Subjective expected lexicographic utility with infinite state sets

Journal of Mathematical Economics 30 Ž1998. 323–346

Subjective expected lexicographic utility with infinite state sets Peter C. Fishburn b

a,)

, Irving H. LaValle

b

a AT&T Research, Florham Park, NJ 07932, USA A.B. Freeman School of Business, Tulane UniÕersity, New Orleans, LA 70118-5669, USA

Accepted 2 January 1997

Abstract This paper extends a theory of subjective expected lexicographic utility for decision under uncertainty with finite state sets to an infinite state set S. Prior axioms yield a lexicographic utility representation with matrix probabilities for preferences between mixtures of acts that are constant on finite partitions of S. Two new axioms extend the lexicographic expectation representation to mixtures of all acts. q 1998 Elsevier Science S.A. All rights reserved. JEL classification: D81 Keywords: Nonarchimedean decision theory; Lexicographic utility; Infinite state sets; Matrix probabilities

1. Introduction The theory of subjective expected utility ŽSEU. is widely regarded as the pre-eminent normative paradigm of preference and choice in decision under uncertainty. LaValle and Fishburn Ž1992., LaValle and Fishburn Ž1996a. recently generalized SEU for finite state sets by relaxing its Archimedean condition while retaining the ordering and linearity axioms that characterize the normative core of the theory and allow preferences to be represented by ordered expected utilities. )

Corresponding author.

0304-4068r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 0 7 - 4

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Our generalization replaces the real-valued utilities and subjective probabilities of Archimedean SEU representations ŽSavage, 1954; Anscombe and Aumann, 1963; Pratt et al., 1964, Pratt et al., 1965; Fishburn, 1967, Fishburn, 1970. by vector utilities ordered lexicographically and matrix probabilities. The present paper extends the generalization to infinite state sets. An overview that begins with Savage Ž1954. offers perspective. Savage’s primitives are a set S of states, a set C of consequences, and an is preferred to relation % on the act set A of maps from S into C. All subsets of S, called events, are considered relevant. It is assumed that % is a weak order Žirreflexive, negatively transitive., so the induced indifference relation ; , defined on A by a ; b if neither a % b nor b % a, is an equivalence Žreflexive, symmetric, transitive.. Weak order and other axioms, including independence assumptions and an Archimedean condition, imply the existence of a finitely additive probability measure p on 2 S and a bounded utility function u: C ™ R that combine in the following SEU representation: for all a, b g A, a % b m u Ž a Ž s . . dp Ž s . ) u Ž b Ž s . . dp Ž s . .

HS

HS

Savage’s structural assumptions and preference axioms imply that S is infinite, p is unique, there exists an event EX included in E g 2 S for which p Ž EX . s lp Ž E . whenever 0 - l - 1 and p Ž E . ) 0, and u is unique up to a positive affine transformation a u q b with a , b g R and a ) 0. Later axiomatizations of SEU that presume or accommodate a finite S introduce extraneous scaling probabilities ŽAnscombe and Aumann, 1963; Fishburn, 1967, Fishburn, 1970. or canonical lotteries ŽPratt et al., 1964, Pratt et al., 1965. to obtain a unique p and a positively-affine unique u. Our finite-S theory of subjective expected lexicographic utility ŽSELU. and its present extension follow the extraneous probability route. Analogous extensions for real-valued utilities and subjective probabilities from finite S to arbitrary or infinite S occur in going from Fishburn Ž1967. to Fishburn Ž1970, Section 13.3., and from Pratt et al. Ž1964. to their 1965 book. Given S, C and A as above, let PC and PA denote the sets of all finite-support probability distributions on C and A, respectively. The probabilities used for these sets are extraneous scaling probabilities, not to be confused with subjective probabilities p Ž E . or subjective matrix probabilities P Ž E . for E g 2 S. The members of PC are consequence lotteries; the members of PA are act lotteries or mixed acts. We denote by ps the consequence lottery in PC induced on state s by p g PA : ps Ž c . s Ý  p Ž a . : a g A and a Ž s . s c 4 Fishburn Ž1967., Fishburn Ž1970., LaValle and Fishburn Ž1992., LaValle and Fishburn Ž1996a. and our infinite-S extension take % on PA as the primitive

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preference relation. We assume that % is a weak order that is state independent in the sense that, for all p, q g PA , ps s q s for all s g S ´ p ; q. Then the preference relation on PC induced by % on PA , and also denoted by % , is defined unambiguously by x % y if p % q when ps s x and q s s y for all s g S. We write the convex Ž0 F l F 1. combination l x q Ž1 y l. y of consequence lotteries x, y g PC as x l y, and the convex combination l p q Ž1 y l. q of act lotteries as pl q. The SEU representations in Fishburn Ž1967., Fishburn Ž1970. use the Archimedean condition of von Neumann and Morgenstern Ž1944.. It says that, for all p, q, r g PA , if p % q % r then pl r % q and q % pm r for some l, m g Ž0, 1.. When this is joined by axioms similar to A1–A4 in the next section, and S is finite, we obtain a unique probability measure p on 2 S and a linear positively-affine unique u: PC ™ R such that, for all p, q g PA , p % q m Ý p Ž s . u Ž ps . ) Ý p Ž s . u Ž q s . . sgS

sgS

By definition, u is linear if u Ž x l y . s lu Ž x . q Ž 1 y l. u Ž y . for all x, y g PC and all 0 F l F 1. The same definition for linear u applies to our SELU theory with multidimensional utilities. In particular, if u: PC ™ R K with component linear functionals u1 , u 2 , . . . , u K for which uŽ x . s Ž u1Ž x ., . . . , u K Ž x .., then uŽ x l y . s Ž u1Ž x l y ., . . . , u K Ž x l y .. s Ž l u1Ž x . q Ž1 y l. u1Ž y ., . . . , l u K Ž x . q Ž1 y l. u K Ž y .. s lŽ u1Ž x ., . . . , u K Ž x .. q Ž1 y l.Ž u1Ž y ., . . . , u K Ž y .. s l uŽ x . q Ž1 y l. uŽ y .. Lexicographic order for u in this case is defined by u Ž x . ) L u Ž y . if u Ž x . / u Ž y . and u i Ž x . ) u i Ž y . for the minimum i at which u i Ž x . / u i Ž y . . We define ) L similarly on R m for every m G 1: if a s Ž a 1 , . . . , a m . and b s Ž b 1 , . . . , bm . then a ) L b if a / b and a i ) bi for the minimum i at which a and b differ. a G L b means that a ) L b or a s b . The finite-S lexicographic generalization of the preceding SEU representation that is axiomatized in LaValle and Fishburn Ž1992., LaValle and Fishburn Ž1996a. is explicated by discussions of vector utility and matrix probability assessment in LaValle and Fishburn Ž1996b., LaValle and Fishburn Ž1996c.. It retains the ordering and independence axioms of the real-valued representation but relaxes its Archimedean condition to admit finite-dimensional utilities ordered lexicographically. Our basic finite-S SELU representation is p % q m Ý P Ž s . u Ž ps . ) L Ý P Ž s . u Ž q s . , sgS

sgS

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where each P Ž s . is a real J = K matrix that premultiplies the K-dimensional utility vector uŽ ps . or uŽ q s . win column formx with 1 F K F J, and u: PC ™ R K is linear and preserves % on PC as x % y m uŽ x . ) L uŽ y . . The sums in the basic representation are J-dimensional utility vectors. When U: PA ™ R J satisfies U Ž p . s Ý P Ž s . u Ž ps . , sgS

