Continuous representability of homothetic preorders by means of sublinear order-preserving functions

Mathematical Social Sciences 45 (2003) 333–341

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Continuous representability of homothetic preorders by means of sublinear order-preserving functions Gianni Bosi a , *, Magalı` E. Zuanon b a

Universita di Trieste, Dipartimento di Matematica Applicata ‘ Bruno de Finetti’, Piazzale Europa 1, Trieste 34127, Italy b Istituto di Econometria e Matematica per le Decisioni Economiche, Universita` Cattolica del Sacro Cuore, Largo A. Gemelli 1, 20123, Milano, Italy Received 1 July 2001; received in revised form 1 June 2002; accepted 1 September 2002

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.  2002 Elsevier Science B.V. All rights reserved. Keywords: Topological vector space; Preordered real cone; Sublinear order-preserving function; Decreasing scale; Homothetic preorder JEL classification: C60; D80

1. Introduction Necessary and sufficient conditions for the existence of a continuous linear orderpreserving function for a complete preorder on a topological real vector space are ´ 1995; and Neuefeind and already found in the literature (see e.g. Candeal and Indurain, Trockel, 1995). 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 Bultel (2001). Further, some authors *Corresponding author. Tel.: 139-040-558-7115; fax: 139-040-54209. E-mail addresses: [email protected] (G. Bosi), [email protected] (M.E. Zuanon). 0165-4896 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-4896(02)00067-7

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were concerned with the existence of a homogeneous of degree one and continuous order-preserving function for a complete preorder on a real cone in a topological real vector space (see e.g. Bosi, 1998; Bosi et al., 2000; Dow and Werlang, 1992; and Maccheroni, 2001). More recently, Bosi and Zuanon (2000) presented a characterization of the existence of a nonnegative, homogeneous of degree one and continuous orderpreserving 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, 2000; and Candeal et al., 1999). Bosi and Zuanon (2002) 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. 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 (1989a,b) in order to characterize the existence of a continuous order-preserving function for a preorder on a topological space (see also Burgess and Fitzpatrick, 1977; Mehta, 1998 and Bosi and Mehta, 2002). 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 5 h mn : n [ h1, . . . ,n * jj be a finite family of concave capacities on a measurable space (V, ! ), with V the state space, and ! a s -algebra of subsets of V. We recall that a capacity m on ! (i.e., a function from ! into [0,1] such that m (5) 5 0, m (V ) 5 1, and m (A) # m (B) for all A 7 B, A,B [ ! ) is said to be concave if for all sets A,B [ !, f m (A < B) 1 m (A > B) # m (A) 1 m (B) g (see e.g. Chateauneuf, 1996). Let X be a real convex cone of nonnegative real random variables (i.e., measurable real functions) on (V, ! ), and assume that X is contained in L 1 (V, !, mn ) for every n [ h1, . . . ,n * j, where L 1 (V, !, m ) stands for the normed space of all the real random variables x such that the Choquet integral eV x dm 5 e0` m (hx $ tj) dt 1 0 e2` ( m (hx $ tj) 2 1) dt is finite (see e.g. Denneberg, 1994). Define a binary relation A on X as follows:

E

E

V

V

if and only if x dmn # y dmn

x Ay

for all n [ h1, . . . ,n * j.

It is clear that A is a preorder on X, and that A is not complete in general. For every n [ h1, . . . ,n * j, denote by tn the norm topology on X which is associated to mn , and let t be any (vector) topology on X which is stronger than tn for all n [ h1, . . . ,n * j (i.e., tn 7 t for all n [ h1, . . . ,n * j). Then the real-valued function u on X defined by

OEx dm n*

u(x) 5

n51

n

(x [ X)

V

is a nonnegative, sublinear and t -continuous order-preserving function for A. Indeed, it

