Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

SUBLINEAR AND CONTINUOUS ORDER-PRESERVING FUNCTIONS FOR NONCOMPLETE PREORDERS

GIANNI BOSI AND MAGALÌ E. ZUANON

Abstract. We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.

1. Introduction

Necessary and sufficient conditions for the existence of a continuous linear order-preserving function for a complete preorder on a topological real vector space are already found in the literature (see e.g. Candeal and Induráin [10] and Neuefeind and Trockel [17]). It is well known that there are important applications of such results in expected utility theory and collective decision making. A characterization of the existence of a continuous linear utility function for a complete preorder on a convex set in a normed real vector space was presented by Bültel [7]. Further, some authors were concerned with the existence of a homogeneous of degree one and continuous orderpreserving function for a complete preorder on a real cone in a topological real vector space (see e.g. Bosi [2], Bosi, Candeal and Induráin [3], and Dow and Werlang [13]). More recently, Bosi and Zuanon [5] presented a characterization of the existence of a nonnegative, homogeneous of degree one and continuous order-preserving function for a noncomplete preorder on a real cone in a topological real vector space. In a different context, other authors were concerned with the existence of an additive order-preserving function on a completely preordered semigroup (see e.g. Allevi and Zuanon [1], and Candeal, de Miguel and Induráin [9]). Bosi and Zuanon [6] presented a characterization of the existence of a Choquet integral representation for a complete preorder on the space of all continuous real-valued functions on a compact topological space.

2000 Mathematics Subject Classification. 28B15, 06A06, 91B16.

Key words and phrases. Topological vector space; preordered real cone; sublinear orderpreserving function; decreasing scale.

JEL classification. C60; D80.

Corresponding author. Gianni Bosi.

Gianni Bosi and Magalì E. Zuanon, Sublinear and continuous order-preserving functions for noncomplete preorders, Proceedings of the Ninth Prague Topological Symposium (Prague, 2001), pp. 1–8, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204101.

In this paper we provide an axiomatization of the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in a topological real vector space. All the results we present are based upon the notion of a decreasing scale (or linear separable system) which was introduced by Herden [14, 15] in order to characterize the existence of a continuous order-preserving function for a preorder on a topological space (see also Burgess and Fitzpatrick [8], and Mehta [16]).

It should be noted that such a topic can be of some interest in the applications to economics. Consider the following example concerning decision theory under uncertainty. Let M = {μ_{n} : n ∈ {1, …, n^{∗}}} be a finite family of concave capacities on a measurable space (Ω, A), with Ω the state space, and A a σ-algebra of subsets of Ω. We recall that a capacity μ on A (i.e., a function from A into [0, 1] such that μ(∅) = 0, μ(Ω) = 1, and μ(A) ≤ μ(B) for all A ⊆ B, A, B ∈ A) is said to be concave if for all sets A, B ∈ A,

μ(A ∪ B) + μ(A ∩ B) ≤ μ(A) + μ(B)

(see e.g. Chateauneuf [11]). Let X be a real convex cone of nonnegative real random variables (i.e., measurable real functions) on (Ω, A), and assume that X is contained in L^{1}(Ω, A, μ_{n}) for every n ∈ {1, …, n^{∗}}, where L^{1}(Ω, A, μ) stands for the normed space of all the real random variables x such that the Choquet integral

∫_{Ω} xdμ = ∫_{0}^{∞} μ({x ≥ t})dt + ∫_{−∞}^{0}(μ({x ≥ t}) − 1)dt

is finite (see e.g. Denneberg [12]). Define a binary relation ⪯ on X as follows:

x ⪯ y if and only if ∫_{Ω} xdμ_{n} ≤ ∫_{Ω} ydμ_{n} for all n ∈ {1, …, n^{∗}}.

It is clear that ⪯ is a preorder on X, and that ⪯ is not complete in general. For every n ∈ {1, …, n^{∗}}, denote by τ_{n} the norm topology on X which is associated to μ_{n}, and let τ be any (vector) topology on X which is stronger than τ_{n} for all n ∈ {1, …, n^{∗}} (i.e., τ_{n} ⊆ τ for all n ∈ {1, …, n^{∗}}). Then the real-valued function u on X defined by

u(x) = ∑_{n=1}^{n^{∗}} ∫_{Ω} xdμ_{n} (x ∈ X)

is a nonnegative, sublinear and τ-continuous order-preserving function for ⪯. Indeed, it is clear that x ⪯ y implies u(x) ≤ u(y) for all x, y ∈ X. If x ≺ y, then we have u(x) < u(y), since ∫_{Ω} xdμ_{n} ≤ ∫_{Ω} ydμ_{n} for all n ∈ {1, …, n^{∗}}, and there exists at least one index n̄ ∈ {1, …, n^{∗}} such that ∫_{Ω} xdμn̄ < ∫_{Ω} ydμn̄. Further, u is sublinear since the functional x → ∫_{Ω} xdμ_{n} is sublinear for all n ∈ {1, …, n^{∗}}. Finally, u is τ-continuous since the functional x → ∫_{Ω} xdμ_{n} is τ_{n}-continuous, and therefore τ-continuous for all n ∈ {1, …, n^{∗}} (see Denneberg [12, Proposition 9.4]).

