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Subjective probability theory with continuous acts
Journal of Mathematical Economics 32 Ž1999. 121–130
Subjective probability theory with continuous acts Lin Zhou Department of Economics, Duke UniÕersity, Durham, NC 27708-0097, USA Received 16 September 1996; accepted 6 April 1997
1. Introduction It is now generally agreed that Savage’s theory of subjective probability has laid a solid foundation of individual decision theory under uncertainty ŽSavage, 1954.. Since Savage’s original contribution, many extensions have been developed using alternative formulations of his model to make the theory of subjective probability more robust. However, the majority of the existing contributions have kept one of Savage’s assumptions intact that an individual has a complete preference relation over all acts that are measurable functions from the state space to the prize space. 1 This is obviously a very strong assumption from both normative and empirical viewpoints. It is desirable that some basic results in subjective probability theory can be established under different assumptions. Let me motivate my analysis through a simple example. Suppose that the state space is represented by the unit interval w0,1x with the Borel algebra. Any indicator function of a Cantor set in w0,1x is a simple, hence measurable, act. The standard theory thus requires that the individual under investigation is able to rank acts of such in a consistent fashion. But it is doubtful that an individual not well trained in mathematics, nevertheless highly rational, is able to comprehend such acts. It is natural to question to what extent does the validity of the standard theory depend upon the assumption that the individual has well-formed preferences over all acts. 1
Chateauneuf and Wakker Ž1993. and Wakker Ž1993. are among the few exceptions.
0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 2 2 - 0
122
L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
For instance, what if the individual’s preferences are defined over continuous acts only? This is the main problem I will address here. I choose to focus on continuous acts for several reasons. First, it is a tradition that many economic models, as well as decision theory models, adopt various topological structures: individuals’ preferences are continuous with respect to alternatives, portfolio returns are continuous with respect to time, etc. Second, most topological concepts such as a continuous curve have vivid geometric images that many people can relate to, while concepts such as a measurable set are merely abstract objects in certain formal algebraic systems and are difficult to visualize. Third, since continuous acts are measurable, the assumption that only continuous acts are feasible is weaker than the standard one. Thus, it is a sensible alternative to begin with even though it is still quite strong in itself from the empirical viewpoint. If the subjective probability theory is robust as the spaces of feasible acts vary, it should at least have some reasonable extensions in this case. In this sense, this paper serves as a starting point for more future research. The main result of this paper is parallel to a result by Schmeidler Ž1989.. We assume that the state space S is a topological space and that a reference relation K is defined over continuous acts only. We show that K has a ‘nice’ Choquet integral representation ŽChoquet, 1955. if and only if it satisfies all standard axioms as in Schmeidler Ž1989. and an additional continuity axiom. Furthermore, if comonotonic independence is replaced by full independence Žas in the earlier work by Anscombe and Aumann, 1963., then K has a regular expected utility representation as long as K is defined on a set of acts that includes all constant acts and is closed under the mixture operation. There are several papers in recent years that have tried to refine the standard subjective probability theory from some related angles. Wakker Ž1993., and Chateauneuf and Wakker Ž1993. consider representing a preference relation K defined on some restricted act spaces. However, only finite state spaces are considered in their works and restrictions on act spaces are introduced by other means. 2 Bewley Ž1986. considers representing a preference relation K defined over all acts, yet being incomplete. Though his study and mine can be viewed as close complements, the respective conclusions are quite different. A preference relation satisfying his weaker rationality conditions usually does not have a simple integral representation. 3
2
They require that the restricted act space contains some open subset of the universal act space. When the state space is infinite, the set of all continuous acts is not open in the space of all measurable acts endowed with the supremum norm. 3 In Bewley’s work, a preference relation is rationalized by comparing two acts through a set of subjective probability vectors. Therefore, no integral representation even exist for most subspaces of acts.
L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
123
2. The main results Assume that the state space Ž S,t . is a topological space and that the prize space L is the space of all lotteries over finite outcomes, hence a simplex endowed with the Euclidean topology. 4 A function f from S to L is called an act. In this section, only continuous acts are allowed. Suppose that F is the space of all continuous acts and K a given preference relation on F. We first briefly review some basic mathematical notion concerning Choquet integrals. Let C b Ž S . be the space of all bounded and continuous real-valued functions, and S the collection of all upper-contour sets of functions in C b Ž S .. 5 A real-valued function m on S is called a capacity if it satisfies:
m Ž B . s 0, m Ž S . s 1; m Ž A . G m Ž B . for all A, B g S with A : B. A capacity m on S is upper-continuous if it also satisfies:
Ž 1. Ž 2. 6
`
lim m Ž A n . s m
n™`
ž / F An
for any non-increasing sequence of A n 4
ns1
`
with
F An g S .
Ž 3.
ns1
Given a capacity m on S and a bounded S-measurable function a on S, the Choquet integral of a with respect to m is defined as: `
HS a d m sH0
mŽ aGt . dtq
0
Hy` Ž m Ž a G t . y 1. d t ,
Ž 4.
where the integral on the right hand side is in the sense of Riemann. While some basic properties of Choquet integrals are well-known, 7 more recent results on Choquet integrals with respect to upper-continuous capacities can be found in Zhou Ž1998.. We say that a preference relation K on F has a Choquet integral representation if there exist a utility function u on L and a capacity m on S such that, fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5.
