JUAN C. CANDEAL, JUAN R. DE MIGUEL and ESTEBAN INDURÁIN
EXPECTED UTILITY FROM ADDITIVE UTILITY ON SEMIGROUPS
ABSTRACT. In the present paper we study the framework of additive utility theory, obtaining new results derived from a concurrence of algebraic and topological techniques. Such techniques lean on the concept of a connected topological totally ordered semigroup. We achieve a general result concerning the existence of continuous and additive utility functions on completely preordered sets endowed with a binary operation “+”, not necessarily being commutative or associative. In the final part of the paper we get some applications to expected utility theory, and a representation theorem for a class of complete preorders on a quite general family of real mixture spaces. KEY WORDS: Preordered sets, utility funtions, continuous and additive utility, expected utility, semigroups
1. INTRODUCTION
This note introduces some results on additive utility theory derived from a mixture of algebraic and topological techniques. We first state a very general result concerning the existence of continuous and additive utility functions on a set S endowed with both a binary operation “+” and a complete preorder . This is achieved with very mild assumptions on “+” and . In particular we do not demand the operation “+” to be either commutative or associative. The assumption of connectedness with respect to the order topology of S along with the fact that the operation “+” is continuous with respect to the same topology allow us to replace the usual axiom of monotonicity by the cancellativity with respect to the indifference relation ∼. Secondly we present some applications of the previous theorem to expected utility theory. The model we study may be motivated as the choice of lotteries over the time. On the one hand n we reach a representation result of the form F (t, l1, . . . , ln ) = ( i=1 αi li )t, where Theory and Decision 53: 87–94, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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αi ∈ R for all i, (l1 , . . . , ln ) is a lottery defined over n outcomes, and t > 0 stands for time. On the other hand we obtain a representation result of the type u(tc + t c ) = tu(c) + t u(c ), where t, t > 0 and c, c ∈ C, C being a convex cone embedded into a topological real vector space. It is important to emphasize that we do not demand the mixture space either to be finite-dimensional or to have nonempty interior. Note on this point that recent papers on this topic take some of these conditions as a crucial hypothesis to get their representability results (see, e.g., Einy, 1989; Neuefeind and Trockel, 1995; Candeal and Induráin, 1995). A thorough study of additive utility theorems can be seen in Chapters 2 and 3 of Krantz et al. (1971). The main difference of our approach in relation to the results stated there is the topological consideration based on the continuity of the binary operation “+” with respect to the order topology. For some other approaches to utility theory which combine both algebraic and topological methods see Gottinger (1976) and more recently Fuhrken and Richter (1991).
2. THE RESULTS
Let S be a set endowed with both a binary operation “+” and a complete preorder . Associated to we define the strict preference and the indifference relations, respectively denoted by “≺” and “∼”, given by x ≺ y ⇐⇒ ¬(y x) and x ∼ y ⇐⇒ x y , y x (x, y ∈ S). The operation “+” is said to be: • ∼-cancellative if x + y ∼ x + z ⇐⇒ y + x ∼ z + x ⇐⇒ y ∼ z for every x, y, z ∈ S. • ∼-associative if x +(y +z) ∼ (x +y)+z for every x, y, z ∈ S. • -continuous if the map (x, y) ∈ S × S → x + y ∈ S is continuous with respect to the order topology of S. The complete preorder “” on S is said to be representable if there is an order-preserving real-valued function u : S −→ R such that x y ⇐⇒ u(x) ≤ u(y) (x, y ∈ S). Such a function “u”is said to be a utility function. Moreover the function u is said to be additive if u(x + y) = u(x) + u(y) (x, y ∈ S).