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is clear that xAy implies u(x) # u( y) for all x,y [ X. If x a y, then we have u(x) , u( y), since eV x dmn # eV y dmn for all n [ h1, . . . ,n * j, and there exists at least one index n¯ [ h1, . . . ,n * j such that eV x dmn¯ , eV y dmn¯ . Further, u is sublinear since the functional x → eV x dmn is sublinear for all n [ h1, . . . ,n * j. Finally, u is t -continuous since the functional x → eV x dmn is tn -continuous, and therefore t -continuous for all n [ h1, . . . ,n * j (see Denneberg, 1994, Proposition 9.4). Let us now consider another situation which may be interesting in the applications to economics, game theory and insurance mathematics. Denote by A a complete preorder on a real convex cone X of nonnegative real random variables in the normed space L 1 (V, !, 3 ) (i.e., L 1 (V, !, 3 ) is the real vector space of all 3 -integrable random variables on a common probability space (V, !, 3 ), endowed with the L 1 -norm topology), and assume that X contains all positive constants. The constant equal to l will be also denoted by l for every positive real number l. If u is a homogeneous of degree one order-preserving function for A (i.e., xAy is equivalent to u(x) # u( y) for every x,y [ X), and u(1) 5 1, then it is immediate to check that u is a certainty equivalence function for A, in the sense that x | u(x) for all x [ X. In this case, u is said to exhibit constant relative risk aversion (see e.g. Epstein and Zin, 2001). We recall that a probability distortion g is a nondecreasing function g: [0,1] → [0,1] such that g(0) 5 0 and g(1) 5 1. If there exists a concave probability distortion g such that u(x) 5 eV x dg + 3 for all x [ X, then from considerations above the real-valued function u on X is not only a sublinear continuous order-preserving function for A but also a certainty equivalence function for A. Indeed, m 5 g + 3 is a concave capacity on (V, ! ). It should be noted that in this case A is risk loving (see e.g. Suijs, 2000), in the sense that IE(x)Ax for all x [ X, where IE( ? ) is the expectation operator on X (i.e., any stochastic payoff x is weakly preferred to its expectation). More generally, the complete preorder A is risk loving whenever A is represented by a Choquet integral with respect to a distorted probability g + 3 such that g( p) $ p for all p [ [0,1] (see e.g. Wang and Young, 1998). In a context which is slightly different than that considered in this paper, Yaari (1987) showed that convexity of the probability distortion in the representation of a complete preorder by means of a Choquet integral is equivalent to the property of risk aversion of A as defined by Rothschild and Stigliz (1970). Finally, it should be noted that, as a consequence of the Dominating Extension Theorem in functional analysis (see e.g. Fuchssteiner and Lusky, 1981), every sublinear real-valued function u which is increasing with respect to a homothetic preorder on a real convex cone in a vector space is the pointwise maximum of all increasing linear real-valued functions which are dominated by u.

2. Notation and preliminaries A preorder A 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 A will be denoted by a , and respectively | . A preorder A on a set X is said to be complete if for any two elements x,y [ X either xAy or yAx.

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If A is a preorder on a set X, then the pair (X,A) will be referred to as a preordered set. Define, for every x [ X, La (x) 5 hz [ X:z a xj, Ua (x) 5 hz [ X:x a zj. Given a preordered set (X,A), a real-valued function u on X is said to be (i) increasing if u(x) # u( y) for every x,y [ X such that xAy; (ii) order-preserving if it is increasing and u(x) , u( y) for every x,y [ X such that x a y. If (X,A) is a preordered set, and t is a topology on X, then the triple (X,t,A) will be referred to as a topological preordered space. If (X,t,A) is a topological completely preordered space, then the complete preorder A is said to be continuous if La (x) and Ua (x) are open subsets of X for every x [ X. Given a preordered set (X,A), a subset A of X is said to be decreasing if y [ A whenever yAx and x [ A. In the sequel, the symbol Q 11 (R 11 ) will stand for the set of all positive rational ] (real) numbers. If (X,t ) is a topological space, then denote by A the topological closure of any subset A of X. We say that a family & 5 hGr :r [ Q 11 j is a countable decreasing scale (countable linear separable system) in a topological preordered space (X,t,A) if (i) Gr is an open decreasing subset of X for every r [ Q 11 ; ] (ii) Gr 1 7 Gr 2 for every r 1 ,r 2 [ Q 11 such that r 1 , r 2 ; (iii) < Gr 9 5 Gr for every r [ Q 11 , < Gr 5 X. r 9,r, r 9[Q 11

