‘expected utility / subjective probability’ analysis...

DOI: 10.1007/s00199-004-0573-6Economic Theory 26, 1–62 (2005)

Research Articles

‘Expected utility / subjective probability’ analysiswithout the sure-thing principleor probabilistic sophistication

Mark J. Machina

Dept. of Economics, University of California, San Diego, La Jolla, CA 92093, USA(e-mail: [email protected])

Received: May 4, 2004; revised version: October 4, 2004

Summary. The basic analytical concepts, tools and results of the classical ex-pected utility/subjective probability model of risk preferences and beliefs undersubjective uncertainty can be extended to general “event-smooth” preferences oversubjective acts that do not necessarily satisfy either of the key behavioral assump-tions of the classical model, namely the Sure-Thing Principle or the Hypothesis ofProbabilistic Sophistication. This is accomplished by a technique analogous to thatused by Machina (1982) and others to generalize expected utility analysis underobjective uncertainty, combined with an event-theoretic approach to the classicalmodel and the use of a special class of subjective events, acts and mixtures thatexhibit “almost-objective” like properties. The classical expected utility/subjectiveprobability characterizations of outcome monotonicity, outcome derivatives, proba-bilistic sophistication, comparative and relative subjective likelihood, and compara-tive risk aversion are all globally “robustified” to general event-smooth preferencesover subjective acts.

Keywords and Phrases: Subjective uncertainty, Almost-objective uncertainty,Robustness.

JEL Classification Numbers: D81.

This paper presents a considerably improved version of the concept of event-differentiability fromMachina (1992). An alternative definition has been independently developed by Epstein (1999) in hisanalysis of the concept of uncertainty aversion. I am grateful to Kenneth Arrow, Mark Durst, JurgenEichberger, Daniel Ellsberg, Clive Granger, Simon Grant, Edi Karni, Peter Klibanoff, David Kreps,Duncan Luce, Robert Nau, Uzi Segal, Peter Wakker, Joel Watson and especially Larry Epstein, TedGroves and Joel Sobel for helpful discussions and comments. This material is based upon work supportedby the National Science Foundation under Grants No. 9209012 and 9870894.

2 M.J. Machina

1 Introduction

This paper addresses the question:

“To what extent are the analytics of the classical expected utility/subjective prob-ability model of choice under subjective uncertainty robust to departures fromboth the assumption of expected utility risk preferences and the assumption ofprobabilistic beliefs?”

We approach this question by combining the following ideas:

The Calculus Approach to Robustness, which can be used to establish the localand global robustness of any model that exhibits “constant sensitivity” in itskey variables, and which has already been used to establish the robustness ofexpected utility analysis under objective uncertainty.1

An Event-Theoretic Representation of the Classical Model, which shows thatthe expected utility/subjective probability model under subjective uncertaintyexhibits constant sensitivity in the events attached to each uncertain outcome,and is therefore also amenable to the above approach to robustness.

While analogous to the earlier robustification of objective expected utility analysisin both its goals and its overall findings, the subjective analysis of this paper involvesquite different mathematical tools, and does not assume prior knowledge of theearlier approach.

The following section motivates the analysis by a discussion of the classicalmodel, its main empirical violations, existing responses to these violations, andthe argument for robustification. Section 3 sketches out the calculus approach torobustness, and how it has been applied to expected utility analysis under objectiveuncertainty. Section 4 presents the tools needed to extend this approach to subjec-tive uncertainty, namely an event-theoretic representation of the classical model,the notion of “event-smoothness,” and a special class of subjective events, acts andmixtures that exhibit “almost-objective” like properties. Section 5 establishes theglobal robustness of the classical expected utility/subjective probability characteri-zations of outcome-monotonicity, outcome derivatives, probabilistic sophistication,comparative and relative subjective likelihood, and comparative risk aversion. Sec-tion 6 discusses related work of Epstein, and extensions of the present analysis.Proofs are in an Appendix.

2 The classical model of risk preferences and beliefs

By “classical model” we mean the subjective expected utility (SEU) modelproposed by Ramsey (1931) and fully developed by Savage (1954), in whichchoice over subjectively uncertain prospects is characterized by expected utilityrisk preferences and standard probabilistic beliefs.2 The key sense in which this

1 E.g., Machina (1982, 1983, 1984, 1989, 1995), Dekel (1986), Allen (1987), Karni (1987, 1989),Chew, Epstein and Zilcha (1988), Chew and Nishimura (1992), Bardsley (1993) and Wang (1993).

2 See Fishburn (1982, Chs.9-12) or Kreps (1988, Chs.8-10) for modern expositions of this approach.

‘Expected utility/subjective probability’ analysis without the sure-thing principle 3

model differs from the earlier objective expected utility model of Bernoulli (1738)and von Neumann-Morgenstern (1947) is the manner in which the two modelsrepresent uncertainty. In the objective framework, uncertainty comes prepackagedin terms of numerical probabilities, and the objects of choice are probabilitydistributions over outcomes or lotteries P = (x1, p1; . . . ;xn, pn), yieldingoutcome xi with probability pi. But in the more realistic subjective framework,uncertainty is represented by a set S = . . . , s, . . . of mutually exclusive andexhaustive states of nature, or by events E (subsets of S), and the objects of choiceare bets or acts f(·) = [x1 on E1; . . . ;xn on En], which specify the outcomex = f(s) as a function of the state or event that occurs. The SEU model posits asubjective probability measure µ(·) over events, and an expected utility preferencefunction VEU (P) ≡

∑ni=1U(xi) ·pi over lotteries, such that preferences over

subjective acts can be represented by the SEU preference function

WSEU

(x1 on E1; . . . ;xn on En

)≡ VEU

(x1, µ(E1); . . . ;xn, µ(En)

)(1)

≡∑n

i=1U(xi)·µ(Ei) ≡

∫SU(f(s)

)·dµ(s)

The classical SEU form can be represented as the composition of two mappings:[x1 on E1; . . . ;xn on En

]→(x1, µ(E1); . . . ;xn, µ(En)

)→∑n

i=1U(xi)·µ(Ei)

In the first mapping, the subjective probability measure µ(·) is used to determinethe act’s implied lottery over outcomes. In the second mapping, the expected utilitypreference function VEU (·) is used to determine the level of preference of thisimplied lottery, and by implication, the level of preference for the act. Thus, theclassical model is often represented as the joint hypothesis of

Probabilistic Sophistication: Acts are evaluated solely on the basis of theirimplied lotteries over outcomes. This property only involves the individual’sbeliefs.

Expected Utility Risk Preferences: Preferences over implied outcome lotteries arelinear in the probabilities. This property only involves the individual’s attitudestoward risk.

This so-called “separation of beliefs from risk preferences” – with the use of sub-jective probabilities to represent the former and expected utility to represent thelatter – is often viewed as the characteristic feature of the classical SEU model.

2.1 The two orthogonal hypotheses of the classical model

Although the separation of the classical model into the hypotheses of probabilisti-cally sophisticated beliefs and expected utility risk preferences accords with bothintuitive and normative ideals of individual and statistical decision making, it isless than perfect from a scientific or analytical point of view, for the simple reasonthat the two hypotheses are nested: the hypothesis of expected utility risk pref-erences is defined over the output of the probabilistic sophistication hypothesis,namely the implied lottery of each act. Accordingly, we shall organize our analysis

4 M.J. Machina

in terms of an alternative pair of hypotheses, which also jointly characterize theclassical model, but which are orthogonal in the sense that each is defined directlyon preferences over subjective acts, and each can be satisfied independently of theother:

Event-Separability: Preferences over acts are separable across mutually exclusiveevents.

Probabilistic Sophistication: As before, acts are evaluated solely on the basis oftheir implied lotteries over outcomes, via some subjective probability measureµ(·).

Axiomatically, these two hypotheses are respectively equivalent to, and can be rep-resented by:

P2 Sure-Thing Principle (Savage (1954)): For all events E and acts f(·), f*(·),g(·), h(·):[f*(·) on E

g(·) on ∼E

][f(·) on E

g(·) on ∼E

]⇒[f*(·) on E

h(·) on ∼E

][f(·) on E

h(·) on ∼E

]

P4* Strong Comparative Probability Axiom (Machina and Schmeidler, 1992):For all outcomes x∗ x, y∗ y, disjoint events A,B and acts g(·), h(·): x

∗ on Ax on B

g(·) elsewhere

x on Ax∗ on B

g(·) elsewhere

⇒

y

∗ on Ay on B

h(·) elsewhere

y on Ay∗ on B

h(·) elsewhere

Behaviorally, the two hypotheses can be respectively described as:

Stable Subact Preferences: The ranking of any pair of subacts [f*(·) on E ]versus [f(·) on E ] is not affected by identical changes in statewise-identicalpayoffs off of the event E.

Stable Revealed Likelihood Rankings: The preferred assignment of greater versuslesser preferred payoffs to the events A versus B is not affected by identicalchanges in statewise-identical payoffs off of A∪B, or by the values of thegreater/lesser preferred payoffs themselves.

Theoretically, the two hypotheses can be shown to respectively characterize thefollowing forms for the preference function over acts:

State-Dependent Expected Utility: The preference function WSDEU (f(·))≡∫

SU(f(s)|s) ·dµ(s), for some subjective probability measure µ(·) andstate-dependent utility function U(x|s).

Probabilistically Sophisticated Non-Expected Utility: The preference functionWPS(f(·)) ≡ V (x1, µ(E1); . . . ;xn, µ(En)), for some subjective proba-bility measure µ(·) and non-expected utility preference function V (P) ≡V (x1, p1; . . . ;xn, pn) over lotteries.


2.2 Three violations of the classical model

The classical SEU model is violated by three well-known phenomena – Allais-typeviolations of event-separability, state-dependence type violations of probabilisticsophistication, and Ellsberg-type violations – which can be illustrated by the exam-ples in Table 1.3 Each example involves three mutually exclusive and exhaustiveevents, and four acts defined on these events. Outcomes are in dollars, and 1M =1,000,000. The uncertainty in the Allais example comes from an urn known to con-tain 10 red, 1 black, and 89 yellow balls, and the rankings α1 α2 and α3 ≺ α4are the well-known Allais (1953) preferences over these acts’ implied lotteries. Theuncertainty in the state-dependence example involves the individual’s future health,with three events: staying healthy, having a debilitating tumor that is fatal withouta $50,000 operation, or getting the flu. The rankings β1 β2 and β3 ≺ β4 reflectthe greater usefulness of $10,000 when healthy than with the tumor, but the greaterneed for $50,000 with the tumor than when healthy. The uncertainty in the Ellsbergexample comes from an urn known to contain 30 red balls and 60 black and/oryellow balls in unknown proportion, and the rankings γ1 γ2 and γ3 ≺ γ4 arethe well-known Ellsberg (1961) preferences. In each case, the stated preferencesviolate the classical SEU form (1).

Table 1. Three violations of the classical subjective expected utility model

Allais paradox4 State-dependence Ellsberg paradox

10balls

1ball

89balls

healthy tumor flu30 balls 60 balls

︷︸︸︷red

︷︸︸︷black yellow

α1 1M 1M 1M β1 10,000 1,000 100 γ1 100 0 0α2 5M 0 1M β2 1,000 10,000 100 γ2 0 100 0α3 1M 1M 0 β3 50,000 5,000 500 γ3 100 0 100α4 5M 0 0 β4 5,000 50,000 500 γ4 0 100 100

α1 α2 yet α3 ≺ α4 β1 β2 yet β3 ≺ β4 γ1 γ2 yet γ3 ≺ γ4

2.3 Theoretical responses to the violations

Researchers have responded to Allais-type violations by the development andanalysis of non-expected utility preference functions over lotteries – that is, pref-erence functions V (P) = V (x1, p1; . . . ;xn, pn) which drop the expected util-ity property of linearity in the probabilities. Work in this area has proceededalong two lines: the development and analysis of specific functional forms forV (x1, p1; . . . ;xn, pn),5 and the analysis of general smooth V (x1, p1; . . . ;xn, pn)

3 See the excellent reviews of these phenomena by Sugden (1986, 1991), Weber and Camerer (1987),Karni and Schmeidler (1991), Epstein (1992), Kelsey and Quiggin (1992), Camerer and Weber (1992)and Starmer (2000).

4 I have reversed the usual labeling of the last two Allais acts in order to better highlight the structureof the violation.

5 Most notably, the rank-dependent form of Quiggin (1982), the weighted utility form of Chew(1983), the quadratic form of Chew, Epstein and Segal (1991), and the disappointment aversion formof Gul (1991).

6 M.J. Machina

functions via their probability and/or outcome derivatives (as described below). Ineach case, non-expected utility risk preferences are completely compatible withprobabilistically sophisticated beliefs. That is, for any non-expected utility prefer-ence function V (x1, p1; . . . ;xn, pn) and any subjective probability measure µ(·),the pair of mappings[x1onE1;...;xnonEn

]→(x1,µ(E1);...;xn,µ(En)

)→V(x1,µ(E1);...;xn,µ(En)

)yields a probabilistically sophisticated non-expected utility preference function

WPS

(x1 on E1; . . . ;xn on En

)≡ V

(x1, µ(E1); . . . ;xn, µ(En)

)(2)

over subjective acts, which can be shown to be the generalization of the classicalform (1) obtained by retaining the hypothesis of probabilistic sophistication butdropping the hypothesis of event-separability.

Researchers have primarily responded to state-dependence type violationsof probabilistic sophistication by working with the well-known state-dependentexpected utility preference function6

WSDEU

(x1 onE1;. . . ;xn onEn

)≡∫

SU(f(s)|s

)·dµ(s)≡

n∑i=1

∫Ei

U(xi|s)·dµ(s) (3)

which can be shown to be the generalization of the classical form (1) obtainedby retaining the hypothesis of event-separability but dropping the hypothesis ofprobabilistic sophistication.

One may ask why state-dependent preferences constitute a violation of proba-bilistic sophistication, since they can be represented by a form (3) that still contains asubjective probability measure µ(·). The answer is that probabilistic sophisticationdoes not simply mean that subjective probabilities are somehow used in the evalu-ation of an act, but rather, that they serve to completely encode its uncertainty, sothat the only feature of an outcome’s event that matters is its subjective probability.Thus, ifEi has a greater subjective probability thanEj , the individual would alwaysprefer staking the more preferred outcome on Ei and the less preferred outcomeonEj rather than vice versa – such preferences are said to “reveal” the individual’scomparative likelihood ranking of the two events. However, the rankings β1 β2and β3 ≺ β4 are seen to violate this property, and constitute a revealed likelihoodreversal. Although the form (3) involves a subjective probability measure µ(·),it accommodates state-dependence type violations of probabilistic sophisticationby allowing the events and outcomes to interact in an additional manner, via thefunction U(x|s).

Since the Ellsberg rankings γ1 γ2 and γ3 ≺ γ4 are seen to violate both eventseparability and probabilistic sophistication (that is, to violate both P2 and P4*),they cannot be represented by the probabilistically sophisticated form (2) or bythe state-dependent expected utility form (3).7 Thus in contrast with the Allais

6 E.g., Karni (1983, 1985).7 That is, if E1,E2,E3 are the Ellsberg events and we assume monotonicity in wealth, the revealed

likelihood reversal γ1 γ2 and γ3 ≺ γ4 cannot be represented by any preference function of the formV (x1, µ(E1); x2, µ(E2); x3, µ(E3)), or of the form U(x1|E1)· µ(E1) + U(x2|E2)· µ(E2) +U(x3|E3)·µ(E3).


Paradox (which can be accommodated by non-expected utility risk preferences) andstate dependence (which can be accommodated by state-outcome interactions), theEllsberg Paradox seems to strike at the heart of the classical probability-theoreticnotion of beliefs. Although the “explanation” of the paradox in terms of a generalaversion to ambiguity is well known, this observation does not “solve” the problemso much as underscore its key point, namely that, even if we allow for non-expectedutility risk preferences or state-dependence, subjective probabilities do not sufficeto encode all aspects of uncertain beliefs.

Responses to Ellsberg-type violations have primarily consisted of alternativeformulas for the subjective preference function W (·) which replace its subjectiveprobability measure µ(·) by some more general structure for beliefs. One of theseis the Choquet expected utility form

WChoquet

(x1 on E1; . . . ;xn on En

)≡

n∑i=1

U(xi)·[C( i⋃j=1

Ej

)− C

(i−1⋃j=1

Ej

)](4)

for some utility function U(·) and capacity (i.e. monotonic non-additive measure)C(·), where the outcomes are labeled so that x1 . . . xn (e.g., Gilboa, 1987;Schmeidler, 1989; Wakker, 1989, 1990; Gilboa and Schmeidler, 1994). Another isthe maxmin expected utility form

Wmaxmin

(x1 on E1; . . . ;xn on En

)≡ min

τ∈T

∫SU(f(s)

)·dµτ(s)

(5)≡ min

τ∈T

n∑i=1

U(xi) ·µτ(Ei)

for some utility function U(·) and family µτ(·) |τ ∈T of probability measureson S (e.g., Gardenfors and Sahlin (1982, 1983), Cohen and Jaffray (1985), Gilboaand Schmeidler (1989)).8 Several other forms for representing preferences in theEllsberg Paradox have been proposed. 9

The above theoretical and empirical relationships can be summarized as inTable 2.

8 Although both WChoquet(·) and Wmaxmin(·) can accommodate the joint departure from event-separability and probabilistic sophistication in the Ellsberg example of Table 1, each form implies thatpreferences are event-separable and probabilistically sophisticated over large subregions of the spaceof subjective acts, so any departure from event-separability or probabilistic sophistication within any ofthese regions will constitute a violation of these forms.

9 Some models represent beliefs by decision weights which depend on the assignment of outcomes toevents (Hazen, 1987, 1989; Luce, 1988, 1991), by lexicographic probabilities (Blume, Brandenbergerand Dekel, 1991), by second order probabilities (Nau, 2001), or by indeterminate probabilities (Whalley,1991; Nau, 1989, 1992, 2002). Other models allow the incorporation of objective information on someevents but not others (Eichberger and Kelsey, 1999), have ordinal but not cardinal likelihoods (Kelsey,1993), allow the level of ambiguity to affect the utility of an outcome (Sarin and Winkler, 1992), evaluateacts just on the basis of their outcomes (Barbera and Jackson, 1988), or represent uncertainty aversionby incomplete preferences (Bewley, 1986, 1987). Sarin and Wakker (1992) derive a belief measurewhich is only additive over a prespecified subalgebra of events, Fishburn (1991, 1993) axiomatizes aprimitive degree of ambiguity relation, and Fishburn (1989) and Fishburn and LaValle (1987) considerskew-symmetric additive act preferences.

8 M.J. Machina

Table 2. Orthogonal hypotheses, theoretical forms and empirical phenomena

Theoretical forms characterized by: Empirical phenomena consistent with:event

separability∼ event

separabilityevent

separability∼ event

separabilityprobabilistic

sophisticationWSEU (·) WPS(·) probabilistic

sophisticationAllais

preferences∼probabilisticsophistication

WSDEU (·) ∼probabilisticsophistication

state-dependence

Ellsbergpreferences

2.4 The argument for robustification

Although each of the forms WPS(·),WSDEU (·),WChoquet(·) and Wmaxmin(·)succeeds in accommodating its own respective departure from the classical model(Allais, state-dependence or Ellsberg), none of them can be considered a satisfactoryalternative to the classical model, for two reasons.

The first reason is that each form constitutes a partial rather than a generalresponse to the above set of violations, in the sense that it can accommodate someof the violations but not others. Thus, WPS(·) cannot accommodate departuresfrom probabilistic sophistication,WSDEU (·) cannot accommodate departures fromevent-separability, and while WChoquet(·) and Wmaxmin(·) can accommodate thespecific Allais and Ellsberg examples of Table 1, they cannot accommodate moregeneral departures from event separability or probabilistic sophistication (see Note8), nor can they accommodate state-dependence type departures without additionalmodification.

The second reason is that each of the above alternatives consists of a specificfunctional form for the preference function W (·). Each such form comes with itsown component function and component measure (or set of component measures),in addition to (or in place of) the classical von Neumann-Morgenstern utility func-tionU(·) and subjective probability measure µ(·). Thus, each new form involves itsown analytical “startup costs” – namely the costs of determining conditions on itscomponent functions or measures that characterize the standard behavioral prop-erties of comparative risk aversion, comparative likelihood, etc. A second round ofanalytical costs is incurred by the need to revisit each of the important applied topicsin choice under uncertainty (such as portfolio allocation, insurance demand, search,or auctions) under each new form. Finally, if more than one form is deemed accept-able, this leads to a third round of costs, namely the costs of deriving conditionsfor comparative risk aversion, comparative beliefs, etc. across such forms. Giventhe classical model’s tremendous head start in each of these topics, the analyticalcosts of “starting over” with any new functional form seem overwhelming.


Although functional forms are indispensable for empirical estimation, calibra-tion and testing, it is hard to think of any other branch of economics that hasconducted so much of its theoretical analysis by the study of functional forms.Besides the above analytical costs, this approach carries another modeling danger,which can be illustrated by a hypothetical example from regular consumer theoryover nonstochastic commodity bundles: Say we observe violations of “linearity inthe commodities” in the form of diminishing marginal rates of substitution, andrespond by developing and analyzing (say) the Cobb-Douglas functional form forutility. Among the theoretical implications that emerge are zero cross-price elas-ticities of demand. But should we really be predicting zero cross-price elasticitiesfrom the empirical phenomenon of diminishing MRS? Such unintended implica-tions have already occurred at least once in choice under uncertainty: For manyyears [Edwards (1955) through Kahneman and Tversky (1979)], the main formused to represent nonlinearities in the probabilities was the non-expected utilityform V (P) ≡

∑ni=1U(xi) ·π(pi). This form was eventually dropped when it was

found to imply violations of first order stochastic dominance preference in everyneighborhood of every lottery, a phenomenon which was not in the data it soughtto represent, and has not been observed in the data since.

The above considerations call for an approach which can accommodate generaldepartures from both event-separability and probabilistic sophistication, which doesnot rely on a functional form, and which can retain as much of the classical analyticsof risk preferences and beliefs as possible – in other words, an approach which“robustifies” the classical model and its analytics.

3 The calculus approach to robustness

3.1 Local and global robustification: A three-step approach

The following approach is simply an extension of the standard manner in whichcalculus is used to derive local and global generalizations of linear algebra results.In its most general terms, this approach to robustifying a given model – call it the“classical model” – involves three steps:

Step 1. Represent the classical model so that it exhibits “constant sensitivity” tochanges in its key variables. Then represent the model’s analytical concepts,tools and results in terms of its “sensitivity rates” to these variables.