it is linear; and the representational correspondence is p % q m U Ž p . ) LU Ž q . . We choose K and J as small as possible under the relaxed Archimedean condition and refer to them as the parsimonious dimensions of % on PC and PA , or of u and U, respectively. The P Ž s . are state matrix probabilities. Each is a real J = K matrix whose first Hs columns are lexicographically positive in a lexicographically decreasing arrangement, and whose final K y Hs columns are zero vectors in R J. Given the parsimonious choices of K and J, the P Ž s . can always be defined so that their sum is a 0–1 matrix with exactly one 1 in each column, with 1 s j1 - j2 - PPP j K F J when the 1 in column k is in row jk . If K s J, then ÝS P Ž s . is the K = K identity matrix under the noted normalization. We anticipate our extension to infinite S by noting that P for finite S can be defined on 2 S as a J = K matrix probability distribution with the understanding that P Ž s . s P Ž s4. for singletons. Each eÕent matrix probability P Ž E . begins with lexicographically decreasing lexicographically positive columns and ends with zero columns. Moreover, P is additive: P Ž E1 j E2 . s P Ž E1 . q P Ž E2 . whenever E1 l E2 sœ 0.

P Žœ. 0 for the empty event is the zero matrix and, under the preceding normalization, P Ž S . is a 0–1 matrix with the row arrangement 1 s j1 - j2 - PPP - j K F J for its columnar 1’s. Our extension to infinite S is expressed by p % q m d P Ž s . u Ž ps . ) L d P Ž s . u Ž q s . ,

HS

HS

where u: PC ™ R K , P is a J = K matrix probability measure on 2 S, 1 F K F J, and K and J are parsimonious dimensions as described above. The measure P is finitely additive but not necessarily countably additive since we make no assumption that implies countable additivity. This is similar to Savage’s additive feature for p , but we have no correspondent to his p Ž EX . s lp Ž E . property because our lottery structure allows efficient derivation of the representation without it. When U: PA ™ R J satisfies UŽ p . s HS d P Ž s . uŽ ps ., it is linear and we have p % q m UŽ p . ) L UŽ q ..

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The integral HS d P Ž s . uŽ ps . is defined in a manner that does not depend explicitly on lexicographic order. We say that a g A is constant on a partition E of S if, for all E g E , s, t g E ´ aŽ s . s aŽ t .. Then PF is defined as the set of all lotteries on acts that are constant on finite partitions of S. When p g PF has ps s x E for all s g E and all E in a finite partition E , the expectation integral is defined by

HS d P Ž s . u Ž p . s Ý P Ž E . u Ž x s

E

..

Eg E

This is extended from PF to PA _ PF by a bounding process with ever-finer partitions and the use of Cauchy’s convergence criterion. The process converges to a unique limit HS d P Ž s . uŽ ps . for p g PA _ PF if both u and P are bounded, which is assumed for the extension. Our extension of the SELU representation has two main parts. The first part establishes the representation on PF ; the second extends it to all of PA . The next section of the paper sets the stage with definitions, axioms for the preference relation % on PA that suffice for the PF representation, a preliminary lemma for linear and lexicographically order-preserving utility that precedes the introduction of matrix probability, and definitions for matrix probability measures. Section 3 then states and proves the representation on PF . Lemma 1, the preliminary lemma of Section 2, implies that PC , PF and PA have linear, lexicographically order-preserving utility functions u, uˆ and U of parsimonious dimensions K, J and M respectively with 1 F K F J F M. As in LaValle and Fishburn Ž1996a., LaValle and Fishburn Ž1996b., LaValle and Fishburn Ž1996c., we accommodate the possibility that passage from the stateless consequence lottery set PC to the finite-partitions act lottery set PF introduces new levels into the lexicographic hierarchy, in which case K - J. We do not, however, allow an increase in parsimonious dimension in going from PF to PA , and require J s M. A new axiom that implies J s M is given in Section 4, where we also discuss boundedness and integration extension from PF to PA . Part two of our extension is completed in Section 5. Its main task is to formulate an axiom which implies that some admissible transformation of U on PA coincides with H d P u on PA when u and P are bounded. Section 6 concludes the paper with remarks on unbounded utility and relaxations of our structural assumptions.

2. Definitions, lexicographic lemma We assume throughout that S is infinite, < C < G 2, and, until Section 6, that A s C S and 2 S is the algebra of relevant events. In addition, S is the set of all finite partitions of S, and for each E g S , AŽ E . is the set of acts that are constant

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on partition E . The sets of all finite-support probability distributions on C, AŽ E . and A are PC , PAŽ E . and PA , respectively, and PF s

D PAŽ E . .

EgS

PF is a proper subset of PA if and only if C is infinite. We denote the constant act in A that assigns c g C to every state by c, use x and y exclusively as members of PC , and use p, q and r exclusively as members of PA . Without exception, 0 F l F 1, so x l y g PC and pl q g PA . In addition, E always denotes an event in 2 S, and E c s S _ E, the complement of E. The marginal distribution in PC induced by p on state s is ps , p sE x means that ps s x for all s g E, and p sE q means that ps s q s for all s g E. The preference relation % is defined on PA , its symmetric complement Žindifference. is denoted by ; , and p ' q means that either p % q or p ; q. Induced preference on PC is defined from % on PA by x % y if p % q when p Ž c . s x Ž c . and q Ž c . s y Ž c . for all c g C. Under axioms A1 and A3 below, we have x % y if p % q whenever p sS x and q sS y. 2.1. Null eÕents Null events are associated with subjective probability zero. They are defined from indifference. We say that E is F-null, and write E g NF , if p ; q for all p, q g PF for which p sE c q. Similarly, E is null, and E g NA , if p ; q whenever p sE c q. We have NA : NF because PF : PA . It is conceivable that NA ; NF , but this will be excluded by the axioms and boundedness assumptions used in the extension theorem of Section 5. 2.2. Axioms The axioms in LaValle and Fishburn Ž1996a. for our finite-S SELU representation are the initial axioms for our infinite-S analysis. They are: for all p, q, r g PA , all x, y g PC , all E g 2 S, and all 0 - l - 1: A1. % on PA is a weak order, and c % d for some c, d g C. A2. p % q ´ pl r % q l r, and p ; q ´ pl r ; q l r. A3. p sS q ´ p ; q. A4.  p sE x, q sE y, p sE c q, p % q . ´ x % y. A5. % on PA is finite dimensional.

