A representation of partially ordered preferences

Carnegie Mellon UniversityResearch Showcase

Department of Statistics College of Humanities and Social Sciences

12-1-1995

A Representation of Partially Ordered PreferencesTeddy SeidenfeldCarnegie Mellon University, [email protected]

Mark J. SchervishCarnegie Mellon University, [email protected]

Joseph B. KadaneCarnegie Mellon University, [email protected]

This Article is brought to you for free and open access by the College of Humanities and Social Sciences at Research Showcase. It has been acceptedfor inclusion in Department of Statistics by an authorized administrator of Research Showcase. For more information, please [email protected].

Recommended CitationSeidenfeld, Teddy; Schervish, Mark J.; and Kadane, Joseph B., "A Representation of Partially Ordered Preferences" (1995).Department of Statistics. Paper 22.http://repository.cmu.edu/statistics/22

http://repository.cmu.edu

http://repository.cmu.edu/statistics

http://repository.cmu.edu/hss

mailto:[email protected]

The Annals of Statistics1995, Vol. 23, No.6, 2168-2217

A REPRESENTATION OF PARTIALLYORDERED PREFERENCES1

By TEDDY SEIDENFELD, MARK J. SCHERVISH AND JOSEPH B. KADANE

Carnegie Mellon University

This essay considers decision-theoretic foundations for robustBayesian statistics. We modify the approach of Ramsey, de Finetti, Savageand Anscombe and Aumann in giving axioms for a theory of robustpreferences. We establish that preferences which satisfy axioms for robustpreferences can be represented by a set of expected utilities. In thepresence of two axioms relating to state-independent utility, robust preferences are represented by a set of probability/utility pairs, where theutilities are almost state-independent (in a sense which we make precise).Our goal is to focus on preference alone and to extract whatever probability and/or utility information is contained in the preference relation whenthat is merely a partial order. This is in contrast with the usual approachto Bayesian robustness that begins with a class of "priors" or "likelihoods,"and a single loss function, in order to derive preferences from theseprobabilityfutility assumptions.

1. Introduction and overview.

1.1. Robust Bayesian preferences. This essay is about decision-theoreticfoundations for robust Bayesian statistics. The fruitful tradition of Ramsey(1931), de Finetti (1937), Savage (1954) and Anscombe and Aumann (1963)seeks to ground Bayesian inference on a normative theory of rational choice.Rather than accept the traditional probability models and loss functions asgiven, Savage is explicit about the foundations. He axiomatizes a theory ofpreference using a binary relation over acts, Al :$ A 2 , "act Al is not preferred to act A 2 ." Then, he shows that :$ is represented by a uniquepersonal probability (state-independent) utility pair according to subjectiveexpected utility. That is, he shows there exists exactly one pair (p, U) suchthat, for all acts Al and A 2 , Al :$A2 if and only if Ep,u[A I ] ~ Ep,u[A2 ].

[More precisely, in Savage's theory what is needed to justify the assertionthat p is the agent's personal probability is the added assumption that eachconsequence has a constant value in each state. Unfortunately this is ineffa-

Received November 1992; revised June 1995.IThe research of Teddy Seidenfeld was supported by the Buhl Foundation and NSF Grant

SES-92-08942. Mark Schervish's research on this project was supported through ONR ContractN00014-91-J-1024 and NSF Grant DMS-88-05676. Joseph Kadane's research was supported byNSF grants SES-89-0002 and DMS-90-05858, and ONR contract N00014-89-J-1851.

AMS 1991 subject classifications. Primary 62C05; secondary 62A15.Key words and phrases. Robust statistics, axioms of decision theory, state-dependent utility,

partial order.

2168

https://www.researchgate.net/publication/246631129_'La_Prevision_Ses_Lois_Logiques_Ses_Sources_Subjectives'?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=

PARTIALLY ORDERED PREFERENCES 2169

ble in Savage's language of preference over acts. [See Schervish, Seidenfeldand Kadane (1990)]. We discuss this in Section 4, below.

In recent years, either under the headings of Bayesian robustness [Berger(1985), Section 4.7; Hartigan (1983), Chapter 12; Kadane (1984)] or sensitivity analysis [Rios Insua (1990)] it has become an increasingly important issueto show how to arrive at Bayesian conclusions from logically weaker assumptions than are required by the traditional Bayesian theory. Given data and aparticular likelihood from a statistical model, for example, how large a classof prior probabilities leads to a class of posterior probabilities that are inagreement about some event of interest? Our work differs from the commontrend in Bayesian robustness in much the same way that Savage's workdiffers from the traditional use of probability models and loss functions inBayesian decision theory. Our goal is to axiomatize robust preferences directly, rather than to robustify given statistical models. Results in this theoryare strikingly different from those obtained in the existing Bayesian robustness literature.

For an illustration of the difference, suppose two Bayesian agents eachrank the desirability of Anscombe-Aumann ("horse lottery") acts accordingto hisfher subjective expected utility. ("Horse lotteries" are defined in Section 2.) Let (PI' UI) and (P2' U2) be the probabilityfutility pairs representingthese two decision makers and assume they have different beliefs and values:that is, assume PI * P2 and UI * U2. Denote by ~I and ~2 their respective (strict) preference relations, each a weak order over acts. Suppose nowour goal is to find those coherent (Anscombe-Aumann) preference relations:$ corresponding to probabilityfutility pairs (p, U) such that the followingPareto condition applies:

If Ep1,uJAI] < Ep1,uJA2] and Ep2 ,u2[A I] < Ep2 ,u2[A2], then Ep,u[AI] <Ep,u[A2]. In words, when both agents strictly prefer act A 2 to act AI' thenthis shared preference is robust for all efficient, cooperative Bayesian decisions that the pair make together. [We assume that though the two Bayesianagents may discuss their individual preferences, nonetheless, some differences remain in their beliefs and in their values even after such conversations. See DeGroot (1974) for a rival model.] We have the following theorem.

THEOREM 1 [Seidenfeld, Kadane and Schervish (1989)]. Assume thereexists a pair of prizes {r *' r*} which the two agents rank in the same order:r * ~i r* (i = 1,2). Then the set of probability futility pairs, each of whichsatisfies the Anscombe-Aumann theory and each of which agrees with thestrict preferences shared by these two decision makers, consists exactly of thetwo pairs themselves {(PI' UI), (P2' U2)}. There are no other coherent, Paretocompromises. [There is no coherent weak order meeting the strong Paretocondition, which requires that Al ~ A 2 ifAl ~i A 2 (i = 1, 2) and at least oneof these two preferences is strict.]

Thus, with respect to Pareto-robust preferences, the set of probabilityfutility pairs for the problem of two distinct Bayesians is not connected and

https://www.researchgate.net/publication/33776606_Sensitivity_analysis_in_Multi-Objective_Decision_Making_NEMS?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=

https://www.researchgate.net/publication/220688406_Statistical_Decision_Theory_and_Bayesian_Analysis?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=


2170 T. SEIDENFELD, M. J. SCHERVISH AND J. B. KADANE

therefore not convex. Hence, a common method of proof-separating hyperplanes (used to develop expected utility representations)-is not available inour investigation. This is just one way in which our methods differ from theusual robust Bayesian analysis. We want the strict preferences held incommon by two Bayesians to be a special case of robust preferences. Appliedto a class of weak orders, the Pareto condition creates a strict partial order~ : to wit, the binary relation ~ is irreflexive and transitive.

Our view of robustness is that sometimes a person does not have a (strict)preference for act Al over act A 2 nor for A 2 over act Al nor are theyindifferent options. Assume that strict preference is a transitive relation.Then such a person's preferences are modeled by a partial order. We ask,under what assumptions on this partial order is there a set ofprobabilityfutility pairs agreeing with it according to expected utility theory,a set which characterizes that partial order? In this we are exploring thepossibility pointed out by Savage [(1954), page 21].

The general form of our inquiry is as follows. Axiomatize coherent preference ~ as a partial order and establish a representation for it in terms of aset of probabilityfutility pairs. That is, we characterize each coherent, partially ordered preference ~ in terms of the set of coherent weak orders {:s}that extend it. We rely on the usual expected utility theory to depict each.coherent weak order :S by one probabilityfutility pair (p, U). Thus, wemodel ~ by a set of probabilityfutility pairs.

In contrast with Savage's theory, which uses only personal probability, ourapproach is based on Anscombe-Aumann's "horse lottery" theory. Preferences over "horse lotteries" accommodate both personal and extraneous(agent-invariant) probabilities. Also, by characterizing strict preference interms of a set of probabilityfutility pairs, we improve so-called one-wayrepresentations of, for example, Fishburn [(1982), Section 11], as we showmore than existence of an agreeing probabilityfutility pair.

1.2. Overview. In outline, our approach is as follows: In Section 2 weintroduce axioms for a partial order over Anscombe-Aumann horse lotteries(HL). Anscombe-Aumann theory contains three substantive axioms thatincorporate the (von Neumann-Morgenstern) theory of cardinal utility forsimple acts:

1. A postulate that preference (:S) is a weak order-analogous to Savage'sPl.

2. The independence postulate-analogous to Savage's P2, "sure thing."3. An Archimedean condition, which plays an analogous role to Savage's P6.

Ou~ replacement axioms for these are:

HL AxIOM 1. A postulate that strict preference (~) is a strict partialorder.

PARTIALLY ORDERED PREFERENCES

HL AxIOM 2. The independence postulate.

2171

HL AxIOM 3. A modified Archimedean axiom for discrete (not just simple)lotteries.

To avoid triviality, a commonplace assumption of expected utility theory isthat not all acts are indifferent, for example, there exist two acts, Wand Bthat do not satisfy B :S W. Let ~ obey our (three) preference axioms on adomain of horse-lottery acts HR. We show (Theorem 2) how to extend ~ to apreference ~'over a larger domain that includes two new acts B (best) andW (worst) where

Then we establish three related theorems (Theorems 3, 4 and 5): ~ isrepresented by a nonempty (maximal and convex) set r= {V: H R ~ (0, I)} ofbounded, real-valued cardinal utilities V(·) defined for acts. Each V E rinduces a weak order :SV that agrees with ~ on the domain of simple actsand almost agrees (Definition lOb) with ~ on all acts. Moreover, given a set.z of bounded, real-valued cardinal (so-called) "linear" utilities, Z(·) definedon H R , the partial order formed using the Pareto condition with the set .zsatisfies our three axioms for preference.

In the light of the surprising "shape" that the family of agreeing subjective expected utilities can have (Theorem 1), we employ a modificationof Szpilrajn's (1930) transfinite induction for extending a partial order. Weshow how to extend a partial order while preserving the other preferenceaxioms. The proofs of all results appear in the Appendix. Also, we numberdefinitions, lemmas, and corollaries to coincide with their logical order in ourarguments, regardless of whether they appear for the first time in the body ofthe text or in the Appendix.

In Section 4 we turn our attention to the representation of r as a set ofsubjective expected utilities. We discuss when a linear utility V over acts alsois a subjective expected utility for a probabilityjutility pair (p, U). Corollary4.1 gives a representation of r in terms of sets of probabilityjstatedependent utility pairs (p, {Uj: j = 1, ... , n}), where the utility Uj(L) of a(von Neumann-Morgenstern) lottery L may depend upon the accompanyingstate Sj. (This follows up the issue raised in the first paragraph in Section1.1.) In Section 4.3 we' introduce two axioms (HL Axioms 4 and 5) thatparallel the fourth Anscombe-Aumann postulate. That postulate (and ourreplacements for it) permits a representation of preference using a (nearly)state-independent utility: where (with high personal probability) the value ofa lottery L does not depend (by more than amount 8 > 0) upon the state inwhich it is awarded. In Section 4.3 we lean heavily on the proof technique ofSection 3 in order to find a representation for the partial order ~ in terms ofa set of agreeing pairs of probabilities and (nearly) state-independent utilities, Lemma 4.3 and Theorem 6.


Section 5 is about conditional preference. Two theorems (Theorems 7 and8) relate conditional (called-off) partially ordered preferences and Bayesianupdating of the family of unconditional personal probabilities that agree withan unconditional partially ordered preference. We provide an example involving conditional probability that highlights the nonconvexity of the agreeingsets. In Section 6 we conclude with a review of several features that distinguish our results.

2. The formal theory.

2.1. The act space: a domain for the preference relation. We provide arepresentation for a partially ordered strict preference relation over (discrete)Anscombe-Aumann (1963) horse lotteries-acts that generalize vonNeumann-Morgenstern (1947) lotteries to allow for uncertainty over states ofnature.

Let R be a set of rewards. We develop our theory for countable sets R.

DEFINITION 1. A simple (von Neumann-Morgenstern) lottery is a simpleprobability distribution P over R, that is, a distribution with finite support. Adiscrete lottery is a countably additive probability over R (with a countablesupport). Denote a lottery by L and its distribution by P.

Horse lotteries are defined with respect to a finite partition of states. Let 7T

be a finite partition of the sure event S into n disjoint, mutually exhaustivenonempty (sets of) states, 7T = {Sl' ••• ' sn: Si n Sj = 0 iff i =1= j and U j5. n(Sj)

= S}. Strictly speaking, elements of 7T are subsets of S. We take thisapproach rather than supposing S is finite, for example, rather than assuming S = {Sl' ••• ' sn}. Then our analysis allows for elaborations of a givenpreference relation in a larger domain of acts defined over (finite) refinementsof the partition 7T. Having made this point, we allow ourselves the familiarconvention of equating the set state Sj with its elements. That is, fornotational convenience, often we shall use Sj when we intend "members ofS ·"J"

DEFINITION 2. A simple (or discrete) horse lottery is a function from statesto simple (or to discrete) lotteries. Denote a horse lottery by H and denote thespace of (discrete) horse lotteries on the reward set R by HR.

In the tradition where acts are functions from states to outcomes, a horselottery is an act with a lottery outcome. For example, the act that yields a50-50 chance at $10 and $20 provided the Republicans win the next Presidential election, and which yields a 0.25 chance at $5 and a 0.75 chance at$10 if the Republicans do not win, is a horse lottery over a binary partitionwith two states: Republicans win and Republicans do not win. Thus, aconstant horse lottery is just a von Neumann-Morgenstern lottery, and a


proper subset of these are the constant von Neumann-Morgenstern lotteries,that is, the acts which yield a specific reward for certain.

Next, we define the operation of convex combination of two horse lotteries,"+", as the state-by-state mixture of their respective v.N-M lottery outcomes.Thus:

DEFINITION 3. xH1 + (1 - x)H2 = H 3 = {XLlj + (1 - x)L2j : j == 1, ... , n;o ~ x ~ I}.

The mixture of two lotteries is a lottery XLI + (1 - x)L2 = L 3 , whereP 3(r) = xP1(r) + (1 - x)P2(r). For the special cases where each H is a"constant" act, that is, if HI is the lottery L 1 and H 2 is L 2, Definition 3coincides with the von Neumann-Morgenstern operation of"+" for lotteries.

2.2. The axioms for order and independence. The von NeumannMorgenstern theory of preference over (simple) lotteries is encapsulated bythree axioms:

1. The assumption that preference :S is a weak order relation.2. The independence postulate (formulated below).3. An Archimedean condition (discussed below).

These axioms may be applied to horse lotteries also. Then the three axiomsguarantee that (i) preference is represented by a (cardinal) utility V over actswith a property (ii) that utility distributes over convex combinations. To wit,given these three axioms:

(i) There exists a real~value V defined on acts, unique up to positive lineartransformations, where V(H1) ~ V(H2) if and only if HI :S H 2; and

(ii) V[ xH1 + (1 - x)H2] = xV(H1) + (1 - x)V(H2).

DEFINITION 4. When a preference relation over acts satisfies (i) and (ii) wesay it has the expected (or linear) utility property and we call V an agreeingexpected (or, linear) utility for :S.

Anscombe-Aumann theory requires a fourth postulate ensuring the existence of a unique decomposition of V as a subjective expected, state-independent utility. That is, subject to a fourth postulate for preference, there existsa (unique) personal probability p defined on states and a utility U defined onlotteries (independent of states) so that:

(iii)n

V(H) = E p(sj)U(Lj ).j=l

[Recall the notation H(sj) = L j .] Key, here, is that U is a state-independentutility, defined on lotteries independent of the state in which they occur. Tobe precise, let HL be the constant horse lottery that yields lottery L in eachstate, HL(sj) = L.


DEFINITION 5. The utility {Uj: j = 1, ... , n} is state-independent when, foreach lottery L and pair of states Sj and sj"

Uj(L) = Uj,(L) = U(L).

[For our purposes, and following the usual practice, the condition ofDefinition 5 is required only for states Sj and sj' that are not "null," Le., onlywhen p(Sj) =1= 0 is it worth restricting Uj in a decomposition of a linear utilityV.] If the utility is state-independent, for convenience, we drop the subscript(for states) and abbreviate it U. When (iii) obtains for a state-independentutility U, V(HL ) = U(L).

DEFINITION 6. When a preference relation over acts satisfies (i)-(iii) wesay it has the subjective expected (state-independent) utility property and wesay the pair (p, U) agrees with :s.

In contrast to (iii), a decomposition of V by a subjective (possibly) statedependent utility allows

n

(iii*) V(H) = E p(sj)Uj(Lj ).j=l

We examine such state-dependent decompositions in Section 4.1.Next, we propose versions of the first two Anscombe-Aumann axioms to

accommodate our theory of preference as a partial order. We postpone ourdiscussion of the Archimedean axiom to Section 2.4 to allow for a timelyaccount of "indifference" in Section 2.3.

In this paper, a partial order ~ identifies a strict preference relation.

HL AxIOM 1. ~ is a strict partial order. It is a transitive and irreflexiverelation on H R X HR.