Step 2. Define “smoothness” to establish local robustness. Define differentiabilityso that non-classical models that are “smooth” in these key variables will have“local classical approximations” and “local sensitivity rates,” thus establishingthe local robustness of the classical analytics.

Step 3. Use line integrals to establish global robustness. Construct line integralsin the key variables, so that a smooth non-classical model’s response to anyglobal change will be the integrated sum of its “local classical” responses to thischange. Those classical results that do not actually require constant sensitivityrates will then be found to be globally robust.

10 M.J. Machina

Figure 1. Line integral, path integral and line integral approximation

We can illustrate this approach – and an important mathematical issue we shallencounter – for a simple result in ordinal utility theory over nonstochastic (apple,banana) commodity bundles, namely the result that a “classical” (i.e. linear) utilityfunction ULin(a, b) ≡ ca·a+ cb·b will exhibit a global weak preference for addi-tional apples if and only if its sensitivity rate to apples satisfies ca ≥ 0. This resultcan be locally robustified to a general smooth U(·, ·) at a given bundle (a0, b0) bymeans of the analogous condition on its local sensitivity rate to apples, namely thatUa(a0, b0) ≡ ∂U(a0, b0)/∂a ≥ 0. It can also be globally robustified to a generalsmooth U(·, ·), to take the form:

U(·, ·) exhibits a global weakpreference for additional apples

⇔ Ua(a, b) ≥ 0at each bundle (a, b) (6)

The above-mentioned mathematical issue, which poses no problem for this ex-ample but will prove substantial when working with subjective uncertainty, involvesthe method of proving a global robustness result such as (6). The left-hand conditionin (6) is simply the property that U(a+∆a, b)≥U(a, b) for all (a, b) and all∆a≥ 0.If U(·, ·) is smooth, then U(a+∆a, b)− U(a, b) can be exactly expressed in termsof U(·, ·)’s local sensitivity rates (i.e. partial derivatives), either by a straight lineintegral from the bundle (a, b) to (a+∆a, b), or by any smooth path integral be-tween these bundles, as illustrated by the left and middle diagrams of Figure 1. Butof the two, only the line integral can be used to prove the global robustness result(6). The reason is that the right-hand condition of (6) ensures that each differentialmovement along the straight line path has a nonnegative differential effect uponU(·, ·), so that the integral of these effects (that is, the line integral in the figure) willbe nonnegative. But even when the right-hand condition of (6) holds, the curvedmiddle path cannot be used to prove U(a+∆a, b) ≥ U(a, b), since the conditionUa(a, b) ≥ 0 at each (a, b) is not enough to ensure that these differential effects areeverywhere nonnegative along the downward sloping portion of the path, or thatany such negative effects are guaranteed to be outweighed by positive effects alongthe upward sloping portion. Intuitively, if a global robustness result like (6) is to beproven by an exact path integral, it must be along a constant-direction path, suchas the straight-line path in the figure, and not along any curved path in the space ofcommodity bundles.

There is, however, another way to prove an exact global robustness re-sult like (6). Consider the integral in the right diagram in Figure 1. Likethe middle diagram, it involves a curved path, but unlike that diagram, it


does not integrate the differential effect upon U(·, ·) of moving in the direc-tion of the path, but rather, integrates the differential effect of moving in thedirection of the global change. Thus if (a(t),b(t)) |t∈[0, 1] is the curvedpath in the middle and right diagrams, the middle path integral is given by∫ 10 [Ua(a(t),b(t))·a′(t)+ Ub(a(t),b(t))·b′(t)] ·dt, whereas the right line integral

approximation is given by∫ 10 [Ua(a(t),b(t))·∆a]·dt. Since it is not a true path

integral, this expression will not exactly equal U(a+∆a, b)− U(a, b). But withsufficient regularity, as the path (a(t), b(t))|t∈[0, 1] becomes arbitrarily close toa straight line path, the line integral approximation

∫ 10 [Ua(a(t),b(t))·∆a]·dt will

converge to U(a+∆a, b)−U(a, b). Since the right condition of (6) ensures thisintegral is nonnegative for any path, and since any limit of nonnegative values mustbe nonnegative, we obtain U(a+∆a, b)−U(a, b) ≥ 0. Line integral approxima-tions will prove essential for robustifying the classical model of risk preferencesand beliefs under subjective uncertainty, where the key variables will be the events,and constant-direction paths will not exist.

3.2 Robustifying expected utility analysis under objective uncertainty

As noted, this approach has already been used to establish the robustness of muchof standard expected utility analysis under objective uncertainty.10 Since the ex-pected utility form VEU (P) =

∑ni=1U(xi) ·pi exhibits constant sensitivity in the

probabilities, we choose probabilities rather than outcomes as the key variables,and accordingly represent each objective lottery in the form

P =. . . , px′ , px′′ , px′′′ , . . .︸︷︷︸

x∈X

=px | x ∈ X

(7)

that is, as an outcome-indexed list of probabilities (summing to unity). The changebetween any pair of lotteries P and P* can be represented in terms of the changesin these probabilities:

P*−P=. . . , p∗

x′−px′ , p∗x′′−px′′ , p∗

x′′′−px′′′ , . . .︸︷︷︸x∈X

=∆px |x ∈ X

(8)

and changes in expected utility can be represented in terms of these probabilitychanges and VEU (·)’s constant sensitivity rates . . . , U(x′), U(x′′), U(x′′′), . . . to these probability changes:

VEU (P*)− VEU (P) =∑x∈X

U(x) · (p∗x − px) =

∑x∈X

U(x) ·∆px (9)

Given the choice of probabilities as the key variables, a smooth non-expectedutility preference function V (·) is defined as one that is differentiable in the prob-abilities, i.e. one that satisfies

V (P*)− V (P) =∑x∈X

U(x;P)·(p∗x − px) + o

(||P*−P||

)(10)

10 See the references in Note 1. The following results are from Machina (1982).

12 M.J. Machina

where U(x; P) |x ∈ X ≡ ∂V (P)/∂px |x ∈ X are V (·)’s partial derivatives(local sensitivity rates) with respect to the variables px |x ∈ X at P, for somechoice of norm || · ||. Because of the close correspondence of (9) and (10), U(·; P)is termed the local utility function of V (·) at the lottery P.

By replacing standard conditions on the expected utility sensitivity ratesU(x) |x ∈ X with the corresponding conditions on the local sensitivity ratesU(x; P) |x ∈ X of a smooth non-expected utility V (·) at each lottery P, we canobtain the following global robustness results:

V (·) exhibits global weak first orderstochastic dominance preference

⇔ U(x;P) is nondecreasingin x at each P

V (·) exhibits global weak risk aversion(weak aversion to mean-preserving spreads)

⇔ U(x;P) is concave in xat each P

V *(·) is at least as risk averse as V (·)(see Machina (1982, Thm.4) for specifics)

⇔ U*(x;P) is at least as concavein x as U(x;P) at each P

As with the global robustness result (6), such results are proven by means ofline integrals along constant-direction paths, which in this setting consist of theprobability mixture path Pα = (. . . , α·p∗

x + (1−α)·px, . . . ) |α ∈ [0, 1] fromP to P*, and which generate the line integral formula

V (P*)−V (P)=∫ 1

0

dV (Pα)dα

·dα=∫ 1

0

[∑x∈X

U(x;Pα

)·(p∗x − px

)]·dα (11)

Researchers have applied and extended this approach to establish the global ro-bustness of several other key results of objective expected utility theory, includingfirst order conditions for optimization, comparative statics with respect to changesin risk, and many results from insurance theory.

The following section prepares for our application of this robustness approach tosubjective uncertainty, by (i) showing that the classical model exhibits constant sen-sitivity in the events attached to each outcome; (ii) defining a notion of “smoothnessin the events”; and (iii) showing that while it is impossible to construct “constant-direction paths” in the events, we can work with line-integral approximation paths,analogous to those in the right diagram of Figure 1.

4 Robustifying the classical model: Preliminary steps

Although we shall consider more general settings in Section 6.2, the formal analysisof this paper will be conducted within the following framework:

X = . . . , x, . . . arbitrary space of outcomes or conse-quences

S = [s, s ] ⊂ R1 set of states of nature, with uniformLebesgue measure λ(·)

E = . . . , E, . . . algebra of events (each a finite unionof intervals) in S


f(·) = [x1 on E1; . . . ;xn on En]finite-outcome act, for some E-meas-urable11 partition E1, . . . ,En of S

A = . . . , f(·), . . . set of all finite-outcome, E-measura-ble acts

W (·) and preference function and its correspon-ding preference relation on A

We defineW (·)’s outcome ranking by x* //∼ x if and only if W (x* onS)≥/>/=W (x on S).An event E is said to be null for preference function W (·) if W (·) isalways indifferent to the payoff assigned toE, so thatW (x* onE; f(·) elsewhere)= W (x on E; f(·) elsewhere) for all x*, x and f(·).

4.1 Event-theoretic representation of the classical model

Although a subjective act is typically viewed as a mapping f(·): S → X fromstates to outcomes, it can be also represented in the form [x1 on f−1(x1); . . . ;xn on f−1(xn)], that is, in terms of its distinct outcomes x1, . . . , xn and theirassociated events f−1(x1), . . . , f−1(xn). More generally, we can represent eachact in A as an event-valued mapping f−1(·) : X → E over the entire outcome setX , or by analogy with (7), as an outcome-indexed partition of S:

f(·) =. . . , Ex′ , Ex′′ , Ex′′′ , . . .︸︷︷︸

x∈X

=Ex

∣∣ x ∈ X (12)

where each event Ex = f−1(x) is E-measurable, and Ex = ∅ for all but a finitenumber of x ∈ X . Figure 2 illustrates this notation for an act f(·) (the solid line)over the state space S.

Figure 2. Event-theoretic representation of an act f(·) and the change f(·) → f*(·)

We can represent the change from any act f(·) to another act f*(·) in terms ofthe changes – the growth and/or shrinkage – in each outcome’s event Ex. For eachx ∈ X , we define the

11 A partition E1, . . . , En is E-measurable if Ei ∈ E for each i.

14 M.J. Machina

growth set of x : ∆E+x = E∗

x − Ex = f*−1(x)− f−1(x)(13)

shrinkage set of x : ∆E−x = Ex − E∗

x = f−1(x) − f*−1(x)

and represent the change f(·) → f*(·) in terms of this family (∆E+x , ∆E

−x )|

x ∈ X of growth and shrinkage sets, which we to refer to collectively as changesets. Since each state in the growth set of one outcome must also be in the shrinkageset of some other outcome, and vice versa, we have

⋃x∈X

∆E+x =

⋃x∈X

∆E−x =

s ∈ S

∣∣f*(s) = f(s)

(14)

which we term the total change set between f(·) and f*(·). Figure 2 also illustratesthe growth and shrinkage sets for the change from f(·) to the (dashed) act f*(·).

Since we retain Savage’s approach of imposing no structure on the outcomespace X , we cannot define the “distance” between a pair of acts f(·) and f*(·) interms of the “closeness” or “distance” between their outcomes. Instead, we base iton the size (Lebesgue measure) of the total change set (14), and define the distancefunction12

δ(f*, f

)≡λs ∈ S

∣∣f*(s) = f(s)

=λ⋃

x∈X∆E+

x

= λ⋃

x∈X∆E−

x

(15)

and we assume each preference functionW (·) is event-continuous in the sense that

limδ(f∗,f)→0

W(f*(·)

)= W

(f(·))

all f(·) ∈ A (16)

Like Savage’s own continuity axiom P6 (1954, p. 39), event-continuity impliesindifference between any two acts that differ only on a finite set of states, whichfor the classical form WSEU (·) serves to rule out any atoms (mass points) in itssubjective probability measure µ(·). However, neither Savage’s P6 nor the event-continuity condition (16) is strong enough to ensure that µ(·) will be absolutelycontinuous13 with respect to Lebesgue measure λ(·), which is required for it to pos-sess a density function ν(·) such that µ(E) =

∫Eν(s)·ds for allE.We shall ensure

absolute continuity of all measures and signed measures considered in this paper byassuming that every preference functionW (·)overA is absolutely event-continuousin the sense thatW (x1 on E1; . . . ;xn on En) is an absolutely continuous functionof the boundary points of each interval or finite interval unionEi, and similarly forthe first order term in (19) below.

12 Since δ(f*, f) = 0 for any acts f(·) and f*(·) that differ only on a set of Lebesgue measurezero, δ(·, ·) is not a metric. It is, however, a pseudometric, since it will satisfy the triangle inequalityδ(f**, f) ≤ δ(f**, f*) + δ(f*, f).

13 A measure or signed measure on S is absolutely continuous with respect to Lebesgue measure λ(·)if it assigns zero measure to every set E ⊆ S for which λ(E)=0, which rules out atoms as well asnonatomic “singular continuous” measures such as the Cantor measure (e.g. Billingsley, 1986, pp. 427–429; Romano and Siegel, 1986, pp. 27–28).Although Savage’s formulation implied only finitely-additivesubjective probability (1954,Sect. 3.4), we follow Arrow (1970,Ch.2) and others and assume countableadditivity for all measures and signed measures in this paper.


Given the above, we can represent the classical preference function as14

WSEU

(f(·))≡∫

SU(f(s)

)·dµ(s) ≡

∑x∈X

∫f−1(x)

U(x)·dµ(s)

(17)

≡∑x∈X

∫Ex

U(x)·ν(s)·ds ≡∑x∈X

∫Ex

φ(x, s)·ds

for what Myerson (1979) has termed the outcome-state evaluation function φ(x, s)≡ U(x)·ν(s). The function φ(x, s) – which gives the effect of receiving outcomex in state s – is seen to fully characterize WSEU (·)’s risk attitudes (when viewedas a function of x) as well as its beliefs (when viewed as a function of s). Both ofthese types of characterizations will be shown to be robust.

Since WSEU (·) is seen to evaluate each outcome’s event Ex by an additivefunction (i.e. signed measure) Φx(Ex) ≡

∫Exφ(x, s)·ds and then sum these evalu-

ations, it is said to be event-additive. Event-additivity is the subjective analogue oflinearity in the probabilities, and is seen to imply constant sensitivity in the events –that is, the property thatWSEU (·)’s response to any change f(·)→ f*(·) will onlydepend upon f(·) and f*(·) through their change sets (∆E+

x , ∆E−x ) |x ∈ X :

WSEU

(f*(·)

)−WSEU

(f(·))≡∑x∈X

[ ∫E∗

x

φ(x, s)·ds−∫

Ex

φ(x, s)·ds]

(18)

≡∑x∈X

[ ∫∆E+

x

φ(x, s)·ds−∫

∆E−x

φ(x, s)·ds]

where the second line follows by subtracting∫

E∗x∩Ex

φ(x,s)·ds from each integralin the first line. The Allais changes α1→α2 and α3→α4 in Table 1 both involvethe same family of change sets, as do the Ellsberg changes γ1→ γ2 and γ3→ γ4,which is why each example constitutes a test of event additivity/constant sensitivityin the events.

The state-dependent expected utility form WSDEU (·) is also seen to be event-additive, with evaluation function φ(x, s) ≡ U(x|s) ·ν(s) :

WSDEU

(f(·))≡∫

SU(f(s)|s

)·dµ(s) ≡

∑x∈X

∫f−1(x)

U(x|s)·dµ(s)

(17)′

≡∑x∈X

∫Ex

U(x|s)·ν(s)·ds ≡∑x∈X

∫Ex

φ(x, s)·ds

and accordingly, WSDEU (·) also exhibits constant sensitivity in the events:

WSDEU

(f*(·)

)−WSDEU

(f(·))≡∑x∈X

[ ∫E∗

x

φ(x, s)·ds−∫

Ex

φ(x, s)·ds]

(18)′

≡∑x∈X

[ ∫∆E+

x

φ(x, s)·ds−∫

∆E−x

φ(x, s)·ds]

14 Since we restrict attention to finite-outcome acts f(·), all sums of the form∑

x∈X in this and thefollowing sections will only involve a finite number of nonzero terms.

16 M.J. Machina

The following result shows that, under absolute event-continuity, the formsWSEU (·) andWSDEU (·) in fact characterize the properties of event-additivity andconstant-sensitivity in the events:

Theorem 1 (Characterization of event-additivity/constant sensitivity in theevents). The following conditions on an absolutely event-continuous preferencefunction W (·) over A are equivalent:

(a) W (·) takes the classical form WSEU (f(·)) ≡∫

S U(f(s))·dµ(s), or the state-dependent expected utility form WSDEU (f(·)) ≡

∫S U(f(s)|s)·dµ(s), for

some absolutely continuous subjective probability measure µ(·).

(b) W (·) is event-additive: there exists a family of absolutely continuoussigned measures Φx(·) | x∈X such that W (f(·)) ≡

∑x∈X Φx(f−1(x))

≡∑

x∈X Φx(Ex).

(c) W (·) exhibits constant sensitivity in the events: if the changes f1(·)→ f2(·)and f3(·)→ f4(·) involve the same change sets (∆E+

x , ∆E−x ) |x ∈ X, then

W (f2(·))−W (f1(·)) = W (f4(·))−W (f3(·)).

4.2 Event-smoothness and the local evaluation function

Since constant sensitivity in the events is equivalent to event-additivity, a naturaldefinition of “differentiability in the events” is local event-additivity, that is:

Definition. An absolutely event-continuous preference functionW (·)overA is saidto be event-differentiable at an act f(·) if there exists a family Φx(· ; f) |x ∈ Xof absolutely continuous signed measures such that

W(f*(·)

)−W

(f(·))=∑x∈X

Φx

(f*−1(x);f

)−∑x∈X

Φx

(f−1(x);f

)+ o(δ(f*,f)

)(19)

where o(·) denotes a function that is zero at zero and of higher order than itsargument.

Since the signed measures Φx(· ; f) |x ∈ X are absolutely continuous, theycan be represented by a family of signed density functions φ(x, · ; f) |x ∈ X, sowe can express (19) as

W(f*(·)

)−W(f

(·))≡∑x∈X

[ ∫E∗

x

φ(x, s;f)·ds−∫

Ex

φ(x, s;f)·ds]

+ o(δ(f*,f)

)(20)

≡∑x∈X

[ ∫∆E+

x

φ(x, s;f)·ds−∫∆E−

x

φ(x, s;f)·ds]

+ o(δ(f*,f)

)

Comparison with (18)/(18)′ yields that an event-differentiable W (·) will evaluatedifferential changes from an act f(·) in precisely the same manner as the expectedutility forms WSEU (·) and WSDEU (·), with respect to its local evaluation func-tion φ(x, s; f) at f(·). Thus, the differential effect uponW (·) of replacing outcome


x0 by x at a state s is given by φ(x, s; f)− φ(x0, s; f), just as it is given by theexpression φ(x, s)− φ(x0, s) for WSEU (·) or WSDEU (·). Since the local evalua-tion function will thus only enter into (20) through its statewise differences, we canselect arbitrary x ∈ X and additively normalize it, so that without loss of generalitywe shall assume φ(x, s; f) = 0 for all s ∈ S and all f(·) ∈ A.

To obtain our formal results we must impose some regularity on how much thelocal evaluation function φ(x, s; f) can vary in its arguments s and f(·). Althoughthe space of acts A is not compact with respect to the distance function δ(· , ·),our regularity conditions include the properties that a continuous φ(x, · ; ·) functionwould exhibit if its domain S×A was compact:

for each outcome x : φ(x, s; f) is uniformly continuous over S×A

for each outcome x : φ(x, s; f) is bounded above and below on S×A

for each pair x* xand nonnull event E

:∫

E[φ(x*, s;f)−φ(x,s;f)]·ds is both bounded

above and bounded above 0, uniformly in f(·)

(21)

We thus define a general event-differentiable preference function W (·) on A to beevent-smooth if it satisfies these properties, which can be stated more formally as:

for each x ∈ X and ε > 0 there exists δx,ε > 0 such that |s′ − s| < δx,ε

and δ(f ′, f) < δx,ε implies |φ(x, s′; f ′)− φ(x, s; f)| < ε

for each x ∈ X there exist φx and φx such that φx > φ(x, s; f) > φx

for all s ∈ S and all f(·) ∈ A

for each pair x* x and nonnull E ∈ E there exist Φx∗,x,E > Φx∗,x,E > 0such that Φx∗,x,E >

∫E

[φ(x*, s; f)−φ(x, s; f)]·ds > Φx∗,x,E for all f(·)∈A

(21)′

Under event-smoothness, the local evaluation function φ(x, s; f) can be ob-tained by differentiating W (·) in the appropriate event-theoretic manner. Definingthe ε-ballBs,ε = [s− ε, s+ ε] about any interior state s∈S, equation (20), event-smoothness and the normalization φ(x, s; f) ≡ 0 imply15

φ(x, s; f) = limε→0

W

(x on Bs,ε

f(·) on S−Bs,ε

)−W

(x on Bs,ε

f(·) on S−Bs,ε

)

2 ·ε (22)

The (pre-normalized) local evaluation function for the form WSEU (·)at any act f(·) is simply φ(x, s; f) ≡ U(x) ·ν(s), and for WSDEU (·) it isφ(x, s; f) ≡ U(x|s) ·ν(s).The “constant sensitivity in the events” property of theseforms is reflected in the fact that their local evaluation functions do not depend uponthe act f(·).When the other forms we have considered are event-smooth, their pre-normalized local evaluation functions are given by:

15 Not surprisingly, (22) is similar to the classic derivative of a set-functionlim

ε→0[Ω(E∪Bs,ε)−Ω(E −Bs,ε)]/λ(Bs,ε), for a function Ω(·) defined over sets E ⊆ Rn

(e.g., Hahn and Rosenthal, 1948, Ch.V; Jeffery, 1953, pp. 125–127).

18 M.J. Machina

Probabilistically Sophisticated Non-Expected Utility: Writing WPS(f(·)) ≡V (Pf,µ) where Pf,µ ≡ (. . . ;x, µ(f−1(x)); . . . ) is the lottery implied by f(·)under the subjective probability measure µ(·), we have

φ(x, s; f) ≡ U(x; Pf,µ) ·ν(s) (23)

where U(x; P) is the local utility function of V (·) (from (10)).