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Axiom A2 is our linearity assumption for % and ; , and A3 is the state independence axiom. A4 says that if p % q is attributable to the difference between p and q on E, where p produces x and q produces y for every state, then x % y. Together, A1–A2 and A4 are similar in spirit to Savage’s postulates P1–P3 and P5, which comprise his ordering axiom, sure-thing principle, and nontriviality assumption. Axiom A5 is an abbreviated form of a relaxed Archimedean condition that is described in Fishburn Ž1982, p. 38. and LaValle and Fishburn Ž1992, p. 224.. When the von Neumann–Morgenstern Archimedean condition fails for p % q % r, there is a unique l) such that pl r % q whenever l ) l) ; q % pl r whenever l - l) ; and, at l s l) itself, either pl) r % q or q % pl) r. This makes it impossible to represent % by a linear utility functional, which would require q ; pl) r, and creates at least two levels in a hierarchy for PA under which % can be represented lexicographically. Axiom A5 says that the number of hierarchical levels induced by sequences of failures of the Archimedean condition is bounded above. 2.3. Lexicographic lemma The preliminary lemma that lies behind our SELU representations will be stated here for an ordered mixture space Ž M , %., where % is a weak order on M . Axioms that characterize M by means of a mixture operation alb for a , b g M appear in Hausner Ž1954. and Fishburn Ž1982.. For present purposes, it suffices to mention that lottery sets such as PC , PF and PA that are closed under the convexity operation x l y or pl q satisfy those axioms. We say that Õ: M ™ R N is linear if Õ Ž alb . s l Õ Ž a . q Ž1 y l. Õ Ž b . for all a , b g M and all l g w0, 1x, and is lexicographically order-preserÕing if a % b m Õ Ž a . ) L Õ Ž b . for all a , b g M . A linear, lexicographically order-preserving Õ: M ™X R N is parsimonious of dimension N if there is no N X - N and ÕX : M ™ R N with the same properties. An Nth-order T matrix is defined as a real N = N lower triangular matrix in which every diagonal entry is positive. Although we habitually write vectors in row format, they should be viewed as column vectors when premultiplied by a matrix as in the following lemma. Confusion can be avoided if column vectors are presumed throughout. Lemma 1. Suppose ( M , %) is a nontriÕial, weak ordered, linear and finite dimensional mixture space, i.e., (i) %/œ 0 , (ii) % on M is a weak order, (iii) ;a , b , g g M and ;l g (0,1), a % b ´ alg % blg and a ; b ´ alg ; blg , (iÕ) % on M is finite dimensional. Then there is a parsimonious, linear and lexicographically order-preserÕing Õ: M ™ R N for some N G 1 that is unique up to an affine transformation ÕX s T0 Õ q t , where T0 is an Nth-order T matrix and t g R N.

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Proof. See Hausner Ž1954. and Fishburn Ž1982, ch. 4..

I

Because A1, A2 and A5 imply the hypotheses of Lemma 1 for each M g  PC , PF , PA 4 , we apply the lemma to obtain parsimonious, linear and lexicographically order-preserving utility functions u on PC with dimension K uˆ on PF with dimension J U on PA with dimension M. Let PC denote the set of constant act lotteries on C. The obvious correspondence between Ž PC , %. and Ž PC , %., plus PC : PF : PA and parsimony imply 1 F K F J F M. We allow K - J throughout, but align uˆ with u under the admissible transformations of Lemma 1 for the two functions so that each dimensional utility functional u k for u has a clone dimensional utility functional uˆ jk in restriction to PC for u, ˆ with 1 s j1 - j2 - PPP - j K F J. When K - J, the dimensions j for uˆ not in  j1 , j2 , . . . , j K 4 are those introduced by the passage from constant act lotteries or consequence lotteries to lotteries in PF . On the other hand, axiom A6 in Section 4 implies J s M. The transformations of Lemma 1 then allow us to align U with uˆ so that the two are identical on PF . At that point, uˆ will pass out of the picture and, as reflected in Section 1, we deal only with u and U thenceforth. 2.4. Matrix probability measures Matrix probabilities are defined for parsimonious representations as real J = K matrices. We begin with a set X of such matrices that include all P Ž E . that might arise for the Ž PF , %. representation and, eventually, for the Ž PA , %. representation. In the following definition, 0 is the zero vector in R J and w0x is the J = K matrix of JK zeroes. Definition 1. X g X if X is a real J = K matrix such that either Ži. X s w0x, or Žii. there is an H g  1, 2, . . . , K 4 such that

Ž column k of X . ) L 0 Ž column k of X . s 0

if 1 F k F H if H - k F K ,

and 1 F j1 - j2 - PPP - j H F J when jk s min j: X jk / 04 for k F H. Entries below a leading positive in a column of X can be negative, but for any such position the sum of matrix probabilities over the events of every E g S will be zero. When Žii. holds, we refer to Ž j1 , 1., . . . , Ž j H , H .4 as X ’s quasi-diagonal set. In addition, X is a quasi-identity matrix if H s K, j1 s 1, the first nonzero entry in each column is 1, and all other entries are 0.

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Definition 2. P is a matrix probability measure if it is a finitely additive map from 2 S into X for which P Ž S . satisfies Žii. of Definition 1 with H s K, j1 s 1, and has exactly one nonzero Žhence positive. entry in each column. Such a P becomes standard under the appropriate scaling to make P Ž S . a quasi-identity matrix. Proposition 2. in LaValle and Fishburn (1996c) proÕes that P for our finite-S SELU representation can always be put into a standard form. We use the same form in the SELU representation of the next section. MoreoÕer, as was done for the finite-S case, when the 1’s of P (S) are in rows 1 s j1 - j2 - PPP - j K , we will haÕe the clonal correspondence between uˆ on PF and u on PC which says that uˆ jkŽ c . s u k Ž c . for all c g C, k s 1, 2, . . . , K , u j Ž c . s 0 for all c g C whenever j g J _  j1 , . . . , j K 4 . With the PF context in mind for infinite S, for each finite partition E in S we define J Ž E . as the parsimonious dimension of a linear, lexicographically orderpreserving utility function for Ž PAŽ E . , %.. We say that E is full-dimensional when J Ž E . s max  J Ž E X . : E X g S 4 . Finally, when E s  E1 , E2 , . . . , Em 4 with Ei /œ 0 for each i, we denote by P E the matrix w P Ž E1 .: P Ž E2 .: PPP : P Ž Em .x with J rows and mK columns. 3. A finite-partitions representation We now state and prove our standard SELU representation theorem for PF . Theorem 2. Suppose axioms A1–A5 hold for (PA , %) with K and J the parsimonious dimensions of (PC , %) and (PF , %) respectiÕely, 1 F K F J. Then there is a set L s { l 0 , l 1 , . . . , l K } of K q 1 lotteries in PC ; linear, lexicographically order-preserÕing functions u: PC ™ R K and u: ˆ PF ™ R J ; and a standard S matrix probability measure P on 2 such that: (i) u( l 0 ) s 0 g R K and u( l k ) s (0, . . . , 0, 1k , 0, . . . , 0) for k s 1, . . . , K; (ii) J s max{J( E ): E g S }; (iii) for all E g 2 S , P (E) s [0] m E g NF ; (iÕ) for all E1 : E2 : S, E2 g NF ´ E1 g NF ; (Õ) for eÕery E g S , H s K for some E g E , and the rows of P E are linearly independent if E is full-dimensional; (Õi) for all E g S and all p g PA( E ) , if p s E x E for all E g E , then uˆ Ž p . s

Ý P Ž E . uŽ x E . . Eg E

The uniqueness properties of u, uˆ and a not necessarily standard form of P for equivalent representations that do not presume Ži. follow from the admissible

