Canonical utility functions and continuous preference extensions

Journal of Mathematical Economics 67 (2016) 32–37

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Canonical utility functions and continuous preference extensions Igor Kopylov ∗ Institute for Mathematical Behavioral Sciences, University of California, Irvine, CA 92697, United States Department of Economics, University of California, Irvine, CA 92697, United States

article

abstract

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Article history: Received 22 January 2016 Received in revised form 2 August 2016 Accepted 26 August 2016 Available online 21 September 2016

I define canonical utility functions via an explicit formula that inherits semicontinuity, continuity, Cauchy continuity, and uniform continuity from preferences. This construction is used to (i) show Rader’s and Debreu’s theorems in a fast and transparent way, (ii) refine these results for Cauchy and uniformly continuous preferences on a metric space X , (iii) extend such preferences from X to any larger metric domain Y ⊃ X , while preserving Cauchy and uniform continuity respectively.

Keywords: Continuous preferences Debreu Theorem Rader Theorem Cauchy continuity Preference extensions Uniform continuity

By contrast, regular continuity does not guarantee that continuous preference extensions should exist even in standard decision theoretic frameworks. For example, continuous preferences over simple lotteries or finite menus need not have continuous extensions to Borel distributions or compact menus respectively. © 2016 Elsevier B.V. All rights reserved.

1. Introduction

First, I show (Theorems 1 and 2) that canonical representations

The foundational theorems of Debreu (1964) and Rader (1963) characterize continuous and semicontinuous utility functions on second countable topological spaces. I study another broad class of canonical utility representations and refine the results of Debreu and Rader in several ways. Consider a utility function U : X → [0, 1] that represents a weak preference ≽ on an arbitrary domain X . Call U a canonical representation if it satisfies the recursive formula U (en ) =

1 2



 max U (ei ) + min U (ej ) j
i
for some suitable order-dense sequence e = (e1 , e2 , . . .) of elements in X . Here by convention, maxima and minima over empty sets are 0 and 1 respectively. Thus, U (e1 ) =

1 2

U (e2 ) =

1  4

if e2 ≺ e1

 

if e2 ≻ e1

1 2 3 4

if e2 ∼ e1

mains. The classic results of Debreu and Rader follow as easy corollaries. This approach is faster and arguably more transparent than Debreu’s open-gap lemma or Nachbin’s separation theorem (see Jaffray, 1975; Mehta, 1977; Bosi and Mehta, 2002). Next, I show (Theorem 4) that canonical representations inherit Cauchy continuity and uniform continuity from preferences ≽ over metric domains. I use this finding

• to refine Debreu’s Theorem for Cauchy and uniformly continuous preferences, and

• to extend ≽ from any metric domain X to any larger space Y ⊃ X in a way that preserves Cauchy and uniform continuity properties.

···

This recursive construction is standard for utility representations over countable sets (e.g. Kreps, 1988, pp. 20–21). I extend it to arbitrary domains X .

∗ Correspondence to: University of California, Irvine, SSPA 3177, Social Science Plaza, Irvine, CA, 92697, United States. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmateco.2016.08.007 0304-4068/© 2016 Elsevier B.V. All rights reserved.

• exist whenever ≽ has any utility representation at all, • preserve continuity and semicontinuity on topological do-

By contrast, if ≽ is continuous, but not Cauchy continuous, then ≽ need not have continuous extensions even to the metric completion of X . To illustrate, let D be a compact metric set. Let M0 be the space of all finite menus—finite subsets of D—and let M be the space of all compact menus in D. Endow both M0 and M with the Hausdorff metric. Let L0 be the space of all simple lotteries (probability distributions with finite support in D), and let L be the space of all Borel distributions. Endow L0 and L with the metrizable topology of weak convergence. These domains—M0 , M , L0 , L—have many

I. Kopylov / Journal of Mathematical Economics 67 (2016) 32–37

applications in decision theoretic models, such as expected utility (e.g. Kreps, 1988, Theorem 5.15), rank-dependent utility (Wakker et al., 1994), preferences for flexibility and commitment (Kreps, 1979; Gul and Pesendorfer, 2001). Yet in general, continuous preferences ≽ on M0 and L0 need not have continuous extensions to M and L respectively. See examples in Section 4. In general, Cauchy (or uniformly) continuous extensions need not be unique, even if X is dense in Y . For example, if X = (0, 1) ∪ (2, 3) and ≽ is the natural linear order on X , then its Cauchy continuous extension ≽Y to Y = [0, 1] ∪ [2, 3] allows both 2 ≻Y 1 and 2 ∼Y 1. Uniqueness is restored if X is dense in Y , and ≽ is extended together with its canonical representations (see Theorem 8). 2. Canonical representations Let X be an arbitrary consumption space, and X N be the set of all sequences e = (e1 , e2 , . . .) in X . Let ≽ be a weak preference relation on X with the asymmetric and symmetric parts ≻ and ∼. For any x, y ∈ X , let

[x, →) = {z ∈ X : z ≽ x} (x, →) = {z ∈ X : z ≻ x} (y, x) = {z ∈ X : x ≻ z ≻ y}

(←, y] = {z ∈ X : z ≼ y} (←, y) = {z ∈ X : z ≺ y} [y, x] = {z ∈ X : x ≽ z ≽ y}.

Call ≽ a weak order if ≽ is complete and transitive. A sequence e ∈ X N is called order-dense if for all x, y ∈ X , x ≻ y ⇒ x ≽ ei ≻ ej ≽ y for some i, j ∈ N.

