Non-Archimedean subjective probabilities in decision theory and games

Mathematical Social Sciences 38 (1999) 139–156

Non-Archimedean subjective probabilities in decision theory and games Peter J. Hammond Department of Economics, Stanford University, Stanford, CA 94305 -6072, USA

Abstract To allow conditioning on counterfactual events, zero probabilities can be replaced by infinitesimal probabilities that range over a non-Archimedean ordered field. This paper considers a suitable minimal field that is a complete metric space. Axioms similar to those in Anscombe and Aumann [Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability, Annals of Mathematical Statistics 34, 199–205.] and in Blume et al. [Blume, L., Brandenburger, A., Dekel, E., 1991. Lexicographic probabilities and choice under uncertainty, Econometrica 59, 61–79.] are used to characterize preferences which: (i) reveal unique non-Archimedean subjective probabilities within the field; and (ii) can be represented by the non-Archimedean subjective expected value of any real-valued von Neumann–Morgenstern utility function in a unique cardinal equivalence class, using the natural ordering of the field.  1999 Elsevier Science B.V. All rights reserved.

1. Introduction and outline Recent game theory relies on players having conditional probabilistic beliefs after a counterfactual event, like a player deviating from a presumed equilibrium strategy. Suggested formal frameworks for dealing with such counterfactuals include complete conditional probability systems, lexicographic hierarchies of probabilities, and logarithmic likelihood ratio functions. For a survey of these approaches and their relationship, see Hammond (1994). Section 2 of this paper begins by pointing out a fundamental inadequacy of these previous suggestions. This arises because they do not permit a compound lottery to be reduced unambiguously. Following some early steps taken in Hammond (1994, 1999c), this paper presents an alternative approach, using the smallest possible extension of the real number field that allows for infinitesimal probabilities. Such extensions require consideration of a nonE-mail address: [email protected]. (P.J. Hammond) 0165-4896 / 99 / $ – see front matter PII: S0165-4896( 99 )00011-6

 1999 Elsevier Science B.V. All rights reserved.

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Archimedean ordered algebraic field of possible probabilities, of the kind discussed in Section 3. The key part of the paper begins with Section 4. First, for a general ordered field of possible probabilities, expected utility is derived from three axioms – ordering, independence, and an ‘‘algebraic continuity’’ axiom. Next, Section 5 introduces additional axioms, similar to those devised by Anscombe and Aumann (1963) for real-valued probabilities, and by Blume et al. (1991) (henceforth BBD) for lexicographic probabilities. These are used to prove existence of a unique cardinal equivalence class of real-valued von Neumann–Morgenstern utility functions and also, except in the case of universal indifference, unique (possibly infinitesimal) subjective probabilities. Section 6 considers the special case when probabilities belong to R(e ), the smallest ordered field including both the real line and at least one positive ‘‘basic’’ infinitesimal. This is extended to the special field R ` (e ) of all power series in e, as the smallest complete metric space that includes R(e ) when the latter is given a suitable metric. In Section 7 this extension allows the algebraic continuity condition to be replaced by a more intuitive ‘‘extended continuity’’ condition. In this setting, moreover, applying the natural ordering of R ` (e ) to expected utilities induces a preference ordering which corresponds to a familiar lexicographic expected utility criterion, with a real-valued von Neumann–Morgenstern utility function and non-Archimedean subjective probabilities.

2. Non-reduction of compound lotteries ´ For dealing with counterfactuals, Renyi’s (1955) complete conditional probability systems (CCPSs) are intuitively appealing. They simply require P(E9uE) to be specified whenever event E9 , E, even if E has probability zero. And the modified form P(E0uE) 5 P(E0uE9)P(E9uE) of Bayes’ rule is required to hold whenever E0 , E9 , E. A crucial hypothesis of von Neumann and Morgenstern is that it should always be enough to analyse the normal form of a game, because the outcome should be invariant to alterations in the extensive form that leave the normal form unaffected. This invariance hypothesis is actually fundamental to orthodox game theory, and even more to its foundations in consequentialist single person decision theory. For this invariance hypothesis to hold, lotteries whose prizes are tickets for other lotteries must be reducible to equivalent simple lotteries merely by multiplying probabilities. Unfortunately, where CCPS are concerned, this reduction property may fail because there can be infinitely many different ways of compounding two lotteries which both involve zero probability events. A simple example to show this is illustrated in Fig. 1. In this tree, the conditional probabilities associated with the first chance move at the initial node n 0 are assumed to satisfy P(hnjuhc, nj) 5 1, implying that P(ha, bjuha, b, cj) 5 1, whereas the conditional probabilities associated with the second chance move at node n are assumed to satisfy P(hajuha, bj) 5 1. Of course, then the other conditional probabilities given hc, nj and ha, bj must be 0. Compounding these conditional probabilities evidently implies that

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Fig. 1. Non-reduction of compound conditional probabilities

P(hajuha, b, cj) 5 P(hajuha, bj)P(ha, bjuha, b, cj) 5 1 Also, because P(hajuha, b, cj) 5 P(hajuha, cj)P(ha, cjuha, b, cj) 5 1 it must be true that P(hajuha, cj) 5 1. Then all other conditional probabilities given the non-trivial events ha, bj, ha, cj and ha, b, cj must be 0. But so far nothing determines the relative likelihoods of the two different ‘‘trembles’’ which result in states b and c respectively. Accordingly, the two other conditional probabilities P(hbjuhb, cj) and P(hcjuhb, cj) could be any real numbers in the interval [0, 1] which sum to 1. Hence, reduction of compound lotteries can only occur in a richer space allowing more information to be provided about the relative likelihood of the two trembles.