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transformations of u and uˆ in Lemma 1. This is described in detail for the finite-S case in LaValle and Fishburn Ž1996c., which applies equally well to the context of Theorem 2. We now give a proof of Theorem 2 with supporting discussion, then conclude the section with a corollary of the theorem which demonstrates a natural monotonicity correspondence between u and u. ˆ The initial hypotheses of Theorem 2 are assumed throughout the rest of the section. We obtain u and uˆ for Theorem 2 by Lemma 1. Proposition 7 in LaValle and Fishburn Ž1996b. and Section 2 in LaValle and Fishburn Ž1996c. describe the formation of L based on K q 1 consequences and an admissible transformation of u according to Lemma 1 that produce Ži.. We refer to Ži. as the standard form for u and assume it henceforth. Because PAŽ E . : PF , max J Ž E .: E g S 4 F J. Because the parsimonious dimension J of Ž PF , %. is demonstrated by a finite number of act lotteries in PF , which necessarily lie in some one PAŽ E . , we have J F max J Ž E .. This verifies Žii.. Consider uˆ restricted to PC . For l g PC , we write uˆŽ l . with the understanding that l in uˆŽ l . denotes the mixture of constant acts c for the consequences c of l g PC with mixing probabilities given by the consequence lottery. We translate uˆ on PF by subtracting uˆŽ l 0 . from every uˆŽ p . so that uˆ Ž l 1 . ) L uˆ Ž l 2 . ) L PPP ) L uˆ Ž l K . ) L uˆ Ž l 0 . s 0 in R J .

Ž 1.

Let jk denote the minimum component position where uˆŽ l k . is nonzero, so uˆ jkŽ l k . ) 0 by Ž1.. The lexicographic order of Ž1. and mixtures of the l k show that 1 F j1 - j2 - PPP - j K F J. We note later that j1 s 1. We will also transform uˆ into a standard form, which for every l g PC has uˆ jkŽ l . s u k Ž l .

for k s 1, . . . , K

uˆ j Ž l . s 0

for every j g  1, . . . , J 4 _  j1 , . . . , j K 4 .

Ž 2.

Until this is done, we work with a more general version of uˆ that satisfies Ž1.. As an aside at this point, we note that if part Žvi. of Theorem 2 holds, and if P Ž S . is a quasi-identity matrix with quasi-diagonal set Ž1, 1., Ž j2 , 2., . . . , Ž j K , K .4 , then Ž2. follows. Three lemmas will be established before we complete the proof of Theorem 2. The first incorporates every x g PC into an indifference based on the consequence lotteries in the convex hull PL of L : PL s Lemma 3. and

½

K

Ý a k l k : a k G 0 for all k , Ýa k s 1 ks0

5

.

For eÕery x g PC there exists l ) 0 and y, z g PL such that x l y ; z

l u k Ž x . q Ž 1 y l . y Ž l k . u k Ž y . s z Ž l k . u k Ž z . for k s 1, . . . , K .

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Proof. Given x g PC let uŽ x . s Ž r 1 , r 2 , . . . , rK . with u in the standard form as assumed above. We have u Ž x . s r 1 u Ž l 1 . q r 2 u Ž l 2 . q PPP qrK u Ž l K . . Choose l ) 0 to satisfy l q Ý1K < l r k < F 1. Then

lu Ž x . q

Ý Ž yl rk . u Ž l k . s Ý Ž l rk . u Ž l k . .  k : r k-0 4

 k : r k)0 4

Because uŽ l 0 . s 0, we can add fractional amounts of l 0 to each side to obtain l uŽ x . q Ž1 y l. uŽ y . s uŽ z . with y, z g PL . I Lemma 4. For eÕery p g PF there are consequences c p and d p inÕolÕed in p with positiÕe lottery probabilities such that c p ' p ' d p . Proof. Suppose E g S has nonempty parts E1 , E2 , . . . , Em , and p g PAŽ E . has p sE i x i for i s 1, . . . , m. Denote p by Ž x 1 , x 2 , . . . , x m . and let c p be a most-preferred consequence that has positive probability in the consequence lotteries in  x 1 , . . . , x m 4 . Then c p ' x i for all i and, by A4, Ž c p , . . . , c p . ' Ž x 1 , c p , . . . , c p . ' Ž x 1 , x 2 , c p , . . . , c p . ' PPP ' Ž x 1 , x 2 , . . . , x m ., so c p ' p by transitivity. A similar proof gives p ' d p for a least-preferred consequence d p that has positive probability in  x 1 , . . . , x m 4 . I The next lemma takes the first step that leads to P by separating events in E . For p g PAŽ E . , and E g E , we denote by Ž p, E . any q g PAŽ E . for which q sE p and q sE c l 0 . Lemma 5.

For all E g S and all p g PAŽ E . ,

uˆ Ž p . s

Ý uˆ Ž p, E . . Eg E

Proof. Given E s  E1 , . . . , Em 4 with the Ei nonempty and disjoint, we have m 1 my1 1 pq l 0 sS Ý Ž p, E . . m m is1 m

ž / ž

/

ž /

These two members of PAŽ E . are indifferent by A3, so their uˆ vectors are equal. Linearity of uˆ then yields the equation of the lemma. I Lemmas 3 and 4 provide an easy demonstration of j1 s 1 for Ž1.. Otherwise, uˆ 1Ž l k . s 0 for all l k g L so, by Lemma 3 and linearity, uˆ 1 ' 0 on PC . Parsimony for J implies that uˆ 1Ž p . / 0 for some p g PF , and for such a p either c % p for all c g C, or p % c for all c g C, thus contradicting the conclusion of Lemma 4. We now transform uˆ into a standard form that satisfies Ž2.. This is done in the following three steps: Ž1. Given u in standard form and uˆ as in Ž1., we define the matrix probability p Ž E . for every E g 2 S by means of standard acts d k, E for k s 1, . . . , K. We use p in this step to distinguish it from the standard matrix probability measure of Theorem 2.

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Ž2. Using p in place of P , we verify the equation of Theorem 2Žvi. that relates uˆ to u. Ž3. We apply results in Section 4 of LaValle and Fishburn Ž1996c. to transform p into a representationally equivalent standard P that satisfies Žiii., Živ. and Žv. of Theorem 2. Our standard form for uˆ is then defined by the equation of Theorem 2Žvi.. The standard act d k, E for step 1 is defined by d k , E sE l k , d k , E sE c l 0 . The k th column of the J = K matrix p Ž E . is then defined as uˆŽ d k, E ., so p Ž E . s w uˆŽ d1, E ., uˆŽ d 2, E ., . . . , uˆŽ d K , E .x. Clearly, p Žœ. 0 s w0x. Ž . For step 2, the equation of Theorem 2 vi with p is uˆ Ž p . s

Ý p Ž E . uŽ x E . .

Ž 3.

Eg E

In view of Lemma 5, it suffices for Ž3. to show that uˆ Ž x on E, l 0 on E c . s p Ž E . u Ž x . , where Ž x on E, l 0 on E c . denotes a q g PF for which q sE x and q sE c l 0 . Let x l y ; z as in Lemma 3 with y s Ý a k l k and z s Ý b k l k in PL . Then, by A4, Ž x l y on E, l 0 on E c . ; Ž z on E, l 0 on E c ., so uˆ Ž z on E, l 0 on E c . s uˆ Ž x l y on E, l 0 on E c . s l uˆ Ž x on E, l 0 on E c . q Ž 1 y l . uˆ Ž y on E, l 0 on E c . . Then uˆ Ž x on E, l 0 on E c . s

1

l

K

uˆ

ž

Ý bk l k

1yl y 1 s

on E, l 0 on E c

ks0

l

K

uˆ

žÝ

a k l k on E, l 0 on E c

ks0

K

Ý bk uˆ Ž d k , E . y

sÝ ks1

ž

bk

y

l

Ž 1 y l. a k

l

l

Ý a k uˆ Ž d k , E . ks1

/ ˆŽ

u dk , E .

