March 25, 2007

Summary We prove the following facts related to the expected multi-utility representation of an aﬃne preorder: If the prize space is not compact and if the lottery set consists of all probabilities on the prize space, standard independence and continuity axioms do not guarantee the existence of a (continuous) representation. If the prize space is σ -compact and lotteries have compact support, a representation exists. When the preorder in question is bounded, this result extends to the set of lotteries that consists of all probabilities on the prize space. For the case of monetary lotteries, the boundedness assumption in this last result can be dropped, provided that the preference relation at hand is monotone and risk-averse. Keywords and Phrases: Incomplete preference relations, Expected utility JEL Classification Numbers: D11, D81

∗

I am grateful to Efe A. Ok for calling my attention to the subject and for his continuous support. I owe special thanks to Juan Dubra for his suggestions which opened the way to some of the results presented here. Comments of a referee of this journal improved the exposition of the paper and the statement of a theorem. I am also indebted to Raphaël Giraud, Kevin Hasker, Farhad Hüsseinov and participants of the Study Group at Bilkent University for valuable discussions. This research started when I was a graduate student at Bilkent University. I finally thank to this institution.

1

Introduction

A first attempt toward developing an expected utility theory for incomplete preferences is due to Aumann (1962). In this work, Aumann showed that if a binary relation defined on a finite dimensional mixture set satisfies all axioms other than the completeness axiom of the classical expected utility theory, there is an expected utility function that extends the relation. Kannai (1963) gave necessary and suﬃcient conditions for possibility of such a representation for the case of relations defined on infinite dimensional vector spaces. Two recent contributions are those of Shapley and Baucells (1998)1 and Dubra, Maccheroni and Ok (2004) (henceforth, DMO). In these works, the authors obtain a set of expected utility functions that completely characterize a relation instead of a single function that extends it. In their main theorem, DMO show that if the set of prizes is a compact metric space, an (incomplete) preference relation which is defined on the set of all lotteries admits a representation by a set of continuous expected utility functions provided the preference relation in question satisfies the usual independence axiom as well as a topological continuity axiom. The authors leave open whether this theorem can be extended to include more general prize spaces. In this note we attack this problem. At the first hand, we show that unless we impose some additional restrictions, the answer of the above question is negative. Specifically, it will be shown that when the lottery space is the set of all probability measures over the real line, the independence and continuity axioms may not be suﬃcient for existence of a (continuous) expected multiutility. A natural question that follows is, under some additional conditions, if we can prove positive results which may be useful in applications and which are not covered by the theorem of DMO. We present here three such results. Our first positive result requires lotteries to have compact support and the prize space to be a σ-compact metric space.2 A potential area of application of this result may be decision problems that involve only lotteries with finite support, e.g., strategic decisions in a game theoretic environment where players and the nature are confined to play simple strategies. It thus seems that this result may be a useful generalization of that of DMO. We next show that if the preference relation at hand is bounded, that is, if there exist a best and a worst prize, then we can cover the case of a lottery set consisting of all 1

For a revised version of this paper, which focuses to finite dimensional mixture sets, see Baucells and Shapley (2006). 2 To describe the prize space that we are working on, the term “locally compact” is often used together with the term “σ-compact.” For brevity, we omit the term “locally compact.” A formal description of the prize space can be found in Subsection 2.3.

1

probability measures over a σ-compact metric space. In a discrete time model with infinite time horizon, a decision problem which involves lotteries defined over time × money space provides a natural framework for potential applications of this result. In such a problem, if we can put bounds to the amount of monetary prize that can ever occur, the prize space would be σ-compact.3 Moreover, assuming that the upper and lower bounds of the money space do not have the same sign, under some natural restrictions on preferences, now × maximum amount and now × minimum amount would be the required best and worst prizes, respectively. We finally investigate the case of monetary lotteries and show that when the prize space is an open interval which is bounded from below, under the usual monotonicity and risk aversion assumptions an expected multi-utility exists. In passing, perhaps a remark about the technical nature of the existence problem that we address in this paper is in order. It turns out that under relatively weak assumptions an incomplete preference relation over lotteries admits a set of linear utility functions (see Proposition 2 in Subsection 3.3). However, whether these utility functions truly posses an expected utility form is a diﬀerent issue which seems to depend on a variety of factors. Compactness properties of the prize space, boundedness of the supports of lotteries, boundedness of the preference relation, continuity properties of expected utility functions that we want to obtain are identified here as some of the vital aspects of the problem. The main results will be presented in Section 2 and proved in Section 3. Section 3 also contains some technical results which may be of independent interest.

2

Main results

2.1

Preliminaries

Throughout the paper X denotes a metric space of prizes. The vector space of all continuous real functions on X is denoted by C(X). Cb (X) stands for the Banach space (endowed with supremum norm) of bounded elements of C(X). P (X) denotes the set of all countably additive (Borel) probability measures on X. The Dirac measure of x ∈ X is denoted by δ x . Ps (X) will stand for the set of all simple probability measures on X, i.e., Ps (X) := {p ∈ P (X) : ∃ a finite F ⊆ X with p(F ) = 1}. The set of all elements of P (X) that have a compact support is denoted by Pk (X), that is, Pk (X) := {p ∈ P (X) : ∃ a compact K ⊆ X with p(K) = 1}. ca(X) stands for the span of P (X). We 3

Of course, we could think of prizes here as degenerate streams of payments so that the prize space would be compact in the topology of pointwise convergence. On the other hand, this topology is considerably weak and this may lead to discontinuity of preferences.