DEFINITION 7. When neither HI ~ H 2 nor H 2 ~ HI' we say the twolotteries are incomparable by preference, which we denote as HI t-.w H 2 • Whent-.w is transitive-corresponding to a weak order-then the relation :s(standing for "~ or t-.w") identifies a weak preference relation. Hereafter, weshall mean by "preference" the strict preference relation.

HL AxIOM 2 (Independence). \:J (HI' H 2 and H 3) and for each 0 < x ~ 1,

xHI + (1 - x) H 3 ~ xH2 + (1 - x) H 3 if and only if H I ~ H 2 .

This version of independence may be used, also, as the second axiom in theAnscombe-Aumann theory or in the von Neumann-Morgenstern theory.

2.3. Indifference (~). Next, we define a (transitive) relation of indifference, ~, based on ~,whichwill playa central role in our extension of ~ toa weak order. [See Fishburn (1979), Exercises 9.1 and 9.4, pages 126-127, foradditional discussion.]


DEFINITION 8 (Indifference). HI ~ H 2 iff \:J H 3 , H 4 (0 < x =:; 1),

xHI + (1 - x)H3 ~ H 4 iff xH2 + (1 - x)H3 ~ H 4 •

2175

We establish several useful corollaries of the HL Axioms 1 and 2 aboutindifference.

That is, when two acts are indifferent, neither is preferred to the other.

COROLLARY 2.2. ~ is an equivalence relation.

COROLLARY 2.3. HI ~ H 2 if and only if \:J H 3 , H 4 , and 0 < x =:; 1,

xHI + (1 - x)H3 -< (>- )H4 iff xH2 + (1 - x)H3 -< (>- )H4 •

COROLLARY 2.4.

\:J 0 < x ~ 1, H, H I ~ H 2 iff xHI + (1 - x) H ~ xH2 + (1 - x) H.

Corollaries 2.3 and 2.4 establish important substitution properties forelements of the same indifference equivalence class.

2.4. The Archimedean axiom: continuity of preference. First, define convergence for acts. Let {Hn } be a denumerable sequence of horse lotteries.

DEFINITION 9. {Hn} converges to a lottery H*, denoted by {Hn } => H*,justin case the respective discrete lottery distributions {Pjn(.)} converge (pointwise) to the lottery distribution Pj*(.).

The third (Archimedean) axiom precludes infinitesimal degrees of preference. As we show in Theorem 4, it suffices for representing preferences bysets of agreeing real-valued utilities.

HL AxIOM 3. Let {Hn } => Hand {Mn} => M.

(a) If \:J n(Hn -< M n ) and (M -< N), then (H -< N).(b) If\:J n(Hn -< M n ) and (J -< H), then (J -< M).

The familiar Archimedean condition from Anscombe-Aumann theory (alsofrom von Neumann-Morgenstern theory), denoted ,here as Axiom 3*, is this:

AxIOM 3*. Whenever HI -< H 2 -< H 3 , 3 (0 < x, Y < 1), yHI + (1 - y)H3 -<H2 -< xHI + (1 - x)H3 •

However, Axiom 3* is overly restrictive for our purposes, as a simpleexample shows.


EXAMPLE 2.1. Consider a set of three rewards R = {rw ' r*, rb } and aminimal, one element partition comprising the single sure state 1T = {s}. ThenH R is the set of von Neumann-Morgenstern lotteries on R. (Denote by r thehorse lottery with constant prize r.) Let r= {Vx : 0 < x < I} be a (convex) setof linear utilities, where Vx(rw ) = 0, Vx(r*) = x and Vx(r b ) = 1. Figure 1graphs these utilities.

Let -<r be the partial order on lotteries generated by this set of utilitiesaccording to the weak Pareto condition. That is, L 1 -<r L 2 iff (\;f V E r)Ev [L 1 ] < Ev [L 2 ]. By Theorem 4 (below), -<r satisfies HL Axioms 1 and 2.However, it fails Axiom 3*. Specifically, r w -<r r* -<r r b , but \;f (0 < y < 1),yrw + (1 - y)r b "'r r*. Of course, -<r is represented by the (convex) set ofagreeing (real-valued) utilities r.

Next, we provide a connection between the Archimedean condition HLAxiom 3 and ~ -indifference.

COROLLARY 2.5. Let {Hn }, {H~} ~ H; {Mn }, {M~} ~ M. If \;f n(Hn -< M n

and M~ -< H~), then H :::: M.

We conclude this discussion of HL Axiom 3 by showing that the familiarArchimedean axiom, Axiom 3*, follows from our replacement HL Axiom 3when preference over lotteries is a weak order.

LEMMA 2.1. If ~ is a weak order over discrete horse lotteries meetingconditions HL Axioms 2 and 3, then ::5 satisfies Axiom 3*.

2.5. Utility for discrete lotteries. The theory of von Neumann and Morgenstern addresses preference over simple lotteries, that is, those with finitesupport. These constitute the subdomain of constant, simple horse lotteries.However, there are weakly ordered preferences over lotteries which satisfythe expected utility hypothesis (Definition 4) for simple lotteries, that is,which are represented by a cardinal, linear utility V over the domain ofsimple lotteries, but which fail the expected utility hypothesis over the largerdomain of discrete lotteries. [See Fishburn's (1979), Section 10, discussion;

1

Vx (r)

o

o < X

FIG. 1.< 1

PARTIALLY ORDERED ,PREFERENCES 2177

also Fishburn (1982), Section 11.3. A similar problem arises in Savage's(1954) theory, as shown by Seidenfeld and Schervish (1983). In a relatedmatter, Aumann's (1962, 1964) argument about a utility for a partiallyordered preference does not apply when the set of rewards is denumerablerather than finite, even though all lotteries are simple. Kannai (1963) showedthat, and strengthened Aumann's Archimedean condition to remedy theproblem.] We address this problem with an extended dominance condition.

Let r denote the simple horse lottery with constant prize r. Recall that H Ldenotes the constant horse lottery that yields L in each state. Consider thefollowing dominance principle:

DOMINANCE. 'V (r,HL ) if for each r n E supp(L), r n ~ r, then it is not thecase that (r ~ H L ) [or, alternatively, if universally, r ~ r n , then not (HL ~ r)].

This weak dominance condition contrasts each reward r with the lottery Lthrough the (countably many) constant horse lotteries r n taken from L'ssupport. The condition precludes a preference for Lover r if it occurs for rover each r n .

Our first three axioms yield dominance, as the next lemma establishes.

LEMMA 2.2. HL Axioms 1-3 entail dominance.

Based on Lemma 2.2, we may apply Fishburn's [(1979), page 139] Theorem10.5 to argue that a weakly ordered preference :s over discrete (horse)lotteries which satisfies our HL Axioms 2 and 3 has the expected utilityproperty. (See Remark 1.) The import for our representation theorems isgiven in terms of agreeing and almost agreeing utilities for a partial order:

DEFINITION lOa. A utility V agrees with a partial order ~ iff V[HI ] <V[H2 ] whenever (HI ~ H 2 )·

DEFINITION lOb. A utility V almost agrees with a partial order ~ iffV[HI ] ~ V[H2 ] whenever (HI ~ H 2 ).

Thus, when our strategy for extending a partial order ~ to a weak order:s succeeds, it induces a linear utility V that agrees with ~ for discretehorse lotteries, not just for simple ones.

Unfortunately, our argument for extending a partial order ~ produces aset of expected utilities {V} each of which agrees with ~ for simple acts, andonly almost agree with it for discrete acts. Of course, by itself the condition of"almost agreeing" is quite weak. A utility V that makes all options indifferentalmost agrees with every partially ordered preference. The point, however, isthat we consider an almost agreeing utility for a partial order ~ only in thecase in which it agrees with ~ on all simple acts. [This idea parallels asimilar distinction between a qualitative probability (a weak order on events)and a quantitative probability that agrees or almost agrees with it. See


Savage (1954), Section 3.3.] Through Corollary 3.2, we provide a sufficientcondition for the existence of a (convex) set of utilities that agree with ~ onall of HR.

REMARK 1. Fishburn's Theorem 10.5 uses the traditional Archimedeanaxiom Axiom 3*. However, by Lemma 2.2, Axiom 3* follows from the assumptions that :s is a weak order satisfying HL Axioms 2 and 3. Also, dominanceis equivalent to Fishburn's [(1979, page 138] Axiom 4c, given the other threeaxioms and our structural assumptions about the domain of lotteries.

2.6. Bounded preferences. Next in our discussion of the axioms, we settlethe question whether a partial order ~ satisfying HL Axioms 1-3 admits anunbounded utility that agrees with it or agrees with it on simple lotteries. Itis well known that utilities for von Neumann-Morgenstern lotteries arefinite. In light of our assumption that all discrete lotteries are acts, utilitiesthat agree with ~ are bounded as well.

COROLLARY 2.6. Let ~ satisfy HLAxioms 1 and 2. IfV agrees with ~,it

is bounded. Hence, all utilities that agree with ~ are bounded.

As noted above, we sometimes construct a utility V that agrees with apartial order ~ for all simple lotteries but (merely) almost agrees with ~

for discrete lotteries. Thus, as V may fail to agree with ~ on nonsimple acts,it is worthwhile to show (appealing only to simple acts) that each utility V weconstruct is bounded. For this purpose we formalize a condition that apartially ordered preference is bounded, and establish it as a corollary of twoof our axioms, HL Axioms 1 and 3. From the fact that a partial order ~ isbounded, we show that a utility V agreeing with it on simple acts also isbounded.

Call a countable (finite or denumerably infinite) sequence of lotteries {Hn :

n = I, ... } an increasing (decreasing) chain if Hi ~ Hj (Hj ~ Hi) wheneveri < j. The following concepts deal with chains of strict preference.

DEFINITION 11a. Say ~ is bounded above if, for each increasing chain{Hn },

lim sup{x: (H2 ~ xHl + (1 - x)Hn )} < 1.n-+ oo

DEFINITION lIb. Say ~ is bounded below if, for each decreasing chain{Hn },

lim sup{x: (xHl + (1 - x)Hn ~ H 2 )} < 1.n-+ 00

DEFINITION llc. Call ~ bounded if it is bounded both above and below.


LEMMA 2.3. If ~ satisfies HL Axioms 1 and 3, then ~ is bounded, thatis, all ~ -chains are bounded.

Also, Lemma 2.3 yields the following claim about utilities for rewards:

COROLLARY 2.7. Let W be a (real-valued) utility and assume that, in thedomain of simple lotteries, its strict order ~w satisfies HL Axioms 1 and 3.Then suPRIW(r)1 < 00.

Recall that a linear utility V is defined only up to a positive lineartransformation. We use the facts reported by Corollaries 2.6 and 2.7 tostandardize the units (0 and 1) for each V in a set of agreeing utilities r(agreeing with ~ on simple acts, at least).

DEFINITION lId. A set r= {V} of utilities is bounded if, for some standardization of its elements,

sup IV(H)I < 00.

W",H R

The problem we face is this. Though each V E r is bounded, there existwhat are for our purposes undesirable standardizations of the V's which failto satisfy Definition lId. For an illustration, recall Example 2.1. There, thedomain of (simple) lotteries H R is generated by three rewards R = {rw' r*, rb}using a partition of one (sure) state. That is, Example 2.1 is about preferencesover von Neumann-Morgenstern lotteries. A partially ordered preference ~r

over H R arises (by the Pareto rule) from the convex set of utilities r={Vx: 0 < x < I}, where Vx(rw ) = 0, Vx(r*) = x and Vx(rb) = 1. That is, HL1 ~r

HL2 iff V(HL1 ) < V(HL2 ) for each V E r.Obviously, the two constant acts (the rewards) r w and r b bound the partial

order ~r, that is, for each act HL different from r w and r b , r w ~r H ~r r b .

Moreover, in this standardization of r, sUPr,HRIV(H)1 = 1. Hence, it is abounded set of utilities. However, we may standardize the elements of r sothat it fails the condition in question. Rewrite each Vx , instead, so thatVx(rw ) = 0, Vx(r*) = 1 and VX(rb) = 1jX. Then limx~oVX(rb) = 00.

To ensure a simple standardization which establishes our rs are, indeed,bounded sets of utilities, we verify that (without loss of generality) we mayintroduce two rewards Wand B (analogous to rwand rb in Example 2.1) thatserve to bound the preferences for all other acts: Theorem 2. Then, the sets rare bounded sets of utilities since we standardize all V E r with V(W) = 0and V(B) = 1.

2.7. Standardizing ~ -preferences with "best" and "worst" acts. In thissection we show how to extend the domain of a partially ordered preferenceby bounding it with "worst" and "best" acts. First, however, we review twoconcepts of "null" events.


DEFINITION 12. An event e is the set of states in a subset T of 11': (\;f e) 3(T c 11'), e = U sET[S].

DEFINITION 13. Call HI and H 2 a pair of e-called-off acts when HI(s) =H 2(s) if s $. e.

Distinguish two senses of "null" events.

DEFINITION 14a. An event e is potentially null iff for each pair of e-calledoff acts HI and H 2 , HI I--.J H 2 •

DEFINITION 14b. Event e is essentially null iff for each pair of e-called-offacts HI and H 2 , HI z H 2 •

It is evident that when event e is essentially null, so too is each state thatcomprises it. Denote by n the union of the essentially null states. It follows(as is proven next) that the union of essentially null states is an essentiallynull event. Hence, n is the maximal essentially null event.

COROLLARY 2.8. Let N c 11' be the subset of all essentially null statesN = {Sjl"'" Sjk}' with n = U sE N(s). Then n is essentially null.

THEOREM 2. Assume « is a partially ordered preference (satisfying HLAxioms 1-3) over a set of discrete horse lotteries H R , defined on the partition1T n = {Sj: j = 1, ... , n}. Let R' = R U {W, B}, where neither W nor B is anelement of R. Then we may extend « to a partially ordered preference «'over H R , so that:

1. «'/HR = « . That is, «' restricted to H R is just «.2. \;f (H E H R ), W « 'H « , B.3. «' satisfies HL Axioms 1-3.

Since n = S iff « is trivial, that is, iff \;f (HI' H 2 ), HI ~ H 2 [also, iff \;f

(HI' H 2 ), HI I--.J H 2 ], without loss of generality, by Theorem 2, assume preference is not trivial by including rewards Wand B. (This proposition, warranted by Theorem 2, is the counterpart in our theory to Savage's P5.)

3. Extending strict partial orders: the inductive argument.

3.1. An overview. Let R = {r l , r 2 , ••• } be a countable (finite or denumerable) set of rewards and let « be a preference over-HR satisfying HL Axioms1-3. Based on Theorem 2, without loss of generality, assume the existence oftwo distinguished rewards not in R: reward W, where W is the worst act, andreward B, where B is the best act. Acts Wand B are to serve as the commonoand 1 in a (convex) set r of bounded utility functions V that agree with «.Hence for all H E H R , W « H « B.

Let us highlight the major results in this section of our essay.


Our strategy is to use a transfinite induction to extend the preference «(a partial order) to a weak order ::) over simple horse lotteries in HR. Let «(= «0) serve as the basis for the induction. The induction at the ith stageextension of «, «i' obtains by assigning a utility Vi to act iii' V(iii) = Vi' sothat iii ::::-i viB + (1 - vi)W. The quantity Vi is chosen (in accord with Definitions 20 and 25 in the Appendix) from a (convex) set of target utilities foriii' .9i(iii). The sequence {ii) is chosen (see Definition 26 in the Appendix) sothat the limit stage «w is a weak order over HR. We use Wand B as the 0and 1 of our utility, as follows.

Assume {Hn} ~ Hand H n E HR. The general target sets .9i(H) are defined through endpoints that bound the candidate utilities:

DEFINITION 17. Let v;(H) be the lim inf of the quantities X n for whichH n «i-I xnB + (1 - xn)W.

DEFINITION 18. Let Vi *(H) be the lim sup of the quantities X n for whichxnB + (1 - xn)W «i-I H n·

[The "utility" bounds V*(H) and V*(H) do not depend upon which sequence {Hn } ~ H is used, as explained in the Appendix.] Next, define the(closed) target set of utilities for an act H E H R :

DEFINITION 19. .9'(H) = {v: v*(H) ~ V ~ v*(H)}.

We report two key properties of .9'(H) with the following lemma.

LEMMA 3.1. Assume « satisfies the three axioms. Then:

(i) V*(H) ~ v *(H); and(ii) v *(H) = v*(H) = VH iffH ::::- vHB + (1 - VH)w.

Our plan succeeds because «i extends «i-I, it satisfies the three HLaxioms and it preserves the ::::-i_I-indifference relations. Each weak order::)y =::)w (corresponding to the limit stage relation" «w or ::::-w") is definedby inequalities in expected V-utility, based on the utilities Vi for each act iiiin a (finite or) denumerable class~ c HR. (As explained below,~ is finite ordenumerable depending upon whether R is.) We choose~ to form a basis for::)y, that is, each H E HR is a limit point of simple acts and each simple acthas its utility fixed by some finite stage of the transfinite induction. Then, ::)y

extends « on simple acts in HR. Also, the utility V almost agrees with «over the discrete lotteries HR. That is, if HI « H 2 , then HI::)y H 2 • InCorollary 3.1, we provide sufficient conditions under which ::)y extends «for all the acts in HR. (See Remark 2.)

We show in Theorem 4 that each set .z of (bounded, standardized)real-valued utility functions over R induces a partial order, «.z, according tothe Pareto preference relation, and «.z satisfies our axioms. Of course, each


utility Z E Y agrees with -<yo That is, Y is a subset of the set of all utilitiesagreeing with -<y. However [Seidenfeld, Schervish and Kadane (1990), E.3],distinct convex sets of bounded utilities may induce the same strict partialorder. Thus, our representation of the partial order -< is in terms of thelargest convex set of agreeing linear utilities-the union of all sets of utilitieswhere each set induces -< according to the Pareto condition.