K

s Ý u k Ž x . uˆ Ž d k , E . ks1

s p Ž E . uŽ x . . Hence Ž3. holds.

/

K

1yl

l ks1 K

/

w Lemma 3 x

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335

It follows from Proposition 2 in LaValle and Fishburn Ž1996c. that p is representationally equivalent for every E g S and hence for all of PF to a standard matrix probability measure P on 2 S. Representational equivalence means that, for all p, q g PF , when p g PAŽ E . and q g PAŽ E X . with p sE x E and q sEX yE X , p%qm

Ý P Ž E . uŽ x E . ) L Ý X

Eg E

E gE

X

P Ž EX . u Ž yE X . .

As indicated for step 3, we define our new uˆ on PF by uˆ Ž p . s

Ý P Ž E . u Ž x E . when

p sE x E for all E g E .

Eg E

Linearity for the new uˆ follows from linearity of u, and uˆ is lexicographically order-preserving by its definition, so it must relate to uˆ for Ž1. by an admissible transformation as specified in Lemma 1. We use only the Theorem 2 version of uˆ henceforth. The parts of the theorem not yet verified are Žiii., Živ. and Žv.. Part Žv. follows from LaValle and Fishburn Ž1996a., especially Propositions 5 and 6. By analogy with the finite-S case, for every finite partition E , we must have H of Definition 1 equal to K for some event in E , and if E is full-dimensional then the rows of P E are linearly independent. For Žiii., it is obvious that E g NF if P Ž E . s w0x. Conversely, if P Ž E . / w0x then uˆ Ž d k, E . / 0 s uˆ Ž l 0 . for some k g  1, . . . , K 4 , so E f NF . For Živ., suppose E1 ; E2 and P Ž E2 . s w0x. Then P Ž E1 . s w0x, for otherwise uˆŽ l k on E1 , l 0 on E1c . / 0 for some l k , whereas uˆŽ l k on E1 j Ž E2 _ E1 ., l 0 on E2c . s 0, and it follows that the leading nonzero and necessarily positive entry in column k of P Ž E1 . is a leading negative entry in column k of P Ž E2 _ E1 ., contrary to membership of P Ž E2 _ E1 . in X . I The following corollary of Theorem 2 presumes its hypotheses and the representation described in the theorem. Corollary 6. Suppose E g S consists of m nonempty and mutually disjoint eÕents E1 , E2 , . . . , Em , and p, q g PA( E ) are such that p sE i x i and q sE i yi for i s 1, 2, . . . , m, with x i , yi g PC for each i. If u(x i ) G L u(yi ) for all i for which Ei f NF , then uˆ(p) G L uˆ(q); if, in addition, u(x i ) ) L u(yi ) for some Ei f NF for which min{t: u t (x i ) / u t (yi )} F H(Ei ), where H(Ei ) is the Õalue of H in Definition 1 (ii) that obtains for P (Ei ), then uˆ(p) ) L uˆ(q). Proof. Given E , p and q as in the hypotheses of the corollary, we have m

uˆ Ž p . y uˆ Ž q . s

Ý is1 E if NF

P Ž Ei . u Ž x i . y u Ž y i .

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by Theorem 2 Žiii. and Žvi.. Let I s  i: Ei f NF 4 , and for each i g I let Di s P Ž Ei .w uŽ x i . y uŽ yi .x. If uŽ x i . G L uŽ yi . then uŽ x i . y uŽ yi . G L 0. If uŽ x i . y uŽ yi . s 0, then Di s 0 g R J. If uŽ x i . y uŽ yi . ) 0, let t i s min k: u k Ž x i . y u k Ž yi . / 04 so that u t iŽ x i . y u t iŽ yi . ) 0. It is easily seen that if t i F H Ž Ei . then the first nonzero component of Di is positive, and if t i ) H Ž Ei . then Di s 0. Hence uŽ x i . G L uŽ yi . ´ Di G L 0, with Di ) L 0 whenever u t iŽ x i . y u t iŽ yi . ) 0 and t i F H Ž Ei .. Summation of the Di yields the conclusions of the corollary, given the hypotheses about uŽ x i . versus uŽ yi .. I

4. Dimensions and expectation Two axioms will be added to A1–A5 to extend Theorem 2 to an expected utility representation on PA with S infinite. They are necessary as well as sufficient for the extension. The first, A6, assures that J s M, i.e., that the parsimonious dimensions of % on PF and on PA are equal. We present it in this section and then define the expectation HS d P Ž s . uŽ ps ., denoted by V Ž p ., for p g PA _ PF . The other new axiom implies that V s U when U is aligned with uˆ of Theorem 2 on PF . It is described in the next section. Both sections presume A1–A5 and the Theorem 2 representation. 4.1. Dimensions We remarked earlier that the transition from the stateless consequence lottery set PC to the finite-partitions act lottery set PF might introduce new levels into the lexicographic hierarchy, in which case K - J. This seems natural in view of the uncertainty aspect for PF that is absent from PC . On the other hand, it seems unnatural to increase parsimonious dimensionality in going from PF to PA since the only essential difference between them is one of degree, of finite versus infinite partitions. We have seen in Theorem 2 that PF already embraces full expression of subjective matrix probabilities on 2 S, and anything beyond this for PA would be strange. Despite this, axioms A1–A5 do not preclude J - M because PA _ PF can be considered separately from PF with its own independent basis as a vehicle for introducing one or more new dimensions beyond those needed for PF . We therefore require a new axiom to ensure J s M, and will use the following. A6. For all p,q g PA , if p % q then there are r, t g PF and l, m g Ž0, 1. such that pl r % q l t and q m t % pm r. This says that every level in a parsimonious hierarchy for Ž PA , %. at which p % q might be resolved has a corresponding level in a parsimonious hierarchy for Ž PF , %. at which some comparison between acts in PF is resolved.

P.C. Fishburn, I.H. LaValler Journal of Mathematical Economics 30 (1998) 323–346

Lemma 7.

337

A6 holds if and only if J s M.

Proof. Suppose J s M. We can assume without loss of generality that the restriction of U: PA ™ R J to PF equals u: ˆ PF ™ R J. Suppose p and q in PA Ž . Ž . have p % q, or U p ) L U q , and that Uj Ž p . ) Uj Ž q . with Ui Ž p . s Ui Ž q . for i - j. Let r, t g PF be such that Uj Ž t . ) Uj Ž r . with Ui Ž r . s Ui Ž t . for i - j. The existence of r and t is guaranteed by J being the parsimonious dimension of % on PF . Linearity then implies that Ui Ž pa r . s Ui Ž q a t . for all a g w0, 1x and all i - j, that Uj Ž pl r . ) Uj Ž q l t . for some l g Ž0, 1., and that Uj Ž q m t . ) Uj Ž pm r . for some m g Ž0, 1.. Thus UŽ pl r . ) L UŽ q l t . and UŽ q m t . ) L UŽ pm r ., so pl r % q l t and q m t % pm r, verifying A6. Suppose A6 holds. For m s 1, . . . , M, let pm , qm g PA have pm % qm resolved at level m of U so that U Ž pm . y U Ž qm . s Ž 0, . . . , 0, a m , . . . . ) L 0, where a m ) 0 is the mth component of UŽ pm . y UŽ qm .. Let rm , t m g PF satisfy A6 for m:

l m U Ž pm . y U Ž q m . ) L Ž 1 y l m . U Ž t m . y U Ž r m .