2

endow ca(X) with the usual setwise order ≥. µ+ (resp. µ− ) will denote the positive (resp. negative) part of a point µ in ca(X). A Riesz subspace of ca(X) refers to a linear subspace Y such that µ+ , µ− ∈ Y whenever µ ∈ Y . A Riesz subspace Y is said to be an ideal if for any µ ∈ ca(X), 0 ≤ µ ≤ η and η ∈ Y imply µ ∈ Y . Given a linear subspace Y of ca(X), we define PY := P (X) ∩ Y . In the remainder of the paper P denotes a non-empty, convex subset of P (X) which represents the set of all lotteries. spanP stands for the linear space spanned by P . We define [P ] to be the linear space of all elements of C(X) which are integrable with respect © R ª to (w.r.t.) all p in P , i.e., [P ] := u ∈ C(X) : ∀p ∈ P, X |u| dp < ∞ . It is important to note that [Pk (X)] = C(X) and [P (X)] = Cb (X). When formulating our continuity axiom, we will endow P with the [P ]-topology.4 As usual, a binary relation on P is a subset of P × P . Preferences of an agent over P will be represented by a preorder (a reflexive and transitive binary relation) on P . Hence, preferences are allowed to be incomplete. Instead of (p, q) ∈ we will often write p q. The next definition introduces the representation notion to be used throughout the paper. Definition 1 We say that a set U ⊆ [P ] represents if, for all p, q ∈ P, Z Z p q if and only if udp ≥ udq for all u ∈ U . X

(1)

X

Throughout the study we shall use the following two axioms. Independence axiom

For all p, q, r ∈ P and all α ∈ (0, 1),

p q implies αp + (1 − α) r αq + (1 − α) r. Continuity axiom is sequentially closed in P × P w.r.t. the product [P ]-topology. That is, for any pair of [P ]-convergent sequences {pn } , {qn } in P pn qn for all n imply

lim pn lim qn . n

n

When [P ] = Cb (X), which is the case if X is compact, the Continuity axiom reduces to the following continuity condition employed by DMO: 4

Given a pair of vector spaces E and R, a set E0 ⊆ E and a bilinear functional h·, ·i on E × R, E0 topology on R refers to the weakest topology in which the function r → he, ri (r ∈ R) is continuous for every e ∈ E0 . In this topology, a net {rα } converges to a point r if and only if limα he, rα i = he, ri for every e ∈ E0 . If R is spanP and E is a subspace of [P ] , the bilinear functional we have in mind will be R he, ri ≡ X edr. If E is a space of linear functionals on R, we will be considering the functional he, ri ≡ e(r).

3

Continuity*

is sequentially closed in P × P w.r.t. the product Cb (X)-topology.

Notice that no matter what the set P is, the space [P ] contains Cb (X). Hence, in general, the Continuity axiom is weaker than Continuity*. The following theorem is the main result of DMO. Theorem 1 (Dubra et al., 2004) Let X be a compact metric space. A binary relation on P (X) is a preorder that satisfies the Independence and Continuity axioms if and only if there is a non-empty set U ⊆ Cb (X) that represents . Our main purpose in the remainder of the paper will be to present alternative versions of this result which do not require a compact prize space. In order to motivate choice of a lottery set and additional restrictions that we impose in these results, we proceed with an example which shows that in Theorem 1 we cannot drop the compactness assumption.

2.2

Example

Let us assume that the prize space X is the real line endowed with the usual metric. We take as the lottery space the set PY , where Y ⊆ ca(X) is an ideal such that Ps (X) ⊆ PY ⊆ P (X). Note that spaces spanned by Ps (X) and Pk (X) are ideals in ca(X). Thus, our example will cover these lottery sets as well as the grand set P (X). For each n ∈ N, pick an increasing sequence of real numbers {an,m }m∈N contained in the open interval (n − 1/2, n) such that supm {an,m } = n. Define the set D ⊆ PY × PY by D := {(δ n , δ ak,n ) : (k, n) ∈ N × N}. Let C be the closure of the convex hull of D relative to PY × PY in the product Cb (X)-topology. Pick any pair of distinct numbers x, y < 0, and put C0 := 1/2 (δ x , δ y ) + 1/2C. Finally, define the binary relation on PY as, for any p, q ∈ PY , p q iﬀ p − q = γ (ω − θ) for a γ ≥ 0, and for some (ω, θ) ∈ C0 .

(2)

We are now ready to state the promised negative result. Proposition 1 The relation defined by (2) is a preorder on PY that satisfies the Independence axiom and Continuity*. However, there does not exist a set U ⊆ Cb (X) that represents . It should be noted that Proposition 1 declares non-existence of a set of bounded utility functions. Of course, if the lottery space under focus is the grand set P (X), we cannot hope to get unbounded utility functions either. On the other hand, as we shall see below, for the lottery set Pk (X) removing this boundedness restriction will provide us with a positive representation result. 4

2.3

Existence results

In this subsection we assume that the prize space is σ-compact, that is, there exists a sequence {Gn } of open subsets of X such that the closure of Gn is compact for all n, and S n Gn = X. σ-compactness of the prize space will allow us to apply Theorem 1 on suitably chosen subspaces successively and prove the following result. Theorem 2 Let X be a σ-compact metric space. A binary relation on Pk (X) is a preorder that satisfies the Independence and Continuity axioms if and only if there exists a non-empty set U ⊆ C(X) that represents . In view of Theorem 2, if the domain of is P (X), the restriction of to Pk (X) will admit a set of expected utility functions. It turns out that when is bounded so that these utility functions are also bounded, the same utility functions also represent . This is the content of the next result. Theorem 3 Let X be a σ-compact metric space and let be a binary relation on P (X) such that for some a, b ∈ X, δ a δ x δ b for all x ∈ X. Then is a preorder that satisfies the Independence and Continuity axioms if and only if there exists a non-empty set U ⊆ Cb (X) that represents . We now turn to preferences defined over monetary lotteries and show that in this context boundedness condition demanded by Theorem 3 may be redundant. Suppose that X is an interval in R, and is a monotone binary relation on P (X), that is, δ x Â δ y for any x, y ∈ X with x > y. Here, as usual, Â is the strict part of which is defined as Â:= {(p, q) ∈: (q, p) ∈}. / We say that a set of real functions U on X is increasing if for any x, y ∈ X, x > y implies u(x) ≥ u(y) for all u ∈ U and u(x) > u(y) for at least one u ∈ U . Finally, recall that is said to be risk averse if δ αx+(1−α)y αδ x + (1 − α)δ y for any x, y ∈ X, and any α ∈ [0, 1]. We are now ready to state our existence result for the case of monetary lotteries. Theorem 4 Let X ⊆ R be an open interval which is bounded from below. is a risk averse and monotone preorder on P (X) that satisfies the Independence and Continuity axioms if and only if there exists a non-empty, increasing set of concave functions U ⊆ Cb (X) that represents . The method that we use to prove Theorem 4 is as follows. We first show that there is a set of linear functionals T on ca(X) which characterize . We then use openness and risk aversion conditions to show that, in fact, elements of T are expected utility functions 5

with continuous Bernoulli indices. As a matter of fact, if we were willing to allow Bernoulli indices to be discontinuous at the lower bound, we could work with a prize space of the form X = [x0 , x0 ) for some x0 ∈ R and some x0 ∈ (x0 , ∞] .5 For more on this we refer to Remark 3 in Subsection 3.5.