Choquet Expected Utility:16 Writing WChoquet(f(·))≡∑

x∈X U(x)·[C(Ex,f )−

C(E≺x,f )] for f(·)’s weak and strict cumulative payoff sets E

x,f = s ∈ S|f(s) x and E≺

x,f = s ∈ S|f(s) ≺ x, and defining C(·)’s generalizeddensity c(s;E) ≡ lim

ε→0[C(E ∪Bs,ε)−C(E−Bs,ε)]/(2 ·ε), we have17

φ(x, s; f) ≡ U(x)·c(s;E

x,f

)+∑yx

U(y)·[c(s;E

y,f

)−c(s;E≺

y,f

)](24)

Maxmin Expected Utility: Writing Wmaxmin(f(·)) ≡∑

x∈X U(x) ·µτ∗(f(·))(f−1(x)) where τ*(f(·)) ≡ argminτ∈T

∑x∈X U(x) ·µτ (f−1(x)),

Wmaxmin(f(·)) is seen to have a structure similar to that of a support functionfrom convex analysis.18 Thus if the measures µτ (·) |τ ∈ T have densitiesντ (·) |τ ∈ T , Wmaxmin(·)’s smoothness properties about any act f(·) willdepend upon the properties of τ*(·) at that act as follows:

If τ*(·) is constant over some neighborhood of f(·), Wmaxmin(·) takes theclassical form over that region, and we will have φ(x,s;f)≡U(x)·ντ∗(f(·))(s)

If τ*(·) is not constant about f(·) but at least continuous there, a standardenvelope theorem argument implies that we will continue to have φ(x, s; f) ≡U(x)·ντ∗(f(·))(s)

If τ*(·) is not continuous at f(·), Wmaxmin(·) is “kinked in the events” and notevent-differentiable at f(·), although it may have directional event-derivatives

Given a pair of constant acts f(·) = [x on S] and f*(·) = [x* on S],we definethe single-sweep path fω(·) |ω ∈ [s, s] from f(·) to f*(·) by19

fω(·) ≡[x* on [s, ω] ; x on (ω, s ]

]ω ∈ [ s, s ] (25)

This path is illustrated in Figure 3. As ω runs from s to s, the outcome x is replacedby x* over an expanding interval [s, ω] whose right edge sweeps across the statespace S at a uniform rate. Since δ(fω, fω′) ≡ |ω − ω′|, event-continuity impliesthat W (fω(·)) is continuous in ω.

16 For notational simplicity, this derivation will assume U(x*) = U(x) whenever x* = x.17 When working with (24), recall that since C(·) is nonadditive we will typically have

∫E c(s; E) ·

ds = C(E), in other words, the generalized densities c(s;Ex,f ) and c(s;E≺

x,f ) typically do not

“integrate back” to C(Ex,f ) or C(E≺

x,f ).18 A function on Rn is a support function if it is the pointwise minimum or the pointwise maximum

of some family of linear functions (e.g., Rockafellar (1970, Ch.13)).19 Note that the initial act on this path is not actually f(·) = [x on S], but rather fs(·) =

[x* on [s, s ]; x on (s, s ]], which by event-continuity will satisfy W (fs(·)) = W (f(·)).


Figure 3. Single-sweep path from the constant act f(·) = [x on S] to f*(·) = [x*on S]

Given an arbitrary ω ∈ [s, s ], event-differentiability implies

W(fω(·)

)−W

(fω(·)

)=∫ ω

ω

φ(x*, s; fω)·ds−∫ ω

ω

φ(x, s; fω)·ds+o(|ω−ω|

)(26)

for all ω > ω, with a corresponding formula for ω < ω, so that W (fω(·)) is dif-ferentiable in ω, with

dW(fω(·)

)dω

∣∣∣∣∣ω=ω

= φ(x*, ω; fω)− φ(x, ω; fω) (27)

By event-smoothness this derivative is seen to be continuous in ω, so that

W(f*(·)

)−W

(f(·))=∫ s

s

dW(fω(·)

)dω

·dω=∫ s

s

[φ(x*, ω;fω)−φ(x, ω;fω)

]·dω (28)

which illustrates howW (·)’s ranking of f(·) versus f*(·) can be exactly expressedin terms of its local evaluation function along the path fω(·) |ω ∈ [s, s ].

Single-sweep paths can also be constructed between pairs of nonconstant actsf(·) and f*(·), by defining fω(·) ≡ [f*(·) on [s, ω]; f(·) on (ω, s ]]. In this case,as ω runs from s to s the interval [s, ω] sweeps rightward, replacing the outcomef(ω) by f*(ω) at its right edge, and similar derivations yield that W (fω(·)) iscontinuous in ω, with dW (fω(·))/dω = φ(f*(ω), ω; fω)−φ(f(ω), ω; fω) at eachpoint ω ∈ [s, s] where both f*(·) and f(·) are continuous. Since f(·) and f*(·) caneach have only a finite number of discontinuities, we obtain the general comparisonformula

W(f*(·)

)−W

(f(·))

=∫ s

s

[φ(f*(ω), ω; fω

)− φ(f(ω), ω; fω

)]·dω (28)′

Although single-sweep paths suffice to establish the exact comparison formu-las (28) and (28)′, they cannot serve as a general engine for establishing globalrobustness results, for the simple reason that they are not “constant-direction paths”in the space of subjective acts. To see this, recall that an inherent property of anyconstant-direction path (such as the left-hand path in Figure 1) is that the variablechanges in going from the starting point to the midpoint of the path are identical to

20 M.J. Machina

the variable changes in going from the midpoint to the final point, which impliesthat any constant-sensitivity function will exhibit an equal response along each halfof the path.20

However for single-sweep paths, say the path fω(·)≡ [x* on [s, ω];x on (ω,s ]]of Figure 3, the variable changes in going from its starting point fs(·) to its mid-point f(s+s)/2(·) are the change sets ∆E+

x∗ = ∆E−x = [s, (s+ s)/2], whereas the

variable changes in going from f(s+s)/2(·) to its final point fs(·) are the distinct(in fact, disjoint) change sets ∆E+

x∗ = ∆E−x = ((s+ s)/2, s ]. Accordingly, the

response of any constant-sensitivity function WSEU (f(·)) ≡∫

S U(f(s))·dµ(s)over the first half of this path will be [U(x*)−U(x)]·µ([s, (s+ s)/2]), whereasits response over the second half of the path will be the generally distinct value[U(x*)−U(x)]·µ(((s+ s)/2, s ]).21

In fact, in our framework of purely subjective acts, where the key variables arethe events Ex |x ∈ X and their change sets (∆E+

x , ∆E−x ) |x ∈ X, constant-

direction paths do not exist, for the simple reason that no family of event changes(∆E+

x , ∆E−x ) |x ∈ X can ever be applied more than once in succession. In other

words, once an event Ex has expanded by ∆E+x to become Ex∪∆E+

x , it cannotexpand by∆E+

x again, and onceEx has shrunk by∆E−x to becomeEx−∆E−

x , itcannot shrink by∆E−

x again. Thus no path fα(·) |α ∈ [α, α ] inA – single sweepor otherwise – can have the property that its change sets from fα(·) to f(α+α)/2(·)are the same as its change sets from f(α+α)/2(·) to fα(·). Since we cannot proveglobal robustness results by constant-direction paths as in the left diagram of Figure1, we must use line integral approximations along paths that come arbitrarily closeto the constant-direction property, as in the right diagram of the figure.

The nonexistence of constant-direction paths between subjective acts is in dis-tinct contrast to the case of objective lotteries considered in Section 3.2, where theprobability mixture path Pα = (. . . , α ·p∗

x + (1−α) ·px, . . . ) |α ∈ [0, 1] fromP to P* does constitute a constant-direction path,22 and hence allows for the useof the line integral formula (11) in establishing global robustness results. In orderto obtain a path fα(·) |α∈ [α,α ] from f(·) to f*(·) that comes arbitrarily closeto the constant-direction property – as in the right diagram of Figure 1 – we mustwork with subjective acts fα(·) ∈ A that come arbitrarily close to being “objec-tive probability mixtures” of f*(·) and f(·). We present such “almost-objective”mixtures and paths in the following section.

20 More generally, if a constant-direction path is divided into k equal portions, the variable changesfrom the beginning to the end of each portion will be identical, so any constant-sensitivity function willexhibit an equal response along each portion.

21 The response of any constant-sensitivity form WSDEU (f(·))≡ ∫S U(f(s)|s)·dµ(s) on these

two halves will be the generally distinct values∫[s,(s+s)/2][U(x*|s) − U(x|s)] · dµ(s) and∫

((s+s)/2,s][U(x*|s)−U(x|s)] ·dµ(s).22 For this path, the changes from P0 to P1/2 are the same as from P1/2 to P1, namely the probability

changes . . . , (p∗x′− px′)/2 , (p∗

x′′−px′′)/2 , (p∗x′′′−px′′′)/2 , . . . . The constant-sensitivity pref-

erence function VEU (·) also exhibits the same response from P0 to P1/2 as from P1/2 to P1, namely[VEU (P*)−VEU (P)]/2.


4.3 Almost-objective events, acts and mixtures under subjective uncertainty

In the purely subjective framework of this paper, a given event E could be as-signed different subjective likelihoods by different classical preference functionsWSEU (f(·))≡

∫S U(f(s))·dµ(s) and W*

SEU (f(·))≡∫

S U*(f(s))·dµ*(s), andmay not be assigned any well-defined likelihood at all by a preference functionsuch asWChoquet(·), Wmaxmin(·), or a general event-smoothW (·). However, ev-ery Euclidean state space contains a special class of events – termed almost objectiveevents – that approximate, and in the limit attain, the property of unanimous, well-defined revealed likelihoods for all event-smooth W (·). Examples of such eventsdate back at least to Poincare (1912), and as shown in Machina (2004), they willarbitrarily closely approximate virtually all of the belief and betting properties ofobjectively uncertain events for every event-smooth W (·).

Given our subjective state space S = [s,s ] ⊂ R1, define λS = λ(S) = s− s.For arbitrary positive integer m, partition S into the m equal-length intervals[

s , s+ λS

m

), . . . ,

[s+ i·λS

m , s+(i+1)·λS

m

), . . . ,

[s+(m−1)·λS

m , s]

(29)

and for any interval (or finite union of intervals) ℘ ⊆ [0, 1], define the almost-objective event

(30)℘×mS =

⋃m−1

i=0

s+ (i+ω)·λS

m

∣∣ω ∈ ℘that is, as the union of the linear images of ℘ into each of the m intervals in (29).Thus, [0 , 12 ]×

mS would be the union of the left halves of these intervals, [ 13 ,

23 ]×

mS

would be the union of their middle thirds, etc. It is straightforward to show thatsuch events will satisfy the limiting measure property

limm→∞

µ(℘×

mS)

= λ(℘) (31)

for every continuous-density subjective probability measure µ(·) on S, where λ(·)is Lebesgue measure over [0,1]. In this sense, the events ℘×

mS can be said to have a

limiting likelihood of λ(℘) for each such subjective probability measure µ(·), andhence for every probabilistically sophisticated preference function WPS(·).

As shown in Machina (2004), this unanimous limiting likelihood property ex-tends to every event-smooth preference function W (·), whether or not it is prob-abilistically sophisticated, or even based on an underlying probability measure orcapacity at all. In other words, for any pair of disjoint finite interval unions ℘, ℘′

⊆ [0, 1] with λ(℘) > (=)λ(℘′), every event-smooth W (·) over A will satisfy

limm→∞

W

x∗ on ℘×

mS

x on ℘′×mS

f(·) elsewhere

> (=)

all x∗ xall f(·)∈A

limm→∞

W

x on ℘×

mS

x∗ on ℘′×mS

f(·) elsewhere

(32)

Besides unanimous revealed likelihoods, almost-objective events arbitrarilyclosely approximate another property of objective rather than subjective events– or more precisely, a property of the objective events in any Anscombe-Aumann(1963) type objective×subjective setting – namely the property that their revealed

22 M.J. Machina

likelihoods will be invariant to conditioning on any fixed subjective eventE.Specif-ically, for disjoint finite interval unions ℘, ℘′ ⊆ [0, 1] with λ(℘) > (=)λ(℘′) andany fixed E ∈ E , every event-smooth W (·) will satisfy

limm→∞

W

x∗ on (℘×

mS)∩E

x on (℘′×mS)∩E

f(·) elsewhere

> (=)


limm→∞

W

x on (℘×

mS)∩E

x∗ on (℘′×mS)∩E

f(·) elsewhere

(33)

We accordingly define the almost-objective subevents of a subjective event E by℘×

mE = (℘×

mS)∩E.

Let ℘1, . . . , ℘n be a partition of [0,1] where each ℘i is a finite interval union.For each set of outcomes x1, . . . , xn ∈ X , we can define the almost-objective act[

x1 on ℘1×mS ; . . . ; xn on ℘n×mS]∈ A (34)

and for each set of subjective acts f1(·), . . . , fn(·) ∈ A, we can define the almost-objective mixture[

f1(·) on ℘1×mS ; . . . ; fn(·) on ℘n×mS]∈ A (35)

which yields outcome x on the eventEx =(℘1×m f

−11 (x)

)∪ · · · ∪

(℘n×m f−1

n (x)).

Even though almost-objective acts and mixtures are elements of the subjectiveact space A, preferences over them correspond more closely to preferences overobjective lotteries and mixtures than to preferences over general subjective acts, intwo respects. First, although a general event-smooth W (·) will not be probabilis-tically sophisticated over general subjective acts f(·), as m→∞ it will be prob-abilistically sophisticated over almost-objective acts. That is, each event-smoothW (·) has an associated risk preference function VW (·) over objective lotteriesP = (x1, p1; . . . ;xn, pn), such that

limm→∞

W(x1 on ℘1×mS; . . . ;xn on ℘n×mS

)≡ VW

(x1, λ(℘1); . . . ;xn, λ(℘n)

)(36)

so that as m→∞, each event-smooth W (·) evaluates an almost-objective actsolely according to its outcomes x1, . . . , xn and the limiting likelihood valuesλ(℘1), . . . , λ(℘n) of their events. The second property is that as m→∞, allevent-smooth WSEU (·) as well as all event-smooth WSDEU (·)23 will be linear inalmost-objective likelihoods and in almost-objective mixture likelihoods:

limm→∞

WSEU

(x1 on ℘1×mS; . . . ;xn on ℘n×mS

)≡

n∑i=1

λ(℘i)·U(xi)(37)

limm→∞

WSDEU

(x1 on ℘1×mS; . . . ;xn on ℘n×mS

)≡

n∑i=1

λ(℘i)·∫

S

U(xi|s)·dµ(s)

and

23 Recall that while state-dependent expected utility is inherently linear in purely objective likelihoods,and also in objective mixtures of subjective acts, it is generally not linear in the subjective likelihoodsof a general subjective act.


limm→∞

WSEU

(f1(·) on ℘1×mS;... ;fn(·) on ℘n×mS

)≡

n∑i=1

λ(℘i)·∫

S

U(fi(s)

)·dµ(s)

(38)

limm→∞

WSDEU

(f1(·) on ℘1×mS;... ;fn(·) on ℘n×mS

)≡

n∑i=1

λ(℘i)·∫

S

U(fi(s)

∣∣s)·dµ(s)

To summarize, almost-objective events, acts and mixtures exhibit the followingproperties:24

• as m → ∞, all event-smooth W (·) assign unanimous revealed likelihoods toalmost-objective events, which are independent of conditioning on fixed sub-jective events

• as m → ∞, all event-smooth W (·) are probabilistically sophisticated overalmost-objective acts

• as m → ∞, all event-smooth WSEU (·) and WSDEU (·) are linear in almost-objective event likelihoods and in almost-objective mixtures of subjective acts

Recall that global robustness results can only be proven by line integrals alongconstant-direction paths (as in the left diagram of Figure 1), or by line integralapproximations along paths that converge to the constant-direction property (asin the right diagram). Although we have seen that constant-direction paths do notexist in the space of subjective acts, as m→∞ the almost-objective mixture pathbetween acts f(·) and f*(·) – that is, the path fm

α (·) |α ∈ [0, 1] defined by

fmα (·) ≡

[f*(·) on [0, α]×

mS; f(·) on (α, 1]×

mS]

(39)

will converge to the constant-direction property, and hence allow for global robust-ness proofs.

To see this, consider the almost-objective mixture path from the con-stant act f(·) = [x on S] to f*(·) = [x* on S], that is, the path definedby fm

α (·) ≡ [x* on [0, α]×mS ; x on (α, 1]×

mS ] and illustrated in Figure 4. Al-

though the change sets in going from the starting point of this path toits midpoint (namely ∆E+

x∗ = ∆E−x = [0, 1

2 ]×mS) are distinct (and even dis-

joint) from the change sets in going from its midpoint to its final point(namely ∆E+

x∗ = ∆E−x = ( 1

2 , 1]×mS), the above results imply that as m→∞

they will be viewed as being virtually equivalent to each other by everycontinuous-density probability measure µ(·) and every event-smooth W (·).25

Since (38) implies limm→∞WSEU (fmα (·)) ≡ α ·U(x*) + (1−α) ·U(x) and

limm→∞WSDEU (fmα (·)) ≡ α ·

∫S U(x*|s) ·dµ(s) + (1−α)·

∫S U(x|s) ·dµ(s),

as m→∞ all preference functions that exhibit constant sensitivity in the events(namely, all WSEU (·) and WSDEU (·)) are seen to respond at a constant rate alongthis path.

24 See Machina (2004) for proofs of these properties and other aspects of almost-objective uncertainty.25 More generally, as m → ∞ the change sets from the act fm

α (·) to fmα+∆α(·) will be viewed as

being equivalent to the change sets from fmα+∆α(·) to fm

α+2·∆α(·), and similarly for all equally-spacedacts along such a path.

24 M.J. Machina

Figure 4. Almost-objective mixture path from f(·) = [x on S] to f*(·) = [x*on S]

Since the above properties will also hold for the almost-objective mixture path(39) between any pair of nonconstant acts f(·), f*(·) ∈ A, almost-objective mix-ture paths can be said to arbitrarily closely approximate the property of being“constant-direction in the events.” The following result from Machina (2004) showsthat line integral approximations along such paths – that is, integrals that eval-uate (in the same manner as (18)/(18)′) the effect of the full global change sets(∆E+

x , ∆E−x ) |x ∈ X at each point along the path – will indeed converge to

W (·)’s exact global evaluation W (f*(·))−W (f(·)). Accordingly, such integralscan serve as the engine for establishing the global robustness of the classical ex-pected utility/subjective probability model:26

Line Integral Approximation Theorem (Machina, 2004). If W (·) is event-smooth, then for any acts f(·), f*(·) ∈ A and any ε > 0 there exists mε such thatfor each m ≥ mε, W (·)’s path derivative along the almost-objective mixture pathfm

α (·) |α∈ [0, 1] = [f*(·) on [0, α]×mS; f(·) on (α, 1]×

mS ] |α ∈ [0, 1] from

f(·) to f*(·) exists and satisfies∣∣∣∣∣dW(fm

α (·))

dα−∑x∈X

[∫∆E+

x

φ(x, s; fm

α

)·ds−

∫∆E−

x

φ(x, s; fm

α

)·ds]∣∣∣∣∣<ε (40)

at all but a finite set of values of α. This implies the line integral approximationformula

W(f*(·)

)−W

(f(·))

= limm→∞

∫ 1

0

∑x∈X

[ ∫∆E+

x

φ(x,s; fm

α

)·ds−

∫∆E−

x

φ(x, s; fm

α

)·ds]·dα (41)

= limm→∞

∫ 1

0

∑x∈X

[ ∫E∗

x

φ(x,s; fm

α

)·ds−

∫Ex

φ(x, s; fm

α

)·ds]·dα

26 In Machina (2004), the bracketed terms in (40)/(41) are equivalently expressed as[Φx(f*−1(x); fm

α ) − Φx(f−1(x); fmα )].


5 Global robustness of the classical analytics

In this section we apply the robustness approach of Section 3.1 and the Line In-tegral Approximation Theorem to show that the fundamental analytical results ofthe classical expected utility/subjective probability model are globally robust togeneral event-smooth departures from both the expected utility hypothesis (event-separability) and the hypothesis of probabilistic sophistication. Section 5.1 estab-lishes the property of outcome-monotonicity and robustifies the classical formulafor outcome derivatives. Section 5.2 robustifies the classical characterization ofglobal probabilistic sophistication. Section 5.3 robustifies the classical character-izations of comparative subjective likelihood and relative subjective likelihood toindividuals who are not necessarily either expected utility maximizers or proba-bilistically sophisticated. Section 5.4 robustifies the classical characterization ofcomparative risk aversion, again to individuals who are not necessarily expectedutility maximizers or probabilistically sophisticated.

5.1 Outcome-monotonicity and outcome derivatives

A preference function W (·) is said to be outcome-monotonic if the outcome rank-ing x* x implies W (x* on E; f(·) on ∼E) > W (x on E; f(·) on ∼E) for allf(·) and all nonnullE.27 For the classical formWSEU (f(·)) ≡

∫S U(f(s))·dµ(s)

this property follows automatically, since x* x implies U(x*) > U(x), so that∫EU(x*)·dµ(s) +

∫∼E

U(f(s))·dµ(s) >∫

EU(x)·dµ(s)+

∫∼E

U(f(s))·dµ(s)for all f(·) and all nonnull E. For the state-dependent form WSDEU (f(·)) ≡∫

S U(f(s)|s)·dµ(s), although the condition “x* x implies U(x*|s) > U(x|s)for all s” is sufficient to imply outcome-monotonicity, it is not necessary forthis property, since an absolutely event-continuous WSDEU (·) can be outcome-monotonic yet still satisfy U(x*|s) = U(x|s) at isolated states s, even isolatedstates s with positive subjective density ν(s). For a general event-smooth W (·),the third of the event-smoothness conditions (21)/(21)′ – namely that for eachx* x and nonnull E the local evaluation term

∫E

[φ(x*, s; f)−φ(x, s; f)]·dsexceeds some positive Φx∗,x,E for all f(·) – is likewise sufficient to ensure thatW (·) is outcome-monotonic, though again it is not necessary. To prove sufficiency,consider arbitrary x* x, f(·), and nonnull E, and let fm

α (·) |α ∈ [0, 1]∞m=1be the almost-objective mixture paths from the act [x on E; f(·) on ∼E] to[x* on E; f(·) on ∼E]. The Line Integral Approximation Theorem and the thirdcondition of (21)/(21)′ then imply

W(x* on E; f(·) on ∼E

)−W

(x on E; f(·) on ∼E

)=

(42)

limm→∞

∫ 1

0

∫E

[φ(x*, s; fm

α )− φ(x,s; fmα )]·ds·dα > Φx∗,x,E > 0

27 We adopt this strict preference/strict inequality definition of outcome monotonicity, rather thana weak preference/weak inequality version, in order to obtain a tighter connection between bettingpreferences and beliefs.