Ž 1 y m m . U Ž t m . y U Ž r m . ) L m m U Ž pm . y U Ž q m . . It follows that UŽ t m . y UŽ rm . s Ž0, . . . , 0, bm , . . . . ) L 0 with mth component bm ) 0. Then, for m s 1, 2, . . . , M y 1, t m a rmq1 % rm a t mq1 for all a g Ž 0, 1 x , so t m % rm must be resolved at a prior level than t mq1 % rmq1 in u. ˆ Hence J G M, and because J F M we have J s M. I We assume A6 henceforth and transform U on PA as in Lemma 1 so that UŽ p . s uˆŽ p . for all p g PF . We use U instead of uˆ henceforth. Thus U: PA ™ R J with p % q m U Ž p . ) L U Ž q . , for all p, q g PA , and, for all E g S and all p g PAŽ E . with p sE x E for all E g E , UŽ p. s

Ý P Ž E . uŽ x E . , Eg E

as in Theorem 2Žvi.. By earlier definitions, NA : NF for null events. It is now possible to state an axiom which implies NA s NF , but we defer alignment of null events until later in the next section because it will be implied by our final axiom, A7. 4.2. Boundedness Our application of A7 in the next section assumes that both u and P are bounded. This means that there is a positive number N such that uŽ PC . : wyN, N x K and that every entry in P Ž E . for all E g 2 S lies in wyN, N x. The

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assumption of bounded P seems unexceptional and is not an issue in SEU theory where real-valued probabilities lie in w0, 1x. It is also not an issue in our matrix probabilities for cell positions Ž jk , k ., where P Ž S . has value 1 in standard form and P Ž E . g w0, 1x for all E g 2 S. However, for cell positions Ž j, k . with j ) jk , where P Ž S . equals 0, it is conceivable that the P Ž E . values are unbounded. We have no example which shows that this is possible and leave the matter open. Suppose C is finite. Then linearity implies that our u is bounded, but we have only passing interest in this case because it implies PF s PA . In particular, p g PA is a finite mixture of acts in A, and each act for which p has positive probability is constant on a partition in S , so the intersection of these partitions is another partition in S and it follows that p g PF . Several axiomatizations of expected utility or SEU, including those in Blackwell and Girshick Ž1954., Savage Ž1954., and some in Fishburn Ž1970., Fishburn Ž1982. imply that u is bounded when C is infinite. Other contributions ŽFishburn, 1970, Fishburn, 1975; Toulet, 1986; Wakker, 1993. consider aspects of unbounded utility. Because PC is confined to simple lotteries on consequences, our present axioms do not preclude unbounded u when C is infinite. Techniques described in LaValle and Fishburn Ž1996b. for assessing u can be used to determine when one or more u k is unbounded, but in our main expectation extension we shall simply assume that u is bounded. Comments on unbounded u are included in Section 6. 4.3. Multidimensional expectation We assume in the rest of this section that u and P are bounded. In addition, V Ž p . denotes the expected multidimensional utility HS d P Ž s . uŽ ps .. By prior definitions and Theorem 2, V Ž p . s U Ž p . for all p g PF . For p g PA _ PF we define V Ž p . s Ž V1Ž p ., V2 Ž p ., . . . , VJ Ž p .. by a standard limit process based on finite-valued functions that converge uniformly to p in the sense of u. Suppose uŽ PC . : wyN, N x K and < P jk Ž E .< F T for all Ž j, k . g  1, . . . , J x =  1, . . . , K 4 and all E g 2 S, where P jk Ž E . is the entry of P Ž E . in row j and column k. Let R 1, R 2 , . . . be a sequence of increasingly finer finite partitions of wyN, N x K into rectangles such that the supremum of the edge lengths of all rectangles in R i is d i with d i ™ 0. For each R g R i let E Ž R . s  s g S: u Ž ps . g R 4 , and let E i denote the partition of S in S defined by the nonempty EŽ R . for R g R i. Also let f i s Ž f 1i , . . . , f Ki . map E i into R K such that, for all E g E i , < f ki Ž E . y u k Ž ps . < F d i for k s 1, . . . , K .

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339

If we extend f i to S by defining f i Ž s . as f i Ž E . for all s g E, then f 1 , f 2 , . . . is a sequence of simple functions on S that converge uniformly to p in the sense of u. Let P j denote the j th row of P . The approximation of Vj Ž p . s HS d P j Ž s . uŽ ps . for R i is K

Vj i Ž p . s

Ý Ý P jk Ž E . f ki Ž E . . Eg E i ks1

To check convergence of Vj 1 Ž p ., Vj 2 Ž p ., . . . , consider the absolute difference of two terms in the sequence, say Vj a Ž p . and Vj b Ž p .. Let E a b be the partition in S whose events are the nonempty E l EX for E g E a and EX g E b , and for such events set f a Ž E l EX . s f a Ž E . and f b Ž E l EX . s f b Ž EX .. Then K

K

< Vj a Ž p . y Vj b Ž p . < s <Ý Ý P jk Ž E . f ka Ž E . y Ý Ý P jk Ž E . f kb Ž E . < E a ks1

E b ks1

K

F Ý <ÝP jk Ž E . f ka Ž E . y ÝP jk Ž E . f kb Ž E . < ks1 E a

Eb

s Ý< Ý P jk Ž E . f ka Ž E . y f kb Ž E . < k E ab

F Ý Ž d a q d b . Ý < P jk Ž E . < E ab

k

s Ž d a q d b . Ý P jk k



D Eg E ab P jk Ž E .)0

0 

E y P jk

D Eg E ab P jk Ž E .-0

E

0

F 2 KT Ž d a q d b . . Because K and T are fixed, and d i ™ 0, the difference approaches 0 as a and b get large. It then follows from Cauchy’s convergence criterion ŽTaylor, 1955, p. 489. that the sequence Vj 1 Ž p ., Vj 2 Ž p ., . . . converges to a value that we define as HS d P j Ž s . uŽ ps ., or Vj Ž p .. The same limit obtains for every sequence R 1, R 2 , . . . as defined above with d i ™ 0 because the inequalities written above give < Vj a Ž p . y Vj b Ž p .< F 2 KT Ž d a q d b . when Vj a Ž p . applies to one sequence and Vj b Ž p . applies to another with d values of d a and d b respectively. Hence the limit Vj Ž p . is not partition specific and is therefore well defined. This is true for each j, so

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V Ž p . is well defined as the multidimensional expectation of uŽ ps . with respect to P.