2.4

Uniqueness

We will conclude this section with a straightforward generalization of the uniqueness theorem of DMO. This general form will cover relations which are defined on an arbitrary convex set of lotteries. In the sequel, U ⊆ [P ] is a linear space such that 1X ∈ U. For a non-empty set U ⊆ U, hUiU stands for the P -closure of the set Υ(U) := cone(U) + {θ1X : θ ∈ R} relative to U. Here cone(U) is the convex cone6 generated by U. The uniqueness theorem reads as follows. Theorem 5 Let X be a metric space, and let P ⊆ P (X) be a non-empty, convex set. Then two non-empty subsets U and V of a linear space U ⊆ [P ] with 1X ∈ U satisfy Z Z Z Z udp ≥ udq ∀u ∈ U =⇒ vdp ≥ vdq ∀v ∈ V, (3) X

X

X

X

for each p, q ∈ P , if and only if hV iU ⊆ hUiU .7 Note that if δ x ∈ P for all x ∈ X, then the weak topology induced by P on [P ] is Hausdorﬀ. This in turn implies that for a u ∈ [P ] the set Υ({u}) = {αu + θ1X : (α, θ) ∈ R+ × R} is P -closed. Hence, if the set of utility functions is a singleton, and if the domain P contains all Dirac measures, Theorem 5 reduces to the uniqueness result of the classical expected utility theory. 5

The following additional assumption would restore continuity at x0 : there exists a point x ∈ X with x > x0 such that for all α ∈ (0, 1], (1 − α)δ x0 + αδ x δ yα for some yα > x0 . To see this point, suppose that U is an increasing set of functions which satisfy (1). Now, if a u ∈ U was discontinuous at x0 , for any feasible x with x > x0 , we could find a small α ∈ (0, 1) such that (1−α)u(x0 )+αu(x) < inf y>x0 u(y). This, in turn, would imply that for any y suﬃciently close to x0 , δ y is incomparable with (1 − α)δ x0 + αδ x (since (1 − α)δ x0 + αδ x Â δ x0 , there cannot be a sequence yn ↓ x0 with δ yn (1 − α)δ x0 + αδ x for all n). It is worth to note that any preorder defined by a finite increasing subset of Cb (X) via the rule (1) would satisfy the assumption that we suggested. Intuitively, this assumption reduces the level of incompleteness about x0 by postulating that for a suﬃciently large prize, occurrence of that prize with a positive probability is preferred to extremely small certain outcomes. 6 Throughout the paper, the term convex cone refers to a convex subset of a linear space that is closed under non-negative scalar multiplication. The convex cone generated by a set Q is the smallest convex cone that contains Q. 7 I owe this statement of Theorem 5 to a referee, which relates two sets of utility functions which represent two preorders one of them being a possibly proper subset of the other.

6

3

Proofs

For convenience, the order of proofs will be diﬀerent than that of results. For future use, we first state a variant of Theorem 1. In the sequel, given a subset K of X, P (K, X) denotes the set {p ∈ P (X) : p(K) = 1}. Theorem 1a Let K be a compact subset of a metric space X. A binary relation on P (K, X) satisfies the Independence and Continuity axioms if and only if there is a non-empty set U ⊆ Cb (X) that represents . Note that any continuous real function on a compact K can be extended to a continuous R bounded function on X. Moreover, for any u ∈ C(X), and any p ∈ P (K, X), X udp = R u |K dp |K , where p |K is the restriction of p to K. Combining these observations with K the fact that the mapping p → p |K is an aﬃne bijection between P (K, X) and P (K), Theorem 1a can easily be proved using Theorem 1.

3.1

Proof of Theorem 5

Here, we will prove a version of the uniqueness theorem that covers arbitrary linear functionals on spanP . This general version will be useful when proving Theorem 3. Let us denote the algebraic dual of spanP, i.e., the vector space of all linear functionals R on spanP , with spanP + . For any u ∈ [P ] , the functional η → X udη (η ∈ spanP ) will eX ∈ E. Given a be denoted by u e. We take a linear subspace E of spanP + such that 1 non-empty set T ⊆ E, with a slight abuse ofonotation, we define hT iE to be the P -closure n e of the set Υ(T ) := cone(T ) + θ1X : θ ∈ R in E. The modified version of Theorem 5 is given next. Theorem 5a Let X be a metric space, and let P ⊆ P (X) be a non-empty convex set. eX ∈ E satisfy Then two non-empty subsets F and T of a linear space E ⊆ spanP + with 1 F (p) ≥ F (q) ∀F ∈ F =⇒ T (p) ≥ T (q) ∀T ∈ T ,

(4)

for each p, q ∈ P , if and only if hT iE ⊆ hFiE . Note that since we can embed any given space U ⊆ [P ] into the algebraic dual of spanP via the operator u → u e (u ∈ U), Theorem 5 readily follows from Theorem 5a. We shall now present a proof of Theorem 5a, which is essentially an abstraction of the proof of the uniqueness result of DMO. Proof of Theorem 5a.

Since it is obvious, we omit the “if” part. To prove the “only 7

if” part, suppose that the set hT iE \ hFiE is non-empty. Obviously, we can pick a point T from the set T \ hFiE . Now, since hFiE is a P -closed convex cone, and since E endowed with P -topology is a locally convex space, from Hahn-Banach separation theorem and a well known duality result (Megginson, 1998, Theorems 2.2.28 and 2.4.11) it follows that there exists a µ ∈ spanP \ {0} such that T (µ) > 0 ≥ F (µ) for all F ∈ hFiE .