(1)

Call ≽ order-separable if there is an order-dense sequence e ∈ X N . Equivalently, order-separability can be defined in terms of orderdense countable sets, as in Jaffray (1975) and Herden (1989), but sequences are more convenient for my objectives. By convention, extremal values of any function f on an empty domain are max f (x) = sup f (x) = 0

min f (x) = inf f (x) = 1.

By (2), un ∈ (0, 1) and U (x) ∈ [0, 1]. By induction, for all n, k ∈ N, e n ≽ e k ⇔ un ≥ uk .

(6)

Thus, U (en ) = un for all n, and (3) holds. Fix any x, y ∈ X . If x ≽ y, then U (x) ≥ U (y). Assume x ≻ y. By (1), x ≽ ei ≻ ej ≽ y for some i, j. By (6), U (x) ≥ U (ei ) = ui > uj = U (ej ) ≥ U (y). Thus U represents ≽, and U ∈ C (≽). By definition, (ii) implies (iii). Finally, let ≽ have some utility representation V : X → R. Let (q1 , r1 ), (q2 , r2 ), . . . enumerate all pairs of rational numbers qi , ri ∈ Q for which there exist ai , bi ∈ X such that for all a, b ∈ X , V (a) ≥ ri > qi ≥ V (b) ⇒ a ≽ ai ≻ bi ≽ b.

(7)

For each i ∈ N, take e2i−1 , e2i ∈ X such that (7) holds for ai = e2i and bi = e2i−1 . Take any x, y ∈ X such that x ≻ y. If (y, x) = ∅, then (7) holds for V (x) > ri > qi > V (y), ai = x and bi = y. Take any z ∈ (y, x). If (z , x) = ∅, then (7) holds for V (x) > ri > qi > V (z ) > V (y), ai = x and bi = z. Take any z ′ ∈ (z , x). Then (7) holds for V (x) > ri > V (z ′ ) > V (z ) > qi > V (y), ai = z ′ and bi = z. In each case, there is i such that V (x) ≥ ri > qi ≥ V (y). By (7), x ≽ e2i ≻ e2i−1 ≽ y. Thus, e is order-dense.  The implication (iii) ⇒ (i) can be derived immediately from the well-known results of Jaffray (1975) and Herden (1989) (see also Mehta, 1998, Theorem 3.1). I provide a direct proof to make my presentation self-contained. The explicit formulas (4) and (5) deliver some canonical representation U ∈ C (≽) for any order-dense sequence e ∈ X N . The main benefits of this construction are obtained when X has a topological structure. 2.1. Canonical representations in topological spaces Given a topology in X , ≽ is called

(2)

• upper semicontinuous (usc) if for all x ∈ X , the set [x, →) is

Call U : X → [0, 1] a canonical representation—or a canonical utility function—if U represents ≽, and there is an order-dense sequence e ∈ X N such that

• lower semicontinuous (lsc) if for all x ∈ X , the set (←, x] is

x∈∅

and

33

x∈∅

x∈∅

U ( en ) =

1 2



max U (ei ) +

i
x∈∅

min U (ej )

j


(3)

Theorem 1. The following statements are equivalent:

2

U ( x) =

 max ui +

i
sup n∈N:x≽en

un

min uj

j
Theorem 2. Let X be a topological space. If ≽ is usc (lsc, continuous), then each canonical representation U ∈ C (≽) is usc (lsc, continuous). Proof. Let ≽ be usc, and U ∈ C (≽). Fix any α ∈ R and b ∈ X such that α > U (b). Show that the set A = {a ∈ X : U (a) ≥ α} is closed. To do so, find an open set B ⊂ X that contains b, but is disjoint from A. If (b, x) = ∅ for some x ≻ b, let B = (←, x) = (←, b]. Otherwise, assume that for any x ≻ b, there exists y ∈ (b, x). By order-density, x ≻ ej ≽ y ≻ b for some j. Take the first k such that ek ∈ (b, x). Similarly, take the first n such that en ∈ (b, ek ). By (3),

≤

Proof. Fix a weak order ≽ and an order-dense sequence e ∈ X N . Let



Equivalently, these properties can be defined via opens sets (←, x) and (x, →).

U (en ) =

(i) ≽ is an order-separable weak order, (ii) there exists a canonical representation U ∈ C (≽), (iii) ≽ is represented by some utility function V : X → R.

1

closed;

• continuous if for all x ∈ X , both sets [x, →) and (←, x] are closed.

for all n ∈ N.

Any canonical representation U must range in [0, 1] on the entire X , and is uniquely determined by (3) on some order-dense sequence e in X . For example, U (e1 ) = 21 , and U (e2 ) = 41 , 12 , 34 if e2 ≺ e1 , e2 ∼ e1 , or e2 ≻ e1 respectively. This recursive construction is standard for utility representations over countable sets (e.g. Kreps, 1988, pp. 20–21)). I show that it can be extended to any domain X . Let C (≽) be the set of all canonical representations for ≽. Note that U ∈ C (≽) if and only if 1 − U ∈ C (≼) because formula (3) is preserved when ≽ and U are replaced by ≼ and 1 − U respectively.

un =

closed;

for all n = 1, 2, . . .

(4)

for all x ∈ X .

(5)

1 2



max U (ei ) +

i
U (b)+U (ek ) 2

≤

min U (ej )



j
U (b)+U (x) 2

.