26 M.J. Machina

In the case of a real-valued outcome space X ⊆ R1, a preference functionW (·) is said to be outcome-differentiable at an act f(·) if it possesses a variationalderivative – that is, an absolutely continuous signed measure Ψ(· ;f) such that

W(f*(·)

)−W

(f(·))

=∫

S

[f*(s)−f(s)

]·dΨ(s; f) + o

(||f*(·)− f(·)||

)(43)

=∫

S

[f*(s)−f(s)

]·ψ(s; f)·ds+ o

(||f*(·)−f(·)||

)

where ||f*(·)−f(·)|| = sups∈S |f*(s)−f(s)|, and ψ(· ;f) is the density ofΨ(· ;f). When an absolutely event-continuous WSEU (·) or WSDEU (·) is out-come-differentiable, its variational derivative is linked to its evaluation functionφ(·, ·) by the following formulas, which hold at each continuity point s of f(·) :

for WSEU (·) : ψ(s; f)

= U ′(f(s))·ν(s) = φx

(f(s), s

)(44)

for WSDEU (·) : ψ(s; f)

= U ′(f(s)|s)·ν(s) = φx

(f(s), s

)where φx(x, s) ≡ ∂φ(x, s)/∂x. This generates the outcome-derivative formulas

∂WSEU

(x on Ef(·) on ∼E

)

∂x

∣∣∣∣∣∣x=x

≡∫

E

U ′(x)·ν(s)·ds

∂WSDEU


)

∂x

∣∣∣∣∣∣x=x

≡∫

E

U ′(x|s)·ν(s)·ds

≡∫

E

φx

(x, s)·ds (45)

An event-smooth, outcome-differentiable W (·) is said to be jointly outcome-event smooth at f(·) if the expressionW (f(s)+γ on [s−εa, s+εb] ; f(·) elsewhere)is twice continuously differentiable in (γ, εa, εb) about (0,0,0) at each continuitypoint s of f(·). The following result globally robustifies the classical outcome-derivative formulas (44) and (45), by showing that they extend to the local evaluationfunction φ(·, ·; f) of any jointly outcome-event smooth W (·).

Theorem 2 (Outcome derivatives). If an event-smooth W (·) is jointly outcome-event smooth at an act f(·) ∈ A, then its variational derivative and outcome deriva-tives at f(·) are linked to its local evaluation function by the formulas

ψ(s; f)

= φx

(f(s), s; f

) at each continuitypoint s of f(·) (44)′

∂W


)

∂x

∣∣∣∣∣∣x=x

=∫

E

φx

(x, s; f

)·ds all x∈X

all E ⊆ f−1(x) (45)′

where φx(x, s; f) ≡ ∂φ(x, s; f)/∂x.


5.2 Characterization of probabilistic sophistication

An absolutely event-continuous expected utility preference functionWSEU (·) or WSDEU (·) will take the probabilistically sophisticated formWPS(x1 on E1; . . . ;xn on En) ≡ V (x1, µ(E1); . . . ;xn, µ(En)) for some sub-jective probability measure µ(·) with density ν(·) if and only if it is state-independent – that is, if and only if its x-normalized evaluation function takes themultiplicatively separable form28

φ(x, s) ≡ U(x) ·ν(s) all x ∈ X , s ∈ S (46)

That (46) implies probabilistic sophistication is clear. To see that it is in turn impliedby it, observe that for each x x, x-normalization, probabilistic sophisticationand outcome-monotonicity29 imply∫

E′φ(x,s)·ds>

∫E′′φ(x,s)·ds⇔

[x on E′

x on ∼E′

][x on E′′

x on ∼E′′

]⇔µ(E′)>µ(E′′) (47)

Thus for each x x,∫

Eφ(x, s)·ds is an increasing transformation of µ(E),which

by additivity implies∫

Eφ(x, s)·ds = U(x)·µ(E) for some U(x), yielding (46). A

similar argument holds for each x ≺ x and each x ∼ x.As seen by (23), the classical characterization of global probabilistic sophisti-

cation by a multiplicatively separable evaluation function will extend to a generalevent-smooth preference function W (·) in the following manner:

• W (·)’s local evaluation functionφ(x, s; f) ≡ U(x; Pf,µ)·ν(s) can now dependupon the act f(·), though only through its local utility term U(x; Pf,µ), and notits state-density term ν(s)

• the local utility term U(x; Pf,µ) can only depend upon f(·) through that act’simplied outcome lottery Pf,µ ≡ (. . . ;x, µ(f−1(x)); . . . )

Formally, we have

Theorem 3 (Characterization of probabilistic sophistication). An event-smooth preference function W (·) takes the probabilistically sophisticated formWPS(x1 on E1; . . . ;xn on En) ≡ V (x1, µ(E1); . . . ;xn, µ(En)) for some sub-jective probability measure µ(·) with density ν(·) if and only if its x-normalizedlocal evaluation function takes the outcome-state separable form

φ(x, s; f

)≡ U(x; Pf,µ) · ν(s) all x∈X , s∈S, f(·)∈A (48)

for some function U(x; P), where Pf,µ = (x1, µ(E1); . . . ;xn, µ(En)). In such acase, U(x; P) will be the x-normalized local utility function of V (·) at P.30

28 If the evaluation function φ(x, s) is multiplicatively separable for some choice of normalizationoutcome x, it will be so for any choice of normalization outcome, and similarly for any local evaluationfunction of the form φ(x,s ;f) ≡ U(x;Pf,µ) · ν(s) as studied below.

29 See Grant (1995) for a treatment of probabilistic sophistication without the assumption of outcome-monotonicity.

30 For standard “integrability” reasons, (48) should not be taken to imply that an arbitrary func-tion U∗(x; P) can necessarily serve as the local utility function of some V ∗(·), or as part of

28 M.J. Machina

Note that when an event-smoothW (·) is probabilistically sophisticated, its sub-jective density ν(·) can be recovered from its local evaluation function at any f(·)by the formula ν(s) ≡ [φ(x*, s; f)−φ(x, s; f)]/

∫S [φ(x*, ω; f)−φ(x, ω; f)]·dω

for any pair of outcomes x* x.

5.3 Characterization of comparative and relative subjective likelihood

In the classic objective approach of von Neumann-Morgenstern, additive numericallikelihoods are specified as part of the objects of choice, and the individual isassumed to adopt these beliefs. In the subjective approaches of Savage and Machina-Schmeidler, uncertainty is represented by events or states of nature, but we imposeenough conditions on subjective betting preferences to imply the existence of anadditive numerical subjective likelihood for each event. In an Ellsberg urn, someevents (namely “red”) possess numerical likelihoods, but betting preferences areinconsistent with the existence of numerical likelihoods over other events (“black”or “yellow”).

Although Ellsberg urns illustrate the fact that an individual can possess numer-ical likelihood beliefs for some events but not others, they fall short of displayingthis in a completely subjective setting, since the numerical likelihoods that do existare exogenously specified (“30 of the 90 balls are red”) rather than inferred frombetting preferences. It is thus worth behaviorally characterizing the phenomenon of“partial probabilistic sophistication,” by obtaining a completely subjective charac-terization of classical (i.e. probabilistic) likelihood beliefs, likelihood comparisons,and likelihood ratios over some events or pairs of events, in a setting where additiverevealed likelihoods over other events (i.e. complete probabilistic sophistication)need not exist.

Given an outcome-monotonic preference functionW (·), we define its revealedcomparative likelihood relation A () B over pairs of disjoint events by thebetting property

W

x∗ on A

x on Bf(·) elsewhere

≥ (>)


W

x on A

x∗ on Bf(·) elsewhere

(49)

that is, if for any pair of non-indifferent outcomes,W (·) always prefers staking themore preferred outcome on A and the less preferred outcome on B, rather than theother way around.

Under probabilistic sophistication and event-continuity, the comparative likeli-hood relation will be complete and transitive, andA B will be equivalent to eachof the following betting properties over subjective partitions, or almost-objectivesubevents, of A and B :

the local evaluation function of some W ∗(·), probabilistically sophisticated or otherwise. By anal-ogy, a function G(z1, . . . , zn) on Rn is non-decreasing if and only its partial derivative functionsg1(z1, . . . , zn), . . . , gn(z1, . . . , zn) are nonnegative, although nonnegativity of an arbitrary set offunctions g∗

1(z1, . . . , zn), . . . , g∗n(z1, . . . , zn) does not necessarily imply they are the partials of

some G∗(z1, . . . , zn).


Existence of Dominating Partitions: For each n, there exist partitions A1, . . . ,An of A and B1, . . . , Bn of B such that Ai Bj for each pair Ai, Bj

Nonexistence of Reverse-Dominating Partitions: All partitions A1, . . . , An ofA and B1, . . . , Bn of B satisfy Ai′ Bj′ for at least one pair Ai′ , Bj′

Comparative Likelihood over Almost-Objective Subevents: For each finite intervalunion ℘⊆ [0,1], as m→∞ the almost-objective subevents ℘×

mA and ℘×

mB

satisfy ℘×mA ℘×mB

We can extend the comparative likelihood relation to nondisjoint events bydefining A ()B ⇔ (A−B) () (B−A), in which case it will exhibitthe monotonicity property A⊇B ⇒ AB and the disjoint additivity propertyA ()B ⇒ (A ∪ C) () (B ∪ C) whenever A ∩ C = B ∩ C = ∅.

Under probabilistic sophistication and event-continuity, we can also define therelative likelihood LA,B of a pair of disjoint events A and B by the followingequivalent betting properties, the last of which compares bets on subjective versusalmost-objective partitions of a common event:

Existence of Dominating Partitions: For each rational valueL < LA,B , there existpartitions A1, . . . , Ana

ofA and B1, . . . , Bnb ofB with na/nb = L, such

that Ai Bj for each pair Ai, Bj

Nonexistence of Reverse-Dominating Partitions: For each rational value L <LA,B , all partitions A1, . . . , Ana of A and B1, . . . , Bnb

of B withna/nb = L satisfy Ai′ Bj′ for some pair Ai′ , Bj′

Relative Likelihood of Almost-Objective Subevents: For each pair of finite inter-val unions ℘a, ℘b ⊆ [0, 1] with the reciprocal likelihood ratio λ(℘a)/λ(℘b) =1/LA,B , as m→∞ the almost-objective subevents ℘a×mA and ℘b×mB satisfy℘a×mA ∼ ℘b×mB

Comparison of Subjective versus Almost-Objective Partitions: For all dis-joint ℘a, ℘b ⊂ [0, 1] with the same likelihood ratio λ(℘a)/λ(℘b) = LA,B , asm→∞ the individual is indifferent between betting on the events in the sub-jective partition (℘a∪℘b)×mA, (℘a∪℘b)×mB versus the events in the almost-objective partition ℘a×m(A∪B), ℘b×m(A∪B) of the event (℘a∪℘b)×m(A∪B)

Of course under probabilistic sophistication, the comparative likelihood relationA B is equivalent to µ(A) ≥ µ(B), and relative likelihood LA,B is given byµ(A)/µ(B). Since the local evaluation function of a probabilistically sophisticatedpreference function takes the form φ(x, s; f) ≡ U(x; Pf,µ)·ν(s), the conditionsµ(A) ≥ µ(B) and µ(A)/µ(B) = LA,B are respectively equivalent to∫

A

[φ(x*, s; f)− φ(x, s; f)

]·ds ≥


∫B

[φ(x*, s; f)− φ(x, s; f)

]·ds (50)

∫A

[φ(x*, s;f)−φ(x,s;f)

]·ds/∫

B


]·ds =

allx∗xallf(·)∈A

LA,B (51)

Intuitively, (50) states that the individual is always at least as sensitive to replacingan outcome x by a preferred outcome x* over the eventA than to making the same

30 M.J. Machina

replacement over B, and (51) states that the ratio of these sensitivities takes thesame value LA,B for all such outcomes pairs.

However in the absence of probabilistic sophistication, the comparative likeli-hood relation need not be complete, subjective likelihoods or likelihood ratiosneed not exist for all events, and even when two events do satisfy the betting property(49), neither disjoint additivity nor the associated partition properties necessarilyfollow from this fact.31 Nevertheless the above characterizations are robust, in thesense that a general event-smooth W (·) – even though it may not be additive overgeneral subjective partitions – will exhibit the comparative likelihood condition(50) or the relative likelihood condition (51) for a pair of subjective events A andB if and only if it satisfies the above betting properties over all partitions andsubevents for which event-smooth preferences are inherently additive, namely allalmost-objective subevents32 and all small-event partitions.

The concepts of comparative likelihood (“A is at least as likely asB”) and rela-tive likelihood (“A isLA,B times as likely asB”) are both special cases of the moregeneral comparative condition “A is at least L times as likely asB,” which may bemeaningful even when A and B are both ambiguous and neither possesses a sub-jective likelihood on its own. The local evaluation function conditions (50) and (51)are also both special cases of a common condition, namely condition (52) below.The following result thus serves to jointly robustify the classical characterizationsof comparative subjective likelihood and relative subjective likelihood:

Theorem 4 (Characterization of comparative and relative subjective likeli-hood). For any event-smooth preference function W (·) over A, the followingconditions on a pair of disjoint nonnull events A,B ∈ E and value L ∈ (0,∞) areequivalent:

(a) Local likelihood ratios: W (·)’s local evaluation function satisfies

∫A


]·ds/∫

B


]·ds ≥

all x∗xall f(·)∈A

L (52)

(b) Existence of dominating small-event partitions: For each pair x* x,act f(·) ∈ A, rational value L* < L and ε > 0, there exist ε-partitions33

A1, . . . , Ana of A and B1, . . . , Bnb

of B, with na/nb = L*, such that

31 Thus, in an Ellsberg urn with 10 red balls, 10 black balls, and 22 yellow or purpleballs in unknown proportion, an ambiguity averse individual may exhibit the likelihood rank-ing (yellow ∪ purple) (red ∪ black), but also yellow≺ red, yellow≺ black, purple ≺ red andpurple ≺ black, which violates nonexistence of reverse-dominating partitions.

32 By way of analogy, whereas most individuals would assign the well-defined likelihood rankingA B to the events A = (yellow∪purple) versus B = (red∪black) of the previous note, mostwould probably not satisfy the associated betting properties for the subjective partitions A1, A2= yellow, purple and B1, B2 = red, black, but most probably would satisfy them for the ob-jective partitions AH , AT = A ∩ heads, A ∩ tails and BH , BT = B ∩ heads, B ∩ tailsgenerated by some fair (i.e. 50:50) coin.

33 A partition E1, . . . , En is said to be an ε-partition if λ(Ei) < ε for each i.


W

x∗ on Ai

x on Bj

f(·) elsewhere

> W

x on Ai

x∗ on Bj

f(·) elsewhere

for all Ai, Bj (53)

(c) Nonexistence of reverse-dominating small-event partitions: For each pairx* x, act f(·) ∈ A and rational value L* < L, there exists some ε > 0such that all ε-partitions A1, . . . , Ana of A and B1, . . . , Bnb

of B withna/nb = L* satisfy

W

x∗ on Ai′

x on Bj′

f(·) elsewhere

> W

x on Ai′

x∗ on Bj′

f(·) elsewhere

for some Ai′, Bj′ (54)

(d) Comparison of almost-objective subevents: For all finite interval unions℘a, ℘b ⊆ [0, 1] with λ(℘a)/λ(℘b) = (>)1/L, the almost-objective subevents℘a×mA and ℘b×mB satisfy34

limm→∞

W

x∗ on ℘a×mA

x on ℘b×mBf(·) elsewhere

≥ (>)


limm→∞

W

x on ℘a×mA

x∗ on ℘b×mBf(·) elsewhere

(55)

(e) Comparison of subjective versus almost-objective partitions: For all disjointfinite interval unions ℘a, ℘b ⊂ [0, 1] with λ(℘a)/λ(℘b) = (<) L

limm→∞

W

x∗ on (℘a∪℘b)×mA

x on (℘a∪℘b)×mBf(·) elsewhere

≥ (>)

all x∗ xallf(·)∈A

limm→∞

W

x∗ on ℘a×m (A∪B)

x on ℘b×m (A∪B)f(·) elsewhere

(56)

In such a case, we say that W (·) reveals A to be at least L times as likely as B,and reveals B to be no more than 1/L times as likely as A.

Theorem 4 can be used to generate the following additional characterizationsof subjective beliefs in the absence of probabilistic sophistication:

Conditional likelihood: Given nonnull events A ⊂ B where neither may have awell-defined likelihood, we say that the conditional likelihood of A givenB is atleast L ifW (·) revealsA to be at leastL/(1− L) times as likely asB −A.Thiswill be equivalent to the local evaluation function condition

∫A[φ(x*,s;f)−

φ(x,s;f)]·ds/∫

B[φ(x*,s;f)−φ(x,s;f)]·ds≥L for all x*x and all f(·)∈A,

equivalent to Theorem 4’s almost-objective subevent property (d) provided ℘a

and ℘b are disjoint (so the bets in (55) will be well-defined), and equivalent toTheorem 4’s partition properties (b) and (c) for partitions A1, . . . , Ana of Aand A1, . . . , Ana , B1, . . . , Bnb

of B with na/(na +nb) = L* < L.34 When L ≥ 1, setting ℘a = ℘b = [0, 1] in (55) yields the comparative likelihood betting property

(49).

32 M.J. Machina

Comparative numerical likelihood: Given an event E which is ambiguous andhence without a well-defined likelihood, we can still define it as havinga likelihood of at least L if its conditional likelihood given the event S isat least L. This will be equivalent to local evaluation function condition∫

E[φ(x*, s; f)−φ(x, s; f)]·ds/

∫S [φ(x*, s; f)−φ(x, s; f)]·ds ≥ L for all

x* x and all f(·) ∈ A, and to the corresponding versions of the subeventand partition properties of the previous paragraph.

Interpersonal comparative likelihood: Even if neither W (·) nor W*(·) assignsa well-defined likelihood to an event E, we can still say that W*(·) revealsE to be at least as likely as does W (·) if their local evaluation functionssatisfy

∫E

[φ*(x, s; f)−φ*(x, s; f)]·ds /∫

S [φ*(x, s; f)−φ*(x, s; f)]·ds ≥∫E

[φ(x, s; f)−φ(x, s; f)]·ds /∫

S [φ(x, s; f)−φ(x, s; f)]·ds for all x xand all f(·) ∈ A, which will be equivalent to the interpersonal versions ofthe above betting and partition properties.

5.4 Characterization of comparative risk aversion under subjective uncertainty35

We motivate our characterization of comparative risk aversion under subjec-tive uncertainty by recalling that notion in the objective case: Given threeoutcomes x′′ x′ x, an objective lottery P is said to differ from P by anx← x′ → x′′ probability spread if P is obtained from P by reducing the proba-bility assigned to x′, and increasing the probabilities assigned to both x and x′′.A preference function V *(·) is then said to be at least as risk averse as V (·) ifV (P) ≥ V (P) ⇒ V *(P) ≥ V *(P) whenever P differs from P by such a spread,or equivalently, if V *(P)> V *(P) ⇒ V (P)> V (P) whenever P differs from Pby such a spread. The notion of comparative risk aversion as comparative toleranceof probability spreads (both three-point and more general) underlies the expectedutility and non-expected utility characterizations of Arrow (1965), Pratt (1964),Machina (1982, Thm. 4) and others.

For a pair of classical preference functions WSEU (f(·)) ≡∫

S U(f(s))·dµ(s)andW∗SEU (·) ≡

∫S U*(f(s))·dµ*(s) or state-dependent functionsWSDEU (f(·))

≡∫

S U(f(s)|s)·dµ(s) and W∗SDEU (·) ≡∫

S U*(f(s)|s)·dµ*(s) in a setting ofobjective×subjective uncertainty, comparative risk aversion over objective lotteriesis characterized by the condition that U*(x) is at least as concave a function of xas U(x), or that

∫SU*(x|s) · dµ*(s) is at least as concave in x as

∫SU(x|s) · dµ(s).

Thus in each case, comparative risk aversion is determined by the comparativeconcavity (as a function of x) of the integrated evaluation function

∫S φ(x, s)·ds,

which reduces to U(x) for WSEU (·), and to∫

S U(x|s)·dµ(s) for WSDEU (·).Since the lotteries described in the previous paragraphs were all objective, it

follows that the individuals whose risk attitudes were being compared both agreed

35 Although we continue to assume a completely arbitrary outcome space X , the discussion andresults of this section will be restricted to pairs of preference functions W (·) and W *(·) that share acommon outcome ordering . Other characterizations of risk aversion and comparative risk aversionunder subjective uncertainty include those of Yaari (1969), Montesano (1994a,b, 1999a,b), Grant andQuiggin (2001) and Nau (2003).


on the likelihoods involved in any x← x′ → x′′ spread they were offered. On theother hand, while a subjective act f(·) = [x′′ on A;x on B; f0(·) elsewhere] canbe said to differ from f(·) = [x′ on A∪B; f0(·) elsewhere] by an x← x′ → x′′

subjective spread, there is no guarantee that the more risk averse individual willalways be more averse to such a spread, since the two individuals’ attitudes towardsuch spreads can also be affected by their respective likelihood beliefs over theevents A and B. Thus, in extending the standard characterization of comparativerisk aversion from objective to subjective uncertainty, it will be necessary to workwith subjective spreads whose evaluations only reflect interpersonal differences inrisk attitudes, and do not reflect any interpersonal differences in event likelihoods– including interpersonal differences in the existence of such event likelihoods.

Under event-smoothness, there are two types of subjective spreads whose eval-uations reflect features of risk attitudes but not beliefs. The first are those in whichA and B are taken from the class of events for which all event-smooth prefer-ence functions exhibit identical limiting likelihoods, namely the class of almost-objective events. Recall that asm→∞ all event-smooth preference functions willexhibit limiting likelihoods for the events ℘a×mS and ℘b×mS, corresponding to thevalues λ(℘a) and λ(℘b). Given outcomes x′′ x′ x and disjoint finite intervalunions ℘a, ℘b ⊂ [0, 1],we thus say that the act fm(·) = [x′′ on ℘a×mS;x on ℘b×mS;f0(·) elsewhere] differs from fm(·) = [x′ on (℘a∪℘b)×mS; f0(·) elsewhere] by anx← x′ → x′′ almost-objective spread. Since all event-smooth preference func-tions will agree on the limiting likelihoods involved in such spread, we wouldexpect that if a more risk averse function W*(·) had a limiting preference for sucha spread, so too would any less risk averse W (·).