(5)

n o eX : θ ∈ R ⊆ hFi , (5) implies µ(X) = 1 eX (µ) = 0. Since θ1 E Now, by definition of spanP , there is a non-empty, finite set {pi : i ∈ I} ⊆ P such that P P P i∈I αi pi = µ for some {αi : i ∈ I} ⊆ R. Note that i∈I αi = i∈I αi pi (X) = 0. So, if we set I+ := {i ∈ I : αi > 0} , I− := {i ∈ I : αi < 0} , and β i := −αi for all i ∈ I− , we P P will have c := i∈I+ αi = i∈I− β i . Moreover, since µ is non-zero, the sets I+ and I− are non-empty and c is positive. So, from convexity of the set P it follows that the points P P µ+ := i∈I+ αi pi /c and µ− := i∈I− β i pi /c are in P . Since µ = c(µ+ − µ− ), by (5), (4) cannot be true.

3.2

A technical characterization

We will proceed with a couple of facts which will be crucial for what follows. Let us first introduce an algebraic continuity condition which is weaker than the Continuity axiom. This condition plays a key role in the algebraic approach of Shapley and Baucells (1998). Weak continuity {α ∈ [0, 1] : αp + (1 − α) q αr + (1 − α) w} is a closed subset of [0, 1] for all p, q, r, w ∈ P. We define the cone induced by as C() := {γ(p − q) : p q}. The next lemma is due to Shapley and Baucells (1998). Lemma 1 Let X be a metric space, and let be a preorder on a non-empty, convex set P ⊆ P (X). Assume further that satisfies the Independence axiom and Weak continuity. Then for any p, q ∈ P, pq

if and only if p − q ∈ C().

It is important to note that if is a preorder which satisfies the Independence axiom, then the set C() is a convex cone. The next result shows that closedness of this cone is a necessary and suﬃcient condition for the existence of a linear representation.

8

Lemma 2 Let P and be as in Lemma 1 and let E be a vector subspace of spanP + such eX ∈ E. Then C() is E-closed in spanP if and only if there exists a non-empty set that 1 T ⊆ E such that, for all p, q ∈ P pq

if and only if T (p) ≥ T (q) for all T ∈ T .

(1a)

Proof. The proof of the “only if” part is identical with that of Theorem 1 by DMO. So, it will be suﬃcient to sketch this part of the proof. Lemma 1 ensures that a point of the form p − q for some (p, q) ∈ P × P \ does not belong to the set C(). Then, by following separation arguments similar to those used in the proof of Theorem 5a, we can find a function T ∈ E such that T (p − q) < 0 ≤ T (µ) for all µ ∈ C(). From these observations it easily follows that the dual cone T := {T ∈ E : T (µ) ≥ 0, ∀µ ∈ C()} satisfies (1a). To prove the “if” part suppose there is a set T ⊆ E that satisfies (1a). Simply by using definitions it can easily be seen that C() ∩ (P − P ) ⊆ C(),

(6)

where C() is the E-closure of C() in spanP . Let µ be an arbitrary point in C(). Note that since e 1X ∈ E, and since η(X) = 0 whenever η ∈ C(), we must have µ(X) = 0. Hence, as in the proof of Theorem 5a, we can find two points µ+ , µ− ∈ P and a number c ≥ 0, such that µ = c(µ+ − µ− ). Since C() and C() are closed under non-negative scalar multiplication, by (6) we see that µ belongs to C().

3.3

An abstract representation result

In this subsection we state and prove the following representation result where we work with general utility functions. Proposition 2 Let X be a metric space and Y be a Riesz subspace of ca(X). Assume further that is a binary relation on PY . Then there exists a non-empty set T of (total variation) norm-continuous linear functionals on Y which satisfy (1a) if and only if the following two conditions hold. i. is a preorder that satisfies the Independence axiom. ii. is closed relative to PY × PY in the product norm-topology. Proof. The “only if” part is obvious. To prove the “if” part, we shall first show that C() is norm-closed in Y . To this end, let {µn } be a norm-convergent sequence in C(), and put µ := lim µn ∈ Y. (7) n

9

Since 0 ∈ C(), it is no loss of generality to assume µ 6= 0 and µn 6= 0 for each n ∈ N. Since µn (X) is equal to 0 for all n ∈ N, so is µ (X) . Set γ := µ+ (X) = µ− (X) > 0 and − γ n := µ+ n (X) = µn (X) > 0 (n ∈ N). Since ca(X) endowed with the total variation norm + − − k·k is an AL-space, from (7) it follows that limn kµ+ n − µ k = limn kµn − µ k = 0. Then, we must have limn γ n = γ, and ° + ° − ° ° ° µn ° µn µ+ ° µ− ° ° ° ° ° = 0. = lim ° (8) − − lim ° n n γn γ ° γn γ °

− Since Y is a Riesz subspace, the points µ+ /γ, µ− /γ, µ+ n /γ n , µn /γ n belong to PY (n ∈ N). − + − Now, as µn = γ n (µ+ n /γ n − µn /γ n ), from Lemma 1 it follows that µn /γ n µn /γ n for all n ∈ N. So, (8) and condition (ii) imply µ+ /γ µ− /γ. This implies µ = γ (µ+ /γ − µ− /γ) ∈ C(), and proves norm-closedness of C() in Y . But then, since C() is a convex set, by a well-known fact (Megginson, 1998, Corollary 2.2.29), C() is closed in the weak topology induced by the norm-dual of Y . Since spanPY = Y , the proof follows from Lemma 2.