Thus, for any x ≻ b, there is z = en ≻ b such that U (b) < U (z ) ≤ U (x)+U (b) . Then there exists z ∗ ≻ b such that U (b) < U (z ∗ ) < α . 2 Let B = (←, z ∗ ). Let ≽ be lsc, and U ∈ C (≽). Then 1 − U ∈ C (≼). As ≼ is usc, then 1 − U is usc. Thus U is lsc. Continuity combines usc and lsc.  The classic results of Debreu (1964) and Rader (1963) are easy corollaries.

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I. Kopylov / Journal of Mathematical Economics 67 (2016) 32–37

Corollary 3. Let X be a second countable topological space.

• Rader’s Theorem: ≽ is an usc (lsc) weak order if and only if ≽ has an usc (lsc) utility representation.

• Debreu’s Theorem: ≽ is a continuous weak order if and only if ≽ has a continuous utility representation. Proof. Let X be a topological space with a countable base {Bi }∞ i=1 . Let ≽ be an usc weak order. Write x ≻ Bi if x ≻ b for all b ∈ Bi . Then V (x) =

(8)

2i

represents ≽. Indeed, if x ≽ y, then y ≻ Bi implies x ≻ Bi and hence, V (x) ≥ V (y). If x ≻ y, take Bk such that y ∈ Bk ⊂ (←, x). Then V (x) ≥ V (y) + 21k because x ≻ Bk , but not y ≻ Bk . As ≽ is represented by V , then by Theorem 1, ≽ has a canonical representation U ∈ C (≽). By Theorem 2, U is usc. If ≽ is continuous, then U is continuous. If ≽ is lsc, then ≼ is usc, and there is usc U ∈ C (≼). Thus 1 − U is lsc and represents ≽.  The above proof via canonical representations is self-contained and arguably more transparent than the alternative approaches (Jaffray, 1975; Bosi and Mehta, 2002) that use variations of Debreu’s open-gap lemma and Nachbin’s separation theorem. Besides the added expositional clarity, canonical representations can be used to obtain some novel characterizations for preferences over metric spaces. 2.2. Canonical representations in metric spaces Let the topology in X be induced by a metric ρ . A sequence c = (c1 , c2 , . . .) ∈ X N is fundamental if for any ε > 0, there is m such that for all k, l ≥ m, ρ(ck , cl ) < ε . A function U : X → R is

• Cauchy continuous if for any fundamental sequence c ∈ X N , the sequence U (c1 ), U (c2 ), . . . is fundamental in the real line1 ; • uniformly continuous if for any ε > 0, there exists δ > 0 such that for all a, b ∈ X , ρ(a, b) < δ implies |U (a) − U (b)| < ε. Say that a weak preference ≽ is

• Cauchy continuous if for any x, y ∈ X and fundamental c ∈ X N , c2i−1 ≽ x

and y ≽ c2i

Proof. Suppose that ≽ is Cauchy (uniformly) continuous on a separable X . Then X is second countable, and ≽ is represented by (8). By Theorem 1, there is U ∈ C (≽). By Theorem 4, U is Cauchy (uniformly) continuous.  3. Continuous preference extensions

 1 i:x≻Bi

Corollary 5. Let (X , ρ) be a separable metric space. Then ≽ is a Cauchy (uniformly) continuous weak order if and only if ≽ can be represented by a Cauchy (uniformly) continuous utility function.

for all i = 1, 2, . . . ⇒ y ≽ x.

• uniformly continuous if for any x, y ∈ X such that x ≻ y, there is δ > 0 such that ρ(a, b) ≥ δ for all a ∈ [x, →) and b ∈ (←, y]. These properties are necessary for ≽ to have Cauchy and uniformly continuous utility representations respectively.2 Analogously to the general topological case, canonical representations inherit the stronger metric continuity properties of ≽ as well. Theorem 4. Let (X , ρ) be a metric space. If ≽ is Cauchy (uniformly) continuous, then each canonical representation U ∈ C (≽) is Cauchy (uniformly) continuous. The proof is relegated to Appendix. Theorem 4 delivers two novel refinements of the Debreu Theorem.

1 By the Cauchy criterion, the real sequence U (c ), U (c ), . . . must be convergent 1 2 as well. 2 Assume that U represents ≽. Take x, y ∈ X and a fundamental sequence c ∈ X N such that c2i−1 ≽ x and y ≽ c2i for all i ∈ N. Then U (c2i−1 ) ≥ U (x) and U (y) ≥ U (c2i ). If U is Cauchy continuous, then the sequence U (c1 ), U (c2 ), . . . converges to some α . Thus U (y) ≥ α ≥ U (x) and y ≽ x. If U is uniformly continuous, take any x ≻ y, ε = U (x) − U (y) > 0 and δ > 0 such that ρ(a, b) < δ implies |U (a) − U (b)| < ε. Then ρ(a, b) < δ implies that a ̸∈ [x, →) or b ̸∈ (←, y].