Let us now consider axioms on K that allow a Choquet integral representation of K . 4 It should be mentioned that the assumption that the prize space L is the space of all lotteries over finite outcomes is made only for the ease of exposition. Our analysis actually applies to any metric space L Žsee the comment at the end of this section.. 5 Notice that S is generally not an algebra. 6 The requirement that a capacity be upper-continuous is parallel to the assumption that an additive measure be countably additive ŽZhou, 1998a.. 7 The textbook by Denneberg Ž1994. contains a quite comprehensive treatment of this topic.
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L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
Axiom 1. ŽWeak Order . The preference relation K is complete and transitiÕe. Since L is a lottery space, it induces a mixture operation on F. For all f, g g F and a g w0,1x, the mixture act a f q Ž1 y a . g is defined by: Ž a f q Ž1 y a . g .Ž s . s a f Ž s . q Ž1 y a . g Ž s . for all s g S. In the classical Von Neumann–Morgenstern model, K is assumed to satisfy the independence condition. Following Schmeidler Ž1989., however, we shall consider a weaker axiom. Because all constant acts are continuous, we can derive from K a preference relation on L, also denoted by K , by x K y m c x K c y , where c x and c y are respectively the constant acts that always assume x and y. We call two acts f and g comonotonic if there are no pair of states s and sX such that f Ž s . % f Ž sX . and g Ž sX . % g Ž s .. It is obvious that a constant act and any other act are comonotonic. Also, if f and g are comonotonic, then all mixtures of f and g are comonotonic. Axiom 2. Ž Comonotonic Independence. For all pairwise comonotonic acts f, g, h g F, and all a g Ž0,1., f % g m a f q Ž1 y a . h % a g q Ž1 y a . h. For an elaboration of Axiom 2, readers are referred to Schmeidler’s original paper or many subsequent articles by others. The significance of this axiom is that it allows us to deal with an individual’s attitude towards uncertainties. Axiom 3. Ž Continuity . For all f, g,h g F, f % h % g imply that there are a , b g Ž0,1. such that a f q Ž1 y a . g % h % b f q Ž1 y b . g. Axiom 4. Ž Nondegeneracy . There exist x U , x U g L such that x U % x U . Axiom 5. Ž Monotonicity . For all f, g g F, if f Ž s . K g Ž s . for all s, then f K g. Given Axiom 5, we can assume that x U and x U in Axiom 4 are such that x K f K x U for all f g F. In the case where all measurable acts are feasible, Schmeidler Ž1989. shows that Axioms 1–5 imply that K has a Choquet integral representation. The capacity in the representation can be easily constructed since all indicator functions are in F. However, if continuous acts only are feasible, such a construction is no longer viable and Axioms 1–5 are generally not sufficient for a ‘proper’ Choquet integral representation. Some additional axiom is needed. U
Axiom 6. Ž Strong Continuity . Suppose that x is any lottery in L and f n 4 is a sequence of decreasing acts in the sense that f nŽ s . K f nq1Ž s . for all n at eÕery s g S. If for eÕery s g S, there exists an nŽ s . such that f nŽ s.Ž s . % x, then for any y % x, there exists an n such that f n % c y .
L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
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Axiom 6 is similar to some axioms in the existing literature that are used to ensure that the additive subjective probability measures derived are countably additive Žfor example, Fishburn, 1970., though they are generally not comparable. We can now present the main result in this section. Theorem 1. A preference relation K on F satisfies Axioms 1–6 if and only if there exist a Von Neumann–Morgenstern utility function u on L, unique up to positiÕe linear transformations, and a unique upper-continuous capacity m on S such that, fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5a .
Proof. The necessity of Axioms 1–5 for an integral representation of K is standard. The necessity of Axiom 6 follows from a result in Zhou Ž1998, Theorem 1.. Let us prove that Axioms 1–6 are also sufficient. Consider all sub-mixture spaces of F that satisfy two conditions: first, it contains all constant acts; second, K obeys full independence on it. Let E denote the collection of all such spaces. For each E g E , since K satisfies Axiom 1–4 and independence, we can apply the Von Neumann–Morgenstern theorem ŽJensen, 1967. to obtain a v-N-M utility function UE on E, which is further fixed by requiring UE Ž x U . s 1, UE Ž x U . s y1. For each f g F, there exists at least one E g E with f g E Žfor example, the mixture space generated by f and all constant acts.. Hence, we can define a functional U on F by: U Ž f . s UE Ž f .
for some E g E with f g E.
Ž 6.
Of course, we need to show that U is defined independently of whatever E is used in Ž6.. Since every such E contains x U and x U , and UE is a v-N-M utility function for K on E with UE Ž x U . s 1 and UE Ž x U . s y1, it is direct that UE Ž f . s 2 a y 1 where a is the unique number such that f ; a x U q Ž1 y a . x U . The same argument also shows that U actually rationalizes K , i.e., for all f, g g F, f K g mUŽ f . G UŽ g . .