3.4

Proofs of Theorems 2 and 3

Throughout this subsection {Gn } denotes an increasing (w.r.t. set inclusion) sequence of S open subsets of X such that n Gn = X. We define Kn to be the closure of Gn and assume that it is compact for all n ∈ N. To simplify our notation we set Pn := P (Kn , X) (elements of P (X) with supports in Kn ). It is worth to note that for any compact subset K of X, S there is a suﬃciently large n such that K ⊆ Gn . This, in turn, implies Pk (X) = n Pn . We are now ready to prove Theorem 2. Proof of Theorem 2. The “if” part is trivial. For the “only if” part, put n := ∩ (Pn × Pn ) for all n ∈ N. Since [Pn ] = [Pk (X)] = C(X), n inherits the Continuity and Independence axioms from . Hence, from Theorem 1a it follows that there exists a non-empty set Un ⊆ Cb (X) which represents n . By Theorem 5, without loss of generality we can assume Un = Υ(Un ) for all n. Let n be an arbitrary natural number. Note that for any p, q ∈ Pn , p n q if and only R R if p n+1 q. Hence, for each p, q ∈ Pn , X udp ≥ X udq for every u ∈ Un if and only if R 0 R 0 u dp ≥ u dq for every u0 ∈ Un+1 . Thus, from Theorem 5 it follows that Un ⊆ U n+1 , X X where the closure operator is applied w.r.t. the weak topology of Cb (X) induced by Pn . Since Un+1 is convex, so is the set of restrictions of elements of Un+1 to Kn . Hence, the closure of this latter set in P (Kn )-topology coincides with its (sup-)norm closure, for the norm-dual of Cb (Kn ) is ca(Kn ). From these observations and the fact that Un ⊆ U n+1 10

© ª it obviously follows that for any u ∈ Un there exists a sequence ul in Un+1 such that ° ° liml °u − ul °n = 0, where kf kn := supx∈Kn |f (x)| for all f ∈ C(X). Now fix any u ∈ Un , and any ε > 0. Put un := u and for all m ≥ n pick inductively a function um+1 in Um+1 such that kum+1 − um km < ε2−(m+1) . Obviously, for any m ≥ n, © l ª u |Km l≥n is a Cauchy sequence, and therefore, norm-convergent in Cb (Km ). Moreover, the restriction of liml≥n ul |Km+1 to Km coincides with liml≥n ul |Km . Thus, we can define S a real function ψ εn,u on X by ψ εn,u |Km = liml≥n ul |Km for all m ≥ n. Since X = m Gm and each Gm is open, continuity of ψ εn,u on each Km implies ψ εn,u ∈ C (X). Also note that ° ° © ª °u − ψ εn,u ° < ε. Finally, since ψ εn,u is the uniform limit of the sequence ul on any l≥n n compact set, we have Z Z ε ψ n,u dr = lim ul dr for any r ∈ Pk (X). (9) l≥n

X

X

© ª What remains to show is that the set U := ψ εn,u : ε > 0, n ∈ N, u ∈ Un satisfies (1). © ª To this end, fix an element ψ εn,u of U and let ul l≥n be the sequence that defines ψ εn,u . First notice that if p q, there is some m0 such that p, q ∈ Pm0 and p l q for all R R l ≥ m0 . But then X ul dp ≥ X ul dq for all l ≥ m0 , and from (9) it immediately follows R R that X ψ εn,u dp ≥ X ψ εn,u dq. R To prove the “if” part of (1), let p, q ∈ Pm1 for some m1 and assume X ψ εn,u dp ≥ R ε ψ dq for all n, all u ∈ Un , and all ε > 0. In particular, for any given u ∈ Um1 , if we X n,u R R pick a sequence of positive real numbers εl ↓ 0, we will have X ψεml 1 ,u dp ≥ X ψ εml 1 ,u dq for ° ° all l. Since °u − ψ εml 1 ,u °m < εl for all l, and since p, q ∈ Pm1 , it follows that 1 Z Z Z Z εl εl udp = lim ψ m1 ,u dp ≥ lim ψ m1 ,u dq = udq. X

l

l

X

X

X

Since u ∈ Um1 was arbitrary, from the definition of Um1 it immediately follows that p q. We proceed with a proof of Theorem 3.

Proof of Theorem 3. The “if” part is trivial. To prove the “only if” part first notice that Cb (X)-topology of P (X) is weaker than the norm-topology of P (X), and Cb (X)-topology of Pk (X) is weaker than the C(X)-topology of Pk (X). Hence, we can apply Proposition 2 to and Theorem 2 to the restriction of to Pk (X). Thus, there exist non-empty sets U ⊆ C(X), T ⊆ ca(X)0 such that U satisfies (1) for all p, q ∈ Pk (X), and T satisfies (1a) for all p, q ∈ P (X), where ca(X)0 is the norm-dual of ca(X). Since δ a δ x δb for all x ∈ X, R U must be a subset of Cb (X). Thus, for any u ∈ U and any η ∈ ca(X), X udη =: u e(η) is 0 well defined. Moreover, the set Uemb := {e u(·) : u ∈ U} is a subset of ca(X) . By Theorem 5a, we can assume Υ(Uemb ) = Uemb and Υ(T ) = T so that cl Uemb = cl T , 11

(10)

where the closure operators are applied w.r.t. the weak topology induced by Pk (X) on ca(X)0 . Note that by the Independence axiom and transitivity, δ a p δ b for all p ∈ Ps (X). Since Ps (X) is Cb (X)-dense in P (X), from the Continuity axiom it follows that δa p δb

for all p ∈ P (X).

(11)

For any p ∈ P (X), define conditional probabilities p−n ∈ Pk (X) and p+n ∈ P (X) n) as follows. For any Borel subset E of X, p−n (E) := p(E∩K whenever p(Kn ) > 0, and p(Kn )

p(E\Kn ) p+n (E) := p(X\K whenever p(X\Kn ) > 0. If p(Kn ) is 0, we define p−n := 0, and similarly n) for p+n . Note that p = p(Kn )p−n + p(X\Kn )p+n . Thus, for any linear functional F on ca(X), and any p ∈ P (X)

F (p) − F (p−n ) = F (p) − [p(Kn ) + p(X\Kn )] F (p−n ) = p(X\Kn )(F (p+n ) − F (p−n )).