Let (X , ρ) be a metric space. Its completion (Xˆ , ρ) ˆ is another metric space such that X is a dense subset of Xˆ , ρ = ρˆ on X , and (Xˆ , ρ) ˆ is complete.3 It is well-known (e.g. Aliprantis and Border, 1999, Theorem 3.14) that such a completion exists and is unique up to an isometry. Consider the problem of extending a preference ≽ from (X , ρ) ˆ Xˆ , ρ) to (Xˆ , ρ) ˆ . Say that a triple (≽, ˆ is a continuous completion of (≽ ˆ is a continuous weak order on (Xˆ , ρ) ˆ on X . , X , ρ) if ≽ ˆ , and ≽= ≽ For any x ∈ Xˆ and ε > 0, the ε -neighborhood of x in X ⊂ Xˆ is written as B(x, ε) = {b ∈ X : ρ( ˆ x, b) < ε}. Note that B(x, ε) does not include x itself if x ̸∈ X . Given any nonempty subset B ⊂ X , write B ≽ (≼)x if b ≽ (≼)x for all b ∈ B. For all x, y ∈ Xˆ , let x ≻∗ y ⇔ B(x, ε) ≽ x′ ≻ y′ ≽ B(y, ε) for some ε > 0 and x′ , y′ ∈ X . ∗

(9)

∗

Let y ≽ x if and only if x ≻ y does not hold. Theorem 6. Let (X , ρ) be a metric space. The following statements are equivalent: (i) ≽ is a Cauchy continuous weak order, (ii) (≽∗ , Xˆ , ρ) ˆ is a continuous completion of (≽, X , ρ), ˆ Xˆ , ρ) (iii) there exists a continuous completion (≽, ˆ of (≽, X , ρ). The proof is in the Appendix. Theorem 6 holds even if ≽ is not order-separable and has no utility representations at all. Given any Cauchy continuous weak order ≽, the explicit formula (9) provides some continuous completion ≽∗ . This completion need not be unique. Moreover, Theorem 6 can be combined with mathematical existence results to extend preferences continuously beyond the completion Xˆ . Let (X , ρ) and (Y , µ) be metric spaces such that Y ⊃ X and µ = ρ on X . Say that ≽Y is an extension of ≽ to Y if ≽Y is a weak order on Y and ≽Y =≽ on X . Corollary 7. Let ≽ be an order-separable weak order on a metric space (X , ρ). Then ≽ is Cauchy (uniformly) continuous on X if and only if ≽ has an order-separable extension ≽Y that is Cauchy (uniformly) continuous on Y . Proof. Let U ∈ C (≽). Suppose that ≽ is Cauchy continuous. Let Xˆ be the closure of X in the completion Yˆ . Then Xˆ is the completion of X . By Theorem 6, there is a continuous function U ∗ : Xˆ → [0, 1] that represents ≽ on X . By the Tietze Extension Theorem, U ∗ has a continuous extension V ∗ on Yˆ . As Yˆ is complete, then the

3 The completeness of metric spaces requires that all fundamental sequences are convergent. It should not be confused with the completeness of preferences.

I. Kopylov / Journal of Mathematical Economics 67 (2016) 32–37

continuous V ∗ is Cauchy continuous as well. The preference ≽Y that V ∗ represents on Y is Cauchy continuous and extends ≽. Suppose that ≽ is uniformly continuous. Then U ∈ C (≽) is uniformly continuous. Katětov (1951, Theorem 3) shows that U has a uniformly continuous extension V : Y → R. Then V represents a uniformly continuous extension ≽Y on Y .  Without order-separability, continuous extensions ≽Y need not exist. See Yi (1993) for counterexamples. 4. Discussion Canonical representations and continuous preference extensions have several other noteworthy aspects. 4.1. Uniqueness of canonical representations Given any sequence e ∈ X N , let C (≽, e) be the class of all functions U : X → [0, 1] that represent ≽ and satisfy the recursive equation (3). By definition, U is canonical if and only if U ∈ C (≽, e) for some order-dense e. Theorem 8. For any weak order ≽ and order-dense sequence e ∈ X N , there is a unique canonical representation U ∈ C (≽, e). Moreover, for all x ∈ X , U (x) = sup U (en ) = inf U (en ). n:en ≼x

(10)

n:en ≽x

Finally, if ≽ is Cauchy continuous over a metric (X , ρ), then the sequence e is order-dense for ≽∗ , and for all x ∈ Xˆ , the unique U ∗ ∈ C (≽∗ , e) satisfies U ∗ ( x) =

sup n:x∈[en ,→)

U (en ) =

inf

U (en ).

(11)

n:x∈(←,en ]

The proof is relegated to Appendix. This theorem establishes the uniqueness of a canonical representation U that satisfies (3) for a given order-dense sequence e. Of course, the canonical utility function U ∈ C (≽, e) varies together with e. Even changing the order of elements in e leads to some non-trivial modifications of U. Formulas (10) and (11) describe canonical representations U ∈ C (≽, e) and U ∗ ∈ C (≽∗ , e) explicitly in terms of the values U (en ), which can be determined via the recursive equation (3). Note that U ∗ is the unique continuous extension of U from X to Xˆ . Thus any continuous extension of a canonical representation U ∈ C (≽) to the completion Xˆ must represent the same relation ≽∗ .

35

4.3. Continuous extensions in special frameworks I formulate Cauchy continuity for preferences over any metric space X and use it to extend ≽ continuously to the completion Xˆ and more broadly, to any larger metric domain Y ⊃ X . In more specific contexts, continuous preference completions have been obtained via various conditions other than general Cauchy continuity. These conditions include Extension Continuity for preferences over menus (Evren and Minardi, forthcoming), Strong Monotone Continuity for preferences over Savage’s acts (Kopylov, 2010a,b), Cauchy Continuity over Anscombe–Aumann acts (the early version of Ghirardato and Sinischalchi, 2012). All of these methods rely in part on the additional primitive structure and hence, cannot be compared directly with general Cauchy continuity. Of course, the technical motivation for all of these methods is essentially the same: to guarantee that the utility function U on the original domain X behaves nicely in the neighborhood of any point z ∈ Xˆ . 4.4. Debreu’s order-separability Most of the above arguments can be restated in terms of Debreu’s weaker notions of order-density and order-separability. Say that e ∈ X N is weakly order-dense if for all x, y ∈ X , x ≻ y ⇒ x ≽ ei ≽ y for some i ∈ N.