Ž 7.
We now show that U has a Choquet integral representation with a v-N-M utility function u on L and an upper-continuous capacity m on S. Obviously, u should be defined by: uŽ x . s U Ž c x .
for all x g L.
Ž 8.
Next we try to find m. Each f g F can be associated with a real-valued function a f on S by: a f s u P f , i.e., a f Ž s . s u Ž f Ž s . .
for all s g S.
Ž 9.
L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
126
The function a f is continuous since both u and f are. Also, sup < a f < F 1. Let us consider the following collection of continuous functions on S: C s a a f < for all a G 1 and f g F 4 . We claim that C s C b Ž S .. Take any a g C b Ž S .. When sup < a < F 1 we let f g F be: 1 q aŽ s . ) 1 y aŽ s . f Ž s. s x q x ) for all s g S, 2 2 then a s a f . When sup < a < ) 1, we can find g g F such that a s sup < a < a g . We now construct a functional I on C b Ž S .. Given any a g C b Ž S ., we can find some l G 1 and f g F such that a s l a f . Then, let I Ž a . s lU Ž f . . Ž 10 . Of course, we need to show that I is well-defined. Suppose that a s l a f s u a g for some l G u G 1 and f, g g F. Obviously, f and g are comonotonic. Let E Žg E . be the mixture space generated by f, g and all constant acts, and h s ly1u g q Ž1 y ly1u .Ž 12 x U q 12 x U .. By Ž6., we have UŽ h. s ly1u UŽ g .. On the other hand, since K satisfies Axiom 5, w uŽ f Ž s .. s a f Ž s . s ly1u a g Ž s . s ly1u uŽ g Ž s .. s uŽ hŽ s .. for all s g S x implies UŽ f . s UŽ h.. Therefore, lUŽ f . s u UŽ g .. Similar arguments can be used to verify: Ži.
I is a comonotonically additive functional on Cb Ž S ., i.e., I Ž a q b . s I Ž a. q I Ž b . for all a and aX g C b Ž S . such that Ž aŽ s . y aŽ sX ..Ž aX Ž s . y aX Ž sX .. G 0 for all s and sX g S; and I is non-negative, i.e., I Ž a. G 0 for any a g C b Ž S . with aŽ s . G 0 for all s g S. Next, we use Axiom 6 to show that I also satisfies: I Ž a n .x0 for any sequence a n4 in Cb Ž S . with a nŽ s .x0 for all s g S. 8
Žii.
Žiii.
Since a nŽ s .x0 for all s g S, sup < a n < F sup < a1 < - ` for all n. We may assume, without loss of generality, that sup < a n < F 1. Hence, there exist a sequence of f n 4 in F such that a n s a f n for all n. It is then direct to verify that Axiom 6 implies Žiii.. Because C b Ž S . is a vector lattice with constant functions, 9 and I a functional on C b Ž S . satisfying Ži., Žii., and Žiii., the result in Zhou Ž1998, Theorem 1. implies that there is a unique upper-continuous capacity m on Ž S, S . such that, I Ž a . s a d m for all a g C b Ž S . .
HS
8
Ž 11 .
The notation ‘I Ž a n .x0’ means I Ž a n . approaches zero monotonically. A collection of real functions over a set is a vector lattice if it is both a vector space for the standard addition and scalar multiplication and a lattice for the join and meet operations defined by max and min. 9
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127
Combining Ž7. to Ž11., we obtain a Choquet integral representation of K . The uniqueness of m and u follow from the corresponding conclusions in the Von Neumann–Morgenstern theorem and Zhou’s result quoted above. I Axiom 6 is used in the proof only to show that the functional I satisfies Žiii.. When S is compact, however, Žiii. is satisfied without assuming Axiom 6. This follows from the Dini theorem that every decreasing sequence of continuous functions that converges to zero pointwise converges uniformly on any compact set. Thus, we have: Corollary A. Assume that S is a compact space. A preference relation K on F satisfies Axioms 1–5 if and only if there exist a Õ-N-M utility function u on L, unique up to positiÕe linear transformations, and a unique upper-continuous capacity m on S such that fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5b .
The next corollary concerns with representing a preference relation K by expectation of a v-N-M with respect to an additive measure. A necessary condition is that K satisfies full independence. Axiom 2 U . Ž Independence. For all acts f, g,h g F, and for all a g Ž0,1., f % g m af q Ž1 y a . h % a g q Ž1 y a . h. Corollary B parallels the classical Anscombe–Aumann theorem Ž1963.. Corollary B. Assume that S is a topological space and S is the Baire algebra of S. 10 A preference relation K on F satisfies Axiom 1, 2 U , and 3–6 if and only if there exist a Õ-N-M utility function u on L, unique up to positiÕe linear transformations, and a unique countably additiÕe measure m on S such that, fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5c .
