty, it is possible todefine expected utility preferences under condi-tions of ‘subjective’ uncertainty. In this case,uncertainty is represented by a set S of mutuallyexclusive and exhaustive ‘states of nature,’ whichcan be a finite set {s1,. . ., sn} (as with a horserace), a real interval s, s½ � � R1 (as with tomor-row’s temperature), or a more abstract space. Theobjects of choice are then ‘acts’ a(�): S ! Xwhich map states to outcomes. In the case of afinite state space, acts are usually expressed in theform {x1 if s1;. . .; xn if sn}. When the state space isinfinite, finite-outcome acts can be expressed inthe form a(�) = [x1 on E1;. . .; xm on Em] for somepartition ofS into a family of mutually exclusiveand exhaustive ‘events’ {E1,. . ., Em}. Except forcasino games and state lotteries, virtually all real-world uncertain decisions (including all invest-ment or insurance decisions) are made under con-ditions of subjective uncertainty.

In such a setting, the ‘subjective expected utilityhypothesis’ consists of the joint hypothesis that theindividual possesses probabilistic beliefs, asrepresented by a ‘personal’ or ‘subjective’probability measure m(�) over the state space,and expected utility risk preferences, as representedby a von Neumann–Morgenstern utility functionU(�) over outcomes, and evaluates acts accordinga preference function of the formW x1 if s1; . . . ; xn if snð Þ � Pn

i¼1 U xið Þ � m sið Þ,W x1 on E1; . . . ; xm onEmð Þ � Pm

i¼1 U xið Þ � m Eið Þ ,or more generally, W(a(�)) � Ð

U(a(s))dm(s).Whereas all individuals facing a given objectiveprospect P = (x1, p1;. . .; xn, pn) are assumed to‘see’ the same probabilities (p1,. . ., pn) (thoughthey may have different utility functions), individ-uals facing a given subjective prospect {x1 if s1;. . .;xn if sn} or [x1 on E1;. . .; xm on Em] will generallypossess differing subjective probabilities overthese states or events, reflecting their differentbeliefs, past experiences, and so on.

Researchers such as Arrow (1974), Debreu(1959, ch. 7) and Hirshleifer (1965, 1966) haveshown how the analytics of the objective expected

utility model can be extended to both the positiveand normative analysis of decisions under subjec-tive uncertainty. As a simple example, consider anindividual deciding whether to purchase earth-quake insurance, and if so, how much. A simplespecification of this decision involves thestate space S = {s1, s2} = {earthquake; noearthquakeg, the individual’s von Neumann–Mor-genstern utility of wealth function U(�), their sub-jective probabilities {m(s1), m(s2)} (which sum tounity), and the price g of each dollar of insurancecoverage. An individual with initial wealthw would then purchase q dollars’ worth of cover-age, where q was the solution to

maxq

U w� gqþ qð Þ � m s1ð Þ þ U w� gqð Þ � m s2ð Þ½ �

Note that this formulation does not require thatthe individual and the insurance company agreeon the likelihood of an earthquake.

As in the objective case, subjective expectedutility can be derived from axiomatic foundations.Completeness and transitivity carry over in astraightforward way, and continuity with respectto mixture probabilities is replaced by continuitywith respect to small changes in the events. Theexistence of additive personal probabilities isobtained by the following axiom:

Comparative likelihood: For all events A, B andoutcomes x� x and y� y, [x* on A; x on ~A] � [x* on B; x on ~ B] implies [y* on A; y on~ A] � [y* on B; y on ~ B].

This axiom states that if the individual‘reveals’ event A to be at least as likely as eventB by their preference for staking the preferredoutcome x* on A rather than on B, then this like-lihood ranking will hold for all other pairs ofranked outcomes y� y. Finally, under subjectiveuncertainty the Independence Axiom is replacedby its subjective analogue, first proposed by Sav-age (1954):

Sure-Thing Principle: For all events E and actsa(�), a*(�), b(�) and c(�), [a*(�) on E; b(�) on ~ E]

8 Expected Utility Hypothesis

� [a(�) on E; b(�) on ~ E] implies [a*(�) on E; c(�) on ~ E] � [a(�) on E; c(�) on ~ E].

where [a(�) on E; b(�) on ~ E] denotes the actyielding outcome a(s) for all s � E and b(s) forall s � ~ E.

Under subjective uncertainty, an individual’sutility of outcomes might sometimes dependupon the particular state of nature. Given a healthinsurance decision with a state space of S ={s1, s2} = {cancer; no cancer}, an individual mayfeel a greater need for $100,000 in state s1 than instate s2. This can be modelled by means of a ‘state-dependent’ utility function U �j sð Þj s�Sf g anda‘state-dependent expected utility’ preferencefunction W x1 if s1; . . . ; xn if snð Þ ¼ Pn

i¼1 U xij sið Þ � msið Þor W a �ð Þð Þ ¼ Ð

U a sð Þj sð Þdm sð Þ. The analyticsof state-dependent expected utility preferenceshave been extensively developed by Karni (1985).