(12)

Say that ≽ is weakly order-separable if there is a weakly orderdense sequence e. Say that U : X → [0, 1] is a weak canonical representation if it represents ≽, and satisfies (3) for some weakly order-dense sequence e. Theorems 1, 2 and 4 are all still true under these weaker definitions. The equivalence of order-separability and weak orderseparability is standard (see Mehta, 1998). The proof of Theorem 2 requires the following modification. Assume that for any x ≻ b, there exists y ∈ (b, x). Then x ≻ y ≻ z ≻ b for some y, z ∈ X . By weak order-density, x ≻ y ≽ ej ≽ z ≻ b for some ej . The rest of the proofs remain the same. The proof of Theorem 4 (Lemma 9 in the Appendix) is modified analogously. By contrast, Theorem 8 fails for weak canonical representations. In particular, formulas (4) and (5) do not work for some weakly order-dense sequences √ e. For example, if ≽ is the natural linear order on X = [0, 1] ∪ [ 2, 3], and the sequence e enumerates all rational √ numbers in X , then e is √weakly order-dense, but (5) implies U ( 2) = U (1) even though 2 ≻ 1. Accordingly, the set C (≽, e) is empty in this case.

4.2. Debreu’s open-gap property 4.5. Some equivalences and counterexamples Given any subset S ⊂ R, say that f : S → R has an open-gap property if each maximal gap in the range f (S ) is open. Ouwehand (2010) uses formulas (4) and (5) to construct an increasing function f : S → [0, 1] that satisfies Debreu’s open-gap property.4 After suitable corrections, Ouwehand’s proof can be used to show that canonical representations satisfy the open-gap property, which can then be used to show Theorem 2 and Lemma 9 in the Appendix. My arguments are independent and appear faster than the open-gap route. Of course, the main substantial difference is that I use canonical representations to obtain several novel results, while Ouwehand’s sole objective is to give a more constructive proof of Debreu’s open-gap lemma.

4 While this claim is correct, Ouwehand’s proof does not work as is. On p. 19, he claims that if k is the first index such that k > m, n and en < ek < em , then f (en )+f (em ) f (ek ) = . However, this equality need not hold because there can be i 2 between m and n such that en < ei < em . In this case, f (ek ) can be close to f (en ) or to f (em ), which destroys Ouwehand’s conclusions. A proper correction should use the ‘‘minimal index’’ operation more than once.

The topological and metric continuity conditions for preferences ≽ and functions U : X → R are embedded via Uniform Continuity

⇒

Cauchy Continuity

⇒ Continuity ⇒

Upper or Lower Semicontinuity.

The opposite implications need not hold. For example, Cobb– Douglas preferences are Cauchy continuous, but not uniformly continuous on the set Rn+ . Continuous preferences need not be Cauchy continuous even on standard decision theoretic domains. For example, let D = [0, 1], and M0 (D) be the space of all finite subsets of D metrized by Hausdorff’s distance ρH . Let U1 (a) = cos



π 2ρH (a, D)



represent ≽1 over all finite menus a ∈ M0 (D). Then the function U1 and the preference ≽1 are continuous, but Cauchy continuous  not  1 on M0 (D). To check that, take cn = 0, n+ , 2 , . . . , n+n 1 , 1 . 1 n +1

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I. Kopylov / Journal of Mathematical Economics 67 (2016) 32–37

Then ρH (cn , D) =

1 , 2(n+1)

and the sequence c = (c1 , c2 , . . .) is

fundamental because ρH (ck , cn ) ≤

1 2(k+1)

+ 2(n1+1) . Let x = {0} 1 and y = 4 . Then ρ(x, D) = 1 and ρ(y, D) = 34 . For all i, π  =0 U1 (c2i−1 ) = cos 2iπ = 1 > U1 (x) = cos 2   2π 1 U1 (c2i ) = cos(2i + 1)π = −1 < U1 (y) = cos =− . 3

2

Thus c2i−1 ≻1 x ≻1 y ≻1 c2i for all i, and ≽ is not Cauchy continuous. Alternatively, let L0 (D) be the space of all simple lotteries (probability distributions with finite support in D) metrized by Prohorov’s metric ρP . If lD is the uniform distribution on D = [0, 1], and