Assume -< satisfies our axioms and let Y,7 be the nonempty (convex) setof bounded utilities that agree with -< for simple acts. That is, Y,7 is the setof all bounded utilities with the property that, for simple acts HI and H 2 ,

HI -< H 2 only if for each utility Z in Y,7, the expected Z-utility of H 2 isgreater than that of HI. Let r be the nonempty (convex) set of utilitiescreated for -< by (our method of induction in) Theorem 3. Theorem 5 asserts¢ =1= r=y,7. Last, when the conditions of Corollary 3.1 apply, then (Corollary3.2) r is the nonempty set of all utilities that agree with -<.

REMARK 2. If the Archimedean axiom is ignored and only simple lotteriesare considered, the induction for extending -< to a weak order (in fact, to atotal order) ::5 over H R is elementary and applies without a cardinalityrestriction on the reward set R and without need of the special acts Wand B.See the Appendix to Schervish and Kadane (1990). There, we show thefollowing: Let K be the cardinality of R. Using Hausner's (1954) result, theorder ::5 (= -<K ) is a lexicographic expected utility.

3.2. The central theorem.

THEOREM 3. Let -< be a nontrivial partial order over H R satisfying HLAxioms 1-3. Then:

(i) For simple lotteries in H R , -< can be extended to a weak order::5 w = ~ satisfying HL Axioms 2 and 3. That is, ::5 is uniquely representedby a (bounded) real-valued utility V over R which agrees with -< for simpleacts. In symbols, \;f (simple HI' H 2 E H R ), if (HI -< H 2 ), then Ey[HI ] <E y[H2 ], and if (HI ~ H 2 ), then Ey[HI ] = E y[H2 ].

(ii) V almost agrees with -<. \;f (HI' H 2 E H R ), if (HI -< H 2 ), then Ey[HI ]~ E y [H2 ]·

It is instructive to illustrate how ::5 w may fail to agree with -< for somenonsimple lotteries in HR. The example motivates a condition on -< whichproves sufficient for -<w to extend -<.

EXAMPLE 3.1. Let 7'"= {~: j = I, ... } be a countable set of utilities onR = {ri : i = I, ... } with the two properties that ~(rm) = 0.25 if m =1= 2j,while ~(r2j) = 0.5. According to Theorem 4 (below), under the (weak) Paretorule, Y induces a partial order -<or which satisfies our three horse lotteryaxioms. Define the constant, nonsimple acts H a and H b by H a = {peri) =


112 m if i = 2m - 1, peri) = 0 otherwise} and H b = {peri) = 112 m if i = 2m,peri) = 0 otherwise}. Then, evidently (Ha -<7 H b ). However, at the kth stage-<k in the extension of -<7' we may arrange our choices of utilities forrewards so that V(rk ) = 0.25 (k = 1, ... ). However, then ~w does not extend-<7 as H a ~w H b •

DEFINITION 28. Given two subsets of -< -preferences ~ and 9/, say that ~is a basis for 9/ if every preference, (HI < H 2 ) E9/, is a consequence (underHL Axioms 1-3) of preferences in ~.

COROLLARY 3.1. If there exists a countable basis /I for -<, then thereexists a (bounded) real-valued utility V and corresponding weak order ~

that agrees with -< on all of HR. (Thus, a sufficient condition for theexistence of an agreeing ~ is that -< is a separable partial order.)

Next, we show that our axioms are not overly restrictive for representing apartial order by a (convex) set of agreeing utilities. We investigate relationships between a partial order -<y (formed by the Pareto rule with a set % ofutilities) and the set r of utilities created by induction on -<y. Let % be aset of bounded utilities on R, standardized so that for Z E % and H E H R ,

o = Z(W) < Z(H) < Z(B) = 1. Define the relation -<yon H R by the Paretocondition:

THEOREM 4. -<y satisfies HL Axioms 1-3.

Next, let r be the set of utilities that can be generated from the partialorder -< according to our induction. Let %.7 be the set of all boundedutilities Z that agree with -< on simple lotteries.

Last, assume -< satisfies our axioms, let % be the set of all utilities thatagree with -< and let r be the set of 'utilities created by (our induction in)Theorem 3. We state three immediate corollaries of Theorem 5:

COROLLARY 3.2. When -< satisfies the separability condition of Corollary3.1, then ¢ =1= r= %.

COROLLARY 3.3. The set r does not depend upon the ordering of7?

COROLLARY 3.4. The set r is convex.


4. A representation of « in terms of probabilities and statedependent utilities.

4.1. The underdetermination of personal probability by HL Axioms 1-3.Let r be the set of utilities V, each of which (by Theorem 5) corresponds to alimit stage ::)y in our inductive extensions of the partial order «. Accordingto Theorem 5, r is the set of all and only utilities that agree with « onsimple acts. According to Corollary 3.2, when « is separable, r is the set ofutilities that agree with «. We examine decompositions of V Eras asubjective expected (state-dependent) utility.

Let ::) be a weak order over the discrete horse lotteries HR. Let p(.) be a(personal) probability defined on states in 1T, with pen) = 0 for the set of::) -null states n. Finally, let Uj(.) be a (possibly) state-dependent utility onthe discrete v.N-M lotteries L R , defined for the ::) -nonnull states Sj. That is,for each ::) -nonnull state, Sj $ n, Uj is a v.N-M utility. (For completeness, wemay take Uj to be a constant function when Sj En.)

DEFINITION 30. Say that ::) represented as a subjective expected (statedependent) utility by the pair (p, {Uj: j = 1, ... , n}), whenever

(4.1) HI ::)H2 iff LP(Sj)Uj(LIj ) ~ LP(Sj)Uj(L2j ).j j

For convenience, abbreviate the probabilityj(state-dependent) utility pairs as(p, Uj).

We rely on a result due to Fishburn [(1979), Theorem 13.1] to show thateach ::)y, V E r, bears the subjective expected utility property for a largeclass of (p, Uj) pairs. In fact, for each such ::)y, the (p, Uj) pairs range overall mutually absolutely continuous probabilities defined on the ::)y-nonnullstates. Specifically:

LEMMA 4.1. Let ::) be a (nontrivial) weak order on H R satisfying HLAxioms 2 and 3. For each probability p(.) with support the (nonempty) set of::) -nonnull states, there is a (possibly) state-dependent utility ~(.) on discretelotteries for which ::) has property (4.1) under (p, Uj).

Putting Theorem 5 and Lemma 4.1 together, we have the following corollary:

COROLLARY 4.1. There exists a set ofpairs {(p(.), Uj(.))}, with p a personalprobability defined on the set of ::)y-nonnull states in p and Uj a statedependent utility over the discrete lotteries, where \;f (H E H R ), V(H) =

LjP(Sj) X Uj(Lj).

[Being linear utilities, the Uj have the expected utility property for lotteries. That is, Uj(xL I + (1 - x)L2) = xUj(LI) + (1 - x)UjL2. Moreover, for eachV E r, the set of personal probabilities {p: 3 Uj with ::)y represented by(p,Uj)] is closed under the relation of mutual absolute continuity.]


4.2. State-independent utilities and a counterexample. Anscombe and Aumann's theory of horse lotteries introduces a fourth axiom which suffices for aunique expected utility representation of a weak order ::5 by a pair (p, U),with p a personal probability over states and U a state-independent utilityover rewards.

Recall Definition 5, when U is state-independent, a lottery L has the sameutility independent of the (nonnull) state Sj. Hence [as in Savage's (1954)theory], U assigns a constant utility across (nonnull) states to each "constant"act. In Anscombe and Aumann's theory, then (4.1) is strengthened to read:

(4.2) HI ::5 H 2 iff LP(Sj)U(LIj ):::; LP(Sj)U(L2j )j j

and each ::5 is so represented by a unique (p, U) pair.The existence of a state-independent utility for ::5 is assured through a

contrast between (unconditional) preferences over constant horse lotteriesand preferences over sj-called-off horse lotteries: pairs of acts that differ onlyin one state. Specifically, let HL (i = 1, 2) be two constant horse lotteries thataward, respectively, the v.N-M lottery L i in all states. Let Hi (i = 1,2) be twosj-called-off horse lotteries with Hi(sj) = L i [and HI(s) = H 2(s) for S -:/= Sj].The Anscombe-Aumann (AA) axiom for state-independent utility reads:

AA AxIOM 4. Provided Sj $ n, for each such quadruple of acts, H L1 ::5 H L2

iff HI ::5 H 2·

(Recall, their Axiom 1 stipulates that preferences are weakly ordered, ::5;hence, in their theory there is no difference between "potentially null" and"essentially null" states.)

This axiom requires that ::5 -preferences over "constant" acts (such as theH Ll ) are reproduced by called-off choices (the Hi) given each nonnull Sj. Theunconditional preference for v.N-M lotteries is their conditional (that is,called-off) preference, given a nonnull state. (We discuss conditional partiallyordered preferences in Section 5.)

It is significant to understand that AA Axiom 4, though sufficient to createstate-independent utilities when preference satisfies the usual ordering, independence and Archimedean condition~, does not preclude alternative expected utility representations by state-dependent utilities. Lemma 4.1 continues to apply, even in the presence of the extra axiom for state-independentutilities. Weak orderings that satisfy the independence, Archimedean andstate-independent utility axioms admit a continuum of different probabilityjutility representations, each in accord with (4.1).

What the Anscombe-Aumann fourth axiom achieves, however, is to guarantee that precisely one probabilityjutility pair, among the set of all pairs{(p, Uj)} indicated by Lemma 4.1, satisfies the more restrictive condition,(4.2). In Anscombe and Aumann's theory, as in Savage's theory, this probabilityjutility pair (p, U) is given priority over the others. That is, these theories


select the one (and only one) expected state-independent utility representation of preference, in accordance with (4.2) and, thereby, fix a personalprobability uniquely from :s -preferences.

We are not satisfied with a conventional resolution of the representationproblem indicated by Lemma 4.1. If state-dependent utilities are plausiblecandidates for an agent's values, and we think sometimes they are, then themeasurement question remains open despite the fourth axiom. What justification is there for a convention which gives priority to state-independentvalues? In two essays [Schervish, Seidenfeld and Kadane (1990, 1991)], weexamine the case of weakly ordered preferences without the extra axiom for"state-independent" utility. Here, instead, we adopt the strategy of imposinga modified Axiom 4 and asking which probabilityjstate-independent utilitypairs agree with the partial order «. Unlike the Anscombe-Aumann orSavage theories, ours does not assert that these pairs of probabilityj(stateindependent) utility functions identify the agent's degrees of beliefs andvalues.

We adapt Anscombe and Aumann's final axiom to our construction byrestricting it to states which are not potentially null. This produces thefollowing axiom:

HL AxIOM 4. If Sk is not « -potentially null, then for each quadruple ofacts H L , Hi (i = 1,2) as described above, H L « H L iff HI « H 2.

l 1 2

Suppose, « is a preference on horse lotteries subject to HL Axioms 1-4.Surprisingly, there may not exist a probability and state-independent utilityagreeing with « [according to (4.2)], even for simple acts. Moreover, theproblem has nothing to do with existence of potentially null states. That is,even if no state is potentially null, the fourth axiom (HL Axiom 4) isinsufficient for the existence of a probabilityjstate-independent utility pairagreeing with «.

EXAMPLE 4.1. Let R = {r *, r, r *} be three rewards and consider the setH R of horse lotteries defined on the binary partition {SI' S2}. Next, considertwo probabilityjutility pairs (pi, Ui) (i = 1, 2), where Ui(r *) = 0, Ui(r*) = 1,U 1(r) = 0.1 and U 2(r) = 0.4; also, pl(SI) = 0.1 and p2(SI) = 0.3. DefineHI « H 2 iff pi(SI)U i(L 1 1) + p'i(S2)Ui(L1 2) < pi(SI)U i(L 2 1) +pi(S2)u i(L2 2) (i = 1, 2). Then,' by Theorem 4, « 'satisfies HL Axiom~ 1-3,and we claim it satisfies HL Axiom 4 as well. Moreover neither state ispotentially null under «.

The proof that « satisfies HL Axiom 4 is straightforward. We observe thefollowing (expected utility) bounds on « -preferences for the constant horselottery r. (\;f 0.1 > e > 0), (0.9 + e)r* + (0.1 - e)r* « r « (0.6 - e)r* +(0.4 + e)r*. However, the utilities U i are state-independent and neither stateis null for either pi (i = 1, 2). That is, using conditional preference (seeDefinition 34, (\;f 0.1 > e> 0) (0.9 + e)r* + (0.1 - e)r* «Sj' r «Sj (0.6-


e)r * + (0.4 + e )r* (j = 1, 2). The utility bounds for r reproduce in bothfamilies of sj-called-off acts. Hence, « satis~es ~L Axiom 4.

According to Theorem 1, the two pairs (pl, Ul) are the sole state-independent expected utilities agreeing with « [according to (4.2)]. Next, we assertthat « may be extended to a strict partial order «", also satisfying HLAxioms 2-4, but where «" narrows the expected utility bounds for r, asfollows: 0.9r* + 0.1r* «" r «" 0.6r* + 0.4r*.

We outline a general result for extending « by forcing a new strictpreference HI «'H 2 , when H l ,....; H 2 • This contrasts with the extension created through Definition 20, which, instead, forces a new indifference relation.

Suppose HI and H 2 are elements of H R that satisfy (1) H l ,....; H 2 and (2)there do not exist two sequences {Hi n} => Hi (i = 1, 2), where \;f (n = 1, ... ),H 2, n « HI, n· Create an extension «; of « as follows:

DEFINITION ( « '). \;f (Ha, H b E H R ), Ha «'Hb if and only if either:

(i) H a « H b (so «' extends « )or

(ii) 3 {Ha,n} => Ha and 3 {Hb,n} => Hb and 3 {xn} with limn~oo{xn} =1= 1,

xnHa,n + (1 - x n)H2 « xnHb,n + (1 - x n)H1 •

CLAIM. «' satisfies HL Axioms 1-3, provided « does. Also, HI «'H 2 •

We omit the proof which follows along similar lines for the demonstrationof Lemma 3.3. Regarding HL Axiom 4, it suffices that «' is formed byextending « using a target set endpoint, for example, let HI = v *B + (1 v *)W, whereg-(H2 ) = [v *, v*] and this interval has interior, that is, v * < v*.Then «' satisfies HL Axiom 4 too.

Last, for Example 4.1, apply the claim, twice over, first to force 0.9r * +0.1r* «' r, then to force r «" 0.6r * + 0.4r*.

Consider the convex sets rand r" of agreeing utilities for « and «"provided by Corollary 3.2. (These utilities agree since R is finite.) Because«" extends «, then r" c r. A fortiori, each agreeing expected stateindependent utility model for «" also is one for «. However, by Theorem 1,there does not exist an agreeing expect,ed state-independent utility model forV" E r", since r" excludes all (that is, both) expected state-independentutility models for «. Nonetheless, «" satisfies HL Axioms 1-4. This endsour discussion of Example 4.1.

4.3. Representation of « in terms of (nearly) state-independent utilities.The four axioms HL Axioms 1-4 are insufficient for the existence of anagreeing state-independent utility. However, with the addition of a fifthaxiom to regulate state-dependence for potentially null states, the resultingtheory is sufficient for an agreeing "almost" state-independent utility. First,we make precise the notion of an "almost" state-independent utility.


Consider a set of probabilityjstate-dependent utility pairs {(p, l!i)}, eachpair agreeing with the partial order -< for simple acts, according to (4.1).

DEFINITION 31. Say that -< admits almost state-independent utilities fora set of n-rewards {r l , ... , rn} if, for each 8 > 0, there exists a pair (p, l!i) thatagrees with -< on simple acts (and almost agrees, otherwise), where for a setof states S# = {Sjl' ... ' Sjk}' p(S#) ~ 1 - 8,

max Il!i(ri ) - l!i,(ri)1 ~ 8.SJ,SJ,ES#

lsisn

Say -< admits almost state-independent utilities if it does so for each set ofn-rewards, n = 1, ....

Obviously, if (p, U) agrees with -< and U is state-independent, then -<admits almost state-independent utilities.

There are two problems created by state-dependent utilities. First, giventhe partial order -<, we would like to indicate probability bounds for anevent E by -< -preferences between a constant act of the form Hx(s) = xB +(1 - x)W and the act HE(s) = B if sEE, and HE(s) = W if s $. E. That is,in general, we want the upper probability bound p*(E) to be the l.u.b.{x:HE -< H x } (or 1, if HE ~ B), and we want the lower bound, p*(E), to equalthe g.l.b.{x: H x -< HE} (or 0, if HE ~ W). However, if such preferences are toindicate probability bounds, then we require that the rewards Band W carrystate-independent utilities 1 and 0, respectively. Thus the first problem.

Second, if a state Sj is potentially null under -<, then there are no-< -preferences among pairs of acts called-off in case Sj does not obtain. LetHS j be the family of sj-called-off acts that yield outcome W for all S $. Sj.When Sj is a potentially null state, V (HI' H 2 E Hs j ), HI ~ H 2 · Suppose :::)v(V E r) extends -< (on simple acts) and let {(p, l!i)} be the set of probabilityj(possibly) state-dependent utilities which represent :::)v according to(4.1). Then, if state Sj is potentially null under -<, unfortunately, HL Axioms1-4 impose too few restrictions on l!i (the state-dependent utility, given stateSj) even when p(Sj) > 0. In particular, it may be that V(rl) > V(r 2 ), yet forall the l!i, UI(r l ) ~ UI(r2 )·

To resolve both these problems, we irppose a fifth axiom-a requirement of"stochastic dominance" among lotteries. For each state Sj and each v.N-Mlottery La' define the set of acts {Hj~m: Hj~m(s) = (1- 2-m)W+ (2- m)La, ifS $. Sj; Hj~m(sj) = La for state Sj} (m = 1, ... ). Observe that, (V j)limm~oo{Hj~m}= Hj,a E Hs j . Moreover, Hj,a(sj) = La. Then, we require thefollowing axiom:

HL AxIOM 5. For each two "constant" acts HLa(s) = La and HL/3(s) = L/3'

(j = 1, ... , n; m = 1, ... ).