2. Notation and preliminaries

A preorder ⪯ on an arbitrary set X is a reflexive and transitive binary relation on X. The strict part and the symmetric part of a given preorder ⪯ will be denoted by ≺, and respectively ∼. A preorder ⪯ on a set X is said to be complete if for any two elements x, y ∈ X either x ⪯ y or y ⪯ x.

If ⪯ is a preorder on a set X, then the pair (X, ⪯) will be referred to as a preordered set. Define, for every x ∈ X, L_{≺}(x) = {z ∈ X : z ≺ x}, U_{≺}(x) = {z ∈ X : x ≺ z}.

Given a preordered set (X, ⪯), a real-valued function u on X is said to be

(1) increasing if u(x) ≤ u(y) for every x, y ∈ X such that x ⪯ y;

(2) order-preserving if it is increasing and u(x) < u(y) for every x, y ∈ X such that x ≺ y.

If (X, ⪯) is a preordered set, and τ is a topology on X, then the triple (X, τ, ⪯) will be referred to as a topological preordered space. If (X, τ, ⪯) is a topological completely preordered space, then the complete preorder ⪯ is said to be continuous if L_{≺}(x) and U_{≺}(x) are open subsets of X for every x ∈ X.

Given a preordered set (X, ⪯), a subset A of X is said to be decreasing if y ∈ A whenever y ⪯ x and x ∈ A.

In the sequel, the symbol ℚ^{++} (ℝ^{++}) will stand for the set of all positive rational (real) numbers. If (X, τ) is a topological space, then denote by \overline{A} the topological closure of any subset A of X.

We say that a family G = {G_{r} : r ∈ ℚ^{++}} is a countable decreasing scale (countable linear separable system) in a topological preordered space (X, τ, ⪯) if

(1) G_{r} is an open decreasing subset of X for every r ∈ ℚ^{++};

(2) \overline{G_{r_{1}}} ⊆ G_{r_{2}} for every r_{1}, r_{2} ∈ ℚ^{++} such that r_{1} < r_{2};

(3) ⋃_{r∈ℚ^{++}} G_{r} = X.

If E is a real vector space, then define, for every subset A of E and any real number t, tA = {ta : a ∈ A}. Further, if A and B are any two subsets of a (real) vector space E, then define A + B = {a + b : a ∈ A, b ∈ B}.

A subset X of a real vector space E is said to be

(1) a real cone if tx ∈ X for every x ∈ X and t ∈ ℝ^{++};

(2) a real convex cone if it is a real cone and x+y ∈ X for every x, y ∈ X.

A real-valued function u on a real cone X in a real vector space E is said to be homogeneous of degree one if u(tx) = tu(x) for every x ∈ X and t ∈ ℝ^{++}.

A real-valued function u on a real convex cone X in a real vector space E is said to be sublinear if it is homogeneous of degree one and subadditive (i.e., u(x + y) ≤ u(x) + u(y) for every x, y ∈ X).

Given a topological real vector space E, denote by τ the vector topology for E (i.e., the topology on E which makes the vector operations continuous).

If X is any subset of a topological real vector space E, denote by τ_{X} the topology induced on X by the vector topology τ on E.

If (X, ⪯) is a preordered real cone in a topological real vector space E, then we say that a countable decreasing scale G = {G_{r} : r ∈ ℚ^{++}} in (X, τ_{X}, ⪯) is homogeneous if qG_{r} = G_{qr} for every q, r ∈ ℚ^{++}.

If (X, ⪯) is a preordered real convex cone in a topological real vector space E, then we say that a countable decreasing scale G = {G_{r} : r ∈ ℚ^{++}} in (X, τ_{X}, ⪯) is subadditive if G_{q} + G_{r} ⊆ G_{q+r} for every q, r ∈ ℚ^{++}.

3. Existence of a sublinear continuous order-preserving function

In the following theorem we characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in a topological real vector space.