3. Non-Archimedean probabilities

3.1. Infinitesimals Selten (1975) and many successors (including Myerson, 1978, 1986) have modelled a tremble as a vanishing sequence of positive probabilities – or, equivalently, as vanishing perturbations of the players’ (expected) payoff functions. Such vanishing sequences are obviously closely related to the ‘‘infinitesimals’’ used in the early development of ‘‘infinitesimal calculus’’ by Leibniz and many successors. Formally, a positive infinitesimal is smaller than any positive rational number and so, because the rationals are dense in the reals, smaller than any positive real number. This obviously violates the ‘‘Archimedean’’ property of the real line, requiring that for every r . 0 there should exist an integer n such that rn . 1. So infinitesimals are only allowed in suitable non-Archimedean structures. The development of ‘‘non-standard analysis’’ by Abraham Robinson and others has allowed infinitesimals to be incorporated into modern formal mathematics. See Anderson (1991) for a recent survey, and an explanation of how the ‘‘transfer principle’’ allows standard results to be easily generalized. Non-Archimedean probabilities were briefly mentioned by Chernoff (1954) and then by BBD. They appeared more prominently in Rajan (1998) and in an earlier unpublished paper by Blume on his own. Finally, LaValle and Fishburn (1992, 1996), and also Fishburn and LaValle (1993), have recently

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investigated several variant forms of non-Archimedean expected utility, especially those involving ‘‘matrix’’ probabilities associated with lexicographic utilities.

3.2. What range of probabilities? Originally probabilities were rational numbers, usually with small denominators. Every monetary bet that has ever been made and settled on the precise agreed terms has involved a rational odds ratio. More sophisticated mathematical models of probability involve continuous distributions, which necessitate irrational but real probabilities. To date, real-valued probability measures have served us rather well. Yet the previous discussion suggests the need for infinitesimal probabilities in game theory. This forces us to face the question: How rich should the range of allowable probabilities be? My provisional answer is: As small as possible, provided that some essential requirements are met. If the transfer principle of non-standard analysis is not an essential requirement, then there is no good reason to use every possible ‘‘hyperreal’’ between 0 and 1, as non-standard analysis seems to require. But the range should include some infinitesimal elements, it seems, as well as all real numbers in the interval [0, 1]. Probabilities need adding, because of finite additivity, and also multiplying when compounding lotteries. Furthermore, they need to be divided when calculating conditional probabilities by Bayesian updating. This suggests imposing the algebraic structure of a field. Finally, since the expectation of a real-valued von Neumann– Morgenstern utility function lies within the same field, and expected utilities should represent a complete preference ordering, there should be a linear or total order on the field.1

3.3. General non-Archimedean ordered fields An ordered field kF, 1 , ? , 0, 1, . l is a set F together with: 1. the two algebraic operations 1 (addition) and ? (multiplication); 2. the two corresponding identity elements 0 and 1; 3. the binary relation . which is a linear order on F satisfying 1 . 0. Moreover, F must be closed under the two algebraic operations. The usual properties of real number arithmetic also have to be satisfied – i.e., addition and multiplication both have to be commutative and associative, the distributive law must hold, and every element x [ F must have both an additive inverse 2 x and a multiplicative inverse 1 /x, except that 1 / 0 is undefined. The linear order . must satisfy y . z⇔y 2 z . 0. Also, the set F 1 of positive elements in F must be closed under both addition and multiplication. Any ordered field F has distinct positive integer elements n 5 1,2, . . . defined as the 1

A binary relation . is a linear (or total) order on a domain X if it is transitive and, for every disjoint pair x, y [ X, one has either x . y or y . x.

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sums n 5 1 1 1 1 ? ? ? 1 1 of n copies of the element 1 [ F. Obviously, it follows that F includes all rational numbers. The ordered field F is Archimedean if, given any x [ F 1 , there exists such a positive integer n for which nx . 1. For any x [ F, let uxu denote x if x $ 0 and 2 x if x , 0. Say that any x [ F\h0j is infinitesimal iff uxu , 1 /n for every (large) positive integer n. Evidently, a field is non-Archimedean iff it has infinitesimal elements, and is Archimedean iff it is isomorphic to a subset of the real line R. See Royden (1968), for example. In the rest of the paper, F will denote any ordered field that may or may not be Archimedean. Also, in the ensuing analysis, it will be convenient to use the notation (0, 1) F to indicate the interval hx [ Fu0 , x , 1j. This notation helps to distinguish this possibly non-Archimedean interval from the usual real interval (0, 1), which will often be denoted by (0, 1) R .

