Economic Theory 17, 233–238 (2001)
Utility and entropy Juan C. Candeal1 , Juan R. De Miguel2 , Esteban Indur´ain2 , and Ghanshyam B. Mehta3 1 2 3
Universidad de Zaragoza, Departamento de An´alisis Econ´omico, c/ Doctor Cerrada, 50005 Zaragoza, SPAIN (e-mail:
[email protected]) Universidad P´ublica de Navarra, Departamento de M´atematica e Inform´atica, Campus Arrosadia, 31006, SPAIN (e-mail:
[email protected] and
[email protected]) Department of Economics, University of Queensland, Brisbane, Queensland 4072, AUSTRALIA (e-mail:
[email protected])
Received: May 17, 1999; revised version: October 16, 2000
Summary. In this paper we study an astonishing similarity between the utility representation problem in economics and the entropy representation problem in thermodynamics. Keywords and Phrases: Utility, Entropy. JEL Classification Numbers: C 60.
1 Introduction The purpose of this note is to make known to economists an astonishing similarity between the utility representation problem in utility theory and the entropy representation problem which is related to the second law of thermodynamics. It is quite remarkable that so many of the ideas, methods and results that arise in studying the entropy problem are also of importance and have been extensively studied in the vast economics literature on the representation of preferences by utility functions. However, it should be observed that this close connection does not seem to be known in the voluminous economics literature on the representation of preferences by utility functions (see Bridges and Mehta, 1995; Mehta, 1999 and the references cited there) or, so far as we are aware, in the literature on thermodynamics and the entropy problem (see, for example, Buchdahl, 1966; Cooper, 1967; Giles, 1964; Lieb and Yngvason, 1998; Owen 1984). We are grateful to an anonymous referee for valuable comments on an earlier version of this paper. Correspondence to: G. B. Mehta.
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It will follow from our examination of Cooper’s work and supervenient considerations that the mathematical structure of the entropy representation problem is exactly the same as that of the utility representation problem. Therefore, a conflation of the methods used in utility theory and the theory of entropy will yield significant insights. In particular, economists interested in utility theory will benefit from the knowledge of cognate methods and/or results which may then be applied to prove the existence of continuous and/or additive utility functions. The theory of differentiable utility representations may also be enriched in this manner. There is no reference in Cooper’s paper, or in the other literature on thermodynamics, to the prevenient economics literature that deals with the utility representation problem. Such an omission may be deemed to be venial and is not surprising because on the surface there does not seem to be any connection whatsoever between these two apparently disparate fields. Furthermore, it should be remembered that economics is inveterately perceived by those in the natural and mathematical sciences to be an inexact science. 2 Background in utility theory We now briefly describe certain essential concepts and theorems in utility theory. For further discussion of these ideas and for the proofs of the assertions given below we refer the reader to Bridges and Mehta (1995) and Mehta (1999). Let X be a nonempty set. A preference relation defined on X is a binary, reflexive, transitive and total relation on X . A preference relation, therefore, is a total preorder. Each preference relation on a set X gives rise to two associated relations of strict preference and indifference, denoted respectively by ≺ and ∼, as follows: if x , y ∈ X then x ≺ y ⇐⇒ (x y) ∧ ¬(y x ) and x ∼ y ⇐⇒ (x y) ∧ (y x ). An anti-symmetric total preorder is said to be an order. Let be a preference relation on a set X . A real-valued function f defined on (X , ) is said to be a utility function if x y ⇐⇒ f (x ) ≤ f (y). If f is a utility function on (X , ) then we also say that f represents the preference relation . The problem of finding conditions which imply the existence of a utility function that represents a given preference relation is called a utility representation problem. Does every preference relation on a set X have a utility representation? Until recently, most economists would probably have either fudged or answered this question in the affirmative. For example, Hicks (1956, p. 19) states that “if a set of items is strongly ordered, it is such that each item has a place of its own in the order; it could, in principle, be given a number”. In his classical 1954 paper, Debreu gives a simple and decisive counter-example to show that there does exist a preference relation which is not representable by a utility function. He proves that the lexicographic preference relation L on R2 defined by (a, b) L (c, d ) ⇐⇒ a < c or (a = c) ∧ (b < d ) does not have a utility representation!