π U2 (l) = cos 2ρP (l, lD ) 



represents ≽2 over all l ∈ L(D), then ≽2 is continuous, but not Cauchy continuous. Let M (D) be the set of all closed subsets in D, and L(D) be the set of all Borel distributions. By Theorem 6, the rankings ≽1 and ≽2 do not have continuous extensions to M (D) and L(D) respectively. Partial equivalences between various continuity conditions can be established under additional constraints on the metric domain X . (i) if X is complete, then continuity implies Cauchy continuity, (ii) if X is totally bounded, then Cauchy continuity implies uniform continuity, (iii) if X is compact, then continuity implies uniform continuity. These implications are true both for preferences ≽ and functions U : X → R. Statement (i) holds because completeness requires any fundamental sequence (c1 , c2 , . . .) ∈ X N to converge to some a ∈ X . If ≽ is continuous, then the rankings c2i−1 ≽ x and y ≽ c2i for all i = 1, 2, . . . imply that y ≽ a ≽ x. Statement (ii) follows from the fact that in a totally bounded X , every sequence has a fundamental subsequence. As compactness is the combination of completeness and total boundedness (Aliprantis and Border, 1999, Theorem 3.17), then (iii) follows from (i) and (ii). Acknowledgments I am grateful to Asen Kochov, Ozgur Evren, John Duggan, Marciano Siniscalchi, Jean-Philippe Lefort, Don Saari, Paolo Ghirardato, Stergios Skaperdas for their comments. The anonymous referee gave an extremely helpful report. Appendix Assume that ≽ is a weak order, and U ∈ C (≽). Lemma 9. For any α, β ∈ [0, 1], there are a, b ∈ X such that for all x, y ∈ X , U (x) ≥ α > β ≥ U (y) ⇒ x ≽ a ≻ b ≽ y.

(13)

Proof. Take an order-dense sequence e ∈ X N such that U ∈ C (≽, e). Fix any α, β ∈ [0, 1] such that α > β . Assume that the sets A = {x ∈ X : U (x) ≥ α} and B = {x ∈ X : U (x) ≤ β} are not empty. If A or B is empty, or β ≥ α , then (13) is vacuously true. α+β Show that A ⊂ [a, →) for some a ∈ X . Let γ = 2 and C = {x ∈ X : U (x) > γ }. If C ⊂ [a, →) for some a ∈ C , then A ⊂ C ⊂ [a, →). Assume that C ( [x, →) for all x ∈ C . Then for any x ∈ C , there is y ∈ C such that x ≻ y. By order-density, x ≻ en ≽ y for some n. As y ∈ C , then en ∈ C . Take the first k such

that ek ∈ C and x ≻ ek . Similarly, take the first m such that em ∈ C and ek ≻ em . By (3), U (em ) = um =

≤

1 2

γ + uk 2



 max ui +

i
≤

γ + U (x) 2

min

jγ

uj

. γ +U (x)

Then for any x ∈ C , there is y ∈ C such that U (y) ≤ . As 2 α > γ , then U (a) ∈ (γ , α) for some a ∈ C . Thus A ⊂ [a, →). Repeat the above argument for ≽′ =≼, U ′ = 1 − U , α ′ = 1 − β and β ′ = 1 − α to show that B ⊂ (←, b] for some b ∈ X such that U ′ (b) > 1 − γ , that is, U (b) < γ . Thus (13) holds.  Proof of Theorem 4. Suppose that U is not Cauchy continuous. Then there is a fundamental sequence c = (c1 , c2 , . . .) ∈ X N such that the sequence U (c1 ), U (c2 ), . . . is not convergent in the compact range [0, 1] and hence, has at least two distinct limit points α > β in [0, 1]. For each k = 1, 2, . . . , take

• Nk ∈ N such that ρ(cm , cn ) < 1k for all m, n ≥ Nk , • n2k−1 ∈ N such that n2k−1 ≥ N2k−1 and U (c2k−1 ) ≥ 23 α + 13 β , • n2k ∈ N such that n2k ≥ N2k and U (c2k ) ≤ 31 α + 23 β . For all k, let dk = cnk . The sequence d = (d1 , d2 , . . .) is fundamental because for all l ≥ k, ρ(dk , dl ) = ρ(cnk , cnl ) < 1k . By Lemma 9, there are a, b ∈ X such that U (x) ≥

2 3

α + 31 β > 13 α + 23 β ≥ U (y) ⇒ x ≽ a ≻ b ≽ y.

(14)

Then d2k−1 ≽ a ≻ b ≽ d2k , and ≽ is not Cauchy continuous. By contradiction, (i) holds. Suppose that U is not uniformly continuous. Then there is ε > 0 such that for all n ∈ N, there are cn , dn ∈ X such that U (cn ) − U (dn ) ≥ ε and ρ(cn , dn ) < 1n . Pick indices n1 < n2 < · · · such that the sequence of pairs