4. Algebraic expected utility

4.1. Consequences and states of the world Let Y be a fixed domain of possible consequences, and S a fixed finite set of possible states of the world. No probability distribution over S is specified. An act, according to Savage (1972), is a mapping a: S → Y specifying what consequence results in each possible state. Inspired by the Arrow (1953, 1964) and Debreu (1959) device of ‘‘contingent’’ securities or commodities in general equilibrium theory, I refer instead to contingent consequence functions, or CCFs for short. Also, each CCF will be considered as a list y S 5 ky s l s [S of contingent consequences in the Cartesian product space Y S : 5 s [S Ys , where each Ys is a copy of the consequence domain Y. Anscombe and Aumann (1963) allowed subjective probabilities for the outcomes of ‘‘horse lotteries’’ or CCFs to be inferred from expected utility representations of preferences over compounds of horse and ‘‘roulette lotteries’’. Formally, Savage’s (1972) framework is extended to allow preferences over D(Y S ; R), the space of all (finitely supported) simple roulette lotteries defined on Y S . Each l S [ D(Y S ; R) specifies a real number l S ( y S ) as the objective probability that the CCF is y S [ Y S . Instead of D(Y S ; R), this paper considers the space D(Y S ; F) of F-valued simple lotteries on Y S . Each l S [ D(Y S ; F) satisfies l S ( y S ) 5 0 iff y S is outside the finite support F , Y S of l S . On F each probability l S ( y S ) [ F 1 , and o y S [F l S ( y S ) 5 1. That is, l S ( y S ) . 0 for all y S [ F, even though l S ( y S ) could be infinitesimal.

P

4.2. Objective expected utility Anscombe and Aumann directly assumed expected utility maximization for roulette lotteries. BBD, like Fishburn (1970) and many others, laid out axioms implying expected utility maximization rather than assuming it directly. Apart from being more fundamental, here such an approach is essential because non-Archimedean expected

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utility maximization should be deduced before subjective probabilities are inferred. Accordingly, I assume: • (O) Ordering. There exists a (reflexive, complete and transitive) preference ordering ' S on D(Y S ; F). • (I*) Strong Independence. If l S , m S , n S [ D(Y S ; F) and a [ (0, 1) F , then

l S ' S m S ⇔al S 1 (1 2 a )n S ' S am S 1 (1 2 a )n S • (AC) Algebraic Continuity. 2 Given any three lotteries l S , m S , n S [ D(Y S ; F) satisfying l S s S m S and m S s S n S , there exists a [ (0, 1) F such that al S 1 (1 2 a )n S | S m S . Axioms (O) and (I*) obviously extend standard conditions for real-valued probability distributions in D(Y S ; R) to F-valued probability distributions in D(Y S ; F). Even the version of axiom (AC) for D(Y S ; R) has appeared in the literature. However, the latter is usually derived from a more fundamental continuity assumption like the counterpart of condition (C) in Section 7.2 – see, for example, Hammond (1999a, b). For this reason, condition (AC) is discussed further in the special non-Archimedean framework of Sections 6 and 7. Lemma 1. Suppose that axioms (O) and (I*) are satisfied on D(Y S ; F). Then, for any pair of lotteries l S , m S [ D(Y S ; F) with l S s S m S and any a 9, a 0 [ (0, 1) F with a 9 . a 0, one has

l S s S a 9 l S 1 (1 2 a 9)m S s S a 0 l S 1 (1 2 a 0)m S s S m S Proof. If a 9, a 0 [ (0, 1) F with a 9 . a 0, then there exists d [ (0, 1) F such that a 0 5 da 9. Now one has

d [a 9 l S 1 (1 2 a 9)m S ] 1 (1 2 d )m S 5 da 9 l S 1 (1 2 da 9)m S 5 a 0 l S 1 (1 2 a 0)m S Repeated application of axiom (I*) gives l S s S a 9 l S 1 (1 2 a 9)m S s S m S , and also

a 9 l S 1 (1 2 a 9)m S s Sd [a 9 l S 1 (1 2 a 9)m S ] 1 (1 2 d )m S s Sdm S 1 (1 2 d )m S 5 mS Together these statements clearly imply the result. h Say that the the utility function U S : D(Y S ; F) → F represents ' S on D(Y S ; F) if for every pair of lotteries l S , m S [ D(Y S ; F) one has

l S ' S m S ⇔U S ( l S ) $ U S ( m S ) 2

In the following, s S and | S respectively denote the strict preference and indifference relations corresponding to ' S .

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Say too that U S satisfies the algebraic mixture preservation condition (AMP) provided that, whenever a [ (0, 1) F , then S

S

S

S

S

S

S

U (al 1 (1 2 a )m ) 5 a U ( l ) 1 (1 2 a ) U ( m ) Note how property (AMP) implies that for every l S [ D(Y S ; F) one can write U S ( l S ) in the algebraic expected utility form U S( l S ) 5

O

l S ( y S )v( y S )

y S [Y S S

S

where v( y ): 5 U (1 y S ) is the utility of the degenerate lottery 1 y S which attaches probability 1 to the particular CCF y S [ Y S . Finally, say that the two functions U S , V S : D(Y S ; F) → F are cardinally equivalent provided there exist constants r [ F 1 and d [ F such that V S ( l S ) 5 d 1 r U S ( l S ) for all l S [ D(Y S ; F). Obviously, if U S and V S are cardinally equivalent, they must also be S S S S ]S identical if they satisfy the common normalizations U S (l ] ) 5V (l ] ) 5 0 and U (l ) 5 S ]S S ]S S V (l ) 5 1 for some l ] , l [ D(Y ; F). The following Lemma is a simple adaptation of a result which is familiar for real-valued probabilities. Accordingly, the proof will not be provided here. For details see, for example, Fishburn (1970) or Hammond (1999a). Lemma 2. Suppose that axioms (O), (I*), and (AC) are all satisfied on D(Y S ; F). Then there exists a unique cardinal equivalence class of utility functions U S : D(Y S ; F) → F which represent ' S and satisfy (AMP).