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A preference relation on X is said to be order-separable in the sense of Debreu if there is a countable subset Z of X such that for every x , y ∈ X with x ≺ y there exists z ∈ Z with x z y. It turns out that this property characterizes the representability of a preference relation as we see in the following theorem. Theorem 1 Let (X , ) be a set equipped with a preference relation. Then is representable if and only if it is order-separable in the sense of Debreu. Remark 1 It is not hard to verify that the lexicographic preference relation given above is not order-separable in the sense of Debreu. Let (X , τ ) be a topological space equipped with a preference relation . In general, there is no relationship between the topology and the preference relation. The topology on a set X which is generated by the sub-basis consisting of the family of all sets of the form {x ∈ X : x ≺ y} and {x ∈ X : y ≺ x } for y ∈ X is called the order topology associated with the preference relation. We say that τ is a natural topology on X if it is finer than the order topology. If τ is a natural topology on X then we also say that the preference relation is τ -continuous on X . A preference relation on a topological space (X , τ ) is said to be continuously representable if there exists a real-valued continuous utility function on (X , τ, ). The problem of the existence of a continuous real-valued utility representation of a preference relation on a topological space was definitively solved by Debreu (1954, 1964) in two fundamental papers. Debreu’s method is based on the concept of a gap. Definition. Let R denote the extended real line. A degenerate set in R is a set with at most one point. A lacuna of a subset S of R is a non-degenerate interval that is disjoint from S and has a lower bound and upper bound in S . A gap is a maximal lacuna. Lemma 1. (Debreu’s Open Gap Lemma). If S is a subset of R, there is a strictly increasing function g : S −→ R such that all the gaps of g(S ) are open. Debreu’s Open Gap Lemma is a fundamental result in the theory of orderpreserving functions. There are many known proofs of the fundamental Debreu Open Gap Lemma. The original proof is due to Debreu (1954, 1964). For a discussion of some of the other proofs the reader is referred to Bridges and Mehta (1995) and Mehta (1999). The essence of Debreu’s method of proving the existence of continuous utility functions may now be described. The idea is to first construct a utility function v on the topological space (X , τ, ). The function v need not be continuous. Then the Debreu “gap function” is applied to the image v(X ) of X to get a composite function u(x ) = g(v(x )). The function u has only open gaps in its image. It is
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then quite easily proved that u is the required continuous utility representation of . We now state two fundamental theorems on the existence of continuous utility functions. Theorem 2. (Eilenberg’s Theorem) Let be a preference relation on a topological space (X , τ ) that is connected and topologically separable. If is τ continuous then there exists a real-valued continuous utility function on (X , τ, ). Theorem 3. (Debreu’s Theorem) Let be a preference relation on a topological space (X , τ ) that is second countable. If is τ -continuous then there exists a real-valued continuous utility function on (X , τ, ). 3 Utility and entropy We now want to compare the mathematical structure of the utility representation problem and that of the entropy representation problem. To that end, let us consider a piquant and insightful paper of Cooper (1967). In this paper Cooper studies the foundations of thermodynamics by developing an appropriate mathematical order-theoretic structure. This is done by introducing an accessibility relation on a state space S of states of a thermodynamic system which satisfies certain topological assumptions. This accessibility relation is assumed to be a total preorder on the state space S and the interpretation here is that if x y then a transition from state x to y is possible. If the states are mutually accessible then the states are thermodynamically equivalent. Also, if y is accessible from x then x need not be accessible from y. Indeed, Caratheodory’s version of the second law of thermodynamics states that in any neighbourhood of any state s of any isolated thermodynamical system there exist states which cannot be reached from s by any possible physical process. Given an accessiblity relation on a state space S , the entropy representation problem is to find general conditions on the state space S and the accessibility relation which imply that there is a continuous order-preserving function f on S with the property that x y ⇐⇒ f (x ) ≤ f (y). We see from these considerations that we have adduced that the mathematical structure of the entropy representation problem is exactly the same as that of the utility representation problem. Cooper observes that there have been previous attempts to prove the existence of an entropy function based solely on Caratheodory’s version of the second law of thermodynamics. Cooper shows that this method cannot succeed by defining an accessibility relation by means of the lexicographic relation on the Cartesian plane R2 . Notice that to prove that the Caratheodory condition is satisfied for this example one only needs to observe that for each point x of R2 and each neighbourhood U of x there is a point y ∈ U such that ¬(y L x ), i.e. y is not accessible from x . Clearly, there is a point y ∈ U with this property. Cooper (1967, p.178) then argues that for such a state space there cannot be an entropy function for this accessibility relation because the existence of such