The second category of spreads whose evaluations reflect features of risk atti-tudes but not beliefs are those in whichA andB are created out of small subjectiveevents E1, . . . , En which an individual considers to be “locally-exchangeable”for the outcomes in question. Given outcomes x′′ x′ x, act f(·), and realvalue L∈ (0,∞), we say thatW (·) is willing to accept small-event x← x′→ x′′

spreads about f(·) at any odds ratio greater than L, if for any na, nb withna/nb > L and any ε > 0, there exists an ε-partition E1, . . . , En of S suchthat if A is the union of any na of these events and B is the union of any nb others,then W (x′′ on A;x on B; f(·) elsewhere) > W (x′ on A ∪B; f(·) elsewhere).Although a different preference function W*(·) would generally require a differ-ent locally-exchangeable partition E∗1 , . . . ,E∗n∗ to exhibit such a property, wewould expect that, relative to their respective partitions, if a more risk averseW*(·)is willing to accept small-event x← x′ → x′′ spreads about f(·) at any odds ratiogreater than some value L, so too would any less risk averse W (·).

Under event-smoothness, these three notions of comparative risk aversion –comparative concavity of the integrated local evaluation functions

∫S φ(x,s;f)·ds

and∫

S φ*(x,s;f)·ds, comparative attitudes toward almost-objective spreads, andcomparative attitudes toward small-event spreads – will turn out to be equivalent,and will have two additional implications.

The first additional implication involves the individuals’ risk preference func-tions VW ∗(·) and VW (·), which from (36) represent their limiting preferences overalmost-objective acts [x1 on ℘1×mS; . . . ;xn on ℘n×mS] whether or not the individ-

34 M.J. Machina

uals are probabilistically sophisticated. IfW*(·) is at least as risk averse asW (·) inany of the above three equivalent senses, it will follow that VW ∗(·) will be at least asrisk averse as VW (·), in the standard sense of comparative risk aversion for lotterypreference functions as stated in the first paragraph of this section. But while thisproperty is implied by the above three notions of comparative risk aversion, is it notstrong to imply them. The reason is that the family of almost-objective acts is toosmall (and non-dense) a subset of the family A of subjective acts, so comparativerisk aversion over almost-objective acts is not strong enough to imply comparativerisk aversion over general subjective acts in the above three senses.

The second implication involves spreads in which A and B are generalsubjective events, but where W*(·) and W (·) happen to agree on their likelihoods– even though they may disagree on the likelihoods (or have no likelihoods)for other subjective events. From Theorem 4 and its follow-up definitions,we can say that W*(·) and W (·) each assign a likelihood pa to A and pb

to B if∫

A[φ*(x, s ;f)−φ*(x, s ;f)]·ds/

∫S [φ*(x, s ;f)−φ*(x, s ;f)]·ds and∫

A[φ(x, s ;f)−φ(x, s ;f)]·ds/

∫S [φ(x, s ;f)−φ(x, s ;f)]·ds both equal pa,

and if∫

B[φ*(x, s ;f)−φ*(x, s ;f)]·ds/

∫S [φ*(x, s ;f)−φ*(x, s ;f)]·ds and∫

B[φ(x, s ;f)− φ(x, s ;f)]·ds/

∫S [φ(x, s ;f)− φ(x, s ;f)]·ds both equal pb,

for all x x and all f(·) ∈ A. Given such events A and B and outcomesx′′ x′ x, if the more risk averseW*(·) prefers the subjective spread from f(·)= [x′ on A ∪B ; f0(·) elsewhere] to f(·) = [x′′ on A; x on B; f0(·) elsewhere],then so will the less risk averse W (·). We can generalize this notion by replacingthe identical likelihood requirement by the condition that W*(·) assigns A andB likelihoods of at most pa and at least pb, and W (·) assigns them likelihoodsof at least pa and at most pb. Once again, while this property will be implied bythe above three equivalent notions of comparative risk aversion, it is not strongenough to imply them.

We formalize the above discussion by:

Theorem 5 (Characterization and implications of comparative risk aversionunder subjective uncertainty). The following conditions on a pair of event-smoothpreference functionsW*(·) andW (·) overA with a common outcome ordering are equivalent:

(a) Comparative concavity of integrated local evaluation functions: At eachf(·) ∈ A, W*(·)’s and W (·)’s integrated local evaluation functions∫

S φ*(x, s ;f)·ds and∫

S φ(x, s ;f)·ds satisfy∫Sφ*(x, s ;f)·ds ≡

all x∈Xρf

(∫Sφ(x, s ;f)·ds

)(57)

for some increasing concave function ρf (·).36

(b) Comparative risk aversion over almost-objective spreads: For all x′′ x′ x,all f(·)∈A and all disjoint nondegenerate finite interval unions ℘a,℘b⊂ [0, 1]

36 As in Pratt (1964, Thm. 1), this is equivalent to the condition that [∫

S φ*(x′′, s; f) · ds− ∫

S φ*(x′, s; f) · ds]/[∫

S φ*(x′, s; f) · ds − ∫S φ*(x, s; f) · ds] ≤ [

∫S φ(x′′, s; f) · ds −∫

S φ(x′, s; f) · ds]/[∫

S φ(x′, s; f) · ds − ∫S φ(x, s; f) · ds] for all x′′ x′ x and all f(·).


limm→∞

W*

x′′ on ℘a×mS

x on ℘b×mSf(·) elsewhere

> lim

m→∞W*

(x′ on (℘a∪℘b)×mSf(·) elsewhere

)

⇒ (58)

limm→∞

W

x′′ on ℘a×mS

x on ℘b×mSf(·) elsewhere

> lim

m→∞W

(x′ on (℘a∪℘b)×mSf(·) elsewhere

)

(c) Comparative risk aversion over small-event spreads: For any x′′ x′ x,f(·)∈A andL∈(0,∞), ifW*(·) is willing to accept small-event x←x′→x′′

spreads about f(·) at any odds ratio greater than L, then so is W (·).

In such a case, we say W*(·) is at least as risk averse as W (·). These conditionsin turn imply:

(d) Comparative risk aversion over almost-objective acts: If VW ∗(·) and VW (·) areW*(·)’s and W (·)’s associated preference functions over lotteries as definedin (36), then VW ∗(P)> VW ∗(P)⇒ VW (P)> VW (P) whenever P differs fromP by an x← x′ → x′′ probability spread for some x′′ x′ x.

(e) Comparative risk aversion over likelihood-adjusted spreads: For each pair ofdisjoint events A,B, if W*(·) assigns A and B respective likelihoods of atmost pa and at least pb, and W (·) assigns them respective likelihoods of atleast pa and at most pb, for some pa, pb ∈ (0, 1), then

W*

x′′ on A


> W*

(x′ on A∪Bf(·) elsewhere

)

⇒ (59)

W

x′′ on A


> W

(x′ on A∪Bf(·) elsewhere

)

for all x′′ x′ x and all f(·) ∈ A.

In the case of a real-valued outcome space X = [a, b] ⊆ R1, we can definea preference function W (·) – whether or not it is probabilistically sophisticated– to be weakly risk averse over subjective acts if it is at least as risk averse asany risk neutral preference function over acts, that is, any preference function ofthe form WRN (f(·)) ≡

∫S f(s)·dµ(s) for some absolutely continuous subjective

probability measureµ(·). Since conditions (a) – (e) in Theorem 5 make no referenceto the respective individuals’ beliefs, it turns out that if W (·) is at least as riskaverse as some risk neutral preference function WRN (f(·)) ≡

∫S f(s)·dµ(s), it

will be at least as risk averse as every risk neutral WRN (f(·)) ≡∫

S f(s)·dµ(s).37

37 To see this, observe that for any such WRN (·), its integrated local evaluation function will takethe form

∫S φRN (x, s; f)·ds ≡ ∫

S x·ν(s)·ds ≡ x, independently of its subjective density ν(·).

36 M.J. Machina

This definition of risk aversion over subjective acts is thus “belief-independent” –both in the sense of not requiring W (·) to be probabilistically sophisticated, and inthe sense of being independent of particular risk neutralWRN (·) (i.e., its particularbeliefs µ(·)) used for comparison. By Theorem 5, any such subjectively risk averseW (·) will exhibit the following properties:

• W (·)’s integrated local evaluation function∫

S φ(x, s; f)·ds is concave in x ateach f(·)• As m→∞, W (·) is weakly averse to every almost-objective spread with

nonpositive mean

• If W (·) is willing to accept small-event x← x′ → x′′ spreads at any oddsratio greater than L, it must be that L ≥ (x′− x)/(x′′− x′)38

• W (·)’s associated risk preference function VW (·) is weakly risk averse

• If W (·) assigns likelihoods of pa to A and pb to B, then it is weakly averseto any subjective spread [x′ on A∪B; f0(·) elsewhere]→ [x′′ on A; x on B;f0(·) elsewhere] for which pa · (x′′− x′) ≤ pb · (x′− x)

6 Connections and extensions

6.1 Related work of Epstein

As mentioned in the introductory Note, Epstein (1999) has also proposed a notionof event-differentiability for a subjective preference function W (·), which can bedescribed as follows:39

As in Section 4.1, view each act f(·) ∈ A as a mapping e(·) : X → E fromoutcomes to events that satisfies (i) e(x) = f−1(x) = ∅ at all but a finite numberof x; and (ii) e(x)|x ∈ X is a partition of S, so that ∪x∈X e(x) = S, and x = x*implies e(x) ∩ e(x*) = ∅. Let A denote the larger family of all event-valued map-pings e(·) : X → E that satisfy property (i) but not necessarily property (ii). Epsteindefines event-differentiability for functions W (·) on A, so it will apply naturallyto any subjective preference function W (·) defined over A ⊂ A. Given a gen-eral event-valued mapping e(·) : X → E that satisfies (i) but not necessarily (ii),e+(·) is said to satisfy e+(·) ∩ e(·) = ∅ if e+(x) ∩ e(x) = ∅ for each x, in whichcase define e(·) + e+(·) as the mapping that takes x to e(x) ∪ e+(x). Similarly,e−(·) is said to satisfy e−(·) ⊂ e(·) if e−(x) ⊂ e(x) for each x, in which case de-fine e(·)− e−(·) as the mapping that takes x to e(x)− e−(x). A finite collectione1(·), . . . , en(·) is said to partition e(·) if e1(x), . . . , en(x) is a partition ofe(x) for each x ∈ X .Given the family e1,κ(·), . . . , enκ,κ(·)|κ ∈ K of all suchfinite partitions of e(·), we can define a partial orderK on this family by defininge1,κ′(·), . . . , enκ′,κ′(·) K e1,κ(·), . . . , enκ,κ(·) if the partition of S impliedby the former is a refinement of the partition implied by the latter.

38 As in the objective case, L ≥ (x′ − x)/(x′′ − x′) is only a necessary condition for a risk averseW (·) to be willing to accept such spreads, and it needn’t be sufficient.

39 In the following, Epstein’s notion has been adapted to maintain consistency with the notation usedin this paper.


A general function W (·) over A is then said to be eventwise-differentiableat e0(·) if there exists a family of bounded and convex-ranged event-additivefunctions dWx(·; e0)|x ∈ X such that, for each ε > 0, each e+(·) such thate+(·)∩e0(·) = ∅, and each e−(·) such that e−(·)⊂ e0(·), there exists partitionse+1 (·), . . . , e+n+(·) of e+(·) and e−

1 (·), . . . , e−n−(·) of e−(·) such that

nκ∑j=1

∣∣∣W(e0(·)+e+j (·)−e−j (·))−W

(e0(·)

)−[dW(e+j ;e0

)−dW

(e−j ;e0

)]∣∣∣ < ε (60)

for each e+1 (·), . . . , e+nκ(·) K e+1 (·), . . . , e+n+(·) and e−

1 (·), . . . , e−nκ

(·) Ke−

1 (·), . . . , e−n−(·). A preference function W (·) over subjective acts is then said

to be eventwise-differentiable if it satisfies the above definition over its domainA ⊂ A, in which case its derivative will be unique and satisfy the chain rule.Epstein applies this notion of eventwise-differentiability in his characterization ofthe property of uncertainty aversion, and it has been also been adapted to analyzethe core of nonatomic transferable-utility games by Epstein and Marinacci (2001).

For each j, the expression inside absolute values (60) is analogousto the error-term expression W (f(·))−W (f0(·))− [

∑x∈X Φx(∆E+

x ; f0)−∑x∈X Φx(∆E−

x ; f0)] implied by the event-differentiability formula (19). In thissense, the two notions of event-differentiability have the same structure for theirfirst-order approximation. Epstein (1999, Appendix C) also compares his definitionof event-differentiability with a definition similar to (19), proves a version of theFundamental Theorem of Calculus, and shows that when a preference function isdifferentiable in both senses, the derivatives coincide. The primary difference be-tween the two notions of differentiability lies in their treatment of convergence. Inparticular, by replacing the error-term expression δ(f*(·), f(·)) of (19) with thesum of nκ such terms in (60), Epstein’s definition eliminates the need for any un-derlying reference measure λ(·) on the state space S, as well as the need for anydistance function δ(·, ·) between acts.

6.2 Extension to more general state spaces and preferences

Given this paper’s theme of robustness, it is worth noting how its assumptions ofa uniform reference measure λ(·), a univariate state space S = [s, s], and event-smoothness can each be relaxed.

Although many state spaces possess a natural uniform reference measure, oth-ers do not: For example, if the state of nature is the exchange rate, then a uniformreference measure on the Dollar/Euro rate is not equivalent to a uniform measureon the Euro/Dollar rate. However, it is straightforward to show that if two alterna-tive choices of reference measure λ(·) and λ*(·) are mutually absolutely continu-ous with respect to each other, with Radon-Nikodym derivatives that are boundedaway from both 0 and ∞, then event-differentiability with respect to one refer-ence measure will be equivalent to event-differentiability with respect to another,and more importantly, W (·)’s local evaluations of the growth and shrinkage sets(∆E+

x , ∆E−x )|x ∈ X between any pair of acts f(·) and f*(·) will be invariant

to the choice of reference measure.

38 M.J. Machina

As with the almost-objective uncertainty analysis of Machina (2004), the anal-ysis of this paper can also be extended to more general state spaces – in particular,to multivariate Euclidean states spaces as well as smooth univariate or multivariatemanifolds. The key requirement is that the state space admit of almost-objectiveevents, which will be the case if it can be “tiled” by arbitrarily small but suitablemeasure spaces. Compared with the approach of Savage (1954), the analysis of thispaper is seen to require more structure on the state space S, and hence on the choicespace A of subjective acts, but less structure (neither expected utility nor proba-bilistic sophistication) on beliefs or preferences over this choice space. Of course,from a scientific and observational point of view, verifying structural assumptionson a state space or a choice space is – if anything – much easier than verifying suchassumptions on an agent’s (or a collection of agents’) beliefs or preferences oversuch spaces, under either certainty or uncertainty.

Finally, the assumption of event-smoothness can also be somewhat relaxed.Recall that in standard analysis, the Fundamental Theorem of Calculus continuesto hold for functions that are not everywhere differentiable, so long as they areabsolutely continuous: we simply “integrate over the kinks.” In a similar manner, theanalysis and results of this paper can be extended to subjective preference functionsW (·) that exhibit kinks in the events, so long as these kinks are sufficiently isolated.

The results of this paper have shown that the analytics of the classical expectedutility/subjective probability model of risk preferences and beliefs – which contin-ues to dominate research in choice under uncertainty – are in fact quite robust tosmooth departures from its basic underlying assumptions of the Sure-Thing Prin-ciple and Probabilistic Sophistication. Or as Sir William Gilbert might have said:It is very very general, for a modern major model.


Appendix – Proofs of Theorems40

Proof of Theorem 1. (a)⇒ (b): Given U(·|·) and µ(·), select arbitrary x ∈ Xand for each x ∈ X , define the signed measure Φx(·) by

Φx(E) ≡∫

E

[U(x|s)− U(x|s) +

∫SU(x|ω) ·dµ(ω)

]·dµ(s) (A.1)

For any f(·) ∈ A, we can thus write W (f(·)) ≡∫

SU(f(s)|s)·dµ(s) as

W(f(·))≡∫

S

[U(f(s)|s

)− U(x|s) +

∫S

U(x|ω) ·dµ(ω)]·dµ(s) ≡

(A.2)∑x∈X

∫f−1(x)

[U(x|s)− U(x|s) +

∫S

U(x|ω)·dµ(ω)]·dµ(s) ≡

∑x∈X

Φx

(f−1(x)

)

Absolute continuity of each Φx(·) follows since Φx([s, s]) ≡ W (x on [s, s] ;x on (s, s ]) − Φx((s, s ]) ≡ W (x on [s, s] ; x on (s, s ]) − [

∫S U(x|ω) ·dµ(ω)] ·

µ((s, s ]) is the difference of two absolutely continuous functions of the eventboundary s.

(b) ⇒ (a): Absolute continuity of the signed measures Φx(·) |x ∈ X impliesthey can be represented by a family of signed densities φ(x, ·) |x ∈ X on S.Define µ(·) = λ(·)/λ(S) (where λ(·) is uniform Lebesgue measure), and defineU(x|s) ≡

x,sφ(x, s) ·λ(S). For any f(·) ∈ A, we then have

W(f(·))

=∑x∈X

Φx

(f−1(x)

)=∑x∈X

∫f−1(x)

φ(x, s) ·ds(A.3)

=∫

S

φ(f(s), s

)·ds =

∫S

U(f(s)|s

)·dµ(s)

(b)⇒ (c): This follows since W (f*(·)) − W (f(·)) =∑

x∈X [Φx(E∗x ) −Φx(Ex)] =

∑x∈X [Φx(∆E+

x )− Φx(∆E−x )] for all f(·), f*(·) ∈ A.

(c)⇒ (b): Select arbitrary x ∈ X and for each x ∈ X andE ∈ E , define Φx(E) =W (x on E; x on S −E)− (1−(λ(E)/λ(S)))·W (x on S). To see that each Φx(·)is additive and hence a signed measure, pick arbitrary disjoint E,E′ and observethat constant sensitivity in the events implies

Φx(E∪E′) = W(x on E∪E′; x on S−(E∪E′)

)−(1−λ(E∪E′)

λ(S)

)·W(x on S

)=[W(x on E∪E′; x on S−(E∪E′)

)−W

(x on E′; x on S−E′)+ (A.4)

λ(E)λ(S)

·W(x on S

)]+[W(x on E′; x on S−E′)−(1−λ(E′)

λ(S)

)·W(x on S

)]

=[W(x on E; x on S−E

)−W

(x on S

)+λ(E)λ(S)

·W(x on S

)]+ Φx(E′)

= Φx(E) + Φx(E′)40 In the following proofs, it will often be notationally more convenient to work with a preference

function’s local evaluation measures Φx(E; f) ≡ ∫E φ(x, s; f)·ds as defined in (19), rather than its

local evaluation function φ(x, s; f).

40 M.J. Machina

Absolute continuity of each Φx(·) follows since Φx([s, s]) ≡ W (x on [s, s] ;x on (s, s ]) − (1 − ((s−s)/λ(S))) ·W (x on S) is the difference of two abso-lutely continuous functions of the event boundary s. Given arbitrary f(·) =[x1 on E1; . . . ;xn on En], constant sensitivity in the events thus implies

W(f(·))

= W(f(·))−W

(x on E1; f(·) elsewhere

)+ W

(x on E1; f(·) elsewhere

)−W

(x on E1∪E2; f(·) elsewhere

)+ W

(x onE1∪E2; f(·) elsewhere

)−W

(x onE1∪E2∪E3; f(·) elsewhere

)... (A.5)

+ W(x on E1∪ . . .∪En−1; f(·) elsewhere

)−W

(x on S

)+W

(x on S

)=

n∑i=1

[W(xi on Ei; x on S−Ei

)−W

(x on S

)]+

n∑i=1

λ(Ei)λ(S)

·W(x on S

)

=∑n

i=1Φxi(Ei)

Proof of Theorem 2. Let s be an arbitrary continuity point of f(·), let εa ≥ 0and εb ≥ 0 be small enough so that [s − εa, s + εb] ⊆ f−1(f(s)), and definefγ,εa,εb

(·) ≡ [f(s)+γ on [s − εa, s + εb] ; f(·) elsewhere ]. By (43) we havethat W (fγ,εa,εb

(·)) − W (f(·)) = γ · Ψ([s − εa, s + εb] ; f) + o(|γ|). Byjoint outcome-event smoothness, this expression is twice continuously differ-entiable in (γ, εa, εb) about (0,0,0), which implies that ψ(s; f) is continuousin s about s, and also that ∂2W (fγ,εa,εb

(·))/∂γ∂εb|γ=εa=εb=0 = ψ(s; f).By (19), we have W (fγ,εa,εb

(·))−W (f(·)) = Φf(s)+γ([s− εa, s+ εb]; f) −Φf(s)([s− εa, s+ εb]; f) + o(|εa + εb|), so that ∂W (fγ,εa,εb

(·))/∂εb|εa=εb=0 =φ(f(s)+γ, s ; f) − φ(f(s), s ; f). By joint outcome-event smoothness, we thushave ψ(s; f) = ∂2W (fγ,εa,εb

(·))/∂γ∂εb|γ=εa=εb=0 = φx(f(s), s; f), which is(44)′. Equations (43) and (44)′ then yield (45)′.

Proof of Theorem 3. (48) ⇒ probabilistic sophistication: We first show thateach finite-outcome lottery P can be associated with a unique “basic reference act”fP(·) ∈ A, and thatW (·) is probabilistically sophisticated over the family of suchacts: Well-order the elements of each indifference class of the outcome preferencerelation , and construct the strict order + over outcomes defined by x* + x ifand only if (i) x* x or (ii) x*∼ x and x* is ordered above xwithin their commonindifference class. Express each lottery in the form P = (x1, p1; . . . ; xn, pn) withx1≺+ x2≺+ . . .≺+ xn and p1, . . . , pn > 0, and define its associated basic refer-ence act fP(·) = [x1 on [s, s1]; x2 on (s1, s2]; . . . ; xn on (sn−1, s] ], where s1 isthe smallest state in S such that µ([s, s1]) = p1, s2 is the smallest state such thatµ((s1, s2]) = p2, etc. For each P, define V (P) = W (fP(·)). Since this impliesW (fP(·)) = V (x1, p1; . . . ; xn, pn) = V (x1, µ([s, s1]); . . . ; xn, µ((sn−1, s])) =V (x1, µ(f−1

P (x1)); . . . ; xn, µ(f−1P (xn))), W (·) is probabilistically sophisticated

over basic reference acts, with lottery preference function V (·) and subjective prob-ability measure µ(·).We now show that every act f(·) ∈ A with an implied outcome lottery P will beindifferent to fP(·), so that W (·) is probabilistically sophisticated over A: Since


f(·) implies the lottery P, we have µ(f−1(x)) = µ(f−1P (x)) for each x ∈ X .