5. Algebraic subjective probability

5.1. The algebraic SEU hypothesis The first aim of this section is to provide sufficient conditions like those of Anscombe and Aumann for the preference ordering ' S on D(Y S ; F) to have a subjective expected utility (SEU) representation which, for each l S [ D(Y S ; F), takes the form U S( l S ) 5

O

y S [Y S

l S( y S )

O p v( y ) s

s

(1)

s [S

for a unique cardinal equivalence class of von Neumann–Morgenstern utility functions (NMUFs) v:Y → F, and for suitable unique subjective probabilities ps [ F 1 satisfying o s [S ps 5 1. The difference from Anscombe and Aumann is that the subjective probabilities ps and utilities v( y) may belong to the general ordered field F, as may the objective probabilities l S ( y S ) determining the lottery l S [ D(Y S ; F). In addition, the subjective probabilities ps must be positive rather than merely non-negative. This particular characterization of the preference ordering ' S will be called the algebraic SEU hypothesis. A second aim is to ensure that the utility function v is real-valued, even when the field

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F is non-Archimedean. To do so, I will introduce a new state-independent continuity axiom (SIC). Then only the probabilities, and not the utilities, are allowed to be non-Archimedean.

5.2. Reversal of order In order to derive subjective probabilities, Anscombe and Aumann added three more axioms to their basic hypothesis that ‘‘roulette lotteries’’ would be chosen to maximize objective expected utility. For each y [ Y and s [ S, define Y Ss ( y): 5 hy S [ Y S uy s 5 yj as the set of CCFs yielding the particular consequence y in state s. Given any lottery l S [ D(Y S ; F), any state s [ S and any consequence y [ Y, let

ls ( y): 5

O

l S( y S )

y S [Y Ss ( y)

denote the marginal probability that y occurs in s. Then the probabilities ls ( y) ( y [ Y) specify the marginal distribution in state s [ S. The first of the three additional axioms is: • (RO) Reversal of Order. Whenever l S , m S [ D(Y S ; F) have marginal distributions satisfying ls 5 ms for all s [ S, then l S | S m S . This condition owes its name to the fact that there is indifference between: (i) having the roulette lottery l S determine the random CCF y S before the horse lottery that resolves which state s [ S and which ultimate consequence y s occur; and (ii) resolving the horse lottery first, before its outcome s [ S determines which marginal roulette lottery ls generates the ultimate consequence y.

5.3. The sure thing principle The second of Anscombe and Aumann’s additional axioms concerns any event E , S, together with the product space Y E : 5 s [E Ys of contingent CCFs taking the form y E 5 ky s l s [E [ Y E , and the existence of an associated contingent preference ordering ' E . Here it is natural to assume that ' E is defined on D(Y E ; F), the space of F-valued probability distributions, instead of only on D(Y E ), the space of real-valued probability distributions. So the second extra axiom becomes:

P

• (STP) Sure Thing Principle. Given any event E , S, there exists a contingent preference ordering ' E on D(Y E ; F) satisfying

l E ' E m E ⇔( l E , n S \E )' S ( m E , n S \E ) for all l E , m E [ D(Y E ; F) and n S \E [ D(Y S \E ; F), where ( l E , n S \E ) denotes the combination of the conditional lottery l E if E occurs with n S \E if S\E occurs, and similarly for ( m E , n S \E ).

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However, following an idea due originally to Raiffa (1961) and then used by BBD, it is easy to show: Lemma 3. Suppose that the three axioms (O), (I*), and (RO) are all satisfied on D(Y S ; F). Then so is (STP). Proof. See Hammond (1999b). h Because of this result, (STP) will not be imposed as an axiom, but it will often be used in the ensuing proofs.

5.4. State independence For each s [ S, condition (STP) ensures the existence of a contingent preference ordering ' hs j on D(Ys ; F) 5 D(Y; F). The last axiom is: • (SI) State Independence. Given any state s [ S, the contingent preference ordering ' hs j over D(Y; F) is independent of s. Let ' * denote this state-independent preference ordering. When ' S on D(Y S ; F) satisfies conditions (O), (I*) and (AC), so too must ' * on D(Y; F), because of (STP).

5.5. Subjective probabilities The five axioms (O), (I*), (AC), (RO), and (SI) are assumed throughout the following, as is condition (STP). Lemma 4. (a) Suppose that E , S is any event and that l E , m E [ D(Y E ; F) satisfy ls ' * ms for all s [ E. Then l E ' E m E . (b) If in addition l E | E m E , then ls | * ms for every state s [ E. Proof. By induction on the number of states in E. h From now on, exclude the trivial case of universal indifference by assuming that there ] ] *l. exist two roulette lotteries l, l ] [ D(Y; F) with l s ] Given any event E , S and any lottery l [ D(Y; F), let l1 E denote the lottery in D(Y E ; F) whose marginal distribution in each state s [ E is ls 5 l, independent of s. Lemma 5. The ordering ' * on D(Y; F) is represented by a utility function U * :D(Y; F) → F satisfying U * ( l) 5 U S ( l1 S ) for all l [ D(Y; F). Proof. Because ' S on D(Y S ; F) and ' * on D(Y; F) both satisfy conditions (O), (I*) and (AC), Lemma 2 of Section 4.2 implies that they can be represented by normalized utility functions U S :D(Y S ; F) → F and U * :D(Y; F) → F satisfying (AMP) and also