The proof is virtually the same as the proof of Theorem 1 except that the Stone–Daniell representation theorem is used in the last step. Before closing, let me comment on the assumption that the prize space L is the lottery space over some finite outcomes. This assumption ensures that the derived v-N-M utility function u on L is bounded and continuous. Both properties are needed for the validity of our analysis. When L is a general metric space, we can 10
The Baire algebra of a topological space is the smallest s-algebra generated by all functions in C b Ž S ..
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L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
impose some continuity property on K that will guarantee the boundedness and continuity of u. Since the existing literature contains many such results Žsee Fishburn, 1970 and Kreps, 1988., I shall not repeat any here.
3. Additional remarks In this paper I have proved a result parallel to that by Schmeidler Ž1989. on subjective probabilities in a topological instead of measure theoretical setting. Although it is far still from a fully satisfactory solution to the issue raised at the beginning of the paper, it shows that some important results in the existing literature are robust with alternative assumptions on feasible act spaces. More research should be conducted to see whether similar results can be obtained when feasible act spaces contain even fewer acts. It should be mentioned that a general extension of the Anscombe–Aumann theorem holds when full independence is assumed. Let us assume for simplicity that the state space S is a metric space. 11 Theorem 2. Suppose that the feasible act space F is a mixture space 12 that includes all constant acts. A preference relation K on F has an expected utility representation if and only if K satisfies Axiom 1, 2 U , and 3–5. Theorem 2 can be deduced from a more general Harsanyi-type result on utilitarianism ŽZhou, 1997.. 13 The requirement on F here is minimal for an Anscombe–Aumann type model. It is not clear whether a similar result is true when Axiom 2 U is replaced by Axiom 2. Although there are no conceptual difficulties with such a result, we will need in our proof a Hahn–Banach type theorem on extension of comonotonically additive functionals, which is still an open problem. More research should also be conducted to investigate if similar results can be obtained in some Savage type model that does not use objective probability devices. 14 I wish I could claim that I consider an Anscombe–Aumann type model here solely for the sake of simplicity, but I honestly cannot. In an Anscombe–Aumann model, we can first apply the v-N-M theorem to derive a functional on some function space, and then derive a subjective probability measure by a duality argument. This procedure, however, runs into serious difficulties in Savage type 11
This assumption is made so that a version of Riesz’s theorem can be applied though other weaker assumptions can also serve the same purpose. With this assumption, Axiom 6 also becomes redundant. 12 The definition of a mixture space can be found in textbooks such as Fishburn Ž1970. or Kreps Ž1988.. 13 It also holds in a measure theoretical setting. 14 Vind Ž1990. has also mentioned this problem without providing a solution.
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models, such as Gilboa Ž1987., and Sarin and Wakker Ž1992.. In fact, such a result cannot be obtained in general without restrictions on the state space or prize space. Consider an example in which the state space S is the unit interval w0,1x and the prize space P is the union of intervals w1,2x and w3,4x and only continuous acts from S to P are feasible. Since S is connected, the range of any feasible act is either entirely in w1,2x or entirely in w3,4x. Now look at an individual with the following preferences. There is an increasing utility function u on P so that all acts with ranges in w3,4x dominates acts with ranges in w1,2x. Between acts f with ranges in w3,4x, preferences are determined by the expected values of u P f with respect to a probability measure m 1 over S; between acts f with ranges in w1,2x, a different m 2 . As long as m 1 and m 2 are different, one cannot find a unified expected utility representation for this individual’s preferences even though the preferences satisfy all Savage-type axioms. This example suggests that one need to assume that P is connected, yet it is far from clear if this assumption will be sufficient.
Acknowledgements I would like to thank S.H. Chew, S. Grant, B. Polak, K. Vind, P. Wakker, as well as two anonymous referees of the journal, for many helpful comments. I am particularly grateful to D. Schmeidler for his numerous correspondences. Of course, all errors that remain are mine.
References Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability. Annals of Mathematical Statistics 34, 199–205. Bewley, T., 1986. Knightian decision theory: Part I. Cowles discussion paper. Yale University, New Haven. Chateauneuf, Wakker, P., 1993. From local to global additive representation. Journal of Mathematical Economics 22, 523–545. Choquet, G., 1955. Theory of capacities. Annals Inst. Fourier 5, 131–295. Denneberg, D., 1994. Non-Additive Measure and Integral. Kluwer, Dordrecht. Fishburn, P.C., 1970. Utility Theory for Decision Making. Wiley, New York. Gilboa, I., 1987. Expected utility with purely subjective non-additive probabilities. Journal of Mathematical Economics 16, 65–88. Jensen, N.E., 1967. An introduction to Bernoullian utility theory. I. Utility functions. Swedish Journal of Economics 69, 163–183. Kreps, D.M., 1988. Note on the Theory of Choice. Westview, Boulder. Sarin, R., Wakker, P., 1992. A simple axiomatization of nonadditive expected utility. Econometrica 60, 1255–1272. Savage, L.J., 1954. The Foundation of Statistics. Wiley, New York. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57, 571–587.
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Vind, K., 1990. Additive utility functions and other special functions in economic theory. University of Copenhagen. Wakker, P., 1993. Additive representation on rank-ordered sets: II. The topological approach. Journal of Mathematical Economics 22, 1–26. Zhou, L., 1998. Integral representation of continuous comonotonically additive functionals. Transactions of American Mathematical Society 350 Ž5., 1811–1822. Zhou, L., 1997. Harsanyi’s utilitarianism theorems: General societies. Journal of Economic Theory 72, 198–207.