5. Extension axiom and theorem Our first six axioms do not imply that U s V when C is infinite despite the fact that U s V on PF . The reason is similar to the reason that A6 is needed to prevent J - M. In particular, if P ) is a maximal linearly independent subset of PA _ PF within PA , we could take U s V on PF , define UŽ p . s V Ž p . q 1 on P ) , and extend U linearly to the rest of PA _ PF without violating our earlier axioms. Similar examples are described in Section 10.5 in Fishburn Ž1970.. Our final axiom will imply U s V on PA by connections between p g PA _ PF and lotteries in PF . Given p, we consider a sequence p 1 , p 2 , . . . of acts in PF that converge to p in the sense of u as in the definition of V Ž p .. Thus, since U s V on PF , we have UŽ p i . ™ V Ž p .. But we also want UŽ p i . ™ UŽ p . so that UŽ p . s V Ž p .. To obtain UŽ p i . ™ UŽ p ., we construct mixtures of p and p i with other acts in PF , a i and b i , to produce indifference tradeoffs of the form pl i a i ; p il i b i with 0 - l i F 1. Linearity of U then gives

li U Ž p . y U Ž p i . s Ž 1 y li . U Ž b i . y U Ž ai . , and if the l i converge to l g Ž0, 1. and wUŽ b i . y UŽ a i .x ™ 0 g R J, we conclude that wUŽ p . y UŽ p i .x ™ 0, or UŽ p i . ™ UŽ p .. Convergence of UŽ b i . y UŽ a i . to 0 is obtained by constructing the a i and b i sequences so that a i ™ a g PF and b i ™ b g PF , and then postulating that a ; b, i.e., that UŽ a. s UŽ b .. Three types of convergence are used in the axiom. Ordinary convergence of a real number sequence, denoted by  l i 4 ™ l, has the usual meaning that for every e ) 0 there is an Ne such that < l i y l < F e for all i G Ne . Convergence of the p i to p in the sense of u is based on the norms 5 u Ž x . y u Ž y . 5 s max < u k Ž x . y u k Ž y . < 1FkFK

and 5 q, r 5 u ssup5 u Ž q s . y u Ž r s . 5 S

for q, r g PA . We write

 pi4 ™ F

p if all p i are in PF and  5 p i , p 5 u 4 ™ 0.

The third type of convergence is a specialization of ™ F for the a i and b i sequences with limits in PF . It is denoted by ™ m to emphasize the fact that the

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341

convergence is based on convergence in mixing probabilities with respect to a fixed set of other acts in PF . We write

 a i 4 ™ m a if there are q1 , q2 , . . . , qW g PF , nonnegative m 1 , m 2 , . . . , m W with Ý m w s 1, nonnegative m 1i , m i2 , . . . , m iW for each i G 1 with Ý w m iw s 1 and with  m iw 4 ™ m w for w s 1, . . . , W , such that W

ai s

W

Ý mwi qw and a s Ý m w qw . ws1

ws1 1

A7. Suppose p g PA _ PF ,  p , p 2 , . . . , a1, a 2 , . . . , b 1 , b 2 , . . . , a, b4 : PF and the following hold: Ži. pl i a i ; p il i b i for all i, with 0 - l i F 1; Žii.  p i 4 ™ F p; Žiii.  l i 4 ™ l with 0 - l - 1; Živ.  a i 4 ™ m a and  b i 4 ™ m b. Then a ; b. We can now state our main extension theorem. Its proof is followed by a note on null events and further comments on axioms like A7. Theorem 8. Suppose A1–A6 hold, C is infinite, and the representation of Theorem 2 holds with u and P bounded. Then A7 holds if and only if U s V on PA _ PF or, for all p, q g PA , p%qm

HS d P Ž s . u Ž p . ) HS d P Ž s . u Ž q . . s

L

s

Proof. Necessity. Suppose the hypotheses of A7 hold and U s V on PA _ PF . We have UŽ p . s V Ž p . s lim UŽ p i ., where the latter equality follows from the definition of multidimensional expectation and Žii.. By Ži. and linearity, l i wUŽ p . y UŽ p i .x s Ž1 y l i .wUŽ b i . y UŽ a i .x. In the limit, the left side of this equation goes to 0, and it then follows from Žiii. and Živ. that UŽ b . y UŽ a. s 0, hence that UŽ a. s UŽ b ., or a ; b. Therefore A7 holds. Sufficiency. Assume that A7 holds. To prove U s V, we suppose p g PA _ PF has UŽ p . / V Ž p . and derive a contradiction. Given such a p, we first construct p i g PF so that  p i 4 ™ F p with lim UŽ p i . s V Ž p .. This can be done by a sequence R 1 , R 2 , . . . of finite partitions as in the preceding section with f i Ž E . for E g E i specified as any uŽ x . in the corresponding R g R i. We then use the x choices for the events in E i in a straightforward way to construct p i so that every psi for s g E g E i is the x chosen for that E. This yields Žii. in the hypotheses of A7. Given UŽ p . / V Ž p ., we have UŽ p . / UŽ p i . for all sufficiently large i. For the limit case with UŽ p i . ™ V Ž p . let U Ž p . y V Ž p . s d s Ž d1 , d 2 , . . . , d J . / 0.

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Also select q j, r j g PF for j s 1, 2, . . . , J so that j U Ž q j . y U Ž r j . s d j s Ž 0, . . . , 0, d jj , d jq1 , . . . , d Jj . j with d jj ) 0. There are no restrictions on the signs of d jq1 through d Jj. The parsimonious dimension J of % on PF assures the existence of such q j and r j. We define real numbers w 1 , w 2 , . . . , wJ by

J

Ý wj d j s d. js1

Thus w1 d 11 s d1 determines w1 , w1 d 21 q w 2 d 22 s d 2 determines w 2 , . . . , and Ý < wj < ) 0. The preceding equation in terms of U is J

Ý wj U Ž q j . y U Ž r j .

s UŽ p. y V Ž p. .

js1

To account for the signs of the wj let Jys  j: wj - 04 , Jqs  j: wj ) 04 and J 0 s  j: wj s 04 s  1, . . . , J 4 _ Ž Jyj Jq .. Then U Ž p . q Ý Ž ywj . U Ž q j . q Ýwj U Ž r j . Jy

Jq

s V Ž p . q Ý Ž ywj . U Ž r j . q Ýwj U Ž q j . . Jy

Jq

Let w s Ý < wj < and define a, b g PF by a s Ý Ž ywjrw . q j q Ý Ž wjrw . r j Jy

Jq

b s Ý Ž ywjrw . r j q Ý Ž wjrw . q j . Jy

Jq

Also let l s 1rŽ1 q w . so that the preceding equation in U and V is

lU Ž p . q Ž 1 y l . U Ž a . s lV Ž p . q Ž 1 y l . U Ž b . with 0 - l - 1. This l along with a and b are used in Žiii. and Živ. of A7. Clearly, UŽ a. / UŽ b .. In applying the definition of  a i 4 ™ m a and  b i 4 ™ m b, we assume with no loss of generality that all J q j and all J r j are involved in both a and b with 0 probabilities for members of  q j, r j 4 not explicitly used in the preceding definitions of a and b. The same assumption applies to a i and b i in the next paragraph, where we use the same  q j, r j 4 for every i to construct a i and b i for the indifference pl i a i ; p il i b i of hypothesis Ži..