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] S S ] S * l) 5 0, U * (l) 5 1 U S (l ]1 ) 5 0, U (l1 ) 5 1, and U (]

(2)

Next, Lemma 4(a) implies that l' * m ⇒ l1 S ' S m 1 S . On the other hand, Lemma 4(b) implies that m s * l ⇒ m 1 S s S l1 S . Because ' * and ' S are complete orderings, the reverse implication l1 S ' S m 1 S ⇒ l' * m follows. Hence, l1 S ' S m 1 S ⇔ l' * m. So U S ( l1 S ) and U * ( l) must be cardinally equivalent functions of l on the domain D(Y; F). Because of the two normalizations in (2), the result follows immediately. h Lemma 6. For each l S [ D(Y S ; F) one has U S( l S ) 5

O q U *( l ) s

(3)

s

s [S

S \hs j ] where qs : 5 U S (l ]1 , l) [ F 1 for all s [ S, implying that o s [S qs 5 1.

Proof. The proof by Fishburn (1970, Theorem 13.1, p. 176) of the corresponding result for the space D(Y S ) of real-valued probabilities applies to D(Y S ; F), with no change except that (AMP) replaces the usual mixture preservation property. See also Hammond (1999b). h Lemma 7. There exists a unique cardinal equivalence class of NMUFs v:Y → F and, unless there is universal indifference, unique subjective probabilities ps (s [ S) such that the ordering ' S on D(Y S ; F) is represented by the subjective expected utility function U S( l S ) ;

O p O l ( y)v( y)

s [S

s y [Y

(4)

s

Proof. By Lemma 6, with ps replacing qs in (3), one has S

S

U (l ) 5

O p U *( l ) 5 O p O l ( y)v*( y) s

s [S

s

s

s [S

s

(5)

y [Y

where v * ( y): 5 U * (1 y ) for all y [ Y, while the second equality in (5) follows because ls is the (finite) mixture o y [Y ls ( y)1 y and U * satisfies (AMP). As in Lemma 2, the NMUF v * could be replaced by any cardinally equivalent v: Y → F. But this is equivalent to replacing U S by a cardinally equivalent V S : D(Y S ; F) → F. Any such transformation leaves the ratio S \hs j ] S U S (l l1 S ) ]S1] S , l) 2SU (] ps 5 ]]]]]]] S U (l1 ) 2 U (l ]1 ) of expected utility differences unaffected. This ensures that the subjective probabilities are unique. h

5.6. Real-valued utility Lemmas 5 and 7 imply that the state-independent preference ordering ' * on D(Y; F) is represented by the expected value of each F-valued state-independent NMUF v: Y → F in a unique cardinal equivalence class. Where F is non-Archimedean, v could be

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also. Yet in game theory, the motivation for non-Archimedean probabilities is that players’ deviations from their presumed best responses can be given infinitesimal subjective probability. Where there is no uncertainty of this kind, but only risk in the form of specified objective probabilities, there is no good reason to depart from classical expected utility theory, which requires preferences over real-valued probability distributions to be continuous, and expected utility to be real-valued – as in BBD, for example. To ensure that v is real-valued, ' * on D(Y; R) is assumed to satisfy an extra standard continuity axiom for real-valued probability distributions. • (SIC) State Independent Continuity. 3 For all l, m,n [ D(Y; R) satisfying l s * m s * n, there exist a 9, a 0 [ (0, 1) such that

a 9 l 1 (1 2 a 9)n s * m

and

m s * a 0 l 1 (1 2 a 0)n

Theorem 8. Suppose that the six axioms (O), (I*), (AC), (RO), (SI) and (SIC) are all satisfied on D(Y S ; F). Then, unless there is universal indifference, there exist unique subjective probabilities ps [ D(S; F 1 ) and a unique cardinal equivalence class of real-valued NMUFs v: Y → R such that

l S ' SmS ⇔

O p O l ( y)v( y) $ O p O m ( y)v( y) s

s [S

s

y [Y

s

s[S

s

y [Y

Proof. By Lemma 7, there exist unique subjective probabilities ps and a unique cardinal equivalence class of NMUFs v:Y → F. Because there is not universal indifference, v cannot be constant. So it can be normalized within the same cardinal equivalence class ] 5 1 and v(y) 5 0 for some pair y, ] y [ Y satisfying 1 ] s * 1 . to make v(y) y y S ] ] ] Next, let ' R denote the restriction of the ordering ' S to the space D(Y; R) of real-valued probability distributions. Note that ' SR satisfies axioms (O), (I*), (SIC), (RO) and (SI). By a standard result in utility theory, ' SR also satisfies (AC), and can be represented by the expected value of a real-valued NMUF vR which can be normalized ] 5 1 and v (y) 5 0. The expected values of both v and v represent ' S to satisfy vR (y) R R R ] on D(Y; R), so these two NMUFs are cardinally equivalent. Because of the common normalization, they must be identical. That is, v( y) 5 vR ( y) [ R for all y [ Y, implying that v: Y → R. h

3

This is sometimes called the Archimedean axiom – see, for instance, Karni and Schmeidler (1991, p. 1769). For obvious reasons I avoid this name here. Note that imposing axiom (AC) – or the counterpart of condition (C) as stated in Section 7.2 – on ' S restricted to D(Y S ; R) would imply that all subjective probabilities are real-valued. This explains why ' * , and not ' S , is assumed to be continuous w.r.t. changes in real-valued probabilities.