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After these definitions we introduce the first and main result. THEOREM 1. Let S be a set endowed with both a binary operation “+” and a complete preorder . Suppose that the following conditions are satisfied: (i) S is connected in the order topology. (ii) The operation “+” is ∼-cancellative, ∼-associative and continuous. Then there exists a continuous and additive utility function that represents (continuity refers to the order topology of S and the Euclidean topology of R). Proof. Consider the quotient space under the indifference relation S/ ∼. Then the operation “+” induces a well defined binary operation, denoted by “⊕”, on S/ ∼. Indeed let [x], [y] ∈ S/ ∼ and define [x] ⊕ [y] = [x + y]. To see that this definition is consistent it is enough to prove that for any x, x , y, y ∈ S such that x ∼ x and y ∼ y, it holds that x + y ∼ x + y . But this clearly follows from repeated application of the fact of “+” being ∼-cancellative because x + y ∼ x + y ∼ x + y . Now the two properties of ∼-cancellativity and ∼-associativity of “+” on S translate into cancellativity and associativity of the operation “⊕” on S/ ∼ respectively. So in algebraical terminology (S/ ∼, ⊕) is a cancellative semigroup. Furthermore, the condition of “+” being continuous along with the connectedness of S in the order topology tell us that (S/ ∼, ⊕, ≺) is a topological connected and cancellative totally ordered semigroup. Following Lemma 4 in Candeal et al. (1997b), we get that the strict preference relation ≺ must be translation-invariant, i.e., for every x, y, z ∈ S it holds that x ≺ y implies x +z ≺ y +z: Actually, given a, b ∈ S with a ≺ b, the continuity conditions on “+” and easily imply that the sets P = {z ∈ S : a + z ≺ b + z} and Q = {y ∈ S : b + y ≺ a + y} are both open. Using cancellativity, it follows that P ∪ Q = S, so that by connectedness either P is empty or else Q is empty. But the existence of an element y ∈ Q would imply, again by connectedness and the continuity conditions on the binary operation and the preorder, that the map Ry : S −→ S given by Ry (s) = s + y (s ∈ S) is monotonic and reverses the ordering, i.e: for every s ≺ t (s, t ∈ S) it holds that Ry (t) ≺ Ry (s).
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Thus b + y ≺ a + y would imply that Ry (a + y) ≺ Ry (b + y) so that a + (y + y) ≺ b + (y + y) and P will be also nonempty, which is a contradiction. (Left translation-invariance is proved in a completely analogous way). Let us prove now the existence of a continuous and additive utility function on the structure (S/ ∼, ⊕, ≺): Remember that a semigroup (T , +, ) is said to be super-Archimedean if for every x, y ∈ T with x ≺ x + x, y ≺ y + y and x ≺ y there exists n ∈ N such that (n + 1) · x ≺ n · y, and for every x, y ∈ T with x + x ≺ x, y + y ≺ y and x ≺ y there exists n ∈ N such that n · x ≺ (n + 1) · y. A classical result by Alimov (1950) (see also De Miguel et al., 1996) proved that, under translation-invariance, the existence of an additive utility function that represents (T , +, ) is equivalent to the super-Archimedeaness of such structure. All connected topological totally ordered semigroups are super-Archimedean as proved in Theorem 3 of Candeal et al. (1997a). So in our case we get an additive utility function that represents (S/ ∼, ⊕, ≺). In addition, Theorem 4 in Candeal et al. (1997a) proves that any additive utility function on a connected topological totally ordered semigroup (T , +, ) is forcely continuous (with respect to the order topology on T and the usual topology on the real line R). To conclude the proof of the theorem, let u be a continuous and additive utility function on (S/ ∼, ⊕, ≺) and let ψ denote the projection map x ∈ S −→ [x] ∈ S/ ∼. Then the function u◦ψ : S −→ R is an additive and continuous utility function for because ψ is additive and continuous. REMARK 1. (i) Note that under the conditions of the theorem the binary operation “+” on S need not be either associative or commutative. To illustrate this remark let S = R2 be with the operation “⊗” given by (a, b) ⊗ (c, d) = (a + c, |2b − d|) and the preorder defined as (a, b) (c, d) ⇐⇒ a ≤ c. It is easy to see that S is neither commutative nor associative. Moreover it is not cancellative either. Note however that all the conditions of the Theorem 1 are satisfied, hence, there is a continuous and additive utility function for . The function u(a, b) = a is such a function. (ii) Even though the binary operation “+” on S need not be either associative or commutative, we have seen in the proof of the theorem that it induces a well-defined binary operation, denoted by “⊕”, on