r [Q 11

If E is a real vector space, then define, for every subset A of E and any real number t, tA 5 hta:a [ Aj. Further, if A and B are any two subsets of a (real) vector space E, then define A 1 B 5 ha 1 b:a [ A,b [ Bj. A subset X of a real vector space E is said to be (i) a real cone if tx [ X for every x [ X and t [ R 11 ; (ii) a real convex cone if it is a real cone and x 1 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) 5 tu(x) for every x [ X and t [ R 11 . 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 1 y) # u(x) 1 u( y) for every x,y [ X). Given a topological real vector space E, denote by t 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, then denote by tX the topology induced on X by the vector topology t on E. If (X,A) is a preordered real cone in a topological real vector space E, then we say that a countable decreasing scale & 5 hGr :r [ Q 11 j in (X,tX ,A) is homogeneous if qGr 5 Gqr for every q,r [ Q 11 . If (X,A) is a preordered real convex cone in a topological real vector space E, then we

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say that a countable decreasing scale & 5 hGr :r [ Q 11 j in (X,tX ,A) is subadditive if Gq 1 Gr 7 Gq1r for every q,r [ Q 11 .

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

( i) There exists a nonnegative, sublinear and continuous order-preserving function u for A. ( ii) There exists a homogeneous and subadditive countable decreasing scale & 5 hGr :r [ Q 11 j in (X,tX ,A) such that for every x,y [ X with x a y there exists r [ Q 11 with x [ Gr , y [ ⁄ Gr . Proof. (i)⇒(ii). Assume that there exists a nonnegative, sublinear and continuous order-preserving function u for A. Define Gr 5 u 21 ([0,r[) for every r [ Q 11 . Let us show that & 5 hGr :r [ Q 11 j is a homogeneous and subadditive countable decreasing scale satisfying condition (ii). It is immediate to check that Gr is an open decreasing subset of X for every r [ Q 11 since u is a continuous order-preserving function for A. From the definition of the sets Gr , we have that conditions (ii) and (iii) in the definition of a countable decreasing scale are both satisfied. Using the fact that u is nonnegative and order-preserving, we have that for every x,y [ X such that x a y, there exists r [ Q 11 such that u(x) , r , u( y), and therefore x [ Gr , y [ ⁄ Gr . Further, since u is homogeneous of degree one, qGr 5 qu 21 ([0,r[) 5 u 21 ([0,qr[) 5 Gqr for every q,r [ Q 11 . Hence, & is homogeneous. It only remains to show that & is subadditive. To this aim, consider any two rational numbers q,r [ Q 11 , and let z [ Gq 1 Gr . Then there exist two elements x,y [ X such that z 5 x 1 y, u(x) , q, u( y) , r. Hence, using the fact that u is subadditive, we have that u(z) 5 u(x 1 y) # u(x) 1 u( y) , q 1 r, and therefore z [ Gq1r . (ii)⇒(i). Define, for every x [ X, u(x) 5 inf hr [ Q

11

: x [ Gr j.

First observe that u is well defined since

<

r [Q 11

Gr 5 X. We claim that u is a nonnegative

continuous order-preserving function for A. By using the fact that Gr is a decreasing set for every r [ Q 11 , it is easily seen that u is increasing. Further, u is order-preserving since for every x,y [ X such that x a y there exists r [ Q 11 such that x [ Gr , y [ ⁄ Gr , and therefore we have that u(x) , r # u( y), which obviously implies that u(x) , u( y). Indeed, from the definition of u, it can be easily proved that x [ Gr implies u(x) , r

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(observe that x [ Gr and u(x) 5 r would imply that x [ ⁄ Gr 9 for every r9 , r, r9 [ Q 11 ,and this contradicts the fact that < Gr 9 5 Gr ). r 9,r, r 9[Q 11