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6. The ordered field R ` (e ) as a complete metric space

6.1. The ordered field R(e) In Hammond (1994, 1999c), the search for a minimal suitable range of probabilities suggested the particular ordered field R(e ), originally described by Robinson (1973). This is the smallest algebraic field that contains the real line R as well as a basic positive infinitesimal element e. One can regard e as a particular element of the ordered field of hyperreals used in non-standard analysis. Or perhaps more intuitively, as any vanishing sequence ken l `n51 of positive real numbers. The typical non-zero element of R(e ) is a ratio of two ‘‘polynomial expressions’’ – i.e., a ‘‘rational expression’’ whose normalized form is

O O

n

a k e k 1 i5k 11 a i e i f(e ) 5 ]]]]]] m 1 1 j 51 b j e j

(6)

for some unique integer k and non-zero leading coefficient a k . Define the ordering . on R(e ) so that f(e ) . 0 iff a k . 0. This makes . a lexicographic linear ordering on R(e ), in effect. It also makes R(e ) a non-Archimedean ordered field, with f(e ) infinitesimal iff k . 0.

6.2. A metric Section 7 will be concerned with finding weaker sufficient conditions for the algebraic continuity axiom (AC) of Section 4.2 to hold when probabilities are allowed to range over R(e ). So R(e ) needs a topology such as that induced by Lightstone and Robinson’s (1975) metric. Given the normalized form (6) of f(e ), define k as the infinitesimal order v[ f(e )] of f(e ), with v[0]: 5 ` for the zero element of R(e ). If f(e ) ± 0, note that: 1. if k . 0, then f(e ) is an infinitesimal of order k; 2. if k , 0, then f(e ) is really an infinite number of order 2 k, which can be regarded as an infinitesimal of negative order; 3. if k 5 0, then f(e ) is infinitesimally different from a non-zero real number, in which case it is said to be of infinitesimal order 0. Note too that, for all non-zero pairs f(e ), g(e ) [ R(e ), one has v[ f(e ) ? g(e )] 5 v[ f(e )] 1 v[g(e )] and v[ f(e ) 1 g(e )] $ minhv[ f(e )], v[g(e )]j It follows that the infinitesimal order is an instance of what Robinson (1973) describes as a ‘‘non-Archimedean valuation’’. Now define the function d: R(e ) 3 R(e ) → R so that d( f(e ), g(e )): 5 2 2v[ f (e )2g(e )]

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for every f(e ), g(e ) [ R(e ), with the obvious convention that 2 2` : 5 0. Obviously, d( f(e ), g(e )) 5 d( g(e ), f(e )) and d( f(e ), g(e )) 5 0 if and only if v[ f(e ) 2 g(e )] 5 `, which is true iff f(e ) 5 g(e ). Finally, it is easy to verify that d is a metric because the triangle inequality is satisfied.

6.3. Convergence Consider any infinite sequence f n (e ) (n 5 1, 2, . . . ) in R(e ). Evidently f n (e ) → f(e ) as n → ` iff d( f n (e ), f(e )) → 0, so iff v[ f n (e ) 2 f(e )] → `. But after defining g n (e ): 5 f n (e ) 2 f(e ) for n 5 1, 2, . . . , this holds iff g n (e ) → 0 because v[g n (e )] → `. Ignoring any zero terms of the sequence g n (e ) (n 5 1, 2, . . . ), this requires that the infinitesimal order k n of n

O O

,n

a nk n e k 1 i 5k n 11 a ni e i g (e ): 5 ]]]]]]] mn 1 1 j 51 b nj e j n

should tend to 1 `. But then, for each number M 5 1, 2, . . . , there must exist n(M) such that n . n(M) implies k n . M because a ni 5 0 for all i # M. That is, all coefficients a in of the numerator must become 0 for n sufficiently large. This is necessary and sufficient for convergence. Note one important implication: An infinite sequence of real numbers converges iff it is eventually equal to a real constant. Thus, the (very fine) topology we have defined on R(e ) induces the discrete topology on the subspace R, meaning that every subset of R is open in the subspace topology.

6.4. Completing the metric space Any sequence f n (e ) (n 5 1, 2, . . . ) of elements in R(e ) is a Cauchy sequence iff for every real d . 0 there exists M(d ) such that d( f m (e ), f n (e )) , d whenever m, n . M(d ). Just as ordinary continuity requires completing the space of rationals by going to the real line R in which Cauchy sequences converge, here I will consider a similar completion R ` (e ) of R(e ). In R(e ) many Cauchy sequences do not converge. Indeed, consider any sequence n f n (e ) 5 o k50 a k e k (n 5 1, 2, . . . ) with a non-recurring infinite sequence ka k l k`51 of real n k2 coefficients – for example, o k 50 e (n 5 1, 2, . . . ). An analogy is the non-recurring ] decimal expansion of any irrational real number such as Œ2, which has no limit among the set of rationals. The obvious limit of the sequence f n (e ) should be the power series ] o `k 50 a k e k , just as the limit of the decimal expansion 1.41421356 . . . is Œ2. Yet a ` non-recurring power series o k50 a k e k does not correspond to a rational expression in R(e ), because multiplying by any polynomial expression always leaves an infinite power series, never a polynomial. The obvious way to complete R(e ) is to allow such ‘‘irrational’’ infinite power series. To retain the algebraic structure of an ordered field, the ratios of such power series must also be accommodated. However, the reciprocal of any power series is itself a power