Subjective probability theory with continuous acts Lin Zhou Department of Economics, Duke UniÕersity, Durham, NC 27708-0097, USA Received 16 September 1996; accepted 6 April 1997
1. Introduction It is now generally agreed that Savage’s theory of subjective probability has laid a solid foundation of individual decision theory under uncertainty ŽSavage, 1954.. Since Savage’s original contribution, many extensions have been developed using alternative formulations of his model to make the theory of subjective probability more robust. However, the majority of the existing contributions have kept one of Savage’s assumptions intact that an individual has a complete preference relation over all acts that are measurable functions from the state space to the prize space. 1 This is obviously a very strong assumption from both normative and empirical viewpoints. It is desirable that some basic results in subjective probability theory can be established under different assumptions. Let me motivate my analysis through a simple example. Suppose that the state space is represented by the unit interval w0,1x with the Borel algebra. Any indicator function of a Cantor set in w0,1x is a simple, hence measurable, act. The standard theory thus requires that the individual under investigation is able to rank acts of such in a consistent fashion. But it is doubtful that an individual not well trained in mathematics, nevertheless highly rational, is able to comprehend such acts. It is natural to question to what extent does the validity of the standard theory depend upon the assumption that the individual has well-formed preferences over all acts. 1
Chateauneuf and Wakker Ž1993. and Wakker Ž1993. are among the few exceptions.
0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 2 2 - 0
122
L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
For instance, what if the individual’s preferences are defined over continuous acts only? This is the main problem I will address here. I choose to focus on continuous acts for several reasons. First, it is a tradition that many economic models, as well as decision theory models, adopt various topological structures: individuals’ preferences are continuous with respect to alternatives, portfolio returns are continuous with respect to time, etc. Second, most topological concepts such as a continuous curve have vivid geometric images that many people can relate to, while concepts such as a measurable set are merely abstract objects in certain formal algebraic systems and are difficult to visualize. Third, since continuous acts are measurable, the assumption that only continuous acts are feasible is weaker than the standard one. Thus, it is a sensible alternative to begin with even though it is still quite strong in itself from the empirical viewpoint. If the subjective probability theory is robust as the spaces of feasible acts vary, it should at least have some reasonable extensions in this case. In this sense, this paper serves as a starting point for more future research. The main result of this paper is parallel to a result by Schmeidler Ž1989.. We assume that the state space S is a topological space and that a reference relation K is defined over continuous acts only. We show that K has a ‘nice’ Choquet integral representation ŽChoquet, 1955. if and only if it satisfies all standard axioms as in Schmeidler Ž1989. and an additional continuity axiom. Furthermore, if comonotonic independence is replaced by full independence Žas in the earlier work by Anscombe and Aumann, 1963., then K has a regular expected utility representation as long as K is defined on a set of acts that includes all constant acts and is closed under the mixture operation. There are several papers in recent years that have tried to refine the standard subjective probability theory from some related angles. Wakker Ž1993., and Chateauneuf and Wakker Ž1993. consider representing a preference relation K defined on some restricted act spaces. However, only finite state spaces are considered in their works and restrictions on act spaces are introduced by other means. 2 Bewley Ž1986. considers representing a preference relation K defined over all acts, yet being incomplete. Though his study and mine can be viewed as close complements, the respective conclusions are quite different. A preference relation satisfying his weaker rationality conditions usually does not have a simple integral representation. 3
2
They require that the restricted act space contains some open subset of the universal act space. When the state space is infinite, the set of all continuous acts is not open in the space of all measurable acts endowed with the supremum norm. 3 In Bewley’s work, a preference relation is rationalized by comparing two acts through a set of subjective probability vectors. Therefore, no integral representation even exist for most subspaces of acts.
L. Zhou r Journal of Mathematical Economics 32 (1999) 121–130
123
2. The main results Assume that the state space Ž S,t . is a topological space and that the prize space L is the space of all lotteries over finite outcomes, hence a simplex endowed with the Euclidean topology. 4 A function f from S to L is called an act. In this section, only continuous acts are allowed. Suppose that F is the space of all continuous acts and K a given preference relation on F. We first briefly review some basic mathematical notion concerning Choquet integrals. Let C b Ž S . be the space of all bounded and continuous real-valued functions, and S the collection of all upper-contour sets of functions in C b Ž S .. 5 A real-valued function m on S is called a capacity if it satisfies:
m Ž B . s 0, m Ž S . s 1; m Ž A . G m Ž B . for all A, B g S with A : B. A capacity m on S is upper-continuous if it also satisfies:
Ž 1. Ž 2. 6
`
lim m Ž A n . s m
n™`
ž / F An
for any non-increasing sequence of A n 4
ns1
`
with
F An g S .