Thus, exactly when L 13 is -< -preferred to La (as constant acts), HL Axiom5 imposes a -< -preference on sequences of pairs of lotteries, (Hj~m' HJ~m)which converge to the sj-called-off pair (Hj , a; Hj , f3). Thus, we obtain theconstraint (Definition 21 of the Appendix) " -, (Hj , f3 -< Hj , a)."

LEMMA 4.2. Suppose -< satisfies HL Axioms 1-5. Then, for each V E r(of Theorem 3.1) we may select (exactly) one pair (p v, ~V) from the set ofpairs {(p, ~)} provided by Corollary 4.1-where each pair represents :::)v inaccord with (4.1)-so that acts Wand B have constant value and bound thestate-dependent utilities of other rewards. In symbols,

v (Sj)V (L i , L k E LR-{w,B})' H LL -< HLk

iff °= ~V(W) ~ ~V(Li) ~ ~V(Lk) ~ ~V(B) = 1,

with at least one outside inequality strict for each Sj such that p(Sj) > 0, andall inequalities strict for each sj that is not -< -potentially null.

DEFINITION 32. We call (pV, ~V) the standard representation of V.

Thus, HL Axiom 5 (via HL Axiom 3) constrains state-dependent utilities ofthe rewards Wand B in potentially null states, as desired. In the course ofthe proof of Theorem 6 (below), we explain how HL Axiom 5 also regulatesthe -< -potentially null, state-dependent utilities of all v.N-M lotteries.

Of course, HL Axioms 1-5 are insufficient for guaranteeing existence of astate-independent utility agreeing with -<. Counterexample 4.1 applies, thatis, -<' satisfies all five axioms (since no states are -<' -potentially null).However, as we show next, these axioms suffice for an almost state independent utility.

THEOREM 6. Assume that -< satisfies HL Axioms 1-5.

(i) Then -< admits almost state-independent utilities.(ii) If -< has a countable basis ~, each (p,~) pair in Definition 31

agrees with -<.

There is a sufficient condition for the existence of a state-independentutility over the finite set {W,B,r1, ... ,ri, ... ,rn }, using closure (at one endpoint, at least) of the target sets !fieri) defined in Definition 19.

LEMMA 4.3. If the target sets !fieri) (i = 1, ... , n) are not open intervals,there exists a subset r' c r of expected utilities for -< (agreeing on simpleacts), where each V' E r' is standardly represented by the set of pairs{(p', Uj)} according to (4.1) and where U;(r i ) is state-independent (i =1, ... , n).


Note: r' may fail to be convex. Also, results similar to Lemma 4.3, usingdifferent assumptions, appear in Rios Insua (1992). Related ideas appear inNau (1992).

5. Conditional preference and conditional probabilities. Let e bean event. (Recall, we equate the set state Sj with its elements.) Let HI andH 2 be a pair of e-called-off acts. Suppose -< satisfies HL Axioms 1 and 2.

LEMMA 5.1. Let H~ and H~ be another pair of e-called-offacts which agreewith HI and H 2 (respectively) on e, that is, V (s $. e), [H~(s) = H~(s)] and V(s E e), [HI(s) = H~(s) and H 2(s) = H~(s)]: (i) HI -< H 2 iff H~ -< H~ and(ii) HI ~ H 2 iffH~ ~ H~.

Therefore, a -< -preference (or ~ -indifference) among two e-called-off actsdoes not depend upon how they are called-off, that is, the preference (orindifference) does not depend upon how the acts agree with each other whene fails. This replicates the core of Savage's [(1954), page 23] "sure thing"postulate, P2, as that applies to our concept of a partially ordered preference.

Consider a (maximal) subset of H R , denoted by He' where every twoelements of He form an e-called-off pair. Obviously, each such family He ofe-called-off acts is closed under convex combinations.

DEFINITION 33. Define -<e = -< jH e , the restriction of -< to the family ofe-called-off acts in He. We call -<e the conditional -< -preference relation,given e. (The preceding lemma insures this relation is well defined, that is, itdepends on e but not on how acts are called-off.)

Note: The event eC is essentially null with respect to the conditionalpreference -<e.

DEFINITION 34. Also, for each pair of horse lotteries HI and H 2' say thatH 2 is -< -preferred to HI given e, provided that, for some pair H~ and H~

(and by Lemma 5.1, provided for all pairs) of e-called-off acts agreeing(respectively) with HI and H 2 on e, H~ -<e H~.

In light of Lemma 5.1(i) and because He is a subset of H R , the followingresult is immediate.

THEOREM 7. If -< (over H R ) satisfies (a subset of) HL Axioms 1-5, then-<e (over He) also satisfies the same horse lottery axioms, at least.

Theorem 7 prompts an interesting question: What is the relation between(i) the set of conditional probabilityjutility pairs {pC Ie), Uj E e}, given e, thatarise from the representation of -< over the family of acts H R and (ii) the setof probabilityjutility pairs {pe' Ue , j E e} that represent the conditional preference -<e over the restricted family of acts He?


The following discussion of conditional indifference tells some of theanswer.

DEFINITION 35. Let ~e be the conditional ~ -indifference relation, givene, defined by restricting ~ to acts in the family He. Then, say that horselotteries HI and H 2 are ~ -indifferent, given e, provided that for some pairHi. and H~ (and by Lemma 5.1, provided for all such pairs) of e-called-off actsagreeing (respectively) with HI and H 2 on e, Hi. ~e H~.

It is important, however, to see that ~e is not always the same as the~ -indifference relation (defined by Definition 8) induced by -<e over actssolely in He.

DEFINITION 36. Denote by ~He the ~ -indifference relation over elements of He' induced by -<e.

Of course ~e-indifferenceentails ~H -indifference, but not conversely. HIand H 2 may be two e-called-off acts e from a family He which satisfiesHI ~H H 2, but where, nevertheless, HI ¢ H 2, that is, HI ¢e H 2. We illustrate this phenomenon using a potentially null state which is not essentiallynull.

EXAMPLE 5.1. Consider a binary partition S = {SI' S2} and horse lotteriesdefined over a binary reward set R = {W, B}. Suppose -< is created by thePareto principle applied to expected utility inequalities from the following setof probabilityj(state-independent) utility pairs: S = {(p, U): 1 ~ P(SI) ~ 0.5;U(B) > U(W)}. Then, S2 is potentially null: acts are ~ -incomparable whenever they belong to a common H S2 family. [With P(SI) = 1, all elements ofH S2 have equal expected utility.] Hence, -<S2 is vacuous. So, based on -<S2restricted to a family H s , all pairs of (e-called-off) acts are ~H -indifferent.

2 82

However, the pair of s2-called-off acts (HI' H 2), defined by HI(SI) = H 2(SI) =

W, H I(S2) = Wand H 2(S2) = B, though ~ -incomparable are not ,...,indifferent: HI ~ H 2 and HI ¢ H 2. This is shown as follows. Consider the actH 3 defined by H 3(SI) = Band H 3(S2) = W. Observe that 0.5HI + 0.5H3 ~

0.7W + 0.3B, whereas 0.7W + 0.3B -< 0.5W + 0.5B = 0.5H2 + 0.5H3 • Thisshows that HI ¢ H 2.

Returning to the question, above, we state our central result about conditional probabilities and conditional preferences.

THEOREM 8. (i) If (p,~) belongs to the set of probability jutility pairsrepresenting -<, then the pair (p( Ie), ~ E e) belongs to the set that representthe conditional preference -<e.

(ii) Suppose that the pair (Pe' Ue,j E e) belongs to the set representing theconditional preference -<e with respect to the family He. Then for some pair(p,~) in the set that represents -<, p( Ie) = Pe and ~Ee = Ue,jEe' providedtwo conditions obtain: (1) The event e is not potentially null (with respect


to -<) and (2) the expected utility Vee ) (with arguments from He), corresponding to the pair (Pe' Ue,jE e)' does not use -< -precluded target endpoints,as regulated by Definition 21 (of the Appendix).

Next, we offer an example of Theorem 8, relating Bayes' updating toconditional preferences.

EXAMPLE 5.2. Consider a partition into three states {SI' S2' S3} and actsinvolving the three rewards, W, rand B. Let r denote the constant act, withoutcome r in each state. For j = 1, 2 and 3, define the three acts Hj(sj) = B,Hj(Sk) = W (j =1= k) and also the three acts Hj,r(sj) = rand Hj,r(Sk) = W(j =1= k). Apart from the strict preferences that follow because Wand Bare,respectively, the "worst" and "best" acts, suppose also the agent reports thesepreferences:

0.5W + 0.5H3,r -< HI -< H 3,r -< H 2 -< H 3 -< r -< 0.5H3 + 0.5r.

We investigate the standardized, state-independent utility representationsfor these preferences. That is, with U(W) = 0, U(B) = 1, let u = U(r),independent of the state Sj. If we denote by Pj the probability of state Sj' thenthe preferences above are modeled by each probabilityjutility pair(PI' P2' P3; u) satisfying 0 < 0.5P3 U < PI < P3 U < P2 < u < 0.5P3 + 0.5.

For each 0 < u < 1, it is possible to determine the set g;(u) of all(PI' P2' P3) that satisfy these inequalities. For example, the set g;(0.5) isshown in Figure 2. The union of all sets g;(u) X {u} such that g;(u) =1= 0 isthe set of all probabilityjutility pairs that agree with the strict preferencesabove. From this set, one can determine other preferences not listed abovewhich must also hold if the axioms do. For example, it is required, though notobvious from the reported preferences, that 0.4B + 0.6W -< r. [By contrast, itis obvious from the preferences above that (lj3)B + (2j3)W -< r.]

If we were to learn that, say, the event E = {SI' S2} occurred, we candetermine which preferences are implied in the conditional problem. The setof all pairs (ql' u), where ql is a conditional probability of SI given E, isshown in Figure 3. Observe that, as provided by Theorem 8, the set ofconditional probabilities from Figure 2 is exactly the set represented by thevertical line (at u = 0.5) in Figure 3. However, the set shown in Figure 3is not convex since it contains the points (0.415,0.293) and (0.455,0.379),but does not contain the point (0.435, 0.336) = 0.5(0.415, 0.293) +0.5(0.455, 0.379).

6. Concluding remarks. There is a burgeoning literature dealing withapplications of sets of probabilities. Separate from work on robust Bayesianstatistical analysis, they occur also in the following settings: as a rivalaccount to strict Bayesian theory for representing uncertainty, such as inLevi's (1974, 1980) theory for Ellsberg's (1961) "paradox"; relating to indeterminate degrees of belief, as in Smith's (1961) theory of "medial odds" developed by Williams (1976), Giron and Rios (1980), Walley (1991) and Nau

(1,0,0)


(0,1,0)

0.66

0.5FIG. 2. The set P(O.5) in Example 5.2.

0.8

2193

(0,0,1 )

(1993); and as a method for capturing multiple "expert" opinions [Kadane andSedransk (1980); Kadane (1986)]. In addition, sets of probabilities arise fromincomplete elicitations, where some but not all of an agent's opinions areformalized by inequalities in probabilities and the question is what decisionsare fixed by these partially reported degrees of belief; see Moskowitz, Wongand Chu (1988) and White (1986). Dual to sets of probabilities, the articles byAumann (1962) and Kannai (1963) explore the existence of ("linear") utilitiesfor von Neumann-Morgenstern lotteries when probabilities are completelyspecified.

However, these efforts rely on convexity of the spaces of probabilities andutilities to arrive at their conclusions. From our point of view, this mathemat-

2194

o1.Oo

oC"')

o

oNo

T. SEIDENFELD, M. J. SCHERVISH AND J. B. KADANE

0.45 0.50 0.55 0.60 0.65 0.70 0.75

U

FIG. 3. The set ofconditional probabilities and utilities.

ical convenience is justified under an assumption, for example, that at leastone of the agent's probability and utility is fully determinate. For instance,in light of Corollary 3.4, convexity is appropriate for Bayesian robustnesswhen a loss function is specified but probability is left indeterminate. Likewise, our theory endorses the use of a convex set of utilities given adeterminate probability, as in Aumann's (1962) result concerning existenceof a utility agreeing with a partially ordered preference over simplevon Neumann-Morgenstern lotteries. However, as shown by Theorem 1,


partially ordered preferences that obey a (weak) Pareto condition may notadmit a convex (or even connected) set of agreeing probability/utility pairs.One way to require convexity of the agreeing sets, then, is to restrict thescope of the Pareto condition [see Levi (1990)], but that is a move we are notwilling to make.

Corollary 3.4 prompts a serious question, we think, about the extent towhich our proof technique for extending a partially ordered preference isuseful for the representation theorems of this essay. To wit, since the set rof agreeing "linear" utilities in Theorem 5 is convex, why bother with theelaborate inductive argument only to arrive at what "separating hyperplanes"yields directly? The answer has two parts.

As a first reason, Theorem 5 applies without additional topological assumptions about the relation -< . Specifically, to the best of our knowledge, allthe existing theorems that appeal to "separating hyperplanes" in order toprovide necessary and sufficient conditions for representing a partially ordered strict preference relation -< by a convex set of "linear" utilities or by aconvex set of probabilities, make assumptions regarding the boundaries of-< . Otherwise, for results that are based on a partially ordered weak preference relation :::5, whether preference at the boundary of r is strict or not, isnot determined by such an approach.

For an illustration of the former approach, Walley [(1991), Section 3.7.8,condition R8] requires that strict preference over gambles be "open" so thatso-called strong separation leads to a representation in terms of sets ofprobabilities "closed" with respect to infimums. By avoiding "separatinghyperplanes," we are able to sidestep this artifice. Surfaces of the set r neednot have a simple topological character.

For an illustration of the latter approach, Giron and Rios (1980) use areflexive, partial (quasi-Bayesian) preference relation, denoted in their paperby :::5, which they represent with a closed, convex set of probabilities. Theynote [Giron and Rios (1980), footnote 3, page 20] that their method generatesthe same "quasi-Bayesian preorder" whether the so-called uncertainty set ofprobabilities (which they denote by K*) or its closure (K*) is used. To explainour assertion about the loss of information at the boundary of r, considerthe following example involving preferences over acts using only two prizes,Wand B.

EXAMPLE 6.1. Define act HE as HE(s) = B for sEE, and HE(s) = Wotherwise.

Case 1. The agent reports the strict preferences xB + (1 - x)W -< HE foro < x ~ 0.6 and noncomparability xB + (1 - x)W '" HE for 0.6 < x ~ 1.

Case 2. The agent reports the strict preferences xB + (1 - x)W -< HE foro< x < 0.6 and noncomparability xB + (1 - x)W '" HE for 0.6 ~ x ~ 1.

The (closed) target set of utilities for HE is the same in both cases:!T(HE ) = [0.6,1]. However, in the first case the lower bound is not a "candidate utility" (Definition 24), whereas in the second case it is. Therefore, byour construction, the representation for the agent's strict preferences in Case


1 is the set g; = {P: 0.6 < peE) ~ I} and in the second case it is the closedset e9 = {P: 0.6 ~ peE) ~ I}.

By contrast, the Giron and Rios (1980) theory uses only a weak preferencerelation, :::5. In both Cases 1 and 2 their theory entails

xB + (1 - x)W:::5 HE for 0 < x ~ 0.6 and

xB + (l-x)W",HE for 0.6 <x ~ 1.

[The weak preferences of Case 2 result from applying Giron and Rios' AxiomA5 (continuity). In particular, that axiom yields the conclusion 0.6B +0.4W:::5 HE from the premise xB + (1 - x)W :::5 HE (0 < x < 0.6).] In theirnotation, the weak-preference relation does not distinguish between thesetwo cases: where K* = {p: 0.6 < peE) ~ I}) and K* = {p: 0.6 ~ peE) ~ I},though our strict-preference does.

As a second reason for bypassing proof techniques using "separatinghyperplanes," though the set r is convex, not so for the set of "linear"utilities that admit a decomposition as subjective (almost) state-independentutilities. We do not see how to show the existence of the set of agreeingprobabilityj(almost) state-independent utility pairs, corresponding to Theorem 6, without exploring details about the surface of r. In light of Theorem5, we have no right to assume those surfaces are closed. By contrast, when rhas sufficiently many closed faces, Lemma 4.3 gives a representation of -< interms of sets of probabilityjstate-independent utility pairs. Thus, we feeljustified in our choice of an "inductive" proof technique by the increasedcontent to the theorems reached.

APPENDIX

Proofs of selected results.