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series, but of the form o `k 5k 0 a k e k where the leading power k 0 could be negative. Accordingly, define R ` (e ) as the set of all such power series.4 Of course, any rational expression in R(e ) can be expanded as a power series in R ` (e ), so R ` (e ) does extend R(e ). Each member of R ` (e ) can be conveniently written as o `k 52` a k e k , where there must exist k 0 such that a k 5 0 for all k , k 0 . Then the metric d on R(e ) can obviously be extended to R ` (e ). In fact, the infinite sequence f n (e ) (n 5 1, 2, . . . ) of power series o `k 52` a nk e k in R ` (e ) will converge to the limit f(e ) 5 o `k 52` a k e k in R ` (e ) iff for every k there exists n k such that a kn 5 a k for all n $ n k . Suppose o k`52` a kn e k (n 5 1, 2, . . . ) is any Cauchy sequence in R ` (e ). Then: (i) there exist k 0 , n 0 [ N such that a kn 5 0 whenever k , k 0 and n . n 0 ; (ii) for every k $ k 0 there exist both n k [ N and a k [ R such that a nk 5 a k for all n . n k . Hence, the Cauchy sequence converges to f(e ): 5 o `k 5k 0 a k e k , so R ` (e ) is a complete metric space. ` ` Two Cauchy sequences ko `k 52` a nk e k l n51 and ko `k 52` b nk e k l n51 are said to be ‘‘limit ` ` n n k equivalent’’ in R (e ) whenever o k 52` (a k 2 b k )e converges to 0 in R ` (e ) as n → `. Because the quotient space of limit equivalence classes of Cauchy sequences in R(e ) is easily seen to be an ordered field which is isomorphic to R ` (e ), it follows that R ` (e ) is effectively the smallest complete metric space containing R(e ), just as R is the smallest complete metric space containing the ordered field of rationals.

7. Lexicographic expected utility

7.1. Continuity In the special case when the ordered field F is the complete metric space R ` (e ), the unsatisfactory algebraic continuity (AC) axiom of Section 4.2 is implied by two weaker continuity axioms. Of these, the first is a non-Archimedean version of a familiar continuity axiom for preferences over real-valued lotteries in D(Y S ). • (C*) Continuity. For all lotteries l S , m S ,n S [ D(Y S ; R ` (e )) satisfying l S s S m S and m S s S n S , the two sets A: 5 ha (e ) [ (0, 1) R ` (e ) ual S 1 (1 2 a )n S ' S m S j B: 5 ha (e ) [ (0, 1) R ` (e ) ual S 1 (1 2 a )n S )S m S j are both closed.

7.2. Extended continuity Before presenting the second continuity axiom, let me first re-state axiom (AC) for the case when probabilities take values in R ` (e ). 4

In fact R ` (e ) is a proper subset of the space + described by Levi-Civita (1954) and by Laugwitz (1968), whose members are power series of the form o `k 5k 0 a k e nk where knk l `k 51 is any increasing sequence of real numbers that tends to ` as k → `.

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• (NAC) Non-Archimedean Continuity. Given any three lotteries l S , m S , n S [ D(Y S ; ` R ` (e )) satisfying l S s S m S and m S s S n S , there exists a (e ) 5 o k50 ak e k [ (0, 1) R `(e ) S S S S such that a (e )l 1 [1 2 a (e )]n | m . Then it is evident that axiom (NAC) can only be satisfied if the following logically weaker axiom is as well: • (XC) Extended Continuity. Suppose am (e ): 5 o km50 a k e k [ (0, 1) R ` (e ) has the property that, for all real d 9, d 0 . 0, one has S 1 S S S S 2 S 2 S d1 m (e )l 1 [1 2 d m (e )]n s m s d m (e )l 1 [1 2 d m (e )]n

(7)

m whenever d m1 (e ) 5 am (e ) 1 d 9e m and d 2 m (e ) 5 am (e ) 2 d 0e . Then there must exist a 9m 11 , a 99 m 11 [ R such that

99 (e )l S 1 [1 2 g m11 99 (e )]n S g 9m 11 (e )l S 1 [1 2 g m9 11 (e )]n S s S m S s Sg m11 9 e m 11 and g m9911 (e ) 5 am (e ) 1 a m11 99 e m11 . where g 9m 11 (e ) 5 am (e ) 1 a m11 Clearly, axiom (XC) is specific to the field R ` (e ). When am (e ) satisfies (7), any positive probability perturbation of order e m is strictly preferred, and any negative perturbation of the same order is strictly dispreferred. Then axiom (XC) requires there to be some perturbations of order e m11 that are strictly preferred, and others that are strictly dispreferred. Note the similarity here with the following weakening of axiom (C*) that is familiar when the ordered field is R instead of R ` (e ): • (C) Continuity. For all lotteries l S , m S ,n S [ D(Y S ; R ` (e )) satisfying l S s S m S s S n S , there exist a 9(e ),a 0(e ) [ (0, 1) R ` (e ) such that

a 9( e )l S 1 [1 2 a 9( e )]n S s S m S s S a 0(e )l S 1 [1 2 a 0(e )]n S 7.3. Non-Archimedean continuity It will now be shown that when probabilities take values in R ` (e ), axioms (C*) and (XC) can together replace the questionable (NAC) among the sufficient conditions for the SEU hypothesis. Lemma 9. On the space D(Y S ; R ` (e )) of probability distributions taking values in the complete metric space R ` (e ), the four axioms (O), (I*), (C*) and (XC) together imply (NAC). Proof. Suppose that l S , m S , n S [ D(Y S ; R ` (e )) satisfy l S s S m S and m S s S n S . The following proof involves a recursive construction. This begins at stage m 5 0 with a21 (e ): 5 0 and the two disjoint subsets A 0 : 5 ha [ (0, 1) R ual S 1 (1 2 a )n S s S m S j