Ž 3.
ns1
Given a capacity m on S and a bounded S-measurable function a on S, the Choquet integral of a with respect to m is defined as: `
HS a d m sH0
mŽ aGt . dtq
0
Hy` Ž m Ž a G t . y 1. d t ,
Ž 4.
where the integral on the right hand side is in the sense of Riemann. While some basic properties of Choquet integrals are well-known, 7 more recent results on Choquet integrals with respect to upper-continuous capacities can be found in Zhou Ž1998.. We say that a preference relation K on F has a Choquet integral representation if there exist a utility function u on L and a capacity m on S such that, fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5.
Let us now consider axioms on K that allow a Choquet integral representation of K . 4 It should be mentioned that the assumption that the prize space L is the space of all lotteries over finite outcomes is made only for the ease of exposition. Our analysis actually applies to any metric space L Žsee the comment at the end of this section.. 5 Notice that S is generally not an algebra. 6 The requirement that a capacity be upper-continuous is parallel to the assumption that an additive measure be countably additive ŽZhou, 1998a.. 7 The textbook by Denneberg Ž1994. contains a quite comprehensive treatment of this topic.
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Axiom 1. ŽWeak Order . The preference relation K is complete and transitiÕe. Since L is a lottery space, it induces a mixture operation on F. For all f, g g F and a g w0,1x, the mixture act a f q Ž1 y a . g is defined by: Ž a f q Ž1 y a . g .Ž s . s a f Ž s . q Ž1 y a . g Ž s . for all s g S. In the classical Von Neumann–Morgenstern model, K is assumed to satisfy the independence condition. Following Schmeidler Ž1989., however, we shall consider a weaker axiom. Because all constant acts are continuous, we can derive from K a preference relation on L, also denoted by K , by x K y m c x K c y , where c x and c y are respectively the constant acts that always assume x and y. We call two acts f and g comonotonic if there are no pair of states s and sX such that f Ž s . % f Ž sX . and g Ž sX . % g Ž s .. It is obvious that a constant act and any other act are comonotonic. Also, if f and g are comonotonic, then all mixtures of f and g are comonotonic. Axiom 2. Ž Comonotonic Independence. For all pairwise comonotonic acts f, g, h g F, and all a g Ž0,1., f % g m a f q Ž1 y a . h % a g q Ž1 y a . h. For an elaboration of Axiom 2, readers are referred to Schmeidler’s original paper or many subsequent articles by others. The significance of this axiom is that it allows us to deal with an individual’s attitude towards uncertainties. Axiom 3. Ž Continuity . For all f, g,h g F, f % h % g imply that there are a , b g Ž0,1. such that a f q Ž1 y a . g % h % b f q Ž1 y b . g. Axiom 4. Ž Nondegeneracy . There exist x U , x U g L such that x U % x U . Axiom 5. Ž Monotonicity . For all f, g g F, if f Ž s . K g Ž s . for all s, then f K g. Given Axiom 5, we can assume that x U and x U in Axiom 4 are such that x K f K x U for all f g F. In the case where all measurable acts are feasible, Schmeidler Ž1989. shows that Axioms 1–5 imply that K has a Choquet integral representation. The capacity in the representation can be easily constructed since all indicator functions are in F. However, if continuous acts only are feasible, such a construction is no longer viable and Axioms 1–5 are generally not sufficient for a ‘proper’ Choquet integral representation. Some additional axiom is needed. U
Axiom 6. Ž Strong Continuity . Suppose that x is any lottery in L and f n 4 is a sequence of decreasing acts in the sense that f nŽ s . K f nq1Ž s . for all n at eÕery s g S. If for eÕery s g S, there exists an nŽ s . such that f nŽ s.Ž s . % x, then for any y % x, there exists an n such that f n % c y .
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Axiom 6 is similar to some axioms in the existing literature that are used to ensure that the additive subjective probability measures derived are countably additive Žfor example, Fishburn, 1970., though they are generally not comparable. We can now present the main result in this section. Theorem 1. A preference relation K on F satisfies Axioms 1–6 if and only if there exist a Von Neumann–Morgenstern utility function u on L, unique up to positiÕe linear transformations, and a unique upper-continuous capacity m on S such that, fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5a .
Proof. The necessity of Axioms 1–5 for an integral representation of K is standard. The necessity of Axiom 6 follows from a result in Zhou Ž1998, Theorem 1.. Let us prove that Axioms 1–6 are also sufficient. Consider all sub-mixture spaces of F that satisfy two conditions: first, it contains all constant acts; second, K obeys full independence on it. Let E denote the collection of all such spaces. For each E g E , since K satisfies Axiom 1–4 and independence, we can apply the Von Neumann–Morgenstern theorem ŽJensen, 1967. to obtain a v-N-M utility function UE on E, which is further fixed by requiring UE Ž x U . s 1, UE Ž x U . s y1. For each f g F, there exists at least one E g E with f g E Žfor example, the mixture space generated by f and all constant acts.. Hence, we can define a functional U on F by: U Ž f . s UE Ž f .
for some E g E with f g E.
Ž 6.