(U (cn1 ), U (dn1 )), (U (cn2 ), U (dn2 )), . . . is convergent in the compact [0, 1]×[0, 1]. Let α = limk→∞ U (cnk ) and β = limk→∞ U (dnk ). Then α ≥ β +ε . Take a, b ∈ X that satisfy (14). Then for all sufficiently large k, cnk ≽ a ≻ b ≽ dnk . Thus cnk ∈ [a, →), dnk ∈ (←, b], but ρ(cnk , dnk ) < n1 for all k. Thus ≽ k is not uniformly continuous. By contradiction, (ii) holds. Proof of Theorem 6. Suppose that ≽ is Cauchy continuous. Define ≽∗ on the completion Xˆ via formula (9). Show that ≽∗ =≽ on X . Take any x, y ∈ X such that x ≻∗ y. Then x ≽ x′ ≻ y′ ≽ y for some x′ , y′ ∈ X , and x ≻ y. Conversely, take x, y ∈ X such that x ≻ y. Consider three cases. (1) (y, x) is empty. As ≽ is continuous, there is ε > 0 such that B(x, ε) ≻ y and x ≻ B(y, ε). Then B(x, ε) ≽ x ≻ y ≽ B(y, ε), and x ≻∗ y. (2) There is x′ ∈ X such that x ≻ x′ ≻ y and (y, x′ ) is empty. As ≽ is continuous, there is ε > 0 such that B(x, ε) ≻ x′ ≻ B(y, ε). Then B(x, ε) ≽ x′ ≻ y ≽ B(y, ε) and x ≻∗ y. (3) There are x′ , y′ ∈ X such that x ≻ x′ ≻ y′ ≻ y. As ≽ is continuous, there is ε > 0 such that B(x, ε) ≽ x′ ≻ y′ ≽ B(y, ε). Thus x ≻∗ y. Show that ≽∗ is a continuous weak order on all of Xˆ . Take any x, y ∈ Xˆ . Suppose that x ≻∗ y and y ≻∗ x. By (9), B(x, ε) ≽ x′ ≻ y′ ≽ B(y, ε) for some x′ , y′ ∈ X and ε > 0 B(y, δ) ≽ y′′ ≻ x′′ ≽ B(x, δ) for some x′′ , y′′ ∈ X and δ > 0. As X is dense in Xˆ , there is y∗ ∈ X ∩ B(y, ε) ∩ B(y, δ) and x∗ ∈ X ∩ B(x, ε)∩ B(x, δ). Then x∗ ≽ x′ ≻ y′ ≽ y∗ ≽ y′′ ≻ x′′ ≽ x∗ , which contradicts transitivity of ≽ on X . Thus x ≻∗ y and y ≻∗ x cannot hold together, and ≽∗ is complete.

I. Kopylov / Journal of Mathematical Economics 67 (2016) 32–37

Turn to transitivity. Take any x, y ∈ Xˆ such that x ≻∗ y, that is, B(x, ε) ≽ x ≻ y ≽ B(y, ε) for some x , y ∈ X and ε > 0. ′

′

′

′

∗

Take c2n−1 ∈ B(z , 1n )∩[x′ , →) and c2n ∈ B(z , 1n )∩(←, y′ ]. Then the sequence c ∈ X n N is fundamental, and c2n−1 ≽ x′ ≻ y′ ≽ c2n for all n. It contradicts Cauchy continuity. Therefore, there is δ ∈ (0, ε) such that B(z , δ) ≺ x′ or B(z , δ) ≻ y′ . Consider three cases. (1) (y , x ) is empty. Then B(z , δ) ≺ x or B(z , δ) ≻ y implies that B(z , δ) ≼ y′ or B(z , δ) ≽ x′ respectively. Thus either B(x, ε) ≽ x′ ≻ y′ ≽ B(z , δ) and x ≻∗ z, or B(z , δ) ≽ x′ ≻ y′ ≽ B(y, ε) and z ≻∗ y. (2) There is x′′ ∈ X such that x′ ≻ x′′ ≻ y′ and (y′ , x′′ ) is empty. Take δ ∈ (0, ε) such that B(z , δ) ≺ x′′ or B(z , δ) ≻ y′ . Then either B(x, ε) ≽ x′′ ≻ y′ ≽ B(z , δ) or B(z , δ) ≽ x′′ ≻ y′ ≽ B(y, ε). Accordingly, either x ≻∗ z or z ≻∗ y. (3) There are x′′ , y′′ ∈ X such that x′ ≻ x′′ ≻ y′′ ≻ y′ . Take δ ∈ (0, ε) such that B(z , δ) ≺ x′′ or B(z , δ) ≻ y′′ . Then B(x, ε) ≽ x′ ≻ x′′ ≽ B(z , δ) or B(z , δ) ≽ y′′ ≻ y′ ≽ B(y, ε). Accordingly, either x ≻∗ z or z ≻∗ y. ′

′

′

′

Thus the preferences y ≽∗ z ≽∗ x are impossible if x ≻∗ y. Transitivity of ≽∗ follows. To show continuity of ≽∗ , take any z ∈ Xˆ such that ρ( ˆ z , x) < 2ε . ε Then B(z , 2 ) ⊂ B(x, ε) and hence, B(z , 2ε ) ≽ x′ ≻ y′ ≽ B(y, 2ε ). By definition, z ≻∗ y. Thus the sets {x ∈ Xˆ : x ≻∗ y} are open in Xˆ for all y ∈ Xˆ . Similarly, the sets {y ∈ Xˆ : x ≻∗ y} are open in Xˆ for all x ∈ Xˆ . The implications (ii) ⇒ (iii) ⇒ (i) are straightforward. Proof of Theorem 8. Suppose that e is order-dense, and U ∈ C (≽ , e). Show that such U is unique. Take U , V ∈ C (≽, e) such that α = V (z ) > U (z ) = β for some z ∈ X . By (13), there are a, b ∈ X such that for all x, y ∈ X , U (x) ≥ V (z ) > U (z ) ≥ U (y) ⇒ x ≽ a ≻ b ≽ y.

(15)

Assume that e1 ≻ z. Note that U (e1 ) = = V (e1 ) > V (z ) > U (z ). Thus e1 ≽ a ≻ b ≽ z. Take en such that a ≻ en ≽ z. Then U (en ) = un = V (en ) ≥ V (z ). By (15), en ≽ a, which is a contradiction. If e1 ≺ z, then repeat for ≽′ =≽, U ′ = 1 − V , and V ′ = 1 − U. If e1 ∼ z, then U (z ) = V (z ) = u1 = 21 . Show (10). Recall from the proof of Theorem 1 that U (x) = supn∈N:x≽en U (en ). As 1 − U ∈ C (≼, e) is unique, then 1 2

1 − U (x) =

sup (1 − U (en )).

n∈N:x≼en

Thus U (x) = supn∈N:x≽en U (en ) = infn∈N:x≼en U (en ).