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B0 : 5 ha [ (0, 1) R ual S 1 (1 2 a )n S a S m S j of the real interval (0, 1) R . Lemma 1 of Section 4.2 implies that a 9 . a 0 whenever a 9 [ A 0 and a 0 [ B0 . So there is a unique a 0 [ (0, 1) R satisfying a 0 5 infA 0 5 supB0 . Suppose recursion steps k 5 0, 1, . . . , m (m $ 0) have already been completed, and that they yield the coefficients a 0 , a 1 , . . . , a m of am (e ): 5 o km50 a k e k [ (0, 1) R `(e ) , together with the associated sets A m : 5 ha [ Ru[am 21 (e ) 1 ae m ] l S 1 [1 2 am21 (e ) 2 ae m ]n S s S m S j Bm : 5 ha [ Ru[am21 (e ) 1 ae m ] l S 1 [1 2 am 21 (e ) 2 ae m ]n S a S m S j which satisfy a 9m . a m . a m99 whenever a 9m [ A m and a 99 m [ Bm . At recursion step m 1 1 (m $ 0), axiom (XC) implies that the two sets A m 11 and Bm11 are non-empty. Applying Lemma 1 again, there must exist a unique a m 11 [ R such that a m11 5 infA m11 5 supBm 11 . Let am 11 (e ): 5 o km11 a k e k [ (0, 1) R `(e ) . Then a m9 11 . 50 9 11 [ A m 11 and a 99m 11 [ Bm 11 . This completes recursion step a m11 . a 99 m 11 whenever a m m 1 1. m m Consider next the two sequences a 1 and a 2 m (e ): 5 am (e ) 1 e m (e ): 5 am (e ) 2 e (m 5 1, 2, . . . ). Then for m 5 1, 2, . . . one has S 1 S S S S 2 S 2 S a1 m (e )l 1 [1 2 a m (e )]n s m s a m (e )l 1 [1 2 a m (e )]n

Also, because R (e ) is complete, both sequences a m (e ) and a m (e ) converge as m → ` to the same power series a (e ): 5 o `k 50 a k e k [ (0, 1) R ` (e ) . By axiom (C*), a (e )l S 1 [1 2 a (e )]n S | S m S . So (NAC) holds. h `

1

2

7.4. Main theorem The following theorem gives sufficient conditions for the non-Archimedean version of the SEU hypothesis to hold, with a real-valued utility function: Theorem 10. Suppose axioms (O), (I*), (C*), (RO), (SI), (SIC) and (XC) are all satisfied on D(Y S ; R ` (e )). Unless there is universal indifference, there exist unique strictly positive non-Archimedean subjective probabilities p( ? ; e ) [ D(S; R `1 (e )) and a unique cardinal equivalence class of real-valued NMUFs v:Y → R such that the preference ordering ' S on D(Y S ; R ` (e )) is represented by the R ` (e )-valued subjective expected utility expression U S( l S ) ;

O p(s; e) O l ( y)v( y)

(8)

s

s [S

y [Y

S

S

S

S

S

S

S

in the sense that l ' m if and only if U ( l ) $ U ( m ) in the natural ordering of R ` (e ). Proof. The result follows trivially from applying Theorem 8 to the field F 5 R ` (e ) and

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then using Lemma 9 to replace the particular axiom (NAC) with the two axioms (C*) and (XC). h

7.5. Lexicographic expected utility For every s [ S, the subjective probability p(s; e ) [ R `1 (e ) can be expressed as the power series o k`50 pk (s)e k . Thus, the SEU expression (8) can be re-written as the power series U S ( l S ) ; o k`50 u kS ( l S )e k where S

S

u k ( l ): 5

O p (s) O l ( y)v( y) k

s [S S

S

s

(k 5 0, 1, 2, . . . )

y[Y

S

But then l ' m if and only if the associated infinite hierarchies of coefficients ` ku Sk ( l S )l k50 and ku kS ( m S )l k`50 satisfy ku Sk ( l S )l k`50 $ L ku Sk ( m S )l `k50 w.r.t. the usual lexicographic total ordering . L on R ` . In this sense, the preference ordering ' S has a lexicographic expected utility representation.

Acknowledgements My thanks to Don Brown for greatly improving my knowledge of non-standard analysis while he was a colleague at Stanford. Also to Giacomo Bonanno and an anonymous referee for helping me clear up some ambiguities, and giving concrete suggestions for how to produce a condensed version for inclusion in the published symposium. A more complete version, including more proofs and some discussion of the intuitive limitations of non-standard analysis, is available as Stanford Economics Department working paper 97-038, whose PDF form can be found on the internet at: http: / / www-econ.stanford.edu / econ / workp / swp97038.pdf

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