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S/ ∼ so that (S/ ∼, ⊕) becomes a cancellative semigroup, and in particular ⊕ is associative. This fact allows us to consider the context of Theorem 1 as a generalization of the functional equation of associativity analyzed in Candeal et al. (1997b). (iii) It should be noted that no monotonicity property (or homogeneity law, or translation-invariance axiom) is required a priori in the statement of Theorem 1, although it follows from the other conditions involved there. (iv) For a discussion of behavioral aspects of the sufficient conditions of Theorem 1 and, in particular, those that have to do with measurement theory, the reader is referred to Chapter 3 of the thorough book of Krantz et al. (1971). Also, Chapter 2 in Narens (1985) explains the need for the consideration of binary operations, associative or not, in contexts as the measurement of physical attributes or certain kinds of qualitative probability, and some psychological example (Weber’s law) in which a psychophysical situation is measured by means of extensive structures that are quite similar to topological ordered semigroups. Let us now see some applications of the previous result to the expected utility context. (For general concepts and results concerning expected utility, see Fishburn, 1982). In order to make things as easy as possible suppose that L is a lottery-space which is embedded into a topological real vector space (E, +, ·R, τ ), where “E” denotes a nonempty set, “+” stands for the internal binary operation on E, “·R ” denotes the external multiplication by real scalars, and “τ ” is the given topology on E. Let t > 0 be a strictly positive real number. In the space S = R++ × L, where each element (t, l) could be interpreted as the choice of the lottery l at time t, we define the following binary operation “⊗”: t t l+ l (t, l) ⊗ (t , l ) = t + t , t + t t + t Note that “⊗” is well defined because t/t + t l + t /t + t l ∈ L. Moreover it is easily checked that the so-defined operation is associative. Now let be a continuous complete preorder defined on S, i.e., for every x ∈ S the sets L(x) = {y ∈ S; y x} and G(x) = {y ∈
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S; x y} are closed in the product topology of R++ × E. Then the following result holds. THEOREM 2. Let (R++ × L, ⊗) be endowed with a continuous complete preorder so that “⊗” is ∼-cancellative and -continuous. Then is represented by a utility function of the form F (t, l) = a(l)t, where a : L −→ R satisfies a(λl + (1 − λ)l ) = λa(l) + (1 − λ)a(l ), (l, l ∈ L, λ ∈ [0, 1]). Proof. It is a direct consequence of Theorem 1 because “⊗” being associative it is ∼-associative. Moreover R++ × L is a connected topological space with respect to the product topology; hence, it is connected in the order topology because is continuous. Thus by Theorem 1 there is a continuous and additive utility function that represents . Let F be such a function. Note that in particular F is continuous with respect to the product topology because is continuous. Let l ∈ L be fixed and define the function g(t) = F (t, l), t (t ∈ R+ ). Then because g(t + t ) = F (t + t , l) = F (t + t , t+t l + t t+t l) = F ((t, l) ⊗ (t , l)) = F (t, l) + F (t , l) = g(t) + g(t ) (for every t, t ∈ R++ ), the function g is of the form g(t) = a(l)t, where a(l) is a constant which depends on l ∈ L. So F (t, l) = a(l)t, t ∈ R++ , l ∈ L. Now from the additivity of F it follows that t t t a(l)t + a(l )t = a( t+t l + t+t l )(t + t ), or equivalently a( t+t l + t t t t+t l ) = t+t a(l) + t+t a(l ), for every t, t ∈ R++ , l, l ∈ L. So by t calling λ = t+t we have that a(λl+(1−λ)l ) = λa(l)+(1−λ)a(l ), (l, l ∈ L, λ ∈ [0, 1]). REMARK 2. (i) In particular if L is a lottery-space over a finite number of outcomes, nsay n, then the function F can be written as F (t, l1, . . . , ln ) = ( i=1 αi li )t, where αi ∈ R for all i. (ii) Note that if all the parameters αi are negative then our model presents the impatience property of the traditional time preference models that appear in the literature (see, e.g., Brown and Lewis, 1981 or Araujo, 1985). Next we obtain a very general result concerning the existence of continuous and additive utility function provided that the mixture space C is a convex cone of a topological real vector space (E, +, ·R, τ ). Note that this theorem does not require either E to be finite-dimensional or C to have nonempty interior.