A. Results from Section 2. Corollaries 2.1, 2.2 and 2.3 have elementaryproofs.

PROOF OF COROLLARY 2.4. From left to right, argue indirectly and applyCorollary 2.3 for a contradiction. In t~e other direction, assume that xHI

+(1 - x)H ~ xH2 + (1 - x)H for some 1 ~ x > 0 and some lottery H. Also,assume yHI + (1 - y)Hs -< (» )H4 , with 1 ~ y > o. Then by HL Axiom 2,V (1 ~ z > 0), z(yHI + (1 - y)Hs) + (1 - z)H -< (» )zH4 + (1 - z)H. Letz ~j(x + Y - xy) > 0 and then 1 > z (unless x = y = 1, in which case we aredone). Last, define the term w = y j( x + y - xy) and we know that 0 <, w < 1.Thus, we have w(xHI + (1 - x)H) + (1 - w)Hs -< (» )zH4 + (1 - z)H. Since(xHI + (1 - x)H) ~ (xH2 + (1 - x)H), by Corollary 2.3 also we havew(xH2 +(1 - x)H) + (1 - w)Hs -< (» )zH4 + (1 - z)H. Again by HL Axiom2, we may cancel the common factor (1 - z)H from both sides, to yieldyH2 + (1 - y)Hs -< (» )H4 • By Corollary 2.3, HI ~ H 2 • D


PROOF OF COROLLARY 2.5. Assume the premises and, by Corollary 2.3,show using HL Axiom 3 that V (0 < x ~ 1, H a , H b ) whenever xM + (1 x)Ha -< (>- )Hb , then xH + (1 - x)Ha -< (>- )Hb • D

Lemma 2.1 has a straightforward proof.

PROOF OF LEMMA 2.2. Without loss of generality, as L is discrete, write Las {P(rn ): P.(ri ) ~ P(rj) for i ~j}. Let

n

x n = L peri)i= 1

and define the simple lotteries L n = {(ljx n )P(ri ): i = 1, ... , n}. Then {HL }

~ HL . If for each rn E supp(L), r n -< r, then by HL Axioms 1 and 2, H L -<;.(or, if r -< r n , then r -< H L ). Thus, we have the desired conclusion; not(r -< H L ) [or, alternatively, 'ilot (HL -< r)]. For if not, by HL Axiom 3 andtransitivity of -<, (HL -< H L ). D

PROOF OF COROLLARY 2.6. On the contrary, if a utility V for acts isunbounded, then there are acts with infinite utility.

Just consider the discrete horse lottery !loo, where Hoo(sj) = Li(2- i )Pi,j fora sequence of acts Hi such that V(Hi) ~ 2 l (i = 1, ... ). Then, by the expectedutility property, V(Hoo ) = 00. The existence of such acts leads to a contradiction with the first two axioms, just as in the St. Petersburg paradox. Assumefor convenience that HI -< H 2 • By axiom HL Axiom 2, 0.5HI + 0.5Hoo -< 0.5H2

+ O.5Hoo . However, V(0.5HI + O.5Hoo ) = V(0.5H2 + 0.5Hoo ) = V(Hoo ) = 00,

which, if V agrees with -<, entails the contrary result that 0.5HI + 0.5Hoo '"0.5H2 + 0.5Hoo . D

PROOF OF LEMMA 2.3. The proof is indirect. Most of the work is done byHL Axiom 3. We present the argument for the case in which -< fails to bebounded above, using Axiom 3(b). By similar reasoning using Axiom 3(a)instead, the result obtains when -< fails to be bounded below.

Let {Hn : n = I, ... } be an increasing chain and suppose it is not boundedabove, that is, limn~oosup{x: (H2 -< xHI + (1 - x)Hn)} = 1. Choose a subsequence, also denoted by {Hn}, so that X n ~I 1 - 1jn and so that H 2 -< xnHI +(1 - xn)Hn. However, {xnHI + (1 - xn)Hn} ~ HI. Trivially, the constant sequence {H2 } ~ H 2 • Also, HI -< H 2 by assumption. Then by HL Axiom 3(b),III -< HI' contradicting HL Axiom 1. D

PROOF OF COROLLARY 2.7. If not, then there is an unbounded increasing(or decreasing) -<w-chain of preferences amongst the set of rewards R. ByI.lemma 2.3, -<w does not satisfy both Axioms 1 and 3. D

PROOF OF COROLLARY 2.8. Let HI and H 2 be a pair of acts which are"called-off' in case n does not obtain, that is, V (s $ n), HI(s) = H 2(s).


(Properties of "called-off' acts are examined in Section 5.) Define k pairs ofacts "called-off' in case Sjz obtains, H li and H 2i (i = 1, ... , k) as follows: Let lbe a lottery. V (s E Sjz)' [Hlz(s) = Hl(s) and H 2i(S) = H 2(s)]; V (s E n & S $.Sjz)' [Hli(S) = H 2z(s) = L]; V (S $. n), [Hlz(s) = H 2z(s) = Hl(s) = H2(s)]. Byassumption, each Sjz E n is essentially null. Therefore, by iteration of Corollary 2.4 (and transitivity of ~) H~ ~ H~, where H~ = E7=1(I/k)Hl andH~ = E7=1(I/k)H2.. However, H~ = (l/k)Hl + (k - l/k)H and, lik~wise,H~ = (l/k)H2 + (k - l/k)H for act H defined by V (s En), [H(s) = L] andV (s $. n} [H(s) = Hl(s) = H2(s)]. Then, by Corollary 2.4, the desired resultobtains, HI ~ H2. D

B. Proof of Theorem 2. The extension from -< to -< I is given in steps, byadding the two new rewards one at a time. First, extend -< to a partial order-< * on H R U {W}' where W is left f'tooI *-incomparable with elements of HR. Thedefinition of -< * is introduced by a lemma that shows the extension isminimal.

LEMMA 2.4. Suppose HI' H 2, H~ and H~ E H R and are related as follows:Hl(s.) = x·L· + (1 - x .)Ll . and H 2(s.) = x·L· + (1 - x .)L2 . while Hl/(S.) =

J J J J,J J J J J ,J' J

xjEj + (1 - xj)Ll,j and H~(sj) = xjEj + (1 - x j)L2,j. Then HI -< H2 iffH~ -<H~.

PROOF. The lemma is immediate by HL Axiom 2 and the identity 0.5Hl +0.5H~ = 0.5H2 + 0.5H~. D

Now, let Hi E HRU{w} be written Hi(sj) = Xi,jW + (1 - xi,j)Li,j' whereLi,j E H R is well defined if and only if Xi,j < 1. Choose a reward r E Randlet Hi# E H R be the act that results by substituting r for W in Hi. Thus,Hi#(sj) = xi,jr + (1 - xi,j)Li,j. Lemma 2.4 shows this choice is arbitrary and,if -< * is to extend -<, it must satisfy the following:

DEFINITION 15 (-< *). Given HI' H2 E H R U{W}, as expressed above, definethe preference -< * from -< by HI -< * H2 iff Xl J' = X2 J' (for all sJ' $. n) and

# # ' ,HI -< H2·

LEMMA 2.5. The order -< * is identical with -< on H R and satisfies HLAxioms 1-3.

PROOF. If HI' H2 E H R , then Xl,j = X2,j = 0, HI = Ht, H2 = H: andthus HI -< * H 2 iff HI -< H 2. Next, we show that -< * satisfies the axioms.Consider all Hi E H R U {W}·

HLAxiom 1 (irreflexivity). If, on the contrary, for some HI' HI -< * HI' thenHt -< Ht, contradicting the irreflexivity of -< .

Transitivity holds because if HI -< * H2 and H 2 -< * H s, then the corresponding three Hi# acts (i = 1,2,3) can be written xj(r) + (1 - xj)Li,j. Since-< is transitive, Ht -< Ht; thus, HI -< * H s·


HL Axiom 2 (independence). V (0 < y ~ 1), V H E H R U{W} HI -< * H 2 iffXI,j = X2,j and Ht -< H: iff YXI,j + (1 - y)XS,j = YX 2,j + (1 - Y)XS,j andyHt + (1 - y)H: -< yH: + (1 - y)H: iff yHI + (1 - y)Hs -< * yH2 +(1 - y)Hs.

HL Axiom 3. Let {Hln -< * H 2n} be an infinite sequence of -< *-preferenceswhere {Hln} ~ HI' {H2n} ~ H 2 and assume H 2 -< * H s. Thus Htn -< H:n,where {Htn} ~ Ht and {H:n} ~ H:. Since H 2 -< *Hs, then H: -< Ht. Byapplying HL Axiom 3 to these -< -preferences, we obtain Ht -< Ht. Wederive XI,j = XS,j from the equalities X;,j = X2,j and X2,j = XS,j. Therefore,HI -< * H s. The argument for HL Axiom 3(b) is similar. D

Lemma 2.5 shows, also, that if -<' is defined on R U {W}, extends -< andsatisfies the axioms, then it extends -< *. That is, -< * is a minimal extensionof -< to the domain R U {W}. Next, we extend -< * to a preference -<w inwhich W serves as a least preferred (worst) act. (The reader is alerted to thefact that, though the partial order -<w makes W a least preferred act withrespect to elements of H R, it does not guarantee that W is, state by state, aleast favorable reward. This feature is addressed in Section 4.1, where weconsider state-dependent utilities for partial orders.)

DEFINITION 16 (-<w). V (HI,H2 E HRU{w}), HI -<w H 2 iff either (a)HI -< *H2 or (b) 3 {Hn E H R}, 3 {Hln} ~ HI' 3 {H2n} ~ H 2 and 3 (qn:0.5 ~ qn < 1) with limn~oo{qn} = q, q < 1, such that V n, qnHln + (1 qn)Hn -< * (or z *) qnH2n + (1 - qn)W.

LEMMA 2.6. (1) The partial order -<w agrees with -< on HR.(2) W bounds -< * from below; that is, V (H E H R ), W -<w H.(3) -<w satisfies HL Axioms 1-3.

PROOF. (1) Let HI and H2 belong to HR. If HI -< H 2, then by Lemma2.4, HI -< * H2, and by clause (a) in Definition 16, HI -<w H 2. For theconverse, if HI -<w H 2, then it is not by clause (b), since V (Hn E H R) and forall sufficiently large n, as 1 > q ~ 0.5, qnHln + (1 - qn)Hn '" * (and ¢ *)qnH2n + (1 - qn)W. Hence, it must be that clause (a) obtains. So, HI ~ * H 2and HI -< H 2.

(2) For each H E H R, recall that 0.5H + 0.5W z * 0.5W + 0.5H. Then, inDefinition 16(b), set HI = {Hln} = W, H2 = {H2n} = H, Hn = Hand qn = 0.5.Thus, W -<w H.

(3) We verify the axioms individually:HLAxiom 1 (irreflexivity). On the contrary, suppose that HI -<w HI. There

are two cases to consider. If this -<w-relation results by Definition 16(a), thenHI -< * HI' contradicting Lemma 2.4. If we hypothesize that HI -<w HIresults by Definitions 16(b), then we derive a contradiction as follows. LetH~n = qnHln + (1 - qn)Hn and H~n = qn H 2n + (1 - qn)W. A necessary condition for the relation H~n -< * (or z *)H~n to obtain is that XH ' = XH '

In 2n


(except on essentially null states). However, as both {HIn } ~ HI and {H2n} ~HI' while limn -HI:) qn = q < 1, this is impossible. That is, limn ~oo xH' =

In

limn~oo qnxH = 0, while xH' ~ (1 - qn); hence, for all sufficiently large n,In 2n

XHin < X H2n '

HLAxiom 1 (transitivity). Suppose both HI -<w H2 and H2 -<w H s obtain.There are four cases to consider depending upon which clause in Definition16 is used for each -<w-preference. The argument is most complicated whenDefinition 16(b) is used twice; hence, we give the details for this case only.Thus, assume there are two sequences of *-relations:

(Bl) H~n -<* (or ~*)H~n and H~n -<* (or ~*)H~n'

where

H~n = qnHIn + (1 - qn)Hn,

H~n = q~H2n + (1 - q~)H~,

H~n = qn H 2n + (1 - qn)W,

H~n = q~Hsn + (1 - q~)W

and where {HIn } ~ HI' {H2n} ~ H 2, {H2n} ~ H 2, {Hsn } ~ H s andlimn~oo{qn} = q, limn~oo{q~} = q', with 0.5 ~ q, q' < 1. Then V (0 < rn < 1),rnH~n + (1 - rn)H~n -< * (or ~ *) rnH~n + (1 - rn)H~n' This is an -< *preference, unless both equations of (Bl) are ~ *-indifferences. Choose rn sothat rnqn = (1 - rn)q~. Since {H2n} and {H2n} both converge to H 2, by HLAxiom 2, cancel the common acts in H~n and H~n (also common with acts inH 2). Then apply clause Definition 16(b) to obtain HI -<w H s.

HLAxiom 2 (independence). V H E HRU{w}, V 0 < x ~ 1:Case (a). HI -< * H2 iff xHI + (1 - x)H -< * xH2 + (1 - x)H iff xHI +

(1 - x)H -<w xH2 + (1 - x)H.Case (b)-(i). If qnHIn + (1 - qn)Hn -< * (or ~ *) qnH2n + (1 - qn)W, then

rn[qnHIn + (1 - qn)Hn] + (1 - rn)H -< * (or ~ *) rn[qnH2n + (1 - qn)W] +(1 - rn)H. Write rn = x/(qn + (1 - qn)x). Then xHI + (1 - x)H -<w xH2 +(1 - x)H by Definition 16(b).

Case (b)-(ii). Suppose xHI + (1 - x)H -<w xH2 + (1 - x)H. Let {HSn } ~xHI + (1 - x)H and {H4n} ~ xH2 + (1 - x)H. Assume qnHsn + (1 - qn)Hn-< * (or ~ *) qnH4n + (1 - qn)W, Apply HL Axiom 2 to cancel acts in H Sn andH 4n common with H. Regroup the remainders to yield a -< *-relation of thedesired form for Definition 16(b): q~HIn + (1 - q~)Hn -< * (or ~ *) q~H2n +(1 - q~)W, where {HIn } ~ HI and {H2,{} ~ H2. (A simple calculation showsthat limn~oo{q~} = q' ~ 0.5.) Thus HI -< * H2.

Next, we give the details for HL Axiom 3(a) [Axiom 3(b) follows similarly.]HL Axiom 3(a) (Archimedes). Assume Hn -<w Mn and M -<w N, where

{Hn } ~ Hand {Mn } ~ M. We are to show that H -<w N. Again, there arefour cases to consider, depending upon how (infinitely many of) the first andthe second -<w-preferences arise through Definition 16. The argument ismost complicated in case clause 16(b) is used throughout.

That is, assume 3 {R n ,Sn E H R}, 3 {H~ }, 3 {M~ }, 3 {M~} and 3 {N~}

such that, V n, as m ~ ~, {H~m} ~ Hn andm

{M~m} ~mMn' while as n ~ 00,


(B2)

{M~} ~ M and {N~} ~ N. Also assume 3 {qn ,sn ~ 0.5}, so V n, limm-HI){qn }= qn < 1 and limn~oo{sn} = s < 1. By Definition 16, m

qnmH~m + (1 - qnm)R nm -< * (or ~ *) qnmM~m + (1 - qnm)W,

snM~ + (1 - sn)Sn -< * (or ~ *) snN~ + (1 - sn)W.

Since {Hn} ~ Hand {Mn} ~ M, V n, 3 (m* = men)) so that, as n ~ 00, both{H~mJ ~ H and {M~mJ ~ M. Moreover, we may choose (a subsequence of)these m~ so that limn~oo{qnmJ = q, 0.5 ~ q < 1. Thus we have

(B3) qnm~H~m* + (1 - qnm*)R nm* -< * (or ~ *) qnm*M~m* + (1 - qnm*)W.

An application of the first two axioms to (B2) and (B3) yields

(B4) X n [left side(B2)] + (1 - x n ) [left side(B3)]

-< * (or ~ *) X n[right side(B2)] + (1 - x n) [right side(B3)] .

Let x n = sn/(sn + qnm*). Then as both {M~mJ ~ M and {M~} ~ M, we maycancel acts common to M on both sides of (B4) to yield znH~ + (1 - zn)Tn -< *(or ~ *) znN~ + (1 - zn)W, where {H~} ~ H, (N:) ~ N, Tn E H R andlimn~oo{zn} = z = sq/(s + q - sq). Last, 0.5 ~ z < 1 because 0.5 ~ s, q < 1.Therefore, by Definition 16(b), H -<w N as required. D

Finally, Theorem 2 is concluded by repeating this construction in a dualized form: extend the preference -<w to -< I by introducing the act Bandmaking it most preferred in H R U {W}. D

C. Proof of Theorem 3. We show (by induction) how to extend -< (= -<0 )while preserving Axioms 2 and 3 over simple lotteries until the desired weakorder is achieved. At stage i of the induction, the strategy is to identify autility Vi for act Hi E 7?, where Vi is chosen (arbitrarily) from a (convex) set oftarget utilities for Hi' 9i(Hi). We create the partially ordered preference -<iso that H. ~. v·B + (1 - v·)Wl l l l •

Begin with a function !T which provides a set of target "utilities" for allelements of HR. We use Wand B as the 0 and 1 of our utility. Assume{Hn} ~ Hand Hn E HR. For i = 1, by Definitions 17 and 18, vi(H) is thelim inf of the quantities X n for which Hn -< xnB + (1 - xn)W and VI *(H) isthe lim sup of the quantities X n for which xnB + (1 - xn)W -< Hn. The two"utility" bounds, V *(H) and V *(H), I do not depend upon which sequence{Hn} ~ H is used. We show this for v*(H). The argument for v*(H) is theobvious dual.

CLAIM 1. Let {Hn} ~ H and {H~} ~ H. Then V *(H) is the same for bothsequences.