Defining fmα (·)|α ∈ [0, 1]∞m=1 as the almost-objective mixture paths from fP(·)

to f(·), the Line Integral Approximation Theorem and (48) imply

W(f(·))−W

(fP(·)

)= lim

m→∞

∫ 1

0

∑x∈X

[Φx

(f−1(x);fm

α

)− Φx

(f−1P (x);fm

α

)]·dα

(A.6)

= limm→∞

∫ 1

0

∑x∈X

[U(x;Pfm

α ,µ

)·µ(f−1(x)

)−U(x;Pfm

α ,µ

)·µ(f−1P (x)

)]·dα = 0

Probabilistic sophistication ⇒ (48): Event-smoothness of W (·), probabilisticsophistication, and absolute continuity of µ(·) jointly ensure that V (·) is differen-tiable in the probabilities. For arbitrary x ∈ X , interior state s ∈ S and act f(·) =[x1 on E1; . . . ;xn on En] ∈ A, formulas (22) and (10) imply41

φ(x, s; f) = limε→0

W

(x on Bs,ε

f(·) on S−Bs,ε

)−W

(x on Bs,ε

f(·) on S−Bs,ε

)

2 ·ε =(A.7)

limε→0

[V(x, µ(Bs,ε);x1, µ(E1−Bs,ε); . . . ;xn, µ(En−Bs,ε)

)− V

(x, µ(Bs,ε);x1, µ(E1−Bs,ε); . . . ;xn, µ(En−Bs,ε)

) ]

µ(Bs,ε)· µ(Bs,ε)

2 ·ε

= U(x;Pf,µ) · ν(s)

where U(x; P) is the x-normalized local utility function of V (·) at P. Whenµ(Bs,ε) = 0 for all small Bs,ε about s, so that the left fraction in the second lineof (A.7) is undefined, the numerator in the first line will be zero, as will be ν(s),so we again obtain φ(x, s; f) = 0 = U(x; Pf,µ)· ν(s).

Proof of Theorem 4. (a)⇒ (b): Given arbitrary x* x, f(·) ∈ A, rationalL* <L and ε > 0, defineγ = [Φx∗(A; f)−Φx(A; f)]−L*·[Φx∗(B; f)−Φx(B; f)] > 0.By (19) there exists δ* > 0 such that δ(f(·), f(·)) < δ* implies∣∣∣W (f(·)

)−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣(A.8)

<γ/2

λ(A) + L*·λ(B) · δ(f(·), f(·)

)Select integers na, nb such that na/nb = L* and λ(A)/na + λ(B)/nb <minε, δ*. By Stromquist and Woodall (1985, Thm.1) and (21)′, there exist E-measurable partitions A1, . . . , Ana ofA and B1, . . . , Bnb

ofB with λ(Ai) =λ(A)/na < ε and Φx∗(Ai; f)− Φx(Ai; f) = [Φx∗(A; f)−Φx(A; f)]/na for eachi, and λ(Bj) = λ(B)/nb < ε and Φx∗(Bj ; f) − Φx(Bj ; f) = [Φx∗(B; f) −Φx(B; f)]/nb for each j. For each i, j we thus have

Φx∗(Ai; f)− Φx(Ai; f)− Φx∗(Bj ; f) + Φx(Bj ; f) = (A.9)41 For states on the boundary of S, replace Bs,ε = [s − ε, s + ε] by the appropriate half-ball, and

replace 2 ·ε by ε.

42 M.J. Machina

[Φx∗(A; f)− Φx(A; f)

]/na − L*·

[Φx∗(B; f)− Φx(B; f)

]/na = γ/na

For each i, j, define the acts fLi,j(·) = [x* on Ai; x on Bj ; f(·) elsewhere] and

fRi,j(·) = [x on Ai; x* on Bj ; f(·) elsewhere] from (53), so that we have both

δ(fL

i,j , f)≤ λ(Ai) + λ(Bj) =

λ(A)na

+λ(B)nb

=λ(A) + L*·λ(B)

na< δ*

(A.10)

δ(fR

i,j , f)≤ λ(Ai) + λ(Bj) =

λ(A)na

+λ(B)nb

=λ(A) + L*·λ(B)

na< δ*

For each i, j, (A.8) and (A.9) thus imply

W(fL

i,j(·))−W

(fR

i,j(·))> (A.11)

∑x∈X

Φx

(fL−1

i,j (x);f)−∑x∈X

Φx

(fR−1

i,j (x);f)−

12 ·γ ·

(δ(fL

i,j ,f)+δ(fRi,j ,f)

)λ(A) + L*·λ(B)

≥ Φx∗(Ai; f)− Φx(Ai; f)− Φx∗(Bj ; f) + Φx(Bj ; f)− γ/na = 0

(b)⇒ (a): Say (a) failed, so that γ = Φx∗(A; f)−Φx(A; f)−L*· [Φx∗(B; f)−Φx(B; f)] < 0 for some x* x, f(·) ∈ A and rationalL* ∈ (0,L). By (19) thereexists some ε > 0 such that δ(f , f) < 2 · ε implies∣∣∣W (f(·)

)−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣(A.12)

<|γ|/2

λ(A) + L*·λ(B) · δ(f(·), f(·)

)Given arbitrary ε-partitions A1, . . . , Ana of A and B1, . . . , Bnb

of B withna/nb = L*, additivity of local evaluation measures and the definition of γ imply∑na

i=1

[Φx∗(Ai; f)−Φx(Ai; f)

]− L*·

∑nb

j=1

[Φx∗(Bj ; f)−Φx(Bj ; f)

](A.13)

=γ

λ(A)+L*·λ(B)·[∑na

i=1λ(Ai)+L*·

∑nb

j=1λ(Bj)

]

Substituting na/nb for the first and third occurrence of L* and rearranging yields∑na

i=1

[Φx∗(Ai; f)− Φx(Ai; f)− γ

λ(A) + L*·λ(B)·λ(Ai)

]=

(A.14)na

nb·∑nb

j=1

[Φx∗(Bj ; f)− Φx(Bj ; f) +

γ

λ(A) + L*·λ(B)·λ(Bj)

]

This implies there exists some Ai′ and Bj′ such that

Φx∗(Ai′ ; f) − Φx(Ai′ ; f) − γ

λ(A) + L*·λ(B)· λ(Ai′)

(A.15)≤ Φx∗(Bj′ ; f) − Φx(Bj′ ; f) +

γ

λ(A) + L*·λ(B)· λ(Bj′)


Define fL(·) = [x* on Ai′ ;x on Bj′ ; f(·) elsewhere] and fR(·) = [x on Ai′ ;x* on Bj′ ; f(·) elsewhere], so δ(fL, f) ≤ λ(Ai′)+λ(Bj′) < 2 ·ε and δ(fR, f) ≤λ(Ai′) + λ(Bj′) < 2 · ε. By (A.12) and (A.15),

W(fL(·)

)−W

(fR(·)

)< (A.16)∑

x∈XΦx

(f−1

L (x); f)−∑x∈X

Φx

(f−1

R (x); f)

+|γ|2· δ(fL, f) + δ(fR, f)λ(A) + L*·λ(B)

≤

Φx∗(Ai′ ;f)+Φx(Bj′ ;f)−Φx(Ai′ ;f)−Φx∗(Bj′ ;f)+ |γ|· λ(Ai′)+λ(Bj′)λ(A) + L*·λ(B)

≤ 0

But since A1, . . . , Ana and B1, . . . , Bnb were arbitrary ε-partitions with

na/nb = L*, this contradicts (b).

(a) ⇒ (c): Given arbitrary x* x, f(·) ∈ A and rational L* < L, (52) impliesγ = Φx∗(A; f)− Φx(A; f)− L*· [Φx∗(B; f)− Φx(B; f)] > 0.By (19) there ex-ists some ε > 0 such that δ(f , f) < 2 · ε implies∣∣∣W (f(·)

)−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣(A.17)

<γ/2

λ(A) + L*·λ(B)· δ(f(·), f(·)

)Given arbitrary ε-partitions A1, . . . , Ana of A and B1, . . . , Bnb

of B withna/nb = L*, we have∑na

i=1

[Φx∗(Ai; f)− Φx(Ai; f)

]− L*·

∑nb

j=1

[Φx∗(Bj ; f)− Φx(Bj ; f)

](A.18)

=γ

λ(A) + L*·λ(B)·[∑na

i=1λ(Ai) + L* ·

∑nb

j=1λ(Bj)

]

Substituting na/nb for the first and third occurrence of L* and rearranging yields∑na

i=1

[Φx∗(Ai; f)− Φx(Ai; f)− γ

λ(A) + L*·λ(B)·λ(Ai)

]=

(A.19)na

nb·∑nb

j=1

[Φx∗(Bj ; f)− Φx(Bj ; f) +

γ

λ(A) + L*·λ(B)·λ(Bj)

]

This implies there exists some Ai′ and Bj′ such that

Φx∗(Ai′ ; f)− Φx(Ai′ ; f)− γ

λ(A) + L*·λ(B)· λ(Ai′)

(A.20)

≥ Φx∗(Bj′ ; f)− Φx(Bj′ ; f) +γ

λ(A) + L*·λ(B)· λ(Bj′)

Define fL(·) = [x* on Ai′ ; x on Bj′ ; f(·) elsewhere] and fR(·) = [x on Ai′ ;x* on Bj′ ; f(·) elsewhere], so δ(fL, f) ≤ λ(Ai′)+λ(Bj′) < 2 ·ε and δ(fR, f) ≤λ(Ai′) + λ(Bj′) < 2 · ε. By (A.17) and (A.20), we have

W(fL(·)

)−W (fR

(·))> (A.21)

44 M.J. Machina

∑x∈X

Φx

(f−1

L (x); f)−∑x∈X

Φx

(f−1

R (x); f)−

12 ·γ ·

(δ(fL, f)+δ(fR, f)

)λ(A) + L*·λ(B)

≥

Φx∗(Ai′ ;f)+Φx(Bj′ ;f)−Φx(Ai′ ;f)−Φx∗(Bj′ ;f)−γ·(λ(Ai′)+λ(Bj′)

)λ(A) + L*·λ(B)

≥ 0

(c)⇒ (a): Say (a) failed, so that γ = Φx∗(A; f)−Φx(A; f)−L* · [Φx∗(B; f)−Φx(B; f)] < 0 for some x* x, f(·) ∈ A, and rationalL* ∈ (0,L).By (19) thereexists ε* > 0 such that δ(f , f) < 2 · ε* implies∣∣∣W (f(·)

)−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣(A.22)

<12 · |γ|

λ(A) + L*·λ(B)· δ(f(·), f(·)

)

Select arbitrary ε ∈ (0, ε*) and integers na > λ(A)/ε and nb > λ(B)/ε suchthat na/nb = L*. By Stromquist and Woodall (1985, Thm.1), there exists an E-measurable partition A1, . . . , Ana of A with λ(Ai) = λ(A)/na < ε andΦx∗(Ai; f) − Φx(Ai; f) = [Φx∗(A; f) − Φx(A; f)]/na for each i, and an E-measurable partition B1, . . . , Bnb

of B with λ(Bj) = λ(B)/nb < ε andΦx∗(Bj ; f) − Φx(Bj ; f) = [Φx∗(B; f) − Φx(B; f)]/nb for each j. This impliesthat for all i, j we have

Φx∗(Ai; f)− Φx(Ai; f)− Φx∗(Bj ; f) + Φx(Bj ; f) =(A.23)

Φx∗(A; f)− Φx(A; f)na

− L*·Φx∗(B; f)− Φx(B; f)na

= γ/na

Define fLi,j(·) = [x* on Ai; x on Bj ; f(·) elsewhere] and fR

i,j(·) = [x on Ai;x* on Bj ; f(·) elsewhere], so that we have both

δ(fL

i,j , f)≤ λ(Ai) + λ(Bj) =

λ(A)na

+λ(B)nb

=λ(A) + L*·λ(B)

na< 2 · ε*

(A.24)

δ(fR

i,j , f)≤ λ(Ai) + λ(Bj) =

λ(A)na

+λ(B)nb

=λ(A) + L*·λ(B)

na< 2 · ε*

For each i, j, (A.22) and (A.23) thus imply

W(fL

i,j(·))−W

(fR

i,j(·))< (A.25)

∑x∈X

Φx

(fL−1

i,j (x); f)−∑x∈X

Φx

(fR−1

i,j (x); f)

+|γ|2·δ(fL

i,j , f) + δ(fRi,j , f)

λ(A) + L*·λ(B)

≤ Φx∗(Ai; f)− Φx(Ai; f)− Φx∗(Bj ; f) + Φx(Bj ; f) + |γ|/na = 0

But since A1, . . . , Ana and B1, . . . , Bnb

were ε-partitions with na/nb = L*for arbitrarily small ε > 0, this contradicts (c).


(a)⇒ (d): Consider arbitrary ℘a, ℘b ⊆ [0, 1] with λ(℘a)/λ(℘b) = (>)1/L,x* x, and f(·) ∈ A. By Step 1 of the proof of Theorem 4 in Machina (2004),both limits in (55) exist. For arbitrary ε > 0, event-smoothness (boundedness anduniform continuity of the families φ(x*,·,f)|f(·)∈A and φ(x,·,f)|f(·)∈A)and Theorem 0 of Machina (2004) imply some mε such that both∣∣Φx∗

(℘a×mA;f

)− Φx

(℘a×mA;f

)− λ(℘a)·

(Φx∗(A;f)− Φx(A;f)

)∣∣ < ε/2∣∣Φx∗(℘b×mB;f

)− Φx

(℘b×mB;f

)− λ(℘b)·

(Φx∗(B;f)− Φx(B;f)

)∣∣ < ε/2(A.26)

for all m > mε and all f(·) ∈ A. Event-smoothness (the bound Φx∗(B; f) −Φx(B; f) > Φx∗,x,B > 0 for all f(·)) and (52) then imply

Φx∗(℘a×mA;f

)− Φx

(℘a×mA;f

)− Φx∗

(℘b×mB;f

)+ Φx

(℘b×mB;f

)>

λ(℘a)·(Φx∗(A;f)− Φx(A;f)

)− λ(℘b)·


)− ε ≥[

λ(℘a)·L−λ(℘b)]·(Φx∗(B;f)−Φx(B;f)

)−ε ≥

[λ(℘a)·L−λ(℘b)

]·Φx∗,x,B − ε

(A.27)

for all m > mε and all f(·). The change sets from fmR (·) = [x on ℘a ×m A ; x* on

℘b×m B; f(·) elsewhere] to fmL (·) = [x* on ℘a×mA; x on ℘b×mB; f(·) elsewhere] in

(55) are ∆E+x∗ = ∆E−

x = ℘a×mA and ∆E−x∗ = ∆E+

x = ℘b×mB. For the almost-objective mixture paths fk,m

α (·) ≡ [fmL (·) on [0, α]×

kS; fm

R (·) on (α, 1]×kS] from

fmR (·) to fm

L (·), the Line Integral Approximation Theorem and (A.27) imply

W(fm

L (·))−W

(fm

R (·))

= limk→∞

∫ 1

0

[Φx∗(℘a×mA;fk,m

α

)+ Φx

(℘b×mB;fk,m

α

)−

(A.28)Φx∗(℘b×mB;fk,m

α

)− Φx

(℘a×mA;fk,m

α

)]·dα ≥

[λ(℘a)·L−λ(℘b)

]·Φx∗,x,B − ε

for all m > mε. Since ε > 0 was arbitrary, limm→∞W (fm

L (·)) − limm→∞W (fm

R (·)) ≥[λ(℘a)·L − λ(℘b)] · Φx∗,x,B = (>) 0.

(d)⇒ (a): Say (a) failed, so that γ = Φx∗(A; f)− Φx(A; f)− L · [Φx∗(B; f)−Φx(B; f)] < 0 for some x* x and f(·). By (19) there exists some δ* > 0 suchthat δ(f , f) < δ* implies∣∣∣W (f(·)

)−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣(A.29)

<116 · |γ|

λ(A) + L·λ(B)· δ(f(·), f(·)

)Select ℘a, ℘b ⊂ [0, 1] such that λ(℘a)/λ(℘b) = 1/L and λ(℘a) ·λ(A) + λ(℘b) ·λ(B) < δ*/2.By event-smoothness and Theorem 0 of Machina (2004) there existsm* such that for all m > m* we have the following three relationships

λ(℘a×mA

)+ λ(℘b×mB

)< 2 ·

(λ(℘a)·λ(A) + λ(℘b)·λ(B)

)< δ* (A.30)∣∣Φx∗

(℘a×mA;f

)− Φx

(℘a×mA;f

)− λ(℘a)·

(Φx∗(A;f)−Φx(A;f)

)∣∣ < 14 ·|γ|·λ(℘a)∣∣Φx∗

(℘b×mB;f

)− Φx

(℘b×mB;f

)− λ(℘b)·

(Φx∗(B;f)−Φx(B;f)

)∣∣ < 14 ·|γ|·λ(℘a)

46 M.J. Machina

For all m > m* we thus have

Φx∗(℘a×mA;f

)− Φx

(℘a×mA;f

)− Φx∗

(℘b×mB;f

)+ Φx

(℘b×mB;f

)< (A.31)

λ(℘a)·(Φx∗(A;f)− Φx(A;f)

)− λ(℘b)·


)+ 1

2 ·|γ|·λ(℘a)

= λ(℘a)·[(Φx∗(A;f)− Φx(A;f)

)− L·


)]+ 1

2 ·|γ|·λ(℘a)

= λ(℘a) ·γ + 12 ·|γ|·λ(℘a) = − 1

2 ·|γ|·λ(℘a)

For each m, define fmL (·) = [x* on ℘a×mA; x on ℘b×mB; f(·) elsewhere] and

fmR (·) = [x on ℘a×mA; x* on ℘b×mB; f(·) elsewhere], so the change sets fromfm

R (·) to fmL (·) are ∆E+

x∗ = ∆E−x = ℘a×mA and ∆E−

x∗ = ∆E+x = ℘b×mB. For

each m > m*, (A.30) implies δ(fmL , f) and δ(fm

R , f) are each less than or equalto λ(℘a×mA)+λ(℘b×mB) < 2 · (λ(℘a)·λ(A)+λ(℘b)·λ(B)) = 2 ·λ(℘a) · (λ(A)+L ·λ(B)) < δ*. This yields a contradiction of (d), since λ(℘a)/λ(℘b) = 1/L, yetfor each m > m*, (A.29) and (A.31) imply

W(fm

L (·))−W

(fm

R (·))< (A.32)

∑x∈X

Φx

(fm−1

L (x); f)−∑x∈X

Φx

(fm−1

R (x); f)

+116 · |γ| ·

(δ(fm

L , f)+δ(fmR , f)

)λ(A) + L·λ(B)

< Φx∗(℘a×mA;f

)+Φx

(℘b×mB;f

)−Φx∗

(℘b×mB;f

)−Φx

(℘a×mA;f

)+ 1

4 ·|γ|·λ(℘a)

< − 12 ·|γ|·λ(℘a) + 1

4 ·|γ|·λ(℘a) = − 14 ·|γ|·λ(℘a) < 0

(a)⇒ (e): Consider arbitrary disjoint ℘a,℘b⊂ [0, 1] with λ(℘a)/λ(℘b)=(<)L,x* x, and f(·) ∈ A. By Step 1 of the proof of Theorem 4 in Machina (2004),both limits in (56) exist. For arbitrary ε > 0, event-smoothness and Theorem 0 ofMachina (2004) imply some mε such that both∣∣Φx∗

(℘b×mA;f

)− Φx

(℘b×mA;f

)− λ(℘b)·


)∣∣ < ε/2(A.33)∣∣Φx∗

(℘a×mB;f

)− Φx

(℘a×mB;f

)− λ(℘a)·


)∣∣ < ε/2

for all m>mε and all f(·)∈A. Event-smoothness (the bound Φx∗(B; f) −Φx(B; f) > Φ x∗,x,B > 0 for all f(·)) and (52) then imply

Φx∗(℘b×mA;f

)− Φx

(℘b×mA;f

)− Φx∗

(℘a×mB;f

)+ Φx

(℘a×mB;f

)>

λ(℘b)·(Φx∗(A;f)−Φx(A;f)

)− λ(℘a)·


)− ε ≥ (A.34)

[λ(℘b)·L−λ(℘a)

]·(Φx∗(B;f)−Φx(B;f)

)−ε ≥

[λ(℘b)·L−λ(℘a)

]·Φx∗,x,B − ε

The change sets from fmR (·) = [x* on ℘a×m (A ∪B); x on ℘b×m (A ∪B); f(·)

elsewhere] to fmL (·) = [x* on (℘a∪℘b)×m A;x on (℘a∪℘b)×m B; f(·) elsewhere]


in (56) are ∆E+x∗ = ∆E−

x = ℘b×mA and ∆E−x∗ = ∆E+

x = ℘a×m B. For thealmost-objective mixture paths fk,m

α (·) ≡ [fmL (·) on [0, α]×

kS; fm

R (·) on (α, 1]×kS]

from fmR (·) to fm

L (·), the Line Integral Approximation Theorem and (A.34) yield

W(fm

L (·))−W

(fm

R (·))

= limk→∞

∫ 1

0

[Φx∗(℘b×mA;fk,m

α

)+ Φx

(℘a×mB;fk,m

α

)(A.35)

−Φx∗(℘a×mB;fk,m

α

)− Φx

(℘b×mA;fk,m

α

)]·dα ≥

[λ(℘b)·L−λ(℘a)

]·Φx∗,x,B − ε

for all m > mε. Since ε > 0 was arbitrary, limm→∞W (fm

L (·)) − limm→∞W (fm

R (·)) ≥[λ(℘b)·L − λ(℘a)] ·Φx∗,x,B = (>) 0.