Show (11). For all x, y ∈ Xˆ such that x ≻∗ y, there exists ε > 0 and x′ , y′ ∈ X such that B(x, ε) ≽ x′ ≻ y′ ≽ B(y, ε). As e is orderdense, B(x, ε) ≽ x′ ≽ ei ≻ ej ≽ y′ ≽ B(y, ε) for some i, j. Thus x ≽∗ ei ≻∗ ej ≽∗ y, and e is order-dense for ≽∗ in Xˆ .

Take U ∈ C (≽, e). Let U ∗ ( x) =

Show that for any z ∈ Xˆ , either x ≻ z or z ≻ y must hold. Suppose that for each n, the set B(z , 1n ) overlaps both [x′ , →) and (←, y′ ]. ∗

37

sup n:x∈[en ,→)

U (en ) for all x ∈ Xˆ

(16)

and show that U ∗ ∈ C (≽∗ , e). For all i, j ∈ N, ei ∈ [ej , →) iff ei ≽ ej . Thus U ∗ (ei ) = U (ei ) = ui where ui is defined by (4). Show that U ∗ represents ≽∗ . Take any x∗ , y∗ ∈ Xˆ . I claim that x∗ ≻∗ y∗ ⇒ U ∗ (x∗ ) > U ∗ (y∗ ).

(17)

Assume x ≻ y . Then there are ei ≻ ej and ε > 0 such that B(x∗ , ε) ≽ ej ≻ ej ≽ B(y∗ , ε). As B(x∗ , ε) ≽ ei , then x∗ ∈ [ei , →). As ej ≽ B(y∗ , ε), then y∗ ̸∈ [ek , →) for any ek ≻ ej . Thus U ∗ (x∗ ) ≥ U (ei ) > U (ej ) ≥ U ∗ (y∗ ). ∗

∗ ∗

Take any x∗ , y∗ ∈ Xˆ such that U ∗ (x∗ ) > U ∗ (y∗ ). Take ej such that U ∗ (x∗ ) ≥ U (ej ) > U ∗ (y∗ ). As U (ej ) > U ∗ (y∗ ), then y∗ ̸∈ [ej , →). It follows that ej ≻ b for some b ∈ X . By (13), b ∈ X can be taken so that for all y ∈ X , U (ej ) > U ∗ (y∗ ) ≥ U (y) ⇒ ej ≻ b ≽ y. For each ei , y∗ ∈ [ei , →) implies U ∗ (y∗ ) ≥ U (ei ) and hence, b ≽ ei . Therefore, U (b) ≥ supei :y∗ ∈[ei ,→) U (ei ) = U ∗ (y∗ ). Take ei such that ej ≻ ei ≽ b. Then U ∗ (x∗ ) ≥ U (ej ) > U (ei ) ≥ U (b) ≥ U ∗ (y∗ ). By (17), x∗ ≽∗ ej ≻∗ ei ≽∗ y∗ . By transitivity of ≽∗ , x∗ ≻∗ y∗ . Finally, as 1 − U ∗ ∈ C (≼∗ , e), then (11) holds. References Aliprantis, C., Border, K., 1999. Infinite Dimensional Analysis. Springer. Bosi, G., Mehta, G., 2002. Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. J. Math. Econom. 38, 311–328. Debreu, G., 1964. Continuity properties of Paretian utility. Internat. Econom. Rev. 5, 285–293. Evren, O., Minardi, S., 2016. Warm-glow giving and freedom to be selfish. Econ. J. forthcoming. Ghirardato, P., Sinischalchi, M., 2012. Ambiguity in the small and in the large. Econometrica 80, 2827–2847. Gul, F., Pesendorfer, W., 2001. Temptation and self-control. Econometrica 69, 1403–1435. Herden, G., 1989. On the existence of utility functions. Math. Social Sci. 17, 297–313. Jaffray, J., 1975. Existence of a continuous utility function: An elementary proof. Econometrica 43, 981–983. Katětov, M., 1951. On real-valued functions in topological spaces. Fund. Math. 38, 85–91. Kopylov, I., 2010a. Simple axioms for countably additive subjective probabilities. J. Math. Econom. 46, 867–876. Kopylov, I., 2010b. Unbounded probabilistic sophistication. Math. Social Sci. 60, 113–118. Kreps, D., 1979. A representation theorem for ‘preference for flexibiility’. Econometrica 47, 565–577. Kreps, D.M., 1988. Notes on the Theory of Choice. Westview Press, Boulder and London. Mehta, G., 1977. Topological ordered spaces and utility functions. Internat. Econom. Rev. 18, 779–782. Mehta, G., 1998. Preference and utility. In: Barberá, S., Hammond, P., Seidl, C. (Eds.), Handbook of Utility Theory, Vol. 1. Kluwer Academic Publishers, Dordrecht. Ouwehand, P., 2010. A simple proof of Debreu’s gap lemma. ORiON 26, 17–20. Rader, J.T., 1963. The existence of a utility function to represent preferences. Rev. Econom. Stud. 30, 229–232. Wakker, P., Erev, I., Weber, E., 1994. Comonotonic independence: The critical test between classical and rank-dependent utility theories. J. Risk Uncertain. 9, 195–230. Yi, G., 1993. Continuous extension of preferences. J. Math. Econom. 22, 547–555.