(12)

For any A ⊆ ca(X)0 , A will stand for the closure of A in ca(X)0 w.r.t. the P (X)topology. If we can show that Uemb = T , the proof will be complete, for then we can apply Theorem 5a and conclude that Uemb satisfies (1a), i.e., U satisfies (1) for all p, q ∈ P (X). To this end, pick any finite set {p1 , ..., pm } ⊆ P (X) and any ε ∈ (0, 1). Fix a function T ∈ T . Define M := 1 + max {|T (δ a )| , |T (δb )|}. Pick a number n such that pi (X\Kn ) < © ª ε(2M)−1 for all i = 1, ..., m. Note that since p1−n , ..., pm −n ⊆ Pk (X), by (10), there exists a function u e ∈ Uemb such that u(δ b ) − T (δ b )| < 1 and |e u(δ a ) − T (δ a )| , |e

(13)

¯ ¯ i ¯u (14) e(p−n ) − T (pi−n )¯ < ε for all i = 1, ..., m. R Now, since u e(δ a ) ≥ u(x) ≥ u e(δb ) for all x ∈ X, u e(δ a ) ≥ X udp ≥ u e(δ b ) for all p ∈ P (X). Hence, (13) implies sup |e u(p)| = max {|e u(δ a )| , |e u(δ b )|} < M. p∈P (X)

Thus, by (12), we conclude that for all i = 1, ..., m ¯ ¯ i ¯ ¯ i ¯u e(pi−n )¯ = pi (X\Kn ) ¯u e(p+n ) − u e(pi−n )¯ ≤ pi (X\Kn )2M < ε. e(p ) − u

(15)

Similarly, by (11) and by definition of T , supp∈P (X) |T (p)| < M, and hence, for all i = 1, ..., m ¯ ¯ ¯ ¯ ¯T (pi ) − T (pi−n )¯ = pi (X\Kn ) ¯T (pi+n ) − T (pi−n )¯ < ε. (16) 12

Combining (14), (15), and (16) gives |T (pi ) − u e(pi )| < 3ε for all i = 1, ..., m. This proves that T ⊆ Uemb . To prove the other inclusion we can simply interchange the roles of u e and T in the above argument. Remark 1 Recall that in a Polish (separable, complete metric) space, every Borel probability measure p is tight, i.e., for any Borel set E, p(E) = sup p(K) where the supremum is taken over all compact subsets of E. In the proof of Theorem 3 σ-compactness of X served for two purposes. First, it gave us tightness of the lotteries in question, and second, it allowed us to apply Theorem 2. Hence, if one can generalize Theorem 2 to the case of a Polish prize space, Theorem 3 could be generalized as well.

3.5

Proof of Theorem 4

In the sequel, X stands for an interval bounded from below and f sd will denote the first order stochastic dominance relation on Pk (X), which is defined as, for any p, q ∈ Pk (X) Z p f sd q if and only if ud(p − q) ≥ 0 for every non-decreasing u ∈ C(X). X

Proof of Theorem 4. The “if” part is obvious and will be omitted. To prove the “only if” part first note that by Proposition 2 there is a non-empty set T ⊆ ca(X)0 which satisfies (1a). For any T ∈ T define the function uT : X → R by uT (x) := T (δ x ) for all x ∈ X. Note that, as the image of the norm-bounded set {δx : x ∈ X} under the norm-continuous linear functional T , the set {uT (x) : x ∈ X} is a bounded subset of the real line. Put U := {uT : T ∈ T }. Obviously, U is increasing and uT is concave for all T ∈ T . Since X is open, it follows that U ⊆ Cb (X). Fix a function T ∈ T . We shall complete the proof by R showing that T (p) = X uT dp for all p ∈ P (X). First assume that p has a compact support and let X0 ⊆ X be a compact interval with p (X0 ) = 1. It is clear that we can construct a pair of sequences of simple probability measures {pn } , {qn } in P (X), such that limn pn = limn qn = p, pn (X0 ) = qn (X0 ) = 1 and pn f sd p f sd qn for all n, where the convergence is in Cb (X)-topology. Following the proof of Proposition 1 in (Dubra and Ok, 2002) one can easily show that the Continuity and Independence axioms together with monotonicity imply f sd ⊆. In particular, we must have pn p qn for all n. Now, since pn and qn are simple, from linearity of T and definition of T it follows that for all n, Z Z uT dpn = T (pn ) ≥ T (p) ≥ T (qn ) = uT dqn . X

X

13

Passing to limits gives Z Z Z Z uT dp = lim uT dpn ≥ T (p) ≥ lim uT dqn = uT dp. X

n

n

X

X

X

R

That is, X uT dp = T (p). To complete the proof for the case of a general p, let {Xn } be an increasing sequence S of compact intervals such that n Xn = X and let pn be p conditioned to Xn . It is clear that {pn } converges to p in the norm-topology. Hence, Z Z T (p) = lim T (pn ) = lim uT dpn = uT dp, n

n

X

X

where the first and the third equalities follow from norm-continuity of corresponding functionals, and the second one follows from the first part of the proof. Remark 2 Related to the above proof, the reader may want to know why we could R not immediately conclude T (p) ≡ X uT dp. The diﬃculty is, Ps (X) is not norm-dense in P (X), and therefore, the corresponding equality for simple probabilities do not propagate to arbitrary probability measures.8 R Remark 3 To prove the identity T (p) ≡ X uT dp, we used the fact that uT is continuous, which followed from risk-aversion and openness conditions. On the other hand, with some more eﬀort it can be shown that the identity is true as long as is monotone, and hence, uT is non-decreasing. Thus, if we were willing to allow discontinuous Bernoulli indices, in Theorem 4 we could drop the risk-aversion and openness conditions. Unfortunately, we could not find a way of solving this discontinuity problem. Even assuming risk-aversion, we could not restore continuity at the lower bound of the prize space without some additional assumptions.

3.6

Proof of Proposition 1

In what follows X denotes the real line. AX stands for the algebra generated by the collection of all closed subsets of X. A finitely additive probability measure p on AX is said to be regular, if for each ε > 0, and for each A ∈ AX , there exist a closed set F and an open set G such that F ⊆ A ⊆ G and p (G\F ) < ε. Pra (X) denotes the set of all regular finitely additive probability measures on AX . ra(X) stands for the span of Pra (X). 8

The following is a simple example which illustrates the problem: Since any p ∈ P (X) has at most P countably many mass points (atoms), the functional T (p) ≡ x∈X p({x}) is well defined on P (X). Now note that T (p) = 1 if and only if p is purely atomic and T (p) = 0 if and only if p is non-atomic. Finally, observe that uT = 1X and T is norm-continuous on P (X).