Theorem 3.1. Let ⪯ be a preorder on a real convex cone X in a topological real vector space E. Then the following conditions are equivalent:

(1) There exists a nonnegative, sublinear and continuous order-preserving function u for ⪯.

(2) There exists a homogeneous and subadditive countable decreasing scale G = {G_{r} : r ∈ ℚ^{++}} in (X, τ_{X}, ⪯) such that for every x, y ∈ X with x ≺ y there exist r_{1}, r_{2} ∈ ℚ^{++} with

r_{1} < r_{2}, x ∈ G_{r_{1}} and y ∉ G_{r_{2}}.

Proof. (1) ⇒ (2). Assume that there exists a nonnegative, sublinear and continuous order-preserving function u for ⪯. Define G_{r} = u^{−1}([0, r[) for every r ∈ ℚ^{++}. Let us show that G = {G_{r} : r ∈ ℚ^{++}} is a homogeneous and subadditive countable decreasing scale satisfying condition (2). Using the fact that u is nonnegative and order-preserving, we have that for every x, y ∈ X such that x ≺ y, there exist r_{1}, r_{2} ∈ ℚ^{++} such that u(x) < r_{1} < r_{2} < u(y), and therefore x ∈ G_{r_{1}}, y ∉ G_{r_{2}}. Further, since u is homogeneous of degree one,

qG_{r} = qu^{−1}([0, r[) = u^{−1}([0, qr[) = G_{qr} for every q, r ∈ ℚ^{++}.

Hence, G is homogeneous. It only remains to show that G is subadditive. To this aim, consider any two rational numbers q, r ∈ ℚ^{++}, and let z ∈ G_{q} + G_{r}. Then there exist two elements x, y ∈ X such that z = x + y, u(x) < q, u(y) < r. Hence, using the fact that u is subadditive, we have u(z) = u(x + y) ≤ u(x) + u(y) < q + r, and therefore z ∈ G_{q+r}.

(2) ⇒ (1). Define, for every x ∈ X,

u(x) = inf{r ∈ ℚ^{++} : x ∈ G_{r}}.

Then u is a nonnegative continuous order-preserving function for ⪯. Indeed, by using the fact that G_{r} is a decreasing set for every r ∈ ℚ^{++}, it is easily seen that u is increasing. Further, u is order-preserving by condition (2) above, and u is continuous since u(x) = inf\{r ∈ ℚ^{++} : x ∈ \overline{G_{r}}\} (see e.g. Theorem 1 in Bosi and Mehta [4] for details). In order to show that u is homogeneous of degree one, it suffices to prove that for no r ∈ ℚ^{++}, and x ∈ X it is u(rx) ≠ ru(x). Then the thesis follows by a standard continuity argument. This part of the proof is already found in Bosi and Zuanon [5, Theorem 1]. Nevertheless, we present all the details here for reader’s convenience. By contradiction, assume that there exist r ∈ ℚ^{++}, and x ∈ X such that u(rx) < ru(x). Then, from the definition of u, there exists r′ ∈ ℚ^{++} such that u(rx) < r′ < ru(x), rx ∈ G_{r′}. Since u(x) > \frac{r′}{r}, it follows that x ∉ G_{\frac{r′}{r}} = \frac{1}{r}G_{r′}, and therefore we arrive at the contradiction

rx ∉ G_{r′}. Analogously it can be shown that for no r ∈ ℚ^{++}, and x ∈ X it is ru(x) < u(rx). It remains to prove that u is subadditive. By contradiction, assume that there exist two elements x, y ∈ X such that u(x) + u(y) < u(x + y). Then, from the definition of u, there exist two rational numbers q, r ∈ ℚ^{++} such that u(x) + u(y) < q + r < u(x + y), x ∈ G_{q}, y ∈ G_{r}. Since the countable decreasing scale G = {G_{r} : r ∈ ℚ^{++}} is subadditive, we have x + y ∈ G_{q+r}, and therefore q + r < u(x + y) is contradictory from the definition of u. This consideration completes the proof. □

If ⪯ is a homothetic complete preorder on a real cone X in a real vector space E (i.e., x ⪯ y entails tx ⪯ ty for every x, y ∈ X, and t ∈ ℝ^{++}), then consider the following subcones of X:

X_{0} = {x ∈ X : x ∼ tx for some positive real number t ≠ 1};
X_{+} = {x ∈ X : x ≺ tx for some real number t > 1};
X_{−} = {x ∈ X : tx ≺ x for some real number t > 1}.