(e)⇒ (a): Say (a) failed, so that γ = Φx∗(A; f) − Φx(A; f) − L · [Φx∗(B; f) −Φx(B; f)] < 0 for some x* x and f(·). By (19) there exists some δ* > 0 suchthat δ(f , f) < δ* implies∣∣∣W (f(·)

)−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣<

116 · |γ|

(L+1) ·λ(A ∪B)· δ(f(·), f(·)

) (A.36)

Select disjoint℘a, ℘b ⊂ [0, 1] such thatλ(℘a)/λ(℘b) = L andλ(℘b)·(L+1)·λ(A∪B) < δ*/2. By event-smoothness and Theorem 0 of Machina (2004) there existsm* such that for all m > m* we have each of the following three relationships

λ((℘a∪℘b)×m(A∪B)

)< 2·λ(℘a∪℘b)·λ(A∪B) = 2·λ(℘b)·(L+1)·λ(A∪B)<δ*

(A.37)∣∣Φx∗(℘b×mA;f

)−Φx

(℘b×mA;f

)−λ(℘b)·


)∣∣ < 14 ·|γ|·λ(℘b)∣∣Φx∗

(℘a×mB;f

)−Φx

(℘a×mB;f

)−λ(℘a)·


)∣∣ < 14 ·|γ|·λ(℘b)

For all m > m*, the second and third relationships in (A.37) imply

Φx∗(℘b×mA;f

)− Φx

(℘b×mA;f

)− Φx∗

(℘a×mB;f

)+ Φx

(℘a×mB;f

)< (A.38)

λ(℘b)·(Φx∗(A;f)− Φx(A;f)

)− λ(℘a)·


)+ 1

2 ·|γ|·λ(℘b)

= λ(℘b)·[(Φx∗(A;f)− Φx(A;f)

)− L·


)]+ 1

2 ·|γ|·λ(℘b)

= λ(℘b)·γ + 12 ·|γ|·λ(℘b) = − 1

2 ·|γ|·λ(℘b)

For eachm, define fmL (·) = [x* on (℘a∪℘b)×m A; x on (℘a∪℘b)×m B; f(·) else-

where] and fmR (·) = [x* on ℘a×m (A∪B); x on ℘b×m (A∪B); f(·) elsewhere], so

the change sets from fmR (·) to fm

L (·) are ∆E+x∗ =∆E−

x =℘b×mA and ∆E−x∗ =

∆E+x = ℘a×m B. For each m > m*, the first relationship in (A.37) implies

δ(fmL , f) and δ(fm

R , f) are each less than or equal to λ((℘a∪℘b)×m (A∪B)) <2·λ(℘b)·(L+1)·λ(A ∪B) < δ*. This yields a contradiction of (e), sinceλ(℘a)/λ(℘b) = L, yet for each m > m*, (A.36) and (A.38) imply

W(fm

L (·))−W

(fm

R (·))<

48 M.J. Machina

∑x∈X

Φx

(fm−1

L (x);f)−∑x∈X

Φx

(fm−1

R (x);f)

+116 ·|γ|·

(δ(fm

L , f)+δ(fmR , f)

)(L+1) ·λ(A∪B)

(A.39)

< Φx∗(℘b×mA;f

)+Φx

(℘a×mB;f

)−Φx∗

(℘a×mB;f

)−Φx

(℘b×mA;f

)+ 1

4 ·|γ|·λ(℘b)

< − 12 ·|γ|·λ(℘b) + 1

4 ·|γ|·λ(℘b) = − 14 ·|γ|·λ(℘b) < 0

Proof of Theorem 5. (a) ⇒ (b): Consider arbitrary x′′ x′ x, f(·) ∈ Aand disjoint nondegenerate ℘a,℘b⊂ [0, 1] that satisfy the upper inequality of(58). For each m define Am = ℘a×mS, Bm = ℘b×mS, fm

L (·) = [x′′ on Am;x on Bm; f(·) elsewhere] and fm

R (·) = [x′ on Am∪Bm; f(·) elsewhere], so thereexist m* and γ > 0 such that 2 ·γ > W*(fm

L (·))−W*(fmR (·)) > γ for all m >

m*. Select ε ∈ (0, 1) less than minλ(℘a), λ(℘b), and small enough so that both(1− 1−ε

1+ε·λ(℘a)−ελ(℘a)+ε

)·[2 ·γ + (1+ε)·

(λ(℘b)+ε

)·Φ∗x′,x,S

]< γ/4

(A.40)(1+ε1−ε ·

λ(℘b)+ελ(℘b)− ε

− 1)·(1+ε)·

(λ(℘b)+ε

)·Φ∗x′,x,S < γ/4

Event-smoothness and Machina (2004, Thm. 0) imply the four relationships∣∣Φx′′(Am; f)− Φx′(Am; f)− λ(℘a)·[Φx′′(S; f)−Φx′(S; f)

]∣∣< ε ·Φx′′,x′,S < ε·

[Φx′′(S; f)− Φx′(S; f)

]∣∣Φx′(Bm; f)− Φx(Bm; f)− λ(℘b)·

[Φx′(S; f)−Φx(S; f)

]∣∣< ε ·Φx′,x,S < ε·

[Φx′(S; f)− Φx(S; f)

](A.41)∣∣Φ∗

x′′(Am; f)− Φ∗x′(Am; f)− λ(℘a)·

[Φ∗

x′′(S; f)− Φ∗x′(S; f)

]∣∣< ε ·Φ∗

x′′,x′,S < ε·[Φ∗

x′′(S; f)− Φ∗x′(S; f)

]∣∣Φ∗

x′(Bm; f)− Φ∗x(Bm; f)− λ(℘b)·

[Φ∗

x′(S; f)− Φ∗x(S; f)

]∣∣< ε ·Φ∗

x′,x,S < ε·[Φ∗

x′(S; f)− Φ∗x(S; f)

]for allm greater than somem** and all f(·) ∈ A , and thus the three relationships

Φx′′(Am; f)− Φx′(Am; f)

Φx′(Bm; f)− Φx(Bm; f)>λ(℘a)− ελ(℘b) + ε

· Φx′′(S; f)− Φx′(S; f)

Φx′(S; f)− Φx(S; f)

Φ∗x′′(Am; f)− Φ∗

x′(Am; f)

Φ∗x′(Bm; f)− Φ∗

x(Bm; f)<λ(℘a) + ε

λ(℘b)− ε· Φ

∗x′′(S; f)− Φ∗

x′(S; f)

Φ∗x′(S; f)− Φ∗

x(S; f)(A.42)

Φ∗x′(Bm; f)−Φ∗

x(Bm; f)

Φx′(Bm; f)−Φx(Bm; f)<λ(℘b)+ελ(℘b)−ε

·Φ∗x′(S; f)−Φ∗

x(S; f)

Φx′(S; f)−Φx(S; f)<λ(℘b)+ελ(℘b)−ε

·Φ∗x′,x,SΦx′,x,S

for all m > m** and all f(·). Select arbitrary m > maxm*,m** and defineγm = W*(fm

L (·)) − W*(fmR (·)), so 2 · γ > γm > γ. For each k, define the


parametrized family of acts fkα,β(·)|α, β ∈ [0, 1] by

fkα,β(·) =

x′′ on [0, α]×kAm

x′ on (α, 1]×kAm

x on [0, β]×kBm

x′ on (β, 1]×kBm

f(·) elsewhere

(A.43)

so that for each k we have fk1,1(·) = fm

L (·) and δ(fk0,0, f

mR ) = 0.

By an argument similar to that in Machina (2004, pp.39-41), we can select largeenough k such that the partial derivatives ∂W (fk

α,β(·))/∂α and ∂W*(fkα,β(·))/∂α

exist and satisfy∣∣∣∂W (fkα,β(·)

)/∂α−

[Φx′′(Am; fk

α,β

)− Φx′

(Am; fk

α,β

)]∣∣∣< ε · Φx′′,x′,Am

< ε ·[Φx′′(Am; fk

α,β

)− Φx′

(Am; fk

α,β

)](A.44)∣∣∣∂W*

(fk

α,β(·))/∂α−

[Φ∗x′′(Am; fk

α,β

)− Φ∗x′

(Am; fk

α,β

)]∣∣∣< ε · Φ∗x′′,x′,Am

< ε ·[Φ∗x′′(Am; fk

α,β

)− Φ∗x′

(Am; fk

α,β

)]at all but a finite set of values of α ∈ [0, 1] (where these values are independent ofβ), and the partials ∂W (fk

α,β(·))/∂β and W*(fkα,β(·))/∂β exist and satisfy

∣∣∣∂W (fkα,β(·)

)/∂β −

[Φx

(Bm; fk

α,β

)− Φx′

(Bm; fk

α,β

)]∣∣∣< ε · Φx′,x,Bm

< ε ·[Φx′(Bm; fk

α,β

)− Φx

(Bm; fk

α,β

)](A.45)∣∣∣∂W*

(fk

α,β(·))/∂β −

[Φ∗x(Bm; fk

α,β

)− Φ∗x′

(Bm; fk

α,β

)]∣∣∣< ε · Φ∗x′,x,Bm

< ε ·[Φ∗x′(Bm; fk

α,β

)− Φ∗x

(Bm; fk

α,β

)]at all but a finite set of values of β (where these values are independent ofα). Since ε < 1 this implies ∂W (fk

α,β(·))/∂α, ∂W*(fkα,β(·))/∂α > 0 and

∂W (fkα,β(·))/∂β, ∂W*(fk

α,β(·))/∂β < 0 except at these values, so W*(fkα,β(·))

and W (fkα,β(·)) are strictly increasing in α and strictly decreasing in β.

For each value of β ∈ [0, 1], let α(β) be the unique solution to W*(fkα(β),β(·)) =

W*(fk0,0(·))+γm·β = W*(fm

R (·))+γm·β, so the mapping α(·) : [0, 1]→ [0, 1]is continuous, strictly increasing, onto, and α′(β) exists with

α′(β)·∂W*(fk

α(β),β(·))/∂α+ ∂W*

(fk

α(β),β(·))/∂β = γm > 0 (A.46)

at all but a finite number of values of β. By (A.45) and (A.41) we have

∂W*(fk

α(β),β(·))/∂β > −(1 + ε)·

[Φ∗x′(Bm;fk

α(β),β

)− Φ∗x

(Bm;fk

α(β),β

)]> −(1 + ε)·

(λ(℘b) + ε

)·[Φ∗x′(S; fk

α(β),β

)− Φ∗x

(S; fk

α(β),β

)](A.47)

> −(1 + ε)·(λ(℘b) + ε

)·Φ∗x′,x,S

50 M.J. Machina

so that at all but a finite number of values of β we also have

α′(β)·∂W*(fk

α(β),β(·))/∂α = γm − ∂W*

(fk

α(β),β(·))/∂β

(A.48)< γm + (1 + ε)·

(λ(℘b) + ε

)·Φ∗x′,x,S

By (A.46), (A.47), (A.48), (A.40) and 2 ·γ > γm > γ, we have

1−ε1+ε

·α′(β) · λ(℘a)−ελ(℘a)+ε

·∂W*

(fk

α(β),β

)∂α

+1+ε1−ε ·

λ(℘b)+ελ(℘b)−ε

·∂W*

(fk

α(β),β

)∂β

= α′(β) ·∂W*

(fk

α(β),β

)∂α

−(

1− 1−ε1+ε

· λ(℘a)−ελ(℘a)+ε

)·α′(β) ·

∂W*(fk

α(β),β

)∂α

+∂W*

(fk

α(β),β

)∂β

+(

1+ε1−ε ·


− 1)·∂W*

(fk

α(β),β

)∂β

(A.49)

> γm −(

1− 1−ε1+ε

· λ(℘a)−ελ(℘a)+ε

)·[γm + (1+ε) ·

(λ(℘b)+ε

)·Φ∗x′,x,S

]

−(

1+ε1−ε ·


− 1)· (1+ε) ·

(λ(℘b)+ε

)·Φ∗x′,x,S

> γ−(1− 1−ε

1+ε·λ(℘a)−ελ(℘a)+ε

)·[2·γ+(1+ε)·

(λ(℘b)+ε

)·Φ∗x′,x,S

]− γ

4>γ

2> 0

The above inequalities and (57) imply that at all but a finite number of values of β,we have

α′(β)·∂W(fk

α(β),β(·))/∂α+ ∂W

(fk

α(β),β(·))/∂β

> (1−ε)·α′(β)·[Φx′′(Am; fk

α(β),β)− Φx′(Am; fkα(β),β)

](A.50)

−(1+ε)·[Φx′(Bm; fk

α(β),β)− Φx(Bm; fkα(β),β)

]

=

(1−ε)·α′(β)·Φx′′(Am; fk

α(β),β)− Φx′(Am; fkα(β),β)

Φx′(Bm; fkα(β),β)− Φx(Bm; fk

α(β),β)− (1+ε)

1/[Φx′(Bm; fkα(β),β)− Φx(Bm; fk

α(β),β)]

>

(1−ε)·α′(β)· λ(℘a)− ελ(℘b) + ε

·Φx′′(S; fk

α(β),β)− Φx′(S; fkα(β),β)

Φx′(S; fkα(β),β)− Φx(S; fk

α(β),β)− (1+ε)


α(β),β)]

≥(1−ε)·α′(β)· λ(℘a)− ε

λ(℘b) + ε·Φ∗

x′′(S; fkα(β),β)− Φ∗

x′(S; fkα(β),β)

Φ∗x′(S; fk

α(β),β)− Φ∗x(S; fk

α(β),β)− (1+ε)


α(β),β)]>

(1−ε)·α′(β)·λ(℘a)−ελ(℘b)+ε

·λ(℘b)−ελ(℘a)+ε

·Φ∗

x′′(Am;fkα(β),β)−Φ∗

x′(Am;fkα(β),β)

Φ∗x′(Bm;fk

α(β),β)−Φ∗x(Bm;fk

α(β),β)−(1+ε)


α(β),β)]


=(1−ε)·α′(β)·λ(℘a)−ε

λ(℘b)+ε· λ(℘b)−ελ(℘a)+ε

·[Φ∗x′′(Am; fk

α(β),β)−Φ∗x′(Am; fk

α(β),β)]

[Φ∗x′(Bm; fα(β),β)−Φ∗

x(Bm; fkα(β),β)]/[Φx′(Bm; fk

α(β),β)−Φx(Bm; fkα(β),β)]

−(1+ε)·[Φ∗

x′(Bm; fkα(β),β)−Φ∗

x(Bm; fkα(β),β)]




>

1−ε1+ε

·α′(β)· λ(℘a)−ελ(℘b)+ε

· λ(℘b)−ελ(℘a)+ε ·

∂W ∗(fkα(β),β)

∂α+

1+ε1−ε ·


∂β




=

1−ε1+ε

·α′(β)· λ(℘a)−ελ(℘a)+ε ·


∂α+

1+ε1−ε ·

λ(℘b)+ελ(℘b)−ε ·


∂β

(λ(℘b)+ε) · [Φ∗x′(Bm; fk

α(β),β)− Φ∗x(Bm; fk

α(β),β)](λ(℘b)−ε) · [Φx′(Bm; fk

α(β),β)− Φx(Bm; fkα(β),β)]

>γ/2[

(λ(℘b) + ε)2· Φ∗x′,x,S

]/[(λ(℘b)− ε)2· Φx′,x,S

] > 0

We thus have

W(fm

L (·))−W

(fm

R (·))

= W(fk1,1(·)

)−W

(fk0,0(·)

)=∫ 1

0

dW (fkα(β),β)

dβ·dβ =

(A.51)∫ 1

0

[α′(β)·

∂W (fkα(β),β)

∂α+∂W (fk

α(β),β)

∂β

]·dβ > γ

2·(λ(℘b)−ε)2·Φx′,x,S

(λ(℘b)+ε)2·Φ∗x′,x,S

> 0

Since m was an arbitrary integer greater than maxm*,m**, the valueslim

m→∞W (fmL (·)) and lim

m→∞W (fmR (·)), which by Machina (2004, Thm.2) both exist,

satisfy limm→∞W (fm

L (·)) > limm→∞W (fm

R (·)).

(b) ⇒ (a): Say (a) fails, so there exist x′′ x′ x, f(·) and γ > 0 with[Φ∗x′′(S; f)−Φ∗x′(S; f)]/[Φ∗x′(S; f)−Φ∗x(S; f)] > γ > [Φx′′(S; f)−Φx′(S; f)]/[Φx′(S; f)−Φx(S; f)]. This implies some ε > 0 such that both[Φx′′(S; f)− Φx′(S; f)

]− γ ·


]< −ε·(1 + γ)

(A.52)[Φ∗x′′(S; f)− Φ∗x′(S; f)

]− γ ·

[Φ∗x′(S; f)− Φ∗x(S; f)

]> ε·(1 + γ)

By (19) there exists some δ* > 0 such that δ(f , f) < δ* implies both∣∣∣W (f(·))−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣<( 1

8 ·ε/λ(S))·δ(f(·), f(·)

)(A.53)∣∣∣W*

(f(·))−W*

(f(·))−[∑

x∈XΦ∗x(f−1(x); f

)−∑x∈X

Φ∗x(f−1(x); f

)]∣∣∣<( 1

8 ·ε/λ(S))·δ(f(·), f(·)

)

52 M.J. Machina

Define τ = min 14 ·δ*/λ(S), 1 and the disjoint intervals ℘a = [0, τ/(1 + γ)] and

℘b = (τ/(1 + γ), τ ], so λ(℘b) = τ ·γ/(1 + γ) = γ ·λ(℘a) and λ(℘a×m S) +λ(℘b×m S) = λ(℘a)·λ(S)+λ(℘b)·λ(S) = τ ·λ(S) < δ*/2. By event-smoothnessand Machina (2004,Thm.0) there exists m* such that for all m > m* we have∣∣Φx′′

(℘a×mS;f

)−Φx′

(℘a×mS;f

)−λ(℘a)·

[Φx′′(S;f)− Φx′(S;f)

]∣∣ < 14 ·τ ·ε∣∣Φx′

(℘b×mS;f

)−Φx

(℘b×mS;f

)−λ(℘b)·

[Φx′(S;f)− Φx(S;f)

]∣∣ < 14 ·τ ·ε∣∣Φ∗x′′

(℘a×mS;f

)−Φ∗x′

(℘a×mS;f

)−λ(℘a)·

[Φ∗x′′(S;f)− Φ∗x′(S;f)

]∣∣ < 14 ·τ ·ε∣∣Φ∗x′

(℘b×mS;f

)−Φ∗x

(℘b×mS;f

)−λ(℘b)·

[Φ∗x′(S;f)− Φ∗x(S;f)

]∣∣ < 14 ·τ ·ε

(A.54)

For all m > m*, (A.54) and (A.52) thus imply

Φx′′(℘a×mS;f

)− Φx′

(℘a×mS;f

)− Φx′

(℘b×mS;f

)+ Φx

(℘b×mS;f

)< λ(℘a)·

[Φx′′(S;f)− Φx′(S;f)

]− λ(℘b)·


]+ 1

2 ·τ ·ε

= λ(℘a)·[[Φx′′(S;f)− Φx′(S;f)

]− γ·


]]+ 1

2 ·τ ·ε

< −λ(℘a)·ε·(1+γ) + 12 ·τ ·ε = −τ ·ε+ 1

2 ·τ ·ε = − 12 ·τ ·ε

(A.55)

and similarly

Φ∗x′′(℘a×mS;f

)− Φ∗x′

(℘a×mS;f

)− Φ∗x′

(℘b×mS;f

)+ Φ∗x

(℘b×mS;f

)> λ(℘a)·

[Φ∗x′′(S;f)− Φ∗x′(S;f)

]− λ(℘b)·


]− 1

2 ·τ ·ε(A.55)′

= λ(℘a)·[[Φ∗x′′(S;f)− Φ∗x′(S;f)

]− γ·


]]− 1

2 ·τ ·ε

> λ(℘a)·ε·(1+γ)− 12 ·τ ·ε = τ ·ε− 1

2 ·τ ·ε = 12 ·τ ·ε

Define fmL (·) = [x′′ on ℘a×m S;x on ℘b×m S; f(·) elsewhere] and fm

R (·) = [x′ on(℘a∪℘b)×mS; f(·) elsewhere], so that both δ(fm

L , f) ≤ λ((℘a∪℘b)×mS) = τ ·λ(S)< δ*/2 and δ(fm

R , f) ≤ λ((℘a ∪ ℘b)×m S) = τ ·λ(S) < δ*/2. By (A.53), (A.54)and (A.55), we have that for all m > m*

W(fm

L (·))−W

(fm

R (·))< (A.56)

∑x∈X

Φx

(fm−1

L (x); f)−∑x∈X

Φx

(fm−1

R (x); f)

+18 ·ελ(S)

·[δ(fm

L , f) + δ(fmR , f)

]≤

Φx′′(℘a×mS;f

)−Φx′

(℘a×mS;f

)−Φx′

(℘b×mS;f

)+Φx

(℘b×mS;f

)+τ ·ε4<−τ ·ε

4< 0

and by (A.53), (A.54) and (A.55)′ we have that for all m > m*

W*(fm

L (·))−W*

(fm

R (·))> (A.56)′∑

x∈XΦ∗x(fm−1

L (x); f)−∑x∈X

Φ∗x(fm−1

R (x); f)−

18 ·ελ(S)

·[δ(fm

L , f) + δ(fmR , f)

]≥

Φ∗x′′(℘a×mS;f

)−Φ∗x′

(℘a×mS;f

)−Φ∗x′

(℘b×mS;f

)+Φ∗x

(℘b×mS;f

)− τ ·ε

4>τ ·ε4> 0


Thus the following four limits – which by Machina (2004, Thm.1) must eachexist – will satisfy lim

m→∞W*(fmL (·)) > lim

m→∞W*(fmR (·)), but lim

m→∞W (fmL (·)) <

limm→∞W (fm

R (·)), which contradicts (b).

(a)⇔ (c): We establish this equivalence by showing thatW (·) is willing to acceptsmall-event x← x′ → x′′ spreads about f(·) at any odds ratio greater than someLif and only if L ≥ [Φx′(S; f)−Φx(S; f)]/[Φx′′(S; f)−Φx′(S; f)], and similarlyfor W*(·). To prove the “if” direction, consider arbitrary x′′ x′ x, f(·) ∈ Aand L ∈ (0,∞) that satisfy this inequality. Given arbitrary na, nb with na/nb > Land arbitrary ε > 0, define γ = na ·[Φx′′(S; f) − Φx′(S; f)] − nb ·[Φx′(S; f) −Φx(S; f)] > 0. By (19) there exists δ* > 0 such that δ(f , f) < δ* implies∣∣∣W (f(·)

)−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣(A.57)

<14 · γ

λ(S)·(na + nb)· δ(f(·), f(·)

)

Select n > maxna+nb, λ(S)/ε, (na+nb)·λ(S)/δ*.By Stromquist and Woodall(1985,Thm.1) and (21)′, there exists an E-measurable partition E1, . . . , En ofS such that for each i, λ(Ei) = λ(S)/n < ε, Φx′′(Ei; f) − Φx′(Ei; f) =[Φx′′(S;f)−Φx′(S;f)]/n and Φx′(Ei;f)−Φx(Ei;f) = [Φx′(S;f)−Φx(S;f)]/n.

