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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 2, No. 1, 2001pp. 27 - 31

Continuous representability of completepreorders on the space of upper-continuouscapacitiesGianni Bosi and Romano IslerAbstract. Given a compact metric space (X;d), and itsBorel �-algebra �, we discuss the existence of a (semi)continuousutility function U for a complete preorder � on a subset M0(X) ofthe space M(X) of all upper-continuous capacities on �, endowedwith the weak topology.2000 AMS Classi�cation: 06A05, 91B16.Keywords: Upper-continuous capacity, compact metric space, weak topology,continuous utility function. 1. IntroductionIn decision theory under uncertainty, it is usual to consider a separable met-ric space (X;d) of possible consequences of a game, endowed with its Borel�-algebra � (i.e., the �-algebra generated by the open subsets of X). A playeris required to choose a probability measure from a set P of �-additive probabil-ity measures on the measurable space (X;�), endowed with the induced weaktopology. The preferences of the player among probability measures in P are ex-pressed by a complete preorder (i.e., a re
exive, transitive and complete binaryrelation) � on P . This is the usual model for expected utility (see Grandmont[5]). In a more general setting, it may be assumed that player's uncertainty isre
ected by capacities better than probability measures (see e.g. Epstein andWang [4]). In particular, the notion of an upper-continuous capacity general-izes the notion of a �-additive probability measure in the case of additive setfunctions. Topological properties of the space M(X) of all upper-continuouscapacities on the measurable space (X;�), endowed with the weak topology,have been studied by Zhou [7] in case that (X;d) is a compact (metric) space.In this paper, we use the results proved by Zhou [7] in order to discuss theexistence of a continuous or at least upper semicontinuous utility function U



28 G. Bosi and R. Islerfor a complete preorder � on a subset M 0(X) of the space M(X) of all upper-continuous capacities.2. Notation and preliminariesThroughout this paper, we shall always consider a compact metric space(X;d), endowed with its Borel �-algebra, denoted by �. The space of all upper-continuous capacities on � will be denoted by M(X) (see Zhou [7]). We recallthat a capacity � on � (i.e., a function from � into [0;1] such that �(?) = 0,�(X) = 1, and �(A) � �(B) for all A � B, A;B 2 �) is said to be upper-continuous if limn!1�(An) = �(T1n=1An) for any weakly decreasing sequence ofsets fAng with An 2 � for all n. A sequence f�ng � M(X) is said to convergeweakly to � 2 M(X) ifZX fd�n ! ZX fd� for all f 2 C(X);with C(X) the space of all continuous real-valued functions on (X;d), andRX fd� the Choquet integral of f with respect to �, namelyZX fd� = Z 10 �(f � t)dt + Z 0�1(�(f � t) � 1)dt:The corresponding topology (i.e., the weak topology on M(X)) will be denotedby �w. The reader could refer to the comprehensive book by Denneberg [2] fordetails concerning the basic properties of the Choquet integral. More recentresults on the Choquet integral with respect to upper-continuous capacities arefound in Zhou [7].Given a complete preorder (i.e., a re
exive, transitive and complete binaryrelation) � on a subset M 0(X) of M(X), we are interested in the existence ofa utility function U for � (i.e., a real-valued function on M 0(X) such that, forevery �;� 2 M 0(X), � � � if and only if U(�) � U(�)) which is continuousor at least upper semicontinuous in the topology induced on M 0(X) by theweak topology �w. We recall that a complete preorder � on a subset M 0(X)of M(X) is said to be upper (lower) semicontinuous if f� 2 M 0(X) : � � �g(f� 2 M 0(X) : � � �g) is a closed set for every � 2 M 0(X). Further, a completepreorder � is said to be continuous if it is both upper and lower semicontinuous.3. Continuous representationsIn the following theorem, we are concerned with the existence of a continuousor at least upper semicontinuous utility function for a complete preorder on anarbitrary set M 0(X) of upper-continuous capacities.Theorem 3.1. Let (X;d) be a compact metric space. Then the following state-ments hold:(i) Every upper semicontinuous complete preorder � on every subset M 0(X)of M(X) admits an upper semicontinuous utility function U;(ii) Every continuous complete preorder � on every subset M 0(X) of M(X)admits a continuous utility function U.