PROOF. Suppose v*(H) = limsup{xn: xnB + (1 - xn)W -< Hn}. Then weshow that v*(H) ~ limsup{xn: xnB + (1 - xn)W -< H~}. This suffices, sinceby symmetry with {H~}, when v'*(H) = lim sup{xn: xnB + (1 - xn)W -< H~},


then v'*(H) ~ lim sup{xn: xnB + (1 - xn)W -< Hn}; hence, v *(H) = v'*(H).Since both sequences {Hn} and {H~} converge to act H, we can write each pair(Hn, H~) as the pair (YnKn + (1 - Yn)Mn, ynKn + (1 - Yn)M~), wherelimn~oo Yn = 1. Assume all but finitely many Yn < 1; else we are finished.Acts Mn and M~ belong to H R because Hn and H~ do. Of course, neither ofthe two sequences of acts {Mn } and {M~} need be convergent, but {Kn } => H.For each n, define the act Nn = ynKn + (1 - Yn)W. Clearly, {Nn} => H. Itfollows from the preference W -< Mn that Nn -< Hn and from the preferenceW -< M~ ~hat Nn -< H~. By hypothesis, there exists a sequence {x n } such thatxnB + (1 - xn)W -< Hn and limn~oo{xn} = v*(H). Let an be the maximum ofo and (x n + Yn - 1). Since xnB + (1 - xn)W -< ynKn + (1 - Yn)B, then anB+ (1 - an)W -< YnKn + (1 - Yn)W = Nn. Transitivity of -< yields anB +(1 - an)W -< H~. However, as limn~oo{Yn} = 1 and limn~oo{xn} = v*(H), thenlimn~oo{an} = v*(H). Thus, v*(H) ~ limsup{xn: xnB + (1 - xn)W -< H~}.

D

Observe that if v*(HI) < v*(H2 ), then HI -< H 2 , by Axiom 3 and the factthat W -< B. However, these "utility" bounds are merely sufficient, not necessary, for the -< -preference HI -< H 2 •

PROOF OF LEMMA 3.1. Note that xB + (1 - x)W -< yB + (1 - Y)W whenever x < y.

(i) Suppose, on the contrary, that v*(H) < v*(H). Then by Corollary 2.5applied twice over, v*(H)B + (1 - v*(H))W z H z v*(H)B + (1 v*(H))W. Since v*(H) < v*(H), also v*(H)B + (1 - v*(H))W -< v*(H)B +(1 - v *(H))W, contradicting the z -relation between them, as just derived.

(ii) This is immediate, by similar reasoning. D

Next, we show that -< may be extended to -<H' a strict partial ordersatisfying HL Axioms 2 and 3, in which H ZH vHB + (1 - VH)W and wherethe utility vH may be any value in the interior of the closed target set !T(H).We resolve when an endpoint of the (closed) target set may be a utilityafterward.

DEFINITION 20. For H E H R , let v E int!T(H). [When !T(H) has nointerior, when v*(H) = v*(H) = v, then by Lemma 3.1(ii), H z vB + (1 v)W. Thus it is appropriate that Defi~ition 20 creates no extension of -<.Then act H already has its "utility" fixed by -<.] Define -<H by

(HI -<H H 2 ) iff 3 (0 < x < 1) 3 (G,G'),

xHI + (1 - x)G -< xH2 + (1 - x)G',

where G and G' are symmetric mixtures of H and vB + (1 - v)W.

Specifically, 3 Y with G = yH + (1 - y)[ vB + (1 - V)W] and G' = Y[ vB +(1 - V)W] + (1 - y)H.


LEMMA 3.2. -<H extends -<.

2203

PROOF. Assume HI -< H 2 • Choose y = 0.5 in Definition 20, so G = G'. ByAxiom 2, xHI + (1 - x)G -< xH2 + (1 - x)G', so that (HI -<H H 2 ). D

LEMMA 3.3. >-H satisfies HL Axioms 1-3.

PROOF. We establish Axioms 1-3 separately.HLAxiom 1 (irreflexivity). Assume not (HI -<H HI). Then xHI + (1 - x)G

-< xHI + (1 - x)G', which by Axiom 2 yields G -< G'. By Definition 20 andanother application of Axiom 2, either H -< vB + (1 - v)W or else vB +(1 - v)W -< H. Either contradicts the relation H '" vB + (1 - v )W. That follows from the assumption v * < v < v *.

HL Axiom 1 (transitivity). Assume (HI -<H H 2 ) and (H2 -<H H 3 ). Then wehave

xHI + (1 - x)G -< xH2 + (1 - x)G' and

wH2 + (1 - w)J -< wH3 + (1 - w)J',

where both pairs (G, G') and (J, J') satisfy Definition 20. These equationsmay be combined to create V z, z(xHI + (1 - x)G) + (1 - z)(wH2 +(1 - w)J) -< z(xH2 + (1 - x)G') + (1 - z)(wH3 + (1 - w)J'). Choose zj(1- z) = w jx. By HL Axiom 2, we may cancel the common term zxH2 [= (1 z)wH2 ] from both sides and recombine the pairs (G, J) and (G', J') to yielduHI + (1 - u)K -< uH3 + (1 - u)K'. Thus, HI -<H H 3 ·

HLAxiom 2. Argue that HI -<H H 2 iff yHI + (1 - y)G -< yH2 + (1 - y)G'iff V (0 < z < 1), z(yHI + (1 - y)G) + (1 - z)H3 -< z(yH2 + (1 - y)G') +(1 - z)H3 iff (3 w) w(xHI + (1 - x)H3 ) + (1 - w)G -< w(xH2 + (1 - x)H3 )

+ (1 - w)G' (choose w = zyjx) iff xHI + (1 - x)H3 -<H xH2 + (1 - x)H3 •

HL Axiom 3(a). Assume V n (Mn -<H N n), and (N -<H 0). Then show(M -<H 0).

1. Thus (a) (xnMn + (1 - xn)Gn) -< (xnNn + (1 - xn)G~) and also (b) (yN +(1 - y)J) -< (yO + (1 - y)J').

As Definition 20 applies to the pairs (Gn , G~), (J, J'), we may cancel (byAxiom 2) common terms to create:

2. Either (a) unMn + (1 - un)H -< u~Nn + (1 - un)(vB + (1 - v)W) or (b)unMn + (1 - un)(vB + (1 - v)W) -< unNn + (1 - un)H.

3. Also, in addition, either (a) wN + (1 - w)H -< wO + (1 - w)( vB +(1 - v)W) or (b) wN + (1 - w)(vB + (1 - v)W) -< wO + (1 - w)H, whereUn 2 x nand w 2 y.

At least one of 2(a) or 2(b) occurs infinitely often. Without loss of generality,assume 2(a) does. Since v*(H) < v < v*(H), then liminflun} = u > 0, in thisinfinite subsequence. [Only here do we use the fact that v is an interior pointof :T(H). See Lemma 3.5 for additional remarks.]


Thus, we are justified in considering a convergent sequence of the form2(a), also indexed by n, with coefficients converging to u > 0. We argue bycases: Assume 3(a) obtains. Using Axiom 2, we mix in the act H to both sidesof 2(a) and the act (vB + (1 - v)W) to both sides of 3(a), yielding:

4(a) x[unMn + (l-u n)H] + (l-x)H -< x[unNn + (l-u n)(vB + (l-v)W)]+(1 - x)H.

4(b) z[wN + (1 - w)H] + (1 - z)(vB + (1 - v)W) -< z[wO + (1 - w)x(vB + (1 - v)W)] + (1 - z)(vB + (1 - v)W).

Choose xu = zw = q =1= 0, (1 - x) = z(l - w). Note all of the following occur:the l.h.s. of 4(a) converges to the act (qM + (1 - q )H); the r.h.s. of 4(b) is theact (qO + (1 - q)( vB + (1 - v)W )); the r.h.s. of 4(a) converges to the l.h.s.of 4(b). Then by HL Axiom 3(a), (qM + (1 - q)H) -< (qO + (1 - q)(vB +(1 - v)W)), so by Definition 20, M -<H O.

In case 3(b) obtains, instead, we modify this argument by mixing the term(vB + (1 - v)W) into 2(a) in case u < w or into 3(b) in case w < u, so that(as above) the r.h.s. of 4(a) converges to the l.h.s. of 4(b) and so forth.

HL Axiom 3(b). This is verified just as HL Axiom 3(a) is.Thus, -<H satisfies the axioms. 0

To complete our discussion of9'(H), we explain when -<H may be createdusing an endpoint of the target set. To motivate our analysis, consider when apartial order -< precludes an extension by a particular new preference orindifference.

DEFINITION 21. Say that a preference for act H a over act H b is precludedby the partial order -<, denoted as -, (Hb -< H a ), if there exist two convergentsequences of acts {Ha,n} =:) H a and {Hb,n} =:) H b, where (\:I n) Ha,n -< Hb,n.[Note: H a ~ H b or H a -< H b implies the condition -,(Hb -< H a).]

DEFINITION 22. Say that indifference between acts H a and H b is precluded by the partial order -<, denoted as -,(Hb ~ H a), if assuming therelation (Ha ~ H b ), the three axioms and the preferences -< all yield apreference precluded by -<.

EXAMPLE 3.2. We illustrate -,(Hb ~ H a ). Suppose -< satisfies the axiomsand the following obtain. Let H a ~ Hb./However, there exist two convergentsequences of acts {Mn} =:) M and {Nn} =:) N and coefficients {x n: x n > 0,lim n _H/J x n = O}, where xnMn + (1 - xn)Ha -< xnNn + (1 - xn)Hb. However,for some y > 0, yN + (1 - y)Ha -< yM + (1 - y)Hb. Thus,(Ha ~ H b) entails M n -< N n and N -< M. By Axiom 3, then M -< M, which is a-< -precluded preference since M ~ M obtains whenever -< satisfies theaxioms.

[We sketch a model for these -< -preferences. Let H a = v *B + (1 - v*)Wand H b = H. Suppose r is a set of utilities {Vd: 1 > d > 0; Vd(H) = v * + dand Vd(M) - Vd(N) = do.5}. Consider -<r, when 9'(H) = [v *, v*] yet H a -<r


H b. Let -< be as -<r except that H a f'tW H b is forced. Vd(xM + (1 - x)v*) :::;Vd(xN + (1 - x)H) entails that x/(l - x) :::; dO. 5

• Since d assumes eachvalue in (0.1), x = 0 is a necessary condition for the preferences of Example3.2.]

PROOF. When -,(Ha -< H b ) and -,(Hb -< H a ), then there exist pairs ofconvergent acts {Ha,n}, {H~, n} ~ H a and {Hb , n}, {Hb,n} ~ H b , where (\I n)H a , n -< H~, nand H b,n -< H~, nt Then by Corollary 2.5, H a ~ R b • 0

Example 3.2 illustrates that our axioms are not strong enough to ensurethe preference H a -< H b when, for example -,(Hb -< H a ) and -,(Ha ~ H b ). Itso happens that when both the conditions -,(Hb -< H a ) and -,(Ha ~ H b )

obtain and these two acts do not involve the distinguished rewards Wand B,then each extension -< * of -< which fixes "utilities" for H a and H b (andwhere -< * arises by iteration of Definition 20) has the desired relation H a -< *H b • Our specific problem, however, is with the case when one of these twoacts is a utility endpoint of the (closed) target set :T(H), for example, letH a = v * B + (1 - v *)W and H b = H, as in the model for the -< -preferencessketched in Example 3.2. We require an extra consideration, then, to determine whether, though v * B + (1 - v *)W f'tW H, a combination of -<preferences arises which prohibits an extension -<H of -< that assigns the"utility" v * for H.

Our solution is to show how to extend the partial order -< to a partialorder -<+ that includes all the so-called missing preferences H a -< H b .

DEFINITION 23. Define -<+ from -< by H a -<+ H b iff H a -< H b or, both-,(Hb -< H a ) and -,(Ha ~ H b ).

The next lemma establishes (very weak) conditions under which the-<+ -closure of a partial order -< satisfies all three axioms. In particular, it isnot necessary that -< satisfies HL Axiom 3. [The condition -,(Hb -< H a ) iswell defined according to Definition 23 even though -< is known only tosatisfy HL Axioms 1 and 2. Specifically, the indifference relation ~ is welldefined and satisfies all those properties, e.g., Corollary 2.4, which depend onHL Axioms 1 and 2 alone.]

LEMMA 3.4. The partial order -<+ satisfies all three axioms provided -<satisfies the first two axioms, HL Axioms 1 and 2, and provided closure of -<under all three axioms does not produce a -< -precluded preference.

PROOF. We verify the axioms separately.HL Axiom 1 (irreflexivity). Since H ~ H obtains and -< yields no -<

precluded preference (under the three axioms), no act H satisfies H -<+ H.That is, H ~ H is not -< -precluded.


HLAxiom 1 (transitivity). Assume Ha -<+ H b and H b -<+ He. Each of these-<+ -preferences may arise two ways, according to Definition 23. We examine ageneral case: -,(Hb -< Ha), -,(He -< H b), -,(Ha ~ H b) and -,(Hb ~ He). Weshow that (i) -, (He -< Ha) and (ii) -, (Ha ~ He).

(i) From the two assumptions -,(Hb -< Ha) and -,(He -< H b), we concludethat there exist convergent sequences {Ha n} =:) Ha, {Hb n} and {Hb n} =:) H band {He n} =:) He' with (V n) Ha n -< H b n a~d H b n -< He ~. Thus, by Axioms 1and 2, 6.5Ha n + O.5Hb n -< O.5'Hb n + 'O.5He n· Using HL Axiom 2 to cancelcommon terms in H b ~ and H b'n, we obt~in -< -preferences of the formH~,n -< H~,n' where {H~,n} =:) Ha ~nd {H~,n} =:) He· Thus, -,(He -< H a)·

(ii) Assume Ha ~ He. Because Ha,n -< H b,n we may construct new convergent sequences {H~,n} =:) He and {H;,n} =:) H b, where H~,n -< H;,n.

This exercise is done as follows. From the indifference Ha ~ He conclude(l/n)W + ([ n - l]/n)He -< (l/n)B + ([ n - l]/n)Ha. Then O.5Ha n +O.5[(1/n)W + ([ n - l]/n)He] -< O.5Hb,n + O.5[(1/n)B + ([ n - l]/n)Haf UseAxiom 2 to cancel common terms involving act H a .

We already have assumed H b n -< He n. Then, since we are entitled to useHL Axiom 3 in determining the con~equences of adopting Ha ~ He' byCorollary 2.5, from H a ~ He we derive H b ~ He. -,(Hb ~ He) means thatadding the ~ -indifference H b ~ He yields a -< -precluded preference. Thus,adding Ha ~ He to -< yields the same -< -precluded preference. Hence,-,(Ha ~ He)·

HL Axiom 2 (independence). This axiom is easy to verify, since -< satisfiesHL Axioms 1 and 2. We illustrate the argument from right to left. SupposexHa + (1 - x)H -<+ xHb + (1 - x)H. We are to show that (i) -,(Hb -< Ha)and (ii) -,(Ha ~ H b ).

(i) We know that both -,(xHb + (1 - x)H -< xHa + (1 - x)H) and-,(xHa + (1 - x)H ~ xHb + (1 - x)H). As in previous cases, we may assumeexistence of convergent sequences {HI n} =:) xHa + (1 - x)H) and {H2 n}=:) xHb + (1 - x)H), where HI n -< H 2 n. 'Use HL Axiom 2 to cancel comm~nterms (involving act H) in ea:ch pai~ HI nand H 2 n. The results are -<preferences of the form Ha,n -< Hb,n' with {Ha,n} ~ Ha and {Hb,n} =:) H b.Thus, -,(Hb -< H a ).

(ii) By Corollary 2.4, from the assumption H a ~ H b it follows that xHa +(1 - x)H ~ xHb + (1 - x)H, which yields a -< -precluded preference as-,(xHa + (1 - x)H ~ xHb + (1 - x)H).

HL Axiom 3(a). Assume Mn -<+ N~ and N -< +0, where {Mn} =:) M and{N,) =:) N. We need to show that (a) -,(0 -< M) and (b) -,(M ~ 0).

(a) Thus -,(Nn -< M n), -,(Mn ~ Nn), -,(0 -< N) and -,(N ~ 0). As inprevious cases, assume each of these -< -precluded preferences arises fromcorresponding sequences of -< -preferences. That is, for each n there is a pairof convergent sequences limj~ 00 {Mn,j} =:) Mn and {Nn,j} =:) Nn, where Mn,j -<Nn,j. Also, there is a pair of convergent sequences {N~} =:) N and {On} =:) 0,where N~ -< On. Since {Mn } =:) M and {Nn } =:) N, for each n we may choose avalue in so that limn ~ 00 {Mn,jn} =:) M and {Nn,jn} =:) N. Of course, Mn,jn -< Nn,jn.


Then, 0.5Mn,Jn + 0.5N~ -< 0.5Nn,Jn + 0.50n. Use HL Axiom 2 to cancel termscommon to act N, yielding -< -preferences sufficient for -,(0 -< M).

(b) If we assume M ~ 0, then (because Mn,Jn -< Nn,Jn) there are sequences{O~} ~ 0 and {N;} ~ N, with O~ -< N;. Since N~ -< On' using Axiom 3, byCorollary 2.5, then N ~ O. However, -,(N ~ 0). Hence, assuming M ~ 0entails some -< -precluded preference. Therefore, -,(M ~ 0). HL Axiom 3(b)is demonstrated in the identical fashion. 0

In the- next definition, based on Lemma 3.4, we indicate whether eitherendpoint of :T(H) is eligible as a utility for H when extending -< to form-<H·

DEFINITION 24. Say that v *(H) is a candidate utility for H if v *B +(1 - v*)W ~+ H. Likewise, v*(H) is a candidate utility for H if H ~+

v *B + (1 - v*)W.

We conclude our discussion of the extension -<H for the special case whenit is generated by a target set endpoint provided, of course, the endpoint is acandidate utility for H. The idea behind the extension, is that as it stands,Definition 20 fails with v = v * or v = v* only because the resulting partialorder is incomplete with respect to Axiom 3. (See Lemma 3.5, below.) Then, inlight of Lemma 3.4, the + -closure (using Definition 23) corrects the omissions. (See Lemma 3.6.)