14

As usual, we endow ra(X) with the setwise order ≥. By Riesz representation theorem (see Dunford and Schwartz, 1958, Theorem IV.6.2), the Banach space ra(X) endowed with the total variation norm is isometrically isomorphic to the norm dual of Cb (X) (with the R duality hu, ηi ≡ X udη). In the sequel, we treat ca(X) as if it is a subspace of ra(X). The following facts justify this approach: (a) Any countably additive signed measure on AX can be extended uniquely to the Borel σ-algebra in such a way that the total variation norm and integrals of continuous functions are preserved. (b) This extension is linear and order preserving. (c) Any p ∈ P (X) is regular. Note that once we consider ca(X) as a subspace of ra(X), Cb (X)-topology of ca(X) can simply be seen as the relative weak*-topology inherited from ra(X), denoted by w*topology. Recall that a cofinal function ϕ from a directed set Λ into a directed set A is a function such that, for each α0 ∈ A, there is a λ0 ∈ Λ such that ϕ (λ) ≥ α0 whenever λ ≥ λ0 .

© ª A subnet of a net {zα }α∈A is a net of the form zϕ(λ) λ∈Λ where ϕ is a cofinal function from Λ into A. We precede the proof of Proposition 1 by two claims. Claim 1 Let {µα }α∈A be a net in Y. Let furthermore {pα }α∈A be a net in PY that w*converges to a point in Y . If 0 ≤ µα ≤ pα for each α ∈ A, then the net {µα }α∈A has a subnet that w*-converges to a point in Y . Proof. Let p be the w*-limit of the net {pα }α∈A . Since 0 ≤ µα ≤ pα ∈ PY for each α ∈ A, the net {µα }α∈A is contained in the unit ball B of ra(X). Since, by Banach-Alaoglu theorem the set B is w*-compact (Dunford and Schwartz, 1958, Theorem V.4.2), the net {µα }α∈A has a w*-convergent subnet (Aliprantis and Border, 1994, Theorem 2.29). Now, we will obviously have 0 ≤ µ ≤ p, where µ ∈ B is the w*-limit of this convergent subnet. Since by hypotheses p ∈ Y ⊆ ca(X), and since ca(X) is an ideal in ra(X), µ belongs to ca(X). Finally, since Y is an ideal in ca(X), µ must be in Y . For any {p, q} ⊆ Y with p, q ≥ 0, if inf{p, q} = 0 we will say that p and q are disjoint and write p ⊥ q. The next claim establishes several properties of the sets C and C0 . Claim 2 The sets C and C0 are convex and relatively w* ×w*-closed subsets of PY × PY and satisfy the following additional properties: i. 0 belongs to the w*-closure of the set T C := {p − q : (p, q) ∈ C} ⊆ Y . 15

ii. (p, q) ∈ C implies p ⊥ q. iii. δ x − δy belongs to the w*-closure of the set {ω − θ : (ω, θ) ∈ C0 } ⊆ Y . S iv. δ x − δ y does not belong to the set γ≥0 γT C0 .

S

γ≥0

γT C0 , where we define T C0 :=

v. (ω, θ) ∈ C0 implies ω ⊥ θ.

It is worth to note that if C was w*×w*-compact, parts (i) and (ii) of Claim 2 could not be true at the same time. In fact, these two properties are the essence of the construction, and once they are established the remaining parts of Claim 2 will easily follow. We postpone the proof of Claim 2 to the Appendix and close the discussion with a proof of Proposition 1. Proof of Proposition 1. Note that for any p, q ∈ PY , p q if and only if p − q ∈ S γ≥0 γT C0 , where the set T C0 ⊆ Y is defined as in part (iii) of Claim 2. It is evident that is reflexive, and that it satisfies the Independence axiom. To prove transitivity of , S observe that since the set C0 is convex, the set γ≥0 γT C0 is a convex cone, and therefore it is closed under summation. Now, we shall show that is w*×w*-closed in PY × PY . Since X is a separable metric space, w*-topology of PY is metrizable (Aliprantis and Border, 1994, Theorem 12.11). Thus, it will be suﬃcient to establish sequential closedness. To this end, let {pn } and {qn } be w*-convergent sequences in PY such that pn qn for each n ∈ N. Put p := limn pn and q := limn qn . By definition of , for each n ∈ N, there is a number γ n ≥ 0 and a point (ω n , θn ) ∈ C0 such that pn − qn = γ n (ω n − θn ) . By definitions of p and q, we have p − q = lim(pn − qn ) = lim γ n (ω n − θn ) . n

n

(17)

Moreover, by part (v) of Claim 2, γ n ω n ⊥γ n θn (n ∈ N). Since disjoint decompositions are unique (Aliprantis and Border, 1994, Theorem 6.11) this implies that γ n ω n = (pn − qn )+ = sup {(pn − qn ), 0} ≤ pn , and similarly, γ n θn ≤ qn for each n ∈ N. Thus, by using Claim © ¡ ¢ª 1 twice, we can conclude that there is a subnet γ ϕ(α) ω ϕ(α) , θϕ(α) α∈A of the sequence {γ n (ωn , θn )}n∈N which w*×w*-converges to a point in Y × Y , where A is a directed set and ϕ : A → N is a cofinal function. Put ¡ ¢ (ν, η) := lim γ ϕ(α) ω ϕ(α) , θϕ(α) ∈ Y × Y. α

(18)

This, together with (17) imply

p − q = ν − η. 16

(19)

Put c := ν(X) = η(X). We can assume that c is positive, for otherwise we would have p = q, and the proof would follow from reflexivity of . Now note that, by (18), we have c = ©¡ ¢ª limα γ ϕ(α) ω ϕ(α) (X) = limα γ ϕ(α) . So, from (18) it follows that the net ωϕ(α) , θϕ(α) α∈A converges to (ν, η) /c. Then, as C0 is closed in PY × PY , (ν, η) /c belongs to C0 . Thus, by (19), (p − q) /c is in T C0 , and hence, p q. This shows that is w*×w*-closed in PY ×PY , and therefore, satisfies Continuity*. What remains to show is that there does not exist a set U ⊆ Cb (X) which represents S . But since C() = γ≥0 γT C0 , this is an obvious consequence of parts (iii) and (iv) of Claim 2, and the “if” part of Lemma 2.