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THEOREM 3. Let C be a convex cone of a topological real vector space (E, +, ·R , τ ). Suppose that C is endowed with a continuous complete preorder so that “+” is ∼-cancellative and -continuous. Then is represented by a continuous utility function u : C −→ R which satisfies u(tc + t c ) = tu(c) + t u(c ), (t, t > 0, c, c ∈ C). Proof. First note that, because C is a convex cone, the addition in C is well defined, i.e., it is closed. Also C being a convex set of E, it is a connected topological space. Hence, C is a connected topological space in the order topology of C because is continuous. Furthermore, “+” is obviously ∼-associative and, by hypothesis, it is also ∼-cancellative. Thus, by Theorem 1, there is an order-continuous (hence τ -continuous) additive utility function u : C −→ R for . Let c0 ∈ C be fixed. Because the set {tc0 ; t > 0} is homeomorphic to R++ , and u is additive and τ -continuous, it follows that u(tc0 ) = tu(c0 ). Thus u(tc) = tu(c), for all t > 0, c ∈ C. Finally u(tc + t c ) = u(tc) + u(t c ) = tu(c) + t u(c ), (t, t > 0, c, c ∈ C). REMARK 3. For the case when the mixture set becomes the convex cone of a topological vector space, Theorem 3 is an strengthening of the classical expected utility theorem in Herstein and Milnor (1953), using only cancellativity instead of translation-invariance. This topological vector space context is absent in Herstein and Milnor (1953).
ACKNOWLEDGEMENTS
This work has been partially supported by the research project PB980551 “Estructuras Ordenadas y Aplicaciones”. (Spain, Dec. 1999). Also, the research of coauthors Candeal and Induráin has been supported by the “Integrated Action of Research HI2000-0116 (SpainItaly)”. REFERENCES Alimov, N.G. (1950), On ordered semigroups (In Russian), Izv. Akad. Nauk SSSR Ser. Math. 14, 569–576.
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Araujo, A. (1985), Lack of Pareto optimal allocations in economies with infinitely many commodities: the need for impatience, Econometrica 53(2), 455–461. Brown, D.J. and Lewis, L.M. (1981), Myopic economic agents, Econometrica 49(2), 359–368. Candeal, J.C., De Miguel, J.R. and Induráin, E. (1997), Topological additively representable semigroups, Journal of Mathematical Analysis and Applications 210, 375–389. Candeal, J.C., De Miguel, J.R., Induráin, E. and Olóriz, E. (1997), Associativity equation revisited, Publicationes Mathematicae Debrecen 51(12), 133–144. Candeal, J.C., and Induráin, E. (1995), A note on linear utility, Economic Theory 6, 519–522. De Miguel, J.R., Candeal, J.C., and Induráin, E. (1996), Archimedeaness and additive utility on totally ordered semigroups, Semigroup Forum 52, 335–347. Einy, E. (1989), On preferences relations which satisfy weak independent property, Journal of Mathematical Economics 18, 291–300. Fishburn, P.C. (1982), The Foundations of Expected Utility. Dordrecht, The Netherlands, D. Reidel. Fuhrken, G., and Richter, M.K. (1991), Additive utility, Economic Theory 1, 83– 105. Gottinger, H. (1976), Existence of a utility on a topological semigroup, Theory and Decision 7, 145–158. Herstein, I.N., and Milnor, J. (1953), An axiomatic approach to measurable utility, Econometrica 21, 291–297. Krantz, D.H., Luce, R.D., Suppes, P. and Tversky, A. (1971), Foundations of measurement. Academic Press, New York. Narens, L. (1985), Abstract Measurement Theory. Cambridge, MA: MIT Press. Neuefeind, W. and Trockel, W. (1995), Continuous linear representability of binary relations, Economic Theory 6, 351–356.
Addresses for correspondence: Juan C. Candeal, Departamento de Análisis Económico, Universidad de Zaragoza, Gran Vía 2-4, E-50005, Zaragoza, Spain; email:
[email protected] Juan R. De Miguel, Departamento de Matemática e Informática, Universidad Pública de Navarra, Campus Arrosadía s.n., E-31006, Pamplona, Spain; e-mail:
[email protected] Esteban Induráin, Departamento de Matemática e Informática, Universidad Pública de Navarra, Campus Arrosadía s.n., E-31006, Pamplona, Spain; e-mail:
[email protected] (Corresponding author)