Upper-continuous capacities and utility 29Proof. From Zhou [7, Theorem 3], the space (M(X);�w) is a compact metricspace, and therefore it is in particular a separable metric space (see e.g. Engelk-ing [3, Theorem 4.1.18]). Then every subset M 0(X) of M(X) can be metrized asa separable metric space, and therefore as a second countable metric space (seee.g. Engelking [3, Corollary 4.1.16]). If � is any upper semicontinuous completepreorder on (M 0(X);�wM0(X)), then � admits an upper semicontinuous utilityfunction U by Rader's theorem (see Rader [6, Theorem 1]). If � is any continu-ous complete preorder on (M 0(X);�wM0(X)), then � admits a continuous utilityfunction U by Debreu's theorem (see Debreu [1, Proposition 3]). So the proofis complete. �For an additive capacity � on �, the condition of upper-continuity is equiv-alent to the condition of countable additivity. Therefore, since the space �(X)of all countably additive probability measures on (X;�) is contained in M(X),Theorem 3.1 generalizes Theorem 1 in Grandmont [5] in case that a compactmetric space is considered.Given a compact metric space (X;d), and a complete preorder � on a sub-set M 0(X) of M(X), containing the set D of all probability measures on themeasurable space (X;�) which are concentrated (i.e., D = fp 2 �(X) : p =px for some x 2 Xg with px the probability measure assigning probability 1 tothe Borel set fxg), we can consider the complete preorder �X on X whichis induced by the complete preorder � on M 0(X), in the sense that, for everyx;y 2 X, x �X y if and only if px � py. The following corollary to the previoustheorem concerns the representability of �X by means of a continuous or atleast upper semicontinuous utility function u on (X;d).Corollary 3.2. Let (X;d) be a compact metric space. Then the followingstatements hold:(i) For every upper semicontinuous complete preorder � on every subsetM 0(X) of M(X) containing D, the induced complete preorder �X ad-mits an upper semicontinuous utility function u;(ii) For every continuous complete preorder � on every subset M 0(X) ofM(X) containing D, the induced complete preorder �X admits a con-tinuous utility function u.Proof. Given an upper semicontinuous complete preorder � on any subsetM 0(X) of M(X) containing D, by the previous theorem there exists an up-per semicontinuous utility function U for �. We claim that the real-valuedfunction u on X de�ned by u(x) = U(px) (x 2 X)is an upper semicontinuous utility function for the induced complete preorder�X on X. It is straightforward to show that u is a utility function for �X.Indeed, we havex �X y , px � py , U(px) � U(py) , u(x) � u(y)



30 G. Bosi and R. Islerfor every x;y 2 X. In order to prove that u is upper semicontinuous (i.e., theset fx 2 X : � � u(x)g is closed for every real number �), consider any realnumber �, any point x 2 X, and any sequence fxng � X converging to x suchthat � � u(xn) for every n. Since the sequence fpxng � D converges to px 2 D,U is upper semicontinuous, and from the de�nition of U we have � � U(pxn)for every n, it must be � � U(px) = u(x), and therefore the conclusion follows.This consideration �nishes the �rst part of the proof.If � is a complete preorder on any subset M 0(X) of M(X) containing D, bythe previous theorem there exists a continuous utility function U for �. Then,by analogous considerations it can be shown that the function u de�ned aboveis a continuous utility function for �X. So the proof is complete. �It is almost immediate to check that the statements named (i) in the previoustheorem and corollary are still valid if we replace the terms \upper semicon-tinuous complete preorder" and \upper semicontinuous utility function" by theterms \lower semicontinuous complete preorder" and respectively \lower semi-continuous utility function". Indeed, one can replace functions U and u by �Uand respectively �u, and then apply the previous results by considering the dualcomplete preorders �d and �Xd de�ned by [� �d � , � � �] and respectively[x �Xd y , y �X x].Acknowledgements. We are grateful to an anonymous referee for helpfulsuggestions. References[1] G. Debreu, Continuity properties of paretian utility, International Economic Review 5(1964), 285{293.[2] D. Denneberg, Non-additive measure and integral, Kluwer, Dordrecht, 1994.[3] R. Engelking, General Topology, Polish Scienti�c Publishers, 1977.[4] L.G. Epstein and T. Wang, \Beliefs about beliefs" without probabilities, Econometrica64 (1996), 1343{1373.[5] J.-M. Grandmont, Continuity properties of a von Neumann-Morgenstern utility, Journalof Economic Theory 4 (1972), 45{57.[6] T. Rader, The existence of a utility function to represent preferences, Review of EconomicStudies 30 (1963), 229{232.[7] L. Zhou, Integral representation of continuous comonotonically additive functionals,Transactions of the American Mathematical Society 350 (1998), 1811{1822.Received October 2000Revised version February 2001
Gianni Bosi, Romano IslerDipartimento di Matematica Applicata\Bruno de Finetti", Universit�a di Trieste



Upper-continuous capacities and utility 31Piazzale Europa 134127 TriesteItalyE-mail address: giannibo@econ.univ.trieste.itE-mail address: romano.isler@econ.univ.trieste.it