Appendix A. Proof of Claim 2 Proof of Claim 2. Convexity and closedness of C and C0 are obvious. Let V be any w*-open subset of ra(X) that contains 0. If we can show that V ∩ T C 6= ∅, proof of (i) will be complete. By continuity of the sum operator (p, q) → p + q , there is a pair of w*-open sets V1 , V2 ⊆ ra(X) such that V1 +V2 ⊆ V and 0 ∈ V1 ∩V2 . Since Pra (X) is w*-compact, the © ª sequence {δ n }n∈N has a subnet δ ϕ(α) α∈A that w*-converges to some r ∈ Pra (X), where A is a directed set and the mapping ϕ : A → N is cofinal. By definition of r, there is an index α ∈ A such that, for each α ∈ A, (A-1) α ≥ α implies δ ϕ(α) ∈ r + V1 ∩ r − V2 . © ª Now note that, by construction, the sequence aϕ(α),m m∈N converges to the number ϕ (α) . Hence, δ ϕ(α) is the w*-limit of the sequence {δ aϕ(α),m }m∈N . So, (A-1) implies that there is a number m0 ∈ N such that, for each m ∈ N, m ≥ m0

implies δ aϕ(α),m ∈ r − V2 .

(A-2)

Furthermore, since ϕ is a cofinal function, there is some α e ∈ A such that, for each α ∈ A, α≥α e implies ϕ (α) ≥ m0 .

(A-3)

e , and note that (A-1) implies δ ϕ(α) ∈ r + V1 , whereas Now, pick any α ∈ A with α ≥ α, α (A-2) and (A-3) imply δaϕ(α),ϕ(α) ∈ r − V2 . Thus, if we set n := ϕ (α) and k := ϕ (α) we will have δ n − δ ak,n ∈ V1 + V2 ⊆ V. Since (δ n , δ ak,n ) ∈ D ⊆ C, this shows that V ∩ T C 6= ∅, and completes the proof of (i). We shall now prove (ii). Let (p, q) be a point in C. Since w*-topology of PY is metrizable, by definition of C, there is a w*×w*-convergent sequence {(pl , ql )}l∈N in the convex hull of D such that lim (pl , ql ) = (p, q) . (A-4) l

17

By definition of the convex hull of D, for each l ∈ N, there is a finite set of indices Al ⊆ N×N © ª P and a set of numbers β l(k,n) : (k, n) ∈ Al ⊆ [0, 1] such that (k,n)∈Al β l(k,n) = 1, and (pl , ql ) =

X

β l(k,n) (δ n , δ ak,n ) .

(A-5)

(k,n)∈Al

Hence, pl (N) = 1 for each l ∈ N. Since N is a closed subset of X, from (A-4) it follows that 1 = lim supl pl (N) ≤ p (N). Thus, p (X\N) = 0, and therefore, to complete the proof of (ii), it suﬃces to show that q (N) = 0. Suppose to the contrary that q (N) > 0. By countable additivity of q, we must have γ := q ({b n}) > 0 for some n b ∈ N. So, by (A-4), lim inf ql (Bε (b n)) ≥ q (Bε (b n)) ≥ γ l

for each ε > 0,

(A-6)

where Bε (b n) := {x ∈ X : |x − n b| < ε}. For each m ∈ N set εm := n b − anb ,m > 0. Now note that using (A-6) we can find a subsequence {qlm }m∈N of {ql } such that n)) ≥ γ/2 for each m ∈ N. qlm (Bεm (b

(A-7)

¯ ¯ b¯ < εm if and only if k = n b and n > m. But, by construction, for any m, k, n ∈ N, ¯ak,n − n Hence, by (A-5) and (A-7), for each m ∈ N we must have X m n)) = β l(k,n) δ ak,n (Bεm (b n)) qlm (Bεm (b (k,n)∈Alm

=

X

(k,n)∈Alm , k=b n, n>m

m β l(k,n) ≥ γ/2.

(A-8)

Since δ n ([m, ∞)) = 1 whenever n > m, from (A-5) and (A-8) it follows that plm ([m, ∞)) ≥ γ/2 (n, m ∈ N). Fix an m0 ∈ N, and note that p ([m0 , ∞)) ≥ lim supm plm ([m0 , ∞)) ≥ lim supm plm ([m, ∞)) , where the first inequality follows from (A-4), and the last one follows from the fact that m ≥ m0 implies plm ([m0 , ∞)) ≥ plm ([m, ∞)). So, we see that p ([m0 , ∞)) ≥ γ/2. Since m0 is arbitrary, we then have limm p ([m, ∞)) ≥ γ/2. But this contradicts the hypothesis that p is countably additive, for {[m, ∞)}m∈N is a decreasing sequence of closed sets with empty intersection. This proves (ii). Since T C0 = 1/2(δ x −δ y )+1/2T C, from (i) it follows that 1/2(δ x −δ y ) is in the w*-closure of T C0 , and this proves (iii). Now, note that for any (p, q) ∈ C, p (X+ ) = q (X+ ) = 1, where X+ is the set of all non-negative real numbers. It is easily seen that this observation together with (ii) and the fact that x and y are distinct negative numbers imply (iv) and (v).

18

References Aliprantis, C.D., Border, K.C.: Infinite dimensional analysis. Berlin: Springer-Verlag 1994. Aumann, R.J.: Utility theory without the completeness axiom. Econometrica 30, 445-462 (1962). Baucells, M., Shapley, L.S.: Multiperson utility. Mimeo, IESE Business School (2006). Dubra, J., Ok, E.A.: A model of procedural decision making in the presence of risk. International Economic Review 43, 1053-1080 (2002). Dubra, J., Maccheroni, F., Ok, E.A.: Expected utility theory without the completeness axiom. Journal of Economic Theory 115, 118-133 (2004). Dunford, N., Schwartz, J.T.: Linear operators: Part I. New York: Interscience 1958. Kannai, Y.: Existence of a utility in infinite dimensional partially ordered spaces. Israel Journal of Mathematics 1, 229-234 (1963). Megginson, R.E.: An introduction to Banach space theory. New York: Springer-Verlag 1998. Shapley, L.S., Baucells, M.: A theory of multiperson utility. Working Paper #779, University of California, Los Angeles (1998).

19