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Our construction of a i and b i for large i mimics the process used to define a and b with V Ž p . replaced by UŽ p i .. We let U Ž p . y U Ž p i . s d i s Ž d1i , d 2i , . . . , d iJ . / 0 and define real numbers w 1i , w 2i , . . . , wJi by J

Ý wjid j s d i , js1

i.e., J

Ý wji U Ž q j . y U Ž r j .

s UŽ p. y UŽ pi . .

js1 i q i  4  4 Ž i. Ž . Let Jy i s j: wj - 0 and Ji s j: wj ) 0 . Because U p ™ V p , we have y y q q J : Ji and J : Ji for all sufficiently large i, and presume this henceforth. y 0 q q 0 Hence Jy i _ J : J and Ji _ J : J . The preceding equation for U gives

U Ž p . q Ý Ž ywji . U Ž q j . q Ýwji U Ž r j . Jy i

Jq i

s U Ž p i . q Ý Ž ywji . U Ž r j . q Ýwji U Ž q j . . Jy i

Jq i

Let w i s Ý J < wji < ) 0 and define a i , b i g PF by a i s Ý Ž ywjirw i . q j q Ý Ž wjirw i . r j Jy i

Jq i

b i s Ý Ž ywjirw i . r j q Ý Ž wjirw i . q j . Jy i

Jq i

Also let l i s 1rŽ1 q w i . so that the preceding U equation gives pl i a i ; p il i b i . It follows that Ži. of A7 holds for all large i. Because UŽ p i . ™ V Ž p ., d i ™ d, and it follows for each j that  wji 4 ™ wj and hence that  l i 4 ™ l for hypothesis Žiii. of A7. If j g J 0 then  wji 4 ™ 0, so any terms in the a i and b i sequences for j g J 0 will vanish in the limit. Hence  a i 4 ™ m a and  b i 4 ™ m b, so Živ. holds. It now follows from A7 that a ; b, i.e., that UŽ a. s UŽ b ., in contradiction to UŽ a. / UŽ b . obtained earlier. Therefore our supposition of UŽ p . / V Ž p . leads to a contradiction, and we conclude that U s V on PA _ PF . I

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The following corollary notes the alignment of null events for PF and PA . Corollary 9.

If A7 and the hypotheses of Theorem 8 hold, then NA s NF .

Proof. Suppose p, q g PA and p sE c q. Then, by Theorem 8, UŽ p. y UŽ q. s s

HE d P Ž s .

u Ž ps . y u Ž q s . q

HE d P Ž s .

u Ž ps . y u Ž q s . .

HE d P Ž s . c

u Ž ps . y u Ž q s .

If E g NF then, for all pX , qX g PF ,

HE d P Ž s .

u Ž pXs . y u Ž qXs . s 0,

and because of the limit process that defines V and hence U on PA from acts in PF , it follows that HE d P Ž s .w uŽ ps . y uŽ q s .x s 0. Hence UŽ p . s UŽ q ., so p ; q and NF : NA . Since NA : NF by the definitions, we have NA s NF . I Although A7 is necessary and sufficient for our bounded extension result, it is not as simple as we would like and differs substantially from Savage’s P7 and A4a–A4c in Fishburn Ž1970, p. 138. that extend real-valued expected utility representations from simple lotteries and acts to more general lotteries and acts. We considered simpler conditions than A7 and mention two here. One is sufficient but is too strong. The other is necessary, but we have been unable to determine whether it is sufficient. Let a G b for real vectors a s Ž a 1 , . . . , a m . and b s Ž b 1 , . . . , bm . mean that a i G bi for i s 1, . . . , m. The first condition in terms of utility vectors says that, for all p g PA and all q g PF , u Ž ps . G u Ž q s . for all s g S ´ U Ž p . G U Ž q . ; u Ž q s . G u Ž ps . for all s g S ´ U Ž q . G U Ž p . . This implies UŽ p . s V Ž p . for p g PA _ PF with the use of sequences of acts in PF that effectively bound p from below and from above in the sense of G on u. However, even with J s K s 2, it has undesirable implications. Suppose, for example, that E and its complement have matrix probabilities

P Ž E. s

1yr u

0 r and P Ž E c . s 1yc yu

0 c

with r , c g Ž0, 1.. Let p, q g PF satisfy p sE x, p sE c y and q sS l 0 with uŽ x . s Ž e , e ., uŽ y . s Ž1 y e , e ., 0 - e - 1, and uŽ l 0 . s Ž0, 0.. Then uŽ ps . ) uŽ q s . for all s, UŽ q . s Ž0, 0., and

r q e y 2 re UŽ p. s P Ž E. e q P Ž Ec . 1 y e s . e e 2 ue q e y u

ž/

ž

/

ž

/

P.C. Fishburn, I.H. LaValler Journal of Mathematical Economics 30 (1998) 323–346

345

We do not have UŽ p . G UŽ q . when 2 ue q e y u - 0, i.e., when e - urŽ2 u q 1., so our first condition is violated in this case. If e can vary between 0 and 1 and the condition is imposed, it forces u s 0 and severely restricts the allowable types of matrix probabilities. The other simpler condition is, for all p, q g PA ,

w ps ' q s for all s g S x ´ p ' q. Its necessity for p, q g PF was noted in Corollary 6, and its necessity for p, q g PA can be deduced from the definition of V and U s V. Its sufficiency for U s V remains open. A bounding argument shows without difficulty that the condition implies U1 s V1 , so it is sufficient for the real-valued case of J s 1. However, when J ) 1, we have not been able to show that it implies U2 s V2 , . . . , UJ s VJ .

6. Discussion This paper extends the finite-state SELU representation of LaValle and Fishburn Ž1992., LaValle and Fishburn Ž1996a. to an infinite state set S. Five previous axioms were used in Theorem 2 for a standard form of the representation on the set PF of mixtures of acts that are constant on finite partitions of S. We then added two axioms to extend the expected-utility feature of the representation to all act lotteries in PA . The first new axiom is necessary and sufficient for equal parsimonious dimensions of % on PF and % on PA : see Lemma 7. The second, A7, is then necessary and sufficient for the expectational form on PA when the utility function u on consequence lotteries and the finitely-additive matrix probability measure P on 2 S are bounded: see Theorem 8. The new axioms are relevant only when the consequence set C is infinite, for otherwise PA s PF . The paper’s results can be generalized in several ways, some of which relax basic structural assumptions. One generalization relaxes the assumption that A s C S to A : C S. This was done in our earlier work, where A was supposed only to be rich enough to support the operations needed to derive the SELU representation. Another relaxation replaces the event set 2 S by a Boolean algebra on S, or even by an arbitrary family of subsets of S, and restricts attention to acts that are measurable with respect to the algebra or subset family. This was not done for our finite-S setting, but precedents for its approach in real-valued representations are in Fishburn Ž1970., Fishburn Ž1982., Fishburn Ž1992. and, for subjective probability, in Chateauneuf Ž1985. and Lehrer Ž1991.. Another direction of generalization drops the boundedness assumption for u and considers expectation when some u k functionals are unbounded. Theorem 8 applies to all p for which  uŽ ps .: s g S4 is bounded, but in other cases one needs to extend the definition of expectation, say for Vj Ž p . s HS d P j Ž s . uŽ ps ., to unbounded u sets. The usual way to do this is to consider bounded truncations of u,

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say to wyN, N x K , and then take N ™ `. If all Vj Ž p . are well defined and finite, another new axiom might be needed to obtain UŽ p . s V Ž p .. The final chapters in Fishburn Ž1982., and Wakker Ž1993., are relevant in this regard. We conclude by emphasizing two questions left open in the paper. First, is it possible for P to be unbounded? Second, is the condition in the final paragraph of the preceding section sufficient for U s V, given the initial hypotheses of Theorem 8?

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