Let A be the union of any na of the events E1, . . . , En and B be the union of anynb others, so that λ(A) + λ(B) = (na+nb) ·λ(S)/n < δ*, and also[

Φx′′(A; f)− Φx′(A; f)]−[Φx′(B; f)− Φx(B; f)

]=

(A.58)

na·[Φx′′(S; f)− Φx′(S; f)

]/n − nb·


]/n = γ/n

Define fL(·) = [x′′ on A;x on B; f(·) elsewhere] and fR(·) = [x′ on A ∪ B;f(·) elsewhere] so δ(fL, f) ≤ λ(A)+λ(B)< δ* and δ(fR, f) ≤ λ(A)+λ(B) <δ*. By (A.57) we have

W(fL(·)

)−W

(fR(·)

)> (A.59)

Φx′′(A; f)− Φx′(A; f)− Φx′(B; f) + Φx(B; f)−14 ·γ ·[δ(fL, f)+δ(fR, f)]

λ(S)·(na+nb)

≥ γ/n−12 ·γ ·[λ(A)+λ(B)]λ(S)·(na+nb)

= γ/n− 12 ·γ/n = 1

2 ·γ/n > 0

To prove the “only if” direction, consider arbitrary x′′ x′ x, f(·) and valueL < [Φx′(S; f) − Φx(S; f)]/[Φx′′(S; f) − Φx′(S; f)]. Select na, nb such that[Φx′(S; f) − Φx(S; f)]/[Φx′′(S; f) − Φx′(S; f)] > na/nb > L, and positive ηsuch that τ = na ·[Φx′′(S; f)−Φx′(S; f)+η ·λ(S)]−nb ·[Φx′(S; f)−Φx(S; f)−η ·λ(S)] < 0. By (19) there exists δ** > 0 such that δ(f , f) < δ** implies

54 M.J. Machina

∣∣∣W (f(·))−W

(f(·))−[∑

x∈XΦx

(f−1(x); f

)−∑x∈X

Φx

(f−1(x); f

)]∣∣∣(A.60)

< 12 · η · δ

(f(·), f(·)

)Select positive ε < minδ**/(na + nb), λ(S)/(na + nb).Let E1, . . . , En be an arbitrary ε-partition of S, which implies n > λ(S)/ε> na +nb. By a standard combinatoric argument, there are K = n!/(na! ·nb! ·(n−na−nb)!) pairs (A,B) such thatA is the union of na of the eventsE1, . . . , En

andB is the union of nb others. LetK = (Ak, Bk) |k = 1, . . . ,K be the familyof all such pairs. Since each eventEi will be included inAk for exactly (na/n) ·Kof the pairs inK, and will be included inBk for exactly (nb/n) ·K others, we have∑K

k=1

[Φx′′(Ak; f)− Φx′(Ak; f) + η ·λ(Ak)

]=∑n

i=1(na/n)·K·

[Φx′′(Ei; f)− Φx′(Ei; f) + η ·λ(Ei)

](A.61)

= (na/n)·K·[Φx′′(S; f)− Φx′(S; f) + η ·λ(S)

]∑K

k=1

[Φx′(Bk; f)− Φx(Bk; f)− η ·λ(Bk)

]=∑n

i=1(nb/n) ·K ·

[Φx′(Ei; f)− Φx(Ei; f)− η ·λ(Ei)

](A.61)′

= (nb/n) ·K ·[Φx′(S; f)− Φx(S; f)− η ·λ(S)

]so that

K∑k=1

Φx′′(Ak; f)−Φx′(Ak; f)−Φx′(Bk; f)+Φx(Bk; f)+η ·(λ(Ak)+λ(Bk)

)K

= (na/n) ·[Φx′′(S; f)− Φx′(S; f) + η ·λ(S)

](A.62)

−(nb/n) ·[Φx′(S; f)− Φx(S; f)− η ·λ(S)

]= τ/n

This implies that at least one pair (Ak, Bk) satisfies

Φx′′(Ak;f)−Φx′(Ak;f)−Φx′(Bk;f)+Φx(Bk;f)+η·(λ(Ak)+λ(Bk)

)≤ τ

n(A.63)

Define fL(·) = [x′′ on Ak;x on Bk; f(·) elsewhere] and fR(·) = [x′ on Ak∪Bk; f(·) elsewhere], so δ(fL, f) ≤ λ(Ak) + λ(Bk) < (na + nb) · ε < δ**and δ(fR, f) ≤ λ(Ak) + λ(Bk) < (na +nb) ·ε < δ**. By (A.60) and (A.63) wehave that W (·) is not willing to accept the spread from fR(·) to fL(·), since

W(fL(·)

)−W

(fR(·)

)< (A.64)

Φx′′(Ak;f)−Φx′(Ak;f)−Φx′(Bk;f)+Φx(Bk;f)+ 12 ·η·(δ(fL, f)+δ(fR, f)

)≤ τ/n− η·

(λ(Ak)+λ(Bk)

)+ 1

2 ·η·2·(λ(Ak)+λ(Bk)

)= τ/n < 0

(a)⇒ (d): Say VW ∗(P) > VW ∗(P) where P differs from P by an x ← x′ →x′′ probability spread for x′′ x′ x. We can write P = (x, p ;x′, p′;x′′, p′′;


x1, p1; . . . ;xn, pn) and P = (x, p ;x′, p′;x′′, p′′;x1, p1; . . . ;xn, pn) for p > p,p′ < p′, p′′ > p′′ and p+p′+p′′ = p+p′+p′′ = p. Define the intervals℘ = [0, p ],℘′ = (p, p+ p′], ℘′′ = (p+ p′, p ], ℘ = [0, p ], ℘′ = (p, p+ p′], ℘′′ = (p+ p′, p ],℘1 = (p, p+ p1], ℘2 = (p+ p1, p+ p1 + p2], . . . , ℘n = (p+ p1 + . . .+ pn−1, 1],and define the almost-objective acts

fm(·)=[x on ℘×mS;x′ on ℘′×

mS;x′′ on ℘′′×

mS;x1 on ℘1×mS; . . . ;xn on ℘n×mS]

(A.65)

fm(·)=[x on ℘×mS;x′ on ℘′×

mS;x′′ on ℘′′×

mS;x1 on ℘1×mS; . . . ;xn on ℘n×mS]

Since limm→∞W*(fm(·)) = VW ∗(P) > VW ∗(P) = lim

m→∞W*(fm(·)), there exists

some m* and γ > 0 such that W*(fm(·)) −W*(fm(·)) > γ for each m ≥ m*.Select ε ∈ (0, 1) small enough so that both the following relationships hold(

1− 1− ε1 + ε

)·(

γ

p− p + (1 + ε) ·Φ∗x′,x,S

)<

γ/4p− p(

1 + ε

1− ε − 1)· (1 + ε) ·Φ∗x′,x,S <

γ/4p− p

(A.66)

For each m, define the parametrized familyfm

α,β(·) |α ∈ [p′′, p′′], β ∈ [p, p ]

by

(A.67)fmα,β(·) =

[x on [0, β]×

mS;x′ on (β, p−α]×

mS;x′′ on (p−α, p ]×

mS;

x1 on ℘1×mS; . . . ;xn on ℘n×mS]

so that for each m we have fmp′′, p(·) = fm(·) and fm

p′′, p(·) = fm(·).By an argument similar to that in Machina (2004, pp. 39-41) we can select largeenough m** such that for any m ≥ m**, the partial derivatives ∂W (fm

α,β(·))/∂αand ∂W*(fm

α,β(·))/∂α exist and satisfy∣∣∣∂W (fmα,β(·)

)/∂α−

[Φx′′(S; fm

α,β)− Φx′(S; fmα,β)]∣∣∣

< ε · Φx′′,x′,S < ε ·[Φx′′(S; fm

α,β)− Φx′(S; fmα,β)]

(A.68)∣∣∣∂W*(fm

α,β(·))/∂α−

[Φ∗x′′(S; fm

α,β)− Φ∗x′(S; fmα,β)]∣∣∣

< ε · Φ∗x′′,x′,S < ε ·[Φ∗x′′(S; fm

α,β)− Φ∗x′(S; fmα,β)]

at all but a finite set of values of α (where these values are independent of β), andthe partials ∂W (fm

α,β(·))/∂β and W*(fmα,β(·))/∂β exist and satisfy∣∣∣∂W (fm

α,β(·))/∂β −

[Φx(S; fm

α,β)− Φx′(S; fmα,β)]∣∣∣

< ε · Φx′,x,S < ε ·[Φx′(S; fm

α,β)− Φx(S; fmα,β)]

(A.69)∣∣∣∂W*(fm

α,β(·))/∂β −

[Φ∗x(S; fm

α,β)− Φ∗x′(S; fmα,β)]∣∣∣

< ε · Φ∗x′,x,S < ε ·[Φ∗x′(S; fm

α,β)− Φ∗x(S; fmα,β)]

at all but a finite set of values of β (where these values are independent of α).Since ε < 1 this implies ∂W (fm

α,β(·))/∂α and ∂W*(fmα,β(·))/∂α are both positive

56 M.J. Machina

and ∂W (fmα,β(·))/∂β and ∂W*(fm

α,β(·))/∂β are both negative except at thesevalues, soW*(fm

α,β(·)) andW (fmα,β(·)) are each strictly increasing inα and strictly

decreasing in β.

Select arbitrary m > maxm*,m** and define γm = W*(fm(·))−W*(fm(·))> γ. For each β ∈ [p, p ], define α(β) as the unique solution to W*(fm

α(β),β(·))= W*(fm(·)) + γm·(β − p)/(p − p), so the mapping α(·): [p, p ]→ [p′′, p′′] iscontinuous, strictly increasing, onto, and α′(β) exists and satisfies

α′(β)·∂W*(fm

α(β),β(·))/∂α + ∂W*

(fm

α(β),β(·))/∂β =

γm

p− p > 0 (A.70)

at all but a finite number of values of β. By (A.69) we have

∂W∗(fm

α(β),β(·))/∂β > −(1 + ε)·

[Φ∗x′(S; fm

α(β),β)−Φ∗x(S; fmα(β),β)

](A.71)> −(1 + ε) ·Φ∗x′,x,S

and thus by (A.70), that

α′(β) ·∂W*(fm

α(β),β(·))/∂α = γm/(p− p) − ∂W*

(fm

α(β),β(·))/∂β

(A.72)< γm/(p− p) + (1 + ε) ·Φ∗x′,x,S

at all but a finite number of values of β. By (A.70), (A.71), (A.72), (A.66) andγm > γ, we have

1− ε1 + ε

·α′(β) ·∂W*(fmα(β),β)/∂α +

1 + ε

1− ε ·∂W*(fmα(β),β)/∂β

= α′(β) ·∂W*(fmα(β),β)/∂α + ∂W*(fm

α(β),β)/∂β (A.73)

−(1− 1−ε

1+ε

)·α′(β) ·∂W*(fm

α(β),β)/∂α+(1+ε

1−ε − 1)·∂W*(fm

α(β),β)/∂β

>γm

p−p −(1− 1−ε

1+ε

)·( γm

p−p+(1+ε)·Φ∗x′,x,S

)−(1+ε

1−ε −1)·(1+ε)·Φ∗x′,x,S

>γ

p−p −(1−1−ε

1+ε

)·( γ

p−p + (1+ε)·Φ∗x′,x,S

)− γ/4p−p >

γ/2p−p > 0

The above inequalities and (57) imply that at all but a finite number of values of β,we have:

α′(β)·∂W(fm

α(β),β(·))/∂α+ ∂W

(fm

α(β),β(·))/∂β

> (1−ε)·α′(β)·[Φx′′(S; fm

α(β),β)− Φx′(S; fmα(β),β)

]−(1+ε)·

[Φx′(S; fm

α(β),β)− Φx(S; fmα(β),β)

]=[Φx′′(S; fm


]·[(1−ε)·α′(β)− (1+ε)·

Φx′(S; fmα(β),β)− Φx(S; fm

α(β),β)Φx′′(S; fm


]

≥[Φx′′(S; fm


](A.74)

·[(1−ε)·α′(β)− (1+ε)·

Φ∗x′(S; fmα(β),β)− Φ∗x(S; fm

α(β),β)Φ∗x′′(S; fm

α(β),β)− Φ∗x′(S; fmα(β),β)

]


=(1− ε) · α′(β) · [Φ∗x′′(S; fm

α(β),β)− Φ∗x′(S; fmα(β),β)][

Φ∗x′′(S; fmα(β),β)− Φ∗x′(S; fm

α(β),β)]/[Φx′′(S; fm


]

−(1 + ε) · [Φ∗x′(S; fm

α(β),β)− Φ∗x(S; fmα(β),β)][

Φ∗x′′(S; fmα(β),β)− Φ∗x′(S; fm

α(β),β)]/[Φx′′(S; fm


]

>

1−ε1+ε

·α′(β)·∂W*(fm

α(β),β)

∂α+

1+ε1−ε ·

∂W*(fmα(β),β)

∂βΦ∗x′′(S; fm

α(β),β)− Φ∗x′(S; fmα(β),β)

Φx′′(S; fmα(β),β)− Φx′(S; fm

α(β),β)

>γ/2p− p ·

Φx′′,x′,S

Φ∗x′′,x′,S

We thus have

W(fm(·)

)−W

(fm(·)

)= W

(fm

α(p),p(·))−W

(fm

α(p),p(·))

=

∫ p

p

dW (fmα(β),β)

dβ·dβ =

∫ p

p

[α′(β)·

∂W (fmα(β),β)

∂α+∂W (fm

α(β),β)

∂β

]·dβ

> (γ/2)·(Φx′′,x′,S/Φ

∗x′′,x′,S

)> 0 (A.75)

Since m was an arbitrary integer greater than maxm*,m** we havelim

m→∞W (fm(·)) > limm→∞W (fm(·)), and hence that VW (P) > VW (P).

(a)⇒ (e): Given events A,B satisfying the stated likelihood properties, arbitraryx′′ x′ x and arbitrary f(·), define fL(·) = [x′′ on A; x on B; f(·) else-where], fR(·) = [x′ on A ∪B; f(·) elsewhere] and γ = W*(fL(·))−W*(fR(·))> 0. Select positive ε < min1/4, γ/(16 ·Φ∗x′,x,B), so that 2 · ε/(1 + ε) < 1/2and (4 ·ε/(1− ε)) ·Φ∗x′,x,B < 8 ·ε ·Φ∗x′,x,B < γ/2.

For each m, define the parametrized family of acts fmα,β(·)|α, β ∈ [0, 1] by

fmα,β(·) ≡

[x′′ on [0, α]×

mA; x′ on (α, 1]×

mA;

(A.76)x on [0, β]×

mB; x′ on (β, 1]×

mB; f(·) elsewhere

]so that fm

1,1(·) = fL(·) and δ(fm0,0, fR) = 0 for each m. The likelihood prop-

erties of A and B imply both [Φ∗x′′(A; fmα,β) − Φ∗x′(A; fm

α,β)]/[Φ∗x′′(S; fmα,β) −

Φ∗x′(S; fmα,β)]≤ pa≤ [Φx′′(A; fm

α,β)−Φx′(A; fmα,β)]/[Φx′′(S; fm

α,β)−Φx′(S; fmα,β)]

and [Φ∗x′(B; fmα,β) − Φ∗x(B; fm

α,β)]/[Φ∗x′(S; fmα,β) − Φ∗x(S; fm

α,β)] ≥ pb ≥[Φx′(B; fm

α,β)−Φx(B; fmα,β)]/[Φx′(S; fm

α,β)−Φx(S; fmα,β)] for all m and all α, β.

By an argument similar to that in Machina (2004, pp. 39-41) we can select a largeenoughm such that the partial derivatives ∂W (fm

α,β(·))/∂α and ∂W*(fmα,β(·))/∂α

exist and satisfy∣∣∣∂W (fmα,β(·)

)/∂α−

[Φx′′(A; fm

α,β)− Φx′(A; fmα,β)]∣∣∣

< ε · Φ x′′,x′,A < ε ·[Φx′′(A; fm

α,β)− Φx′(A; fmα,β)]

(A.77)∣∣∣∂W*(fm

α,β(·))/∂α−

[Φ∗x′′(A; fm

α,β)− Φ∗x′(A; fmα,β)]∣∣∣

58 M.J. Machina

< ε · Φ∗x′′,x′,A < ε ·[Φ∗x′′(A; fm

α,β)− Φ∗x′(A; fmα,β)]

at all but a finite number of values of α (where these values are independent of β),and the partials ∂W (fm

α,β(·))/∂β and ∂W*(fmα,β(·))/∂β exist and satisfy∣∣∣∂W (fm

α,β(·))/∂β −

[Φx(B; fm

α,β)− Φx′(B; fmα,β)]∣∣∣

< ε · Φ x′,x,B < ε ·[Φx′(B; fm

α,β)− Φx(B; fmα,β)]

(A.78)∣∣∣∂W*(fm

α,β(·))/∂β −

[Φ∗x(B; fm

α,β)− Φ∗x′(B; fmα,β)]∣∣∣

< ε · Φ∗x′,x,B < ε ·[Φ∗x′(B; fm

α,β)− Φ∗x(B; fmα,β)]

at all but a finite number of values of β (where these values are independent of α).Since ε < 1/4, this implies that ∂W (fm

α,β(·))/∂α and ∂W*(fmα,β(·))/∂α are each

positive and ∂W (fmα,β(·))/∂β and ∂W*(fm

α,β(·))/∂β are each negative except atthese values, so that W*(fm

α,β(·)) and W (fmα,β(·)) are each strictly increasing in α

and strictly decreasing in β.

For each β ∈ [0, 1], let α(β) be the unique solution to W*(fmα(β),β(·)) =

W*(fm0,0(·)) + γ ·β, so the mapping α(·) : [0, 1] → [0, 1] is continuous, strictly

increasing, onto, and α′(β) exists with

α′(β) ·∂W*(fm

α(β),β(·))/∂α + ∂W*

(fm

α(β),β(·))/∂β = γ > 0 (A.79)

at all but a finite number of values of β. Since (A.78) implies ∂W*(fmα(β),β(·))/∂β

>−(1+ε) · [Φ∗x′(B; fmα(β),β)−Φ∗x(B; fm

α(β),β)] > −(1+ε) ·Φ∗x′,x,B ,we also have

α′(β) ·∂W*(fmα(β),β(·))/∂α = γ − ∂W*(fm

α(β),β(·))/∂β < γ+(1+ ε) ·Φ∗x′,x,B

at all but a finite number of values of β.

The above inequalities imply that at all but a finite number of values of β, we have1− ε1 + ε

·α′(β)·∂W*(fm

α(β),β(·))/∂α +

1 + ε

1− ε ·∂W*(fm

α(β),β(·))/∂β

= α′(β)·∂W*(fm

α(β),β(·))/∂α + ∂W*

(fm

α(β),β(·))/∂β

− 2 · ε1 + ε

·α′(β)·∂W*(fm

α(β),β(·))/∂α+

2 · ε1− ε ·∂W*

(fm

α(β),β(·))/∂β

> γ − 2 · ε1 + ε

·[γ + (1 + ε)·Φ∗x′,x,B

]− 2 · ε

1− ε ·(1 + ε)·Φ∗x′,x,B

= γ − 2 · ε1 + ε

·γ − 2 · ε·Φ∗x′,x,B − 2 · ε · 1 + ε

1− ε ·Φ∗x′,x,B

= γ − 2 · ε1 + ε

·γ − 4 · ε1− ε ·Φ

∗x′,x,B > γ − γ/2− γ/2 = 0

(A.80)

The above inequalities, the likelihood properties of A and B, and (57) then implythat at all but a finite number of values of β, we have

α′(β)·∂W(fm

α(β),β(·))/∂α + ∂W

(fm

α(β),β(·))/∂β

> (1− ε)·α′(β)·[Φx′′(A; fmα(β),β)− Φx′(A; fm

α(β),β)]

−(1 + ε)·[Φx′(B; fmα(β),β)− Φx(B; fm

α(β),β)]


=

(1− ε)·α′(β)·Φx′′(A; fm

α(β),β)− Φx′(A; fmα(β),β)

Φx′(B; fmα(β),β)− Φx(B; fm

α(β),β)− (1 + ε)

1/[Φx′(B; fmα(β),β)− Φx(B; fm

α(β),β)]

≥(1− ε)·α′(β)· pa

pb·Φx′′(S; fm


Φx′(S; fmα(β),β)− Φx(S; fm

α(β),β)− (1 + ε)


α(β),β)]

≥(1− ε)·α′(β)· pa

pb·Φ∗

x′′(S; fmα(β),β)− Φ∗

x′(S; fmα(β),β)

Φ∗x′(S; fm

α(β),β)− Φ∗x(S; fm

α(β),β)− (1 + ε)


α(β),β)]

≥(1− ε)·α′(β)·

Φ∗x′′(A; fm

α(β),β)− Φ∗x′(A; fm

α(β),β)

Φ∗x′(B; fm

α(β),β)− Φ∗x(B; fm

α(β),β)− (1 + ε)


α(β),β)](A.81)

=(1− ε) · α′(β) · [Φ∗

x′′(A; fmα(β),β)− Φ∗

x′(A; fmα(β),β)]

[Φ∗x′(B; fm


α(β),β)]/[Φx′(B; fmα(β),β)− Φx(B; fm

α(β),β)]

−(1 + ε) · [Φ∗

x′(B; fmα(β),β)− Φ∗

x(B; fmα(β),β)]

[Φ∗x′(B; fm



α(β),β)]

>

1−ε1+ε

·α′(β)·∂W*(fm

α(β),β)

∂α+

1+ε1−ε ·

∂W*(fmα(β),β)

∂β

[Φ∗x′(B; fm



α(β),β)]> 0

Since α′(β) ·∂W (fmα(β),β(·))/∂α + ∂W (fm

α(β),β(·))/∂β > 0 at all but a finite

number of values of β, we have

W(fL(·)

)−W

(fR(·)

)= W

(fm1,1(·)

)−W

(fm0,0(·)

)=

(A.82)∫ 1

0

dW (fmα(β),β)

dβ· dβ =

∫ 1

0

[α′(β)·

∂W(fmα(β),β)

∂α+∂W(fm

α(β),β)

∂β

]·dβ > 0

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