In order to prove that u is continuous, let us first show that u is upper semicontinuous. Consider any x [ X, and a [ R 11 such that u(x) , a. Then, from the definition of u, there exists r [ Q 11 such that u(x) , r , a, x [ Gr , and therefore Gr is an open subset of X containing x such that u(z) , a for every z [ Gr (observe that u(z) $ a implies u(z) . r which in turn implies z [ ⁄ Gr ). In order to show that u is lower semicontinuous, consider any x [ X, and any nonnegative real number a such that a , u(x). Further, let ] r 1 ,r 2 [ Q 11 be such that a , r 1 , r 2 , u(x). Then we have that x [ ⁄ Gr 1 because ] otherwise x [Gr 1 implies x [ Gr 2 and this contradicts the fact that u(x) . r 2 . Hence, ] ] X\Gr 1 is an open subset of X containing x such that a , u(z) for every z [ X\Gr 1 ] (observe that u(z) # a implies u(z) , r 1 which in turn implies z [Gr 1 since u(x) 5 ] inf hr [ Q 11 :x [Gr j for every x [ X). In order to show that u is homogeneous of degree one, it suffices to prove that for no r [ Q 11 , 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 (2000, Theorem 1). Nevertheless, we present all the details here for reader’s convenience. By contradiction, assume that there exist r [ Q 11 , and x [ X such that u(rx) , ru(x). Then, from the definition of u, there exists r9 [ Q 11 such that u(rx) , r9 , ru(x), rx [ Gr 9 . Since u(x) . r9 /r, it follows that x [ ⁄ Gr 9 / r 5 1 /rGr 9 , and therefore we arrive at the contradiction rx [ ⁄ Gr 9 . Analogously it can be shown that for no r [ Q 11 , 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) 1 u( y) , u(x 1 y). Then, from the definition of u, there exist two rational numbers q,r [ Q 11 such that u(x) 1 u( y) , q 1 r , u(x 1 y), x [ Gq , y [ Gr . Since the countable decreasing scale & 5 hGr :r [ Q 11 j is subadditive, we have that x 1 y [ Gq1r , and therefore q 1 r , u(x 1 y) is contradictory from the definition of u. This consideration completes the proof. h Remark 3.2. It should be noted that if we introduce the concept of a superadditive countable decreasing scale (i.e., a countable decreasing scale & 5 hGr :r [ Q 11 j such that (X\Gq ) 1 (X\Gr ) 7 X\Gq 1r for every q,r [ Q 11 ) in a topological preordered real convex cone (X,tX ,A), then we can characterize the existence of a superlinear continuous order-preserving function u on (X,tX ,A) (i.e., u is homogeneous of degree one and u(x 1 y) $ u(x) 1 u( y) for all x,y [ X). If A is a homothetic complete preorder on a real cone X in a real vector space E (i.e., xAy is equivalent to txAty for every x,y [ X, and t [ R 11 ), then consider the following subcones of X: X0 5 hx [ X: x | tx

for some positive real number t ± 1j;

X1 5 hx [ X: x a tx

for some real number t . 1j;

X2 5 hx [ X: tx a x

for some real number t . 1j.

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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), a subset A of X is said to be an order-dense subset of (X,A) if for every x,y [ X such that x a y there exists a [ A such that x a a a y. Corollary 3.3. Let A be a complete preorder on a real convex cone X in a topological real vector space E, and assume that X0 and X1 are both nonempty, while X2 is empty. Then the following conditions are equivalent:

( i) There exists a nonnegative, sublinear and continuous order-preserving function u for A. ( ii) The following conditions are satisfied: (a) A is homothetic; ( b) A is continuous; (c) The set hqx 1 :q [ Q 11 j is an order-dense subset of (X1 ,A) for every element x 1 [ X1 ; (d) x | y for every x,y [ X0 ; (e) x a x 1 for every x [ X0 , x 1 [ X1 ; ( f ) x 1 y a (q 1 r)x 1 for every x,y [ X, x 1 [ X1 , q,r [ Q 11 such that x a qx 1 , y a rx 1 . Proof. Observe that under our assumptions X is partitioned into the subcones X0 and X1 . Further, it should be noted that if there exists a homogeneous of degree one orderpreserving function u for the complete preorder A, then X2 5 hx [ X:u(x) , 0j. (i)⇒(ii). Assume that there exists a nonnegative, sublinear and continuous orderpreserving function u for A. Then it is clear that A is homothetic and continuous. If we consider any element x 1 [ X1 , then it is necessarily u(x 1 ) . 0, and therefore, using the fact that u is homogeneous of degree one, condition (c) easily follows. Indeed, for every x,y [ X1 such that x a y we have that 0 , u(x) , u( y), and therefore there exists q [ Q 11 such that u(x) , qu(x 1 ) , u( y), which is equivalent to x a qx 1 a y by homogeneity of u. Finally, it is easily seen that conditions (d), (e) and (f) are verified. (ii)⇒(i). Consider any element x 1 [ X such that x 1 a tx 1 for some real number t . 1, and let Gr 5 La (rx 1 ) for every r [ Q 11 . Let us show that the family & 5 hGr :r [ Q 11 j is a countable decreasing scale satisfying condition (ii) of Theorem 3.1. First, it is clear that Gr is an open decreasing subset of X for every r [ Q 11 . For every r 1 ,r 2 [ Q 11 such that r 1 , r 2 we have that r 1 x 1 a r 2 x 1 by homotheticity of A since x 1 [ X1 . Therefore, hz [ X:zAr 1 x 1 j is a closed subset of X which contains Gr 1 and is ] contained in Gr 2 , implying that Gr 1 7 Gr 2 . Further, for every x [ X1 there exists 11 r [ Q , r . 1, such that x a rx 1 , implying that X is the union of the sets Gr (r [ Q 11 ). In order to verify that < Gr 9 5 Gr for every r [ Q 11 , just consider r 9,r, r 9[Q 11 11