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an order-preserving function entails the existence of uncountably many pairwise disjoint non-empty open order intervals of the real line R which is impossible. Of course, it is precisely by this reasoning that Debreu (1954) proves his famous result about the non-representability of the lexicographic preference relation by a utility function! Let us now consider Cooper’s main theorem and method of proof. Cooper’s Theorem 1 (1967, p. 178) states that if a thermodynamic system is a separable topological space (S , τ, ) and there is defined on S an accessibility relation which is a τ -continuous total preorder then there is a real-valued continuous entropy function for this system. Cooper’s proof is quite long and complicated. It is similar to the method of proof used by Debreu (1954) and consists in first constructing a strictly isotone function on S , which may not be continuous, and then modifying it by removing the untoward ”gaps” to get a continuous entropy function on S .1 It is interesting and instructive to compare Cooper’s theorem with the fundamental Eilenberg and Debreu theorems discussed above. To do this, we notice, first, that Cooper says (1967, p. 181) in connection with the lexicographic accessiblity relation that “this counter-example involves a space which is not linear or even arcwise conected, unlike the normal state spaces of physical systems”. We infer that the normal state spaces are arcwise connected and, therefore, connected. It follows, therefore, that for normal state spaces, Cooper’s theorem is the same as the Eilenberg-Debreu theorem. Cooper also says (1967, p. 177) that the “normal state spaces, in addition, have topologies based on a metric”. Now a metrizable separable topology is second countable. Hence, for the normal state spaces for which the topology is induced by a metric we see that Cooper’s theorem is the same as the Debreu theorem. Finally, we advert to the fact that Cooper also studies additive entropy. Cooper’s results in this area contain many ideas which also appear in the modern literature on the existence of continuous utility functions that are also additive. The reader is referred to Candeal, De Miguel, Indur´ain and Mehta (1999b) for further discussion. 4 Conclusion In this note we have argued sedulously that there is a very close connection between the utility representation problem and the entropy representation problem. Indeed, the underlying mathematical structure is the same. We feel that Cooper, who was not aware of this close relationship, deserves much credit for anticipating or independently discovering ideas and methods that have been extensively used in the literature on utility theory. It is hoped that economists, physicists and even mathematicians will benefit from such a cross-fertilization of ideas. 1 Cooper’s proof is vitiated by some errors. It is not true that any continuous total preorder on a separable topological space has a continuous utility representation as may be seen by considering the set [0, 1] × {0, 1} with the lexicographic order and the order topology. For further discussion see Candeal, De Miguel, Indur´ain and Mehta (1999a).
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References Bridges, D. S., Mehta, G. B.: Representations of preference orderings. Berlin Heidelberg New York: Springer 1995 Buchdahl, H. A.: The concepts of thermodynamics. Cambridge: Cambridge University Press 1966 Candeal, J. C., De Miguel, J. R., Indur´ain, E., Mehta, G. B.: On a Theorem of Cooper. Preprint (1999a) Candeal, J. C., De Miguel, J. R., Indur´ain, E., Mehta G. B.: Representations of ordered semigroups and the physical concept of entropy. Preprint (1999b) Cooper, J. L. B.: The foundations of thermodynamics. Journal of Mathematical Analysis and Applications 17, 172–193 (1967) Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall R., Coombs, C., Davis R. (eds.) Decision processes, pp. 159–166. New York: Wiley 1954 Debreu, G.: Continuity properties of Paretian utility. International Economic Review 5, 285–293 (1964) Giles, R.: Mathematical foundations of thermodynamics. Oxford: Pergamon 1964 Hicks, J.: A revision of demand theory. Oxford: Clarendon Press 1956 Lieb, E., Yngvason, J.: A guide to entropy and the second law of thermodynamics. Notices of the American Mathematical Society 45, 571–581 (1998) Mehta, G. B.: Preference and utility. In: Barbera S., Hammond, P., Seidl, C. (eds.) Handbook of utility theory, pp. 1–47. Boston: Kluwer 1999 Owen, D.: A first course on the mathematical foundations of thermodynamics. Berlin Heidelberg New York: Springer 1984