History

The hypothesis that individuals might maximizethe expectation of ‘utility’ rather than of monetaryvalue was proposed independently by mathemati-cians Gabriel Cramer and Daniel Bernoulli, ineach case as the solution to a problem posed byDaniel’s cousin Nicholas Bernoulli (see Bernoulli1738). This problem, now known as the ‘StPetersburg Paradox’, considers the gamblewhich offers a 1/2 chance of $1, a 1/4 chance of$2, a 1/8 chance of $4, and so on. Although theexpected value of this prospect is

1=2ð Þ � $1þ 1=4ð Þ � $2þ 1=8ð Þ � $4þ . . . . . .¼ $0:50þ $0:50þ $0:50þ . . . ¼ $1,

common sense suggests that no one would bewilling to forgo a very substantial certain paymentin order to play it. Cramer and Bernoulli proposedthat, instead of using expected value, individualsmight evaluate this and other lotteries by means oftheir expected ‘utility’, with utility given by afunction such as the natural logarithm or thesquare root of wealth, in which case the certainty

equivalent of the St Petersburg gamble becomes amoderate (and plausible) amount.

Two hundred years later, the St Petersburgparadox was generalized by Karl Menger (1934),who noted that, whenever the utility of wealthfunction was unbounded (as with the natural log-arithm or square root functions), it would be pos-sible to construct similar examples with infiniteexpected utility and hence infinite certainty equiv-alents (replace the payoffs $1, $2, $4 . . . in theabove example by x1, x2, x3 . . ., where U(xi) = 2i

for each i). In light of this, von Neumann–Mor-genstern utility functions are typically (though notuniversally) postulated to be bounded functions ofwealth.

The earliest formal axiomatic treatment of theexpected utility hypothesis was developed byFrank Ramsey (1926) as part of his theory ofsubjective probability, or individuals’ ‘degrees ofbelief’ in the truth of alternative propositions.Starting from the premise that there exists an‘ethically neutral’ proposition whose degree ofbelief is 1/2, and whose validity or invalidity isof no independent value, Ramsey proposed a setof axioms on how the individual would be willingto stake prizes on its truth or falsity, in a mannerwhich allowed for the derivation of the ‘utilities’of these prizes. He then used these utility valuesand betting preferences to determine the individ-ual’s degrees of belief in other propositions. Per-haps because it was intended as a contribution tothe philosophy of belief rather than to the theoryof risk bearing, Ramsey’s analysis did not havethe impact among economists that it deserved.

The first axiomatization of the expected utilitymodel to receive widespread attention was that ofJohn von Neumann and Oskar Morgenstern, pre-sented in connection with their formulation of thetheory of games (von Neumann and Morgenstern1944, 1947, 1953). Although this developmentwas recognized as a breakthrough, the mistakenbelief that von Neumann and Morgenstern hadsomehow mathematically overthrown theHicks–Allen ‘ordinal revolution’ led to some con-fusion until the difference between ‘utility’ in thevon Neumann-Morgenstern and the ordinal (thatis, non-stochastic) senses was illuminated by

Expected Utility Hypothesis 9

writers such as Ellsberg (1954) andBaumol (1958).

Another factor which delayed the acceptanceof the theory was the lack of recognition of therole played by the Independence Axiom, whichdid not explicitly appear in the vonNeumann–Morgenstern formulation. In fact, theinitial reaction of researchers such as Baumol(1951) and Samuelson (1950) was that there wasno reason why preferences over probability distri-butions must necessarily be linear in the probabil-ities. However, the independent discovery of theIndependence Axiom by Marschak (1950), Sam-uelson (1952) and others, and Malinvaud’s (1952)observation that it had been implicitly invoked byvon Neumann and Morgenstern, led to an almostuniversal acceptance of the expected utilityhypothesis as both a normative and positive the-ory of behaviour towards risk. This period alsosaw the development of the elegant axiomatiza-tion of Herstein and Milnor (1953) as well asSavage’s (1954) joint axiomatization of utilityand subjective probability, which formed thebasis of the state-preference approach describedabove.

While the 1950s essentially saw the comple-tion of foundational work on the expected utilitymodel, subsequent decades saw the flowering ofits analytic capabilities and its application to fieldssuch as portfolio selection (Merton 1969), optimalsavings (Levhari and Srinivasan 1969; Flemingand Sheu 1999), international trade (Batra 1975;Lusztig and James 2006), environmental econom-ics (Wolfson et al. 1996), medical decision-making (Meltzer 2001) and even the measurementof inequality (Atkinson 1970). This movementwas spearheaded by the development of theArrow–Pratt characterization of risk aversion(see above) and the characterization, byRothschild–Stiglitz (1970, 1971) and others, ofthe notion of ‘increasing risk’. This latter workin turn led to the development of a general theoryof ‘stochastic dominance’ (for example,Whitmore and Findlay 1978; Levy 1992), whichhas further expanded the analytical powers of themodel.

Although the expected utility model received asmall amount of experimental testing by

economists in the early 1950s (for example,Mosteller and Nogee 1951; Allais 1953) and con-tinued to be examined by psychologists, econo-mists’ interest in the empirical validity of themodel waned from the mid-1950s through themid-1970s, no doubt due to both the normativeappeal of the Independence Axiom and model’sanalytical successes. However, since the late1970s there has been a revival of interest in thetesting of the expected utility model; a growingbody of evidence that individuals’ preferencessystematically depart from linearity in the proba-bilities; and the development, analysis and appli-cation of alternative models of choice underobjective and subjective uncertainty. It is fair tosay that today the debate over the descriptive (andeven normative) validity of the expected utilityhypothesis is more extensive than it has been inover half a century, and the outcome of this debatewill have important implications for the directionof research in the economics of uncertainty.

See Also

▶Bernoulli, Daniel▶Non-Expected Utility Theory▶Ramsey, Frank Plumpton▶Risk▶Risk Aversion▶ Savage’s Subjective Expected Utility Model▶Uncertainty▶Utility

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