the fact that, for every x [ X, and r [ Q , if x a rx 1 then by the above conditions (c) and (e) there exists r9 [ Q 11 , r9 , r such that x a r9x 1 a rx 1 , which implies that x [ Gr 9 . By homotheticity of A, x a rx 1 is equivalent to qx a qrx 1 (q [ Q 11 ), and

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therefore & is homogeneous (see Bosi and Zuanon, 2000, Corollary 2). Further, & is subadditive by condition (f) since, for every q,r [ Q 11 , and x,y [ X, x [ Gq and y [ Gr is equivalent to x a qx 1 and y a rx 1 , which implies that x 1 y a (q 1 r)x 1 or equivalently x 1 y [ Gq 1r . Finally, by condition (c) above, for every x,y [ X such that x a y there exists r [ Q 11 such that x a rx 1 a y, which implies that x [ La (rx 1 ), y[ ⁄ La (rx 1 ). So the proof is complete. h Denote by 0¯ the zero vector in a real vector space E. In the following proposition we are concerned with a sublinear representation of a complete preorder on a real convex cone containing the zero vector. Proposition 3.4. Let A be a complete preorder on a real convex cone X in a topological real vector space E, and assume that X1 is nonempty, while X2 is empty. If in addition 0¯ belongs to X, then there exists a nonnegative, sublinear and continuous order-preserving function u for A if and only if A is homothetic and continuous, and it satisfies condition ( f ) of Corollary 3.3. Proof. The necessity part is immediate from Corollary 3.3. Conversely, from the corollary in Bosi et al. (2000), there exists a nonnegative, homogeneous of degree one and continuous order-preserving function u for A. It should be noted that in our case u is necessarily nonnegative since X2 is empty. Indeed, the complete preorder A 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 (f) of Corollary 3.3. Assume by contraposition that there exist x,y [ X such that u(x) 1 u( y) , u(x 1 y), and consider any element x 1 [ X1 . Then it must be u(x 1 ) . 0 since u is a homogeneous of degree one utility function for A, and there exist two positive rational numbers q and r such that u(x) 1 u( y) , (q 1 r)u(x 1 ) , u(x 1 y), x a qx 1 , y a rx 1 . But here we have a contradiction since it should be u(x 1 y) , (q 1 r)u(x 1 ) by condition (f) of Corollary 3.3. So the proof is complete. h Remark 3.5. We can observe that, if A is a complete, homothetic and continuous ¯ and 0Ax ¯ preorder on a real convex cone X containing the zero vector 0, for every x [ X, then a homogeneous and subadditive countable decreasing scale satisfying condition (ii) of Theorem 3.1 comes into consideration in a very natural way. Indeed, for every x 1 [ X such that 0¯ a x 1 the family & 5 hLa (rx 1 ):r [ Q 11 j is a homogeneous countable decreasing scale such that for every x,y [ X with x a y there exists r [ Q 11 with x [ La (rx 1 ), y [ ⁄ La (rx 1 ). If in addition condition (f) of Corollary 3.3 is verified, then & is also subadditive.

Acknowledgements ´ and three anonymous referees for many valuable We thank Professor Esteban Indurain suggestions. We are particularly grateful to one of the referees for suggesting the first part of condition (iii) in the definition of a countable decreasing scale.

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