In the following corollary, we present a characterization of the existence of a nonnegative, sublinear and continuous order-preserving function for a complete preorder on a real convex cone in a topological real vector space. We recall that, given a preordered set (X, ⪯), a subset A of X is said to be an order-dense subset of (X, ⪯) if for every x, y ∈ X such that x ≺ y there exists a ∈ A such that x ≺ a ≺ y.

Corollary 3.2. Let ⪯ be a complete preorder on a real convex cone X in a topological real vector space E, and assume that X_{0} and X_{+} are both nonempty, while X_{−} is empty. Then the following conditions are equivalent:

(1) There exists a nonnegative, sublinear and continuous order-preserving function u for ⪯.

(2) The following conditions are satisfied:

(a) ⪯ is homothetic;

(b) ⪯ is continuous;

(c) The set {qx_{+} : q ∈ ℚ^{++}} is an order-dense subset of (X_{+}, ⪯) for every element x_{+} ∈ X_{+};

(d) x ∼ y for every x, y ∈ X_{0};

(e) x ≺ x_{+} for every x ∈ X_{0}, x_{+} ∈ X_{+};

(f) x + y ≺ (q + r)x_{+} for every x, y ∈ X, x_{+} ∈ X_{+}, q, r ∈ ℚ^{++} such that x ≺ qx_{+}, y ≺ rx_{+}.

Proof. (1) ⇒ (2). Assume that there exists a nonnegative, sublinear and continuous order-preserving function u for ⪯. Then it is clear that ⪯ is homothetic and continuous. If we consider any element x_{+} ∈ X such that x_{+} ≺ tx_{+} for some real number t > 1, then it is necessarily u(x_{+}) > 0, and therefore, using the fact that u is homogeneous of degree one, condition (2c) easily follows. Finally, it is easily seen that conditions (2d), (2e) and (2f) are verified.

(2) ⇒ (1). Consider any element x_{+} ∈ X such that x_{+} ≺ tx_{+} for some real number t > 1, and let G_{r} = L_{≺}(rx_{+}) for every r ∈ ℚ^{++}. Then it is easy to check that the family G = {G_{r} : r ∈ ℚ^{++}} is a countable decreasing scale satisfying condition (2) of Theorem 3.1. Indeed, by homotheticity of ⪯, x ≺ rx_{+} is equivalent to qx ≺ qrx_{+} (q ∈ ℚ^{++}), and therefore G is homogeneous (see Bosi and Zuanon [5, Corollary 2]). Further, G is subadditive by condition (2f) since, for every q, r ∈ ℚ^{++}, and x, y ∈ X, x ∈ G_{q} and y ∈ G_{r} is equivalent to x ≺ qx_{+} and y ≺ rx_{+}, which implies x + y ≺ (q + r)x_{+} or equivalently x + y ∈ G_{q+r}. Finally, by condition (2c) above, for every x, y ∈ X such that x ≺ y there exist r_{1}, r_{2} ∈ ℚ^{++} such that

r_{1} < r_{2}, x ≺ r_{1}x_{+} ≺ r_{2}x_{+} ≺ y,

or equivalently

x ∈ L_{≺}(r_{1}x_{+}), y ∉ L_{≺}(r_{2}x_{+}).

So the proof is complete. □

Denote by ¯0 the zero vector in a real vector space E. In the following corollary we are concerned with a sublinear representation of a complete preorder on a real convex cone containing the zero vector.

Corollary 3.3. Let ⪯ be a complete preorder on a real convex cone X in a topological real vector space E, and assume that X_{+} is nonempty, while X_{−} is empty. If in addition ¯0 belongs to X, then there exists a nonnegative, sublinear and continuous order-preserving function u for ⪯ if and only if ⪯ is homothetic and continuous, and it satisfies condition (2f) of Corollary 3.2.

Proof. From the corollary in Bosi, Candeal and Induráin [3], there exists a nonnegative, homogeneous of degree one and continuous order-preserving function u for ⪯. Indeed, the complete preorder ⪯ on the real (convex) cone X is homothetic and continuous, and we have in addition ¯0 ∈ X. Let us show that u must be subadditive as a consequence of condition (2f) of Corollary 3.2. Assume by contraposition that there exist x, y ∈ X such that u(x) + u(y) < u(x + y), and consider any element x_{+} ∈ X_{+}. Then it must be u(x_{+}) > 0 since u is a homogeneous of degree one utility function for ⪯, and there exist two positive rational numbers q and r such that

u(x) + u(y) < (q + r)u(x_{+}) < u(x + y), x ≺ qx_{+}, y ≺ rx_{+}.

But here we have a contradiction since it should be u(x + y) < (q + r)u(x_{+}) by condition (2f) of Corollary 3.2. So the proof is complete. □

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