Of course, we need to show that U is defined independently of whatever E is used in Ž6.. Since every such E contains x U and x U , and UE is a v-N-M utility function for K on E with UE Ž x U . s 1 and UE Ž x U . s y1, it is direct that UE Ž f . s 2 a y 1 where a is the unique number such that f ; a x U q Ž1 y a . x U . The same argument also shows that U actually rationalizes K , i.e., for all f, g g F, f K g mUŽ f . G UŽ g . .
Ž 7.
We now show that U has a Choquet integral representation with a v-N-M utility function u on L and an upper-continuous capacity m on S. Obviously, u should be defined by: uŽ x . s U Ž c x .
for all x g L.
Ž 8.
Next we try to find m. Each f g F can be associated with a real-valued function a f on S by: a f s u P f , i.e., a f Ž s . s u Ž f Ž s . .
for all s g S.
Ž 9.
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The function a f is continuous since both u and f are. Also, sup < a f < F 1. Let us consider the following collection of continuous functions on S: C s a a f < for all a G 1 and f g F 4 . We claim that C s C b Ž S .. Take any a g C b Ž S .. When sup < a < F 1 we let f g F be: 1 q aŽ s . ) 1 y aŽ s . f Ž s. s x q x ) for all s g S, 2 2 then a s a f . When sup < a < ) 1, we can find g g F such that a s sup < a < a g . We now construct a functional I on C b Ž S .. Given any a g C b Ž S ., we can find some l G 1 and f g F such that a s l a f . Then, let I Ž a . s lU Ž f . . Ž 10 . Of course, we need to show that I is well-defined. Suppose that a s l a f s u a g for some l G u G 1 and f, g g F. Obviously, f and g are comonotonic. Let E Žg E . be the mixture space generated by f, g and all constant acts, and h s ly1u g q Ž1 y ly1u .Ž 12 x U q 12 x U .. By Ž6., we have UŽ h. s ly1u UŽ g .. On the other hand, since K satisfies Axiom 5, w uŽ f Ž s .. s a f Ž s . s ly1u a g Ž s . s ly1u uŽ g Ž s .. s uŽ hŽ s .. for all s g S x implies UŽ f . s UŽ h.. Therefore, lUŽ f . s u UŽ g .. Similar arguments can be used to verify: Ži.
I is a comonotonically additive functional on Cb Ž S ., i.e., I Ž a q b . s I Ž a. q I Ž b . for all a and aX g C b Ž S . such that Ž aŽ s . y aŽ sX ..Ž aX Ž s . y aX Ž sX .. G 0 for all s and sX g S; and I is non-negative, i.e., I Ž a. G 0 for any a g C b Ž S . with aŽ s . G 0 for all s g S. Next, we use Axiom 6 to show that I also satisfies: I Ž a n .x0 for any sequence a n4 in Cb Ž S . with a nŽ s .x0 for all s g S. 8
Žii.
Žiii.
Since a nŽ s .x0 for all s g S, sup < a n < F sup < a1 < - ` for all n. We may assume, without loss of generality, that sup < a n < F 1. Hence, there exist a sequence of f n 4 in F such that a n s a f n for all n. It is then direct to verify that Axiom 6 implies Žiii.. Because C b Ž S . is a vector lattice with constant functions, 9 and I a functional on C b Ž S . satisfying Ži., Žii., and Žiii., the result in Zhou Ž1998, Theorem 1. implies that there is a unique upper-continuous capacity m on Ž S, S . such that, I Ž a . s a d m for all a g C b Ž S . .
HS
8
Ž 11 .
The notation ‘I Ž a n .x0’ means I Ž a n . approaches zero monotonically. A collection of real functions over a set is a vector lattice if it is both a vector space for the standard addition and scalar multiplication and a lattice for the join and meet operations defined by max and min. 9
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Combining Ž7. to Ž11., we obtain a Choquet integral representation of K . The uniqueness of m and u follow from the corresponding conclusions in the Von Neumann–Morgenstern theorem and Zhou’s result quoted above. I Axiom 6 is used in the proof only to show that the functional I satisfies Žiii.. When S is compact, however, Žiii. is satisfied without assuming Axiom 6. This follows from the Dini theorem that every decreasing sequence of continuous functions that converges to zero pointwise converges uniformly on any compact set. Thus, we have: Corollary A. Assume that S is a compact space. A preference relation K on F satisfies Axioms 1–5 if and only if there exist a Õ-N-M utility function u on L, unique up to positiÕe linear transformations, and a unique upper-continuous capacity m on S such that fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5b .
The next corollary concerns with representing a preference relation K by expectation of a v-N-M with respect to an additive measure. A necessary condition is that K satisfies full independence. Axiom 2 U . Ž Independence. For all acts f, g,h g F, and for all a g Ž0,1., f % g m af q Ž1 y a . h % a g q Ž1 y a . h. Corollary B parallels the classical Anscombe–Aumann theorem Ž1963.. Corollary B. Assume that S is a topological space and S is the Baire algebra of S. 10 A preference relation K on F satisfies Axiom 1, 2 U , and 3–6 if and only if there exist a Õ-N-M utility function u on L, unique up to positiÕe linear transformations, and a unique countably additiÕe measure m on S such that, fKgm
HS Ž u P f . d m GHS Ž u P g . d m
for all f , g g F.