When extending -< with a candidate utility, v = v * or v = v*, that is,using an endpoint of :T(H), we define the extension -<H in two steps, asfollows: Analogous with Definition 20, let G and G' be symmetric mixtures ofH and vB + (1 - v)w.

DEFINITION 25. Define HI -<# H 2 iff 3 (0 < x < 1) 3 (G, G'), xHI +(1 - x)G -< xH2 + (1 - x)G', and let -<H result by closing -<# using Definition 23, that is, -<H = -<~ .

LEMMA 3.5. The partial order -<# extends -< and satisfies axioms HLAxioms 1 and 2.

PROOF. Since v is a candidate utility, vB + (1 - v)W '" H. Then, asDefinition 25 duplicates Definition 20,1 the proofs from Lemmas 3.2 and 3.3apply to show that -<# extends -< and satisfies the first two axioms. 0

LEMMA 3.6. The partial orderaxioms.

-<+# extends -< and satisfies all three

PROOF. In light of Lemma 3.5, the result follows by Lemma 3.4 once weshow that -<# may be closed under the axioms without generating a -<#precluded preference. Note that -<# extends -< by some, but not necessarily


all, preferences entailed (by the three axioms) from the ~ -indifferenceH ~ vB + (1 - v)W. Then, since v is a candidate utility, closing -<# underthe three axioms does not lead to a -< -precluded preference. We claim, next,that it does not lead to a -<#-precluded preference either. Suppose, on thecontrary, it does. Suppose, for example, closing -<# with the axioms resultsin a relation H b -<# H a , where also -,(Hb -<# H a ). The former means thatadding H ~ vB + (1 - v)W to the set of -< -preferences entails (by theaxioms) that H b -< H a . The latter requires that, for two convergent sequences{Ha , n} a~d {Hb , n}, (Ha , n -<# H b , n)· Thus, adding H ~ vB + (1 - v)W to theset of -< -preferences entails (by the axioms) that (Ha n -< H b n). By HLAxiom 3, these lead to a -< -precluded preference (Hb -< lib). The~ v is not acandidate utility for H with respect to -<, a contradiction. 0

Thus, with Definition 20, we have indicated how to extend -< to -<H'

where act H is assigned a utility v from the interior of its target set !T(H),and with Definition 25, how to extend to -<H using a candidate utilityendpoint.

We interject two simple, but useful results about ~H-indifferences.Thefirst confirms that the extension -<H preserves ~ -indifferences. The secondshows that the extension -<H makes act H ~H-indifferent with its assignedutility v.

LEMMA 3.7. If M ~ N, then M ~H N.

PROOF. Suppose M ~ N and that xM + (1 - x)H3 -<H H4 • We are toshow that xN + (1 - x)H3 -<H H4 . By Definition 23, [y(xM + (1 - x)H3 ) +(1 - y)G] -< [yH4 + (1 - y)G']. Mter rearranging terms, by Corollary 23,[y(xN + (1 - x)H3 ) + (1 - y)G] -< [yH4 + (1 - y)G'], so that xN +(1 - x)H3 -<H H4 • 0

LEMMA 3.8. H ~H vB + (1 - v)W.

PROOF. Since W -< B, we have the following:

(ljn)W+ [en -1)jn][0.5H+ 0.5(vB + (1- v)W)]

-< (ljn)B + [en -1)jn][0.5H+ 0.5(vB + (1- v)W)].

This equation may be written as xnHn + (1 - xn)(vB + (1 - v)W) -<xnMn + (1 - xn)H, where {x n} ~ 0.5, {Hn} ~ Hand {Mn} ~ (vB +(1 - v)W). By Definition 20, Hn -<H Mn. Similarly, it may be written xnM~ +(1 - xn)(H) -< xnH~ + (1 - xn)(vB + (1 - v)W), where {x n} ~ 0.5, {H~} ~ Hand {M~} ~ (vB + (1 - v)W). By Definition 20, M~ -<H H~. Then,"by Corollary 2.5, H ~H vB + (1 - v)W. 0

We iterate Definition 20 (or Definition 25) in a denumerable sequence ofextensions of -< .


_ DEFINITION 26. Define the set 7?= {Hik: Hik(Sj) = r l if j =1= k, and

Hik(Sk) = rJ. Let r l denote the constant act that yields reward r l in eachstate, so that r I E 7?

LEMMA 3.9. 7? is countable and finite if R is finite.

The proof is obvious.[7? remains countable even when 7r is a denumerable partition. Then it

follows from HL Axiom 3 that personal probabilities over 7r are O"-additive.That is, HL Axiom 3 entails "continuity": limn~oo p{n En} = p{limn~oo nEn}'In the light of Fishburn's (1979), page 139, result Theorem 10.5, we conjecture that our central theorems, e.g., Theorems 3 and 6, carryover to countably infinite partitions. However, this is not evident, e.g., our proof of Claim 1(for Theorem 3) does not apply when 7r is infinite. Our use of finite partitionsavoids mandating O"-additivity of personal probability.]

Hereafter, we enumerate 7? with a single subscript i. At stage i of theinduction, -<i is obtained by choosing a target utility Vi for act Hi E??',denoted V(Hi) = Vi' Here Vi E .9;(Hi) and .9;(.) identifies sets of target utilities, based on -<i-I' By Lemma 3.7, extensions preserve utilities alreadyassigned, so that all utilities fixed by stage i are well defined over stagesj 2 i. Next, we show that each simple act has its "utility" V determined by afinite subset of 7?

LEMMA 3.10. If H E H R is a simple act, then there is a (finite) stage -<msuch that ~(H) is a unit set, that is, by stage -<m' H is assigned a preciseutility V(H).

PROOF. First we verify that V has the expected utility property overelements of 7? Consider ita' Hb E7? Without loss of generality, let b =

max{a, b}. Both Ha and H b have their respective utilities by stage -<b . Thatis, Ha ::::;b vaB + (1 - va)W and H b ::::;b Vb B + (1 - Vb)W, By Corollary 2.4,xHa + (1 - x)Hb ::::;b x(vaB + (1 - va)W) + (1 - X)(Vb B + (1 - vb)W).Hence, V(xHa + (1 - x)Hb) = xV(Ha) + (1 - x)V(Hb).

Next, write H(sj) = L:7!: IPj(ri). Define the act H;(s) = H(s) if s = Sj;othe:wise H'(s) = r l . Since H is simpl~, each H; _i.s a finite combinationof H· E7? Specifically H~ = "~J IP.(HI) where Hl(s) = r· if s = s· and_. 1,' , J L.J1, = J 1,' 1, 1, J

HI(s) = r l otherwise. Observe that (l/n)H + (n - l/n)r l = 'k(l/n)H).Thus the utility V(H) is determined once V(rl) and the n values V(H;) arefixed, all of which occurs after finitely many elements of7? are assigned theirutilities. 0

We create a weak order :::5 w from the partial orders -<i (i = 1, ... ) usingthe fact that each H E H R is a limit point of simple horse lotteries. ForH E H R consider a sequence {Hn } =:) H, where H n is a simple act. LetV(H) = limn~oo V(Hn ). Then:


LEMMA 3.11. V(H) is well defined.

PROOF. We show that if {Hn} ~ H, then limn~ooV(Hn) exists and isunique. Assume {Hn} ~ H and {H~} ~ H, where all these acts belong to HR.Without loss of generality, since the simple acts form a dense subset of HRunder the topology of pointwise convergence, suppose that each of Hn , H~ issimple. Then write Hn as YnKn + (1 - Yn)Mn and H~ as YnKn + (1 - Yn)M~,

where limn~oo Yn = 1 and each of K n, M n and M~ is a simple act inHR. By Lemma 3.10, V(Hn) - V(H~) = (1 - Yn)[V(Mn) - V(M~)]. Sincelimn ~oo Yn = 1 and V is in the unit interval [0,1], limn ~oo V(Hn) - V(H~) = 0.

D

The next lemma establishes that V has the expected utility property for allHEHR ·

LEMMA 3.12. If Ha, H b E H R , then V(xHa + (1 - x)Hb) = xV(Ha) +(1 - x)V(Hb).

PROOF. Consider two sequences {Ha n} ~ Ha and {Hb n} ~ H a, whereeach of H a nand H b n is simple and belo~gs to HR. Then, f~r each n, the actx(Ha n) + (1 - x)Hb'n is simple and belongs to HR. It is evident that{x(Ha,n) + (1 - x)Hb,n} ~ xHa + (1 - x)Hb. By Lemma 3.10, V(xHa,n +(1 - x)Hb,n) = xV(Ha,n) + (1 - x)V(Hb,n). Then by Lemma 3.11, V(xHa +(1 - x)Hb) = xV(Ha) + (1 - x)V(Hb). D

Last, define the weak order ::SW for H E HR using the utilities fixed by V:

We complete the proof of Theorem 3:(i) That ::SW is a weak order over elements ofHR follows simply by noting

that V is real-valued. By Lemma 3.12, it satisfies the independence axiom.The Archimedean axiom also is a simple consequence of Lemmas 3.10 and3.11, that is, if {Mn} ~ M, {Nn} ~ Nand M n -<w N n, then V(M)::::; yeN).Next, let H a and H b be simple, that is, each with finite support. Suppose(Ha -< H b). According to Lemma 3.10, the utilities V(Ha) and V(Hb) aredetermined by some stage k of the induction, where k is the maximum indexof the (finitely many) elements of t7? in the combined supports of H a and H b •

Lemma 3.2 establishes that -<k extends -<. Then (L l -<k L 2 ) and thusV(Ha) < V(Hb). Therefore, ::SW extends -< for simple lotteries.

(ii) We argue that V almost agrees with -<, that is, if (HI -< H 2 ), then(HI ::Sv H 2 )· Here is a simple lemma about the changing endpoints of targetsets which completes the theorem.

LEMMA 3.13. For every act H E HR and stage j = 2, ... , (i) vj- l *(H) ::::;vj*(H) ::::; vJ(H) ::::; vJ-l(H) and (ii) limj~oo vj*(H) = vJ(H) = V(H).


PROOF. (i) Since -<j extends -<j-l' any sequence of j - 1 stage preferences H n -<j-l xnB + (1 - xn)W also obtain at stage j. Thus, by Definitions20 and 25 and Lemma 3.2(i), vj_1*(H) ~ vj*(H) ~ vj(H) ~ vj_l(H).

(ii) For each act iii E~ V (j > i), Vj*(iii) = Vj(iii) = V(iii) = Vi. Hence,(ii) is obvious for all simple lotteries. Assume H is not simple. It is easyto find a convergent sequence of simple acts in H R , {Kn } ==> H, whereVn *(Kn) = v~(Kn) = V(Kn) and H = YnKn + (1 - Yn)Mn. The sequence ofacts M n , though elements of H R , need not converge. Since H is not simple,Yn < 1. Then, YnKn + (1 - Yn)W -< YnKn + (1 - Yn)Mn -< YnKn + (1 - Yn)B.As each -<n extends -<, we have YnV(Kn) < Vn*(H) ~ v~(H) < YnV(Kn) +(1 - Yn). However, limn~oo Yn = 1 and, by Lemma 3.8, limn~oo V(Kn) = V(H).Thus, limj~oo vj*(H) = limj~oo vj(H) = V(H). D

Finally, if (HI -< H 2), since for each n, -<n extends -<, we have thatvn*(H1) ~ V~(H2). Then by Lemma 3.13, V(H1) ~ V(H2). D

D. Other results from Section 3.

PROOF OF COROLLARY 3.1. The extensions -<i created in Theorem 3 relyon the existence at stage i-I of a nonempty target set !Ji(iii)' only for theacts iii Et?? However, !Ji(.) is defined on all of H R, including the nonsimpleacts. Hence, we can amend the sequence of extensions of -< to fix utilities forany countable set of acts, in addition to fixing utilities for each element of t??Just modify the argument of Theorem 3 to assign utilities to the countable sett7'?U!I. D

In connection with Example 3.1, for instance, we can introduce acts H a

and H b into a well ordering of~ for example, {iiI' H a , ii2 , H b , ii3 , ••• }, sothat by stage 4 of the sequence of extensions, k 1 = V(Ha ) < V(Hb ) = k 2 ,

which precludes the undesired limit stage in which Veri) = 0.25 (i = 1, ... ).

Theorem 4 is easily demonstrated.

PROOF OF THEOREM 5. That c/J =1= r ~%,9 is part (i) of Theorem 4. For theconverse, argue indirectly. If Z E %,9/r then let iik be the first element of t7'?

(that is, let k be the least integer) fori which Z(iik) $!7k(Hk), even thoughVI =Z(ii1), ... ,Vk- 1 =Z(iik- 1) for acts ii1, ... ,iik- 1. Then Z agrees with-< k- l' since -< k- 1 is the result of extending -< by the conditions iii ;::;viB + (1 - Vi)W (i = 1, ... , k - 1). That is, expand each -<k_l-preferenceinto a -< -preference. The former follows from the latter by adding a set ofk - 1 assumptions {iii ~ ViB + (1 - vi)W: (i = 1, ... , k - I)} to -<. But thesek - 1 conditions are satisfied under Z, and Z agrees with -< on simple acts.Hence, it must be that either Z(iik ) = Vk = vk*(Hk) and !7k(Hk) is open atthe lower end or else Z(iik ) = vk = vt(iik) and !7k(iik ) is open at the upperend. However, if the target set is open and if an endpoint Vk of!7k(iik ) is not a


candidate utility for iik, then adding iik ~ Vk B + (1 - vk)W to -<k-1produces a -<k- I-precluded preference. Since Z agrees with -<k -1' Z doesnot agree with any -<k_1-precluded preference. Thus, Z cannot assign act iik

the utility Vk' which contradicts the assumption Z(iik) = Vk. 0

E. Results from Section 4. The proof of Lemma 4.1 is immediate afterTheorem 13.1 of Fishburn (1979).

PROOF OF LEMMA 4.2. Recall the strict preferences W -< H -< B, wheneverW, B $: supp(H). Hence, for each V, we may standardize the (expected)utility of act W as 0 and the (expected) utility of act B as 1, where all otheracts (not involving Wand B) have (expected) cardinal utilities in the openinterval (0, 1). Next, define a set of simple, called-off acts {Hi,j E Hs j }, whichyield the lottery outcome L i E LR-{w, B} in state Sj and outcome W in all otherstates. In keeping with this notation, let HW,j = Wand let H.B,j be the HS j

ac~with outcome B in state 8j • Recall, for each j, limm-->",{H~)= Hi,} andH m ,} = HW,j = W. Then, whenever H Ll < H Lk (by HL Axiom 5), W -< H:n,j -<H~,j -< H;;',j. Hence, by the Archimedean HL Axiom 3 (as in Lemma 2.3), wehave the restriction ,(HB,) -< H k,} -< Hi,j -< W). Moreover, this constraintobtains also for each extension of -<, including all the limit extensions ::)y

since these -< -preferences involve simple acts. Then, for each V, W ::)y

Hk,j ::)y Hi,j ::)y HB,j. Trivially, either W ~y HB,j or else W -<v HB,j. Theupshot is that, for each V, one of two circumstances obtains:

Case 1. IfW ~y HB,j' Ha,j ~y H(3,j and Sj is null under ::)y ,so p(Sj) = o.Case 2. If W -<v HB,j' then Sj is V-nonnull and for each representation of

V as an expected, state-dependent utility [in accord with condition (4.1)],Uj(W) ~ Uj(L i ) ~ Uj(L k ) ~ Uj(B), with at least one of the outside inequalities strict. However, since the Uj are defined only up to a similarity transformation, without loss of generality choose Uj(W) = 0 and Uj(B) = 1 andrescale P accordingly. 0

PROOF OF LEMMA 4.3. Without loss of generality (Corollary 3.3), let thedenumerable sequence 7?= {ii) of simple horse lotteries, used to create theset r of extensions for -<, take {Hr , ... , H r } as its initial segment: the

1 n

constant acts that award ri in each state. Suppose the interval .9;:(r1) is notopen, for example, .9;:(r1) = [VI *, vi). Then 0 < VI *. Extend -< according tothe condition H

r1~1 VI *B + (1 - VI *)W. That is (by Definition 2.3), HI -<1

H 2 iff xH1 + (1 - x)G1 -< xH2 + (1 - X)G'l' where G1 and G'l are constantacts, symmetric mixtures of outcomes r1 and VI * B + (1 - VI * )W.

We show that each V E r which extends -<1 (where V is standardlyrepresented by the set of pairs {(p, Uj)} according to condition (4.1)) carriesonly state-independent utilities for r 1. That is, for each such Uj, Uj(r1 ) = VI *if Sj is p-nonnull. To verify this claim, define act Hj as follows:

H_e,j(sj) =(v1*-s)B+(I-[v1*-s])W and H_e,j(s) =W fors$:sj.

If state Sj is not -< -potentially null then, since VI * is the lower bound of.9;:(r1), by HL Axiom 4, we have \:f (VI * > s > 0), H _e,j -< H 1,j. [Recall,


H 1,j(Sj) = r 1 and H 1,j(s) = W when S $: Sj.] By the Archimedean conditionHL Axiom 3, letting 8 ~ 0, we find that these -< -preferences create theconstraint -,(H1,j -< Hs=o,j), which applies also to each extension of -<.Thus, if Sj is not -< -potentially null, each V which extends -<1 has VI * as alower bound on the state dependent utility ~(rl). Likewise, by appeal to HLAxiom 5 in case Sj is -< -potentially null, it follows that -,(H1,j -< Hs=o,j) andthis applies also to all extensions of -<. So, again, VI * is a lower bound onthe state dependent utility ~(rl) for cases where Sj is -< -potentially null butV-nonnull and ::SV extends -< (on simple acts). (Note: Here we use axiomHL Axiom 5 to regulate the state-dependent utility of lotteries in -<potentially null states.) Because V(r 1) = VI * for each ::SV that extends -<1on simple acts, VI * also is an upper bound on all such V-nonnull statedependent utilities ~(rl). This is so because VI * = V(r 1) is the p-expectationof ~(rl). Hence, each ::SV that so extends -<1 assigns to reward r 1 thestate-independent utility VI *.

Next, assume that 92(r2 ) is not an open interval, for example, let 92(r2 ) =(V 2*, v~], and we know v~ < 1. Thus, H r2 -<1(V~ + 8)B + (1 - [v~ + 8])W.Extend -<1 to -<2 by introducing the ~2 -condition H r2 ~2 v~ B +(1 - v~)W. That is, define -<2 by HI -<2 H 2 iff xH1 + (1 - x)G2 -<1 xH2 +(1 - x)G~, where G2 and G~ are constant horse lotteries, which are symmetric mixtures of acts H r2 and v~B + (1 - v~)W.

To see that all ::Sv-extensions of -<2 impose a state-independent utility onr 2, that is, to show ~(r2) = v~, it suffices to demonstrate that v~B +(1 - v~)W serves as an upper utility bound for r 2 over all -<1' sj-called-offpreferences, called-off if Sj fails. In other words, we are to establish that, foreach state Sj' the constraint -,(Hv~+s,j -<1 H 2,j) applies to -<1 and itsextensions. Then, by the reasoning we used above, since V(r 2 ) = v~ for all::SV which extend -<2 (on simple acts), v~ also is a lower utility bound foreach state-dependent utility ~(r2)' and thus ~(r2) is state-independent.That is, since V(r 2 ) = v~ is the p-expectation of quantities, none of which isgreater than v~, then ~(r2) = v~ if P(Sj) > O.

To establish that v~ is such a state-independent upper bound, expandeach of the relevant -<I-preferences, to wit, V (1 - v~ > 8 > 0) expandH r2 -<1(V~ + 8)B + (1 - [v~ + 8])W, into its respective -< -preference:3 X s > 0, 3 (GIs' G'ls),

x s H r2 + (1 - X s )G1s -< X s [( V~ + 8)BI+ (1 - [V~ + 8 ])W] + (1 - X s )G'ls.

Each pair (GIs' G'ls) is a symmetric mixture of acts H r1 and VI *B +(1 - VI *)W. These -< -preferences are between cons~anthorse lottery acts. Byappeal to HL Axiom 4 in case Sj is not -< -potentially null, or by appeal to HLAxiom 5 in case Sj is -< -potentially null, we arrive at a constraint forcalled-off acts involving the two lottery outcomes x sr2 + (1 - x s)G1s andxs[(v~ + 8)B + (1 - [v~ + 8])W] + (1 - xs)G~s. Specifically, we obtain therestriction -,(Hx+s,j -< H x2 ,j)-a constraint on all extensions of -< -where

H x+ s, j ( S j) = X s [( V~ + 8) B + (1 - [v ~ + 8])W] + (1 - X s ) G'ls and


Hx+e,j(S) = W if S $. Sj'

H x2 ,j(sj) = x e r2 + (1 - x e )G1e and H x2 ,j(s) = W if S $. Sj.

However, each ::SV extension of -<1 (on simple acts) assigns to r 1 thestate-independent utility VI *. Thus, each extension assigns G 1e and G'le thissame state-independent utility V 1*. Then, as the constraint -,(Hx +e,j-<1

H x2 ,j) obtains, so too does the constraint which results at the limit, whenB = 0, an9- terms G 1e and G'le are canceled according to HL Axiom 2. Hence,each ::SV extension of -<1 has the quantity v~ as an upper bound on thestate-dependent utility ~(r2) of r2, provided Sj is not null under ::Sv.Therefore, since V is a weighted average of ~ values, ~(r2) = v~ for each::SV that extends -<2 on simple acts.

Proceed, this way, through the first n stages in the extension of -< (usingTheorem 3), by choosing for the ith stage either the condition H r ::::;i vi*B +(1 - Vi *)W or the condition H r ::::;i vi B + (1 - vi )W, as 9i(ri) is ~losed belowor above (respectively). Then the set r' of extensions for -<n provides therequisite subset of r. [Note: r' may fail to be convex when 9i(ri ) is a closedinterval, as in the example for Theorem 1. Then either endpoint may bechosen, but not values in between.] D

F. Proof of Theorem 6. The proof of Theorem 6(i) is based on the idea ofthe proof of Lemma 4.3. The argument is by induction on the number ofrewards, that is, on the length of the initial segment of {r 1 , r 2 , ••• }. Themethod is a straightforward epsilon-delta technique of fixing the degree ofstate-dependence to be tolerated and then choosing target set values sufficiently close to a boundary of the target sets to force agreement with theallowed tolerance for state-dependent utilities.

The proof of Theorem 6(ii) follows the argument of Corollary 3.1; that is,use the countable set {9/ u go} in forming the extensions of -<, subject to thefollowing modification in the ordering of {9/ Ug}: Fix k, which determinesthe initial segment of 9/, {r1 , ••• , rk}, over which the almost state-independent utilities are to be provided. Given a nonsimple act H Eg, insert it intothe sequence of extensions based on 7? only after these k-many rewards havebeen assigned their utilities. This method ensures that assigning utilities tothe nonsimple acts in g does not interfere with using the boundary regionsof the target sets of the k-many rewards, {r1 , ••• , rk}, to locate their almoststate-independent utilities. For interesting discussion of this point, see Section 5 of Nau (1993). D

Two remarks help to explain the content of Theorem 6. First, in light ofExample 4.1, it may be that for each B > 0, -< admits an almost stateindependent utility, but (corresponding to B = 0) there is no agreeing probabilityjstate-independent utility pair in the limit. That is, the limit (as B ~ 0)of the (nested) sets of agreeing, almost state-independent utilities is empty.Second, Definition 31 requires only that -< admit almost stateindependent utilities for each finite set of n-many rewards. Obviously, by


increasing n, we can form sequences of (nested) sets of probabilityjutilitypairs. However, Definition 31 does not provide for an almost state-independent utility covering infinitely many rewards simultaneously. We do not yetknow whether, given our five axioms, there exists a nonempty limit (asn ~ 00) to these nested sets.

G. Results from Section 5.

PROOF OF LEMMA 5.1. (i) By HL Axiom 2, HI -< H2 iff 0.5HI + 0.5H~ -<0.5H2 + 0.5H~. Regrouping terms on the r.h.s. of the second -< relation, weobtain HI -< H2 iff 0.5HI + 0.5H~ -< 0.5HI + 0.5H~. Another application ofHL Axiom 2 yields the desired result: HI -< H2 iff H~ -< H~.

(ii) Suppose HI Z H2. By Corollary 2.3, it suffices to show that xH~ +(1 - x)H3 -< H4 iff xH~ + (1 - x)H3 -< H4 • By HL Axiom 2, xH~ + (1 - x)H3-< H 4 iff z[ xH~ + (1 - x)H3 ] + (1 - z)HI -< zH4 + (1 - z)HI (0 < z ~ 1).Since HI Z H 2, by Corollary 2.3, substituting H 2 for HI on the l.h.s., thebiconditional reads: iff z[ xH~ + (1 - x)H3 ] + (1 - z)H2 -< zH4 + (1 - z)HI.Let zx = 1 - z, that is, z = (1 + X)-I. Then regrouping terms in H~ andH2, the biconditional reads: iff z[ xH~ + (1 - x)H3 ] + (1 - z)HI -< zH4 +(1 - z)HI . Another application of HL Axiom 2 yields the desired result. D

PROOF OF THEOREM 8. Part (i) is immediate as He is a subset of HR.Specifically, if a weak order :5v (of Theorem 3) agrees with -<, it agreeswith -<e. That is, consider the e-called-off family He' where H(s) = W ifs $. e and -<e is the restriction of -< to He. Let HI and H2 be simple actsthat belong to He. If HI -<e H 2, then HI -< H 2 and therefore V(HI) -< V(H2).Let the expected utility V be given by the probabilityj(state-dependent)utility pair (p, ~). As ~(W) = 0 and Hi(s) = W for s $. e (i = 1,2), thenL s]E e p(Sj)~(LIj) < L s]E e p(Sj)~(L2j)· Hence, (Pe' ~ E e) agrees with -<e·

For part (ii), without loss of generality (Lemma 5.1), continue with thee-called-offfamily He determined by fixing H(s) = W if s $. e. Define the actBe E He by Be(s) = B if sEe. With respect to -<e , Be serves as the "best"act and W serves as the "worst." Thus, for H E He' V(Hle)V(Be) = V(H).Let Ve(·) agree with -<e over the set He. Assume Ve(·) differs from eachconditional expected utility V(·le) (V E r). In particular, with~ ordered forapplying Theorem 3 to -<e' let Hz E~ satisfy the following condition: Foreach V E r such that Ve(Hi) = V(Hile) (i = 1, ... , z - 1), Ve(Hz) =1= V(Hzle).That is, Hz is the first e-called-off act, where Ve differs from each V(·le),V E r. Without loss of generality, according to Corollary 3.3, put the firstz-elements of~ as the initial segment of 7? Thus, Hz is the zth element inthis reordering of7?

By hypothesis, for some V E r, Ve(Hi) = V(Hile) (i = 1, ... , z - 1). Thenmimic the first z - 1 extensions of -<e in the first z - 1 extensions of -<.That is, provided e is not potentially null so that W -< Be' use Definition 20to extend -< to -<z-I with symmetric mixtures of the z - 1 act pairs: Hiand Ve(Hi)Be + (1 - Ve(Hi))W. Also by hypothesis, -<z-I cannot be ex-


tended to -<z using Definition 20 with symmetric mixtures of Hz andVe{Hz)B e + (I - Ve{Hz))W.

Next, we show that Ve{Hz) is an endpoint of the conditional target set.9;{Hz)' defined using mixtures of Be and W. Argue indirectly: either Hz -<z-lVe{Hz)Be + {I - Ve{Hz))W or else Ve{Hz)Be + {I - Ve{Hz))W -<z-l Hz. Wegive the analysis for the former case. (The reasoning for the latter case isparallel.) Expand the -<z_l-preference into its equivalent -< -preference.Thus, for i = 1, ... , Z - 1, there exist Xi ~ 0, Xz > 0, Lz;Xi + Xz = 1, suchthat

x1G1 + ... +xz-1Gz- 1 + xzHz

-< X1G'1 + ... +Xz-IG~-l + xz[Ve(Hz)Be + (1 - Ve(Hz))W] ,

where the pairs (Gi, G~) are symmetric mixtures of Hi and Ve{Hi)Be +(I - Ve{Hi))W. However, as this -< -preference involves elements of~ only,then x1G1 + ... +xz-1Gz- 1 + xzHz -<e x1G'1 + ... +xz-IG~-l + xz[VeCHz)X Be + (I - Ve{Hz))W]. Thus Ve{Hz) $. 9;, z{Hz)-a contraction with the assumption that Ve{·) agrees with -<e. Hence, Ve{Hz) is a precluded endpoint ofthe conditional.9;{Hz) according to preferences -<z-l' but it is not precludedfrom 9; z{Hz) according to the subset of preferences in -<e z-l . D, ,

Acknowledgments. The authors thank Peter Fishburn for his carefulreading of an earlier version of this paper and for providing us with insightfuland extensive feedback. Also, we thank Robert Nau for stimulating exchangesrelating to matters of state-dependent utility. We are exceedingly grateful tothe two Annals referees, whose patient and detailed readings of our longmanuscript resulted in many helpful ideas and useful suggestions.

REFERENCES

ANSCOMBE, F. J. and AUMANN, R. J. (1963). A definition of subjective probability. Ann. Math.Statist. 34 199-205.

AUMANN, R. J. (1962). Utility theory without the completeness axiom. Econometrica 30 445-462.AUMANN, R. J. (1964). Utility theory without the completeness axiom: a correction. Econometrica

32 210-212.BERGER, J. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New

York.DE FINETTI, B. (1937). La prevision: ses lois logiques, ses sources subjectives. Ann. [nst. H.

Poincare 7 1-68.DEGROOT, M. (1974). Reaching a consensus. J. Amer. Statist. Assoc. 69 118-121.ELLSBERG, D. (1961). Risk, ambiguity, and the Savage axioms. Quart. J. Econom. 75 643-669.FISHBURN, P. C. (1979). Utility Theory for Decision Making. Krieger, New York.FISHBURN, P. C. (1982). The Foundations of Expected Utility. Reidel, Dordrecht.GIRON, F. J. and RIOS, S. (1980). Quasi-Bayesian behaviour: A more realistic approach to

decision making? In Bayesian Statistics (J. M. Bernardo, M. H. DeGroot, D. V. Lindleyand A. F. M. Smith, eds.) 17-38. Univ. Valencia Press.

HARTIGAN, J. A. (1983). Bayes Theory. Springer, New York.HAuSNER, M. (1954). Multidimensional utilities. In Decision Processes (R. M. Thrall, C. H.

Coombs and R. L. Davis, eds.) 167-180. Wiley, New York.

https://www.researchgate.net/publication/243779000_Utility_Theory_Without_the_Completeness_Axiom_A_Correction?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=

https://www.researchgate.net/publication/243779000_Utility_Theory_Without_the_Completeness_Axiom_A_Correction?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=

https://www.researchgate.net/publication/232546802_The_Foundations_of_Expected_Utility_Theory?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=

https://www.researchgate.net/publication/232546802_The_Foundations_of_Expected_Utility_Theory?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=






KADANE, J. B., Ed. (1984). Robustness of Bayesian Analysis. North-Holland, Amsterdam.KADANE, J. B. (1986). Progress toward a more ethical method for clinical trials. Journal of

Medicine and Philosophy 11 385-404.KADANE, J. B. and SEDRANSK, N. (1980). Toward a more ethical clinical trial. In Bayesian

Statistics (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.)329-338. Univ. Valencia Press.

KANNAI, Y. (1963). Existence of a utility in infinite dimensional partially ordered spaces. Israel J.Math. 1 229-234.

LEVI, 1. (1974). On indeterminate probabilities. J. Philos. 71 391-418.LEVI, 1. (1980). The Enterprise of Knowledge. MIT Press.LEVI, 1. (1990). Pareto unanimity and consensus. J. Philos. 87 481-492.MOSKOWITZ, H., WONG, R. T. and CHU, P.-Y. (1988). Robust interactive decision-analysis (RID):

Concepts, methodology, and system principles. Paper 948, Krannert Graduate Schoolof Management, Purdue Univ.

NAU, R. F. (1992). Indeterminate probabilities on finite sets. Ann. Statist. 20 1737-1767.NAU, R. F. (1993). The shape of incomplete preferences. Paper 9301, The Fuqua School of

Business, Duke Univ.RAMSEY, F. P. (1931). Truth and probability. In The Foundations of Mathematics and Other

Logical Essays (R. B. Braithwaite, ed.) 156-198. Kegan, Paul, Trench, Trubner andCo. Ltd., London.

RIOS INSUA, D. (1990). Sensitivity Analysis in Multi-Objective Decision Making. Springer, NewYork.

RIOS INSUA, D. (1992). On the foundations of decision making under partial information. Theoryand Decision 33 83-100.

SAVAGE, L. J. (1954). The Foundations of Statistics. Wiley, New York.SCHERVISH, M. J., SEIDENFELD, T. and KADANE, J. B. (1990). State-dependent utilities. J. Amer.

Statist. Assoc. 85 840-847.SCHERVISH, M. J., SEIDENFELD, T. and KADANE, J. B. (1991). Shared preferences and state

dependent utilities. Management Sci. 37 1575-1589.SEIDENFELD, T., KADANE, J. B. and SCHERVISH, M. J. (1989). On the shared preferences of two

Bayesian decision makers. J. Philos. 86 225-244.SEIDENFELD, T. and SCHERVISH, M. J. (1983). A conflict between finite additivity and avoiding

Dutch Book. Philos. Sci. 50 398-412.SEIDENFELD, T., SCHERVISH, M. J. and KADANE, J. B. (1990). Decisions without ordering. In

Acting and Reflecting (W. Sieg, ed.) 143-170. Kluwer, Dordrecht.SMITH, C. A. B. (1961). Consistency in statistical inference and decision. J. Roy. Statist. Soc. Ser.

B 23 1-25.SZPILRAJN, E. (1930). Sur l'extension de l'ordre partiel. Fund. Math. 16 386-389.VON NEUMANN, J. and MORGENSTERN, O. (1947). Theory of Games and Economic Behavior, 2nd

ed. Princeton Univ. Press.WALLEY, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman and Hall,

London.WHITE, C. C. (1986). A posteriori representation,s based on linear inequality descriptions of a

priori and conditional probabilities. IEEE Trans. Systems Man Cybernet. 16 570-573.WILLIAMS, P. (1976). Indeterminate probabilities. In Formal Methods in the Methodology of

Empirical Sciences (M. Przelecki, K. Szaniawski and R. Wojcicki, eds.) 229-246.Reidel, Dordrecht.

DEPARTMENTS OF PHILOSOPHY AND STATISTICSCARNEGIE MELLON UNIVERSITYPITTSBURGH, PENNSYLVANIA 15213



https://www.researchgate.net/publication/227227913_On_the_Foundations_of_Decision_Making_Under_Partial_Information?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=

https://www.researchgate.net/publication/227227913_On_the_Foundations_of_Decision_Making_Under_Partial_Information?el=1_x_8&enrichId=rgreq-9cfc25b8c902a2e07cadacf354c79d2d-XXX&enrichSource=Y292ZXJQYWdlOzM4MzQ4ODM1O0FTOjk3NDcxMDE5NjE4MzE2QDE0MDAyNTAzMTEyMzU=

A representation of partially ordered preferences

Documents

Transcript of A representation of partially ordered preferences