Ž 5c .
The proof is virtually the same as the proof of Theorem 1 except that the Stone–Daniell representation theorem is used in the last step. Before closing, let me comment on the assumption that the prize space L is the lottery space over some finite outcomes. This assumption ensures that the derived v-N-M utility function u on L is bounded and continuous. Both properties are needed for the validity of our analysis. When L is a general metric space, we can 10
The Baire algebra of a topological space is the smallest s-algebra generated by all functions in C b Ž S ..
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impose some continuity property on K that will guarantee the boundedness and continuity of u. Since the existing literature contains many such results Žsee Fishburn, 1970 and Kreps, 1988., I shall not repeat any here.
3. Additional remarks In this paper I have proved a result parallel to that by Schmeidler Ž1989. on subjective probabilities in a topological instead of measure theoretical setting. Although it is far still from a fully satisfactory solution to the issue raised at the beginning of the paper, it shows that some important results in the existing literature are robust with alternative assumptions on feasible act spaces. More research should be conducted to see whether similar results can be obtained when feasible act spaces contain even fewer acts. It should be mentioned that a general extension of the Anscombe–Aumann theorem holds when full independence is assumed. Let us assume for simplicity that the state space S is a metric space. 11 Theorem 2. Suppose that the feasible act space F is a mixture space 12 that includes all constant acts. A preference relation K on F has an expected utility representation if and only if K satisfies Axiom 1, 2 U , and 3–5. Theorem 2 can be deduced from a more general Harsanyi-type result on utilitarianism ŽZhou, 1997.. 13 The requirement on F here is minimal for an Anscombe–Aumann type model. It is not clear whether a similar result is true when Axiom 2 U is replaced by Axiom 2. Although there are no conceptual difficulties with such a result, we will need in our proof a Hahn–Banach type theorem on extension of comonotonically additive functionals, which is still an open problem. More research should also be conducted to investigate if similar results can be obtained in some Savage type model that does not use objective probability devices. 14 I wish I could claim that I consider an Anscombe–Aumann type model here solely for the sake of simplicity, but I honestly cannot. In an Anscombe–Aumann model, we can first apply the v-N-M theorem to derive a functional on some function space, and then derive a subjective probability measure by a duality argument. This procedure, however, runs into serious difficulties in Savage type 11
This assumption is made so that a version of Riesz’s theorem can be applied though other weaker assumptions can also serve the same purpose. With this assumption, Axiom 6 also becomes redundant. 12 The definition of a mixture space can be found in textbooks such as Fishburn Ž1970. or Kreps Ž1988.. 13 It also holds in a measure theoretical setting. 14 Vind Ž1990. has also mentioned this problem without providing a solution.
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models, such as Gilboa Ž1987., and Sarin and Wakker Ž1992.. In fact, such a result cannot be obtained in general without restrictions on the state space or prize space. Consider an example in which the state space S is the unit interval w0,1x and the prize space P is the union of intervals w1,2x and w3,4x and only continuous acts from S to P are feasible. Since S is connected, the range of any feasible act is either entirely in w1,2x or entirely in w3,4x. Now look at an individual with the following preferences. There is an increasing utility function u on P so that all acts with ranges in w3,4x dominates acts with ranges in w1,2x. Between acts f with ranges in w3,4x, preferences are determined by the expected values of u P f with respect to a probability measure m 1 over S; between acts f with ranges in w1,2x, a different m 2 . As long as m 1 and m 2 are different, one cannot find a unified expected utility representation for this individual’s preferences even though the preferences satisfy all Savage-type axioms. This example suggests that one need to assume that P is connected, yet it is far from clear if this assumption will be sufficient.
Acknowledgements I would like to thank S.H. Chew, S. Grant, B. Polak, K. Vind, P. Wakker, as well as two anonymous referees of the journal, for many helpful comments. I am particularly grateful to D. Schmeidler for his numerous correspondences. Of course, all errors that remain are mine.
References Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability. Annals of Mathematical Statistics 34, 199–205. Bewley, T., 1986. Knightian decision theory: Part I. Cowles discussion paper. Yale University, New Haven. Chateauneuf, Wakker, P., 1993. From local to global additive representation. Journal of Mathematical Economics 22, 523–545. Choquet, G., 1955. Theory of capacities. Annals Inst. Fourier 5, 131–295. Denneberg, D., 1994. Non-Additive Measure and Integral. Kluwer, Dordrecht. Fishburn, P.C., 1970. Utility Theory for Decision Making. Wiley, New York. Gilboa, I., 1987. Expected utility with purely subjective non-additive probabilities. Journal of Mathematical Economics 16, 65–88. Jensen, N.E., 1967. An introduction to Bernoullian utility theory. I. Utility functions. Swedish Journal of Economics 69, 163–183. Kreps, D.M., 1988. Note on the Theory of Choice. Westview, Boulder. Sarin, R., Wakker, P., 1992. A simple axiomatization of nonadditive expected utility. Econometrica 60, 1255–1272. Savage, L.J., 1954. The Foundation of Statistics. Wiley, New York. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57, 571–587.
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