A simple proof of a basic result for multiple priors

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1 A simple proof of a basic result for multiple priors Massimo Marinacci August 1998 Abstract In this note we provide a simple and direct proof of the basic result on which rests the axiomatization of the multiple priors model of Gilboa and Schmeidler (1989). 1 Introduction In the past few years the multiple priors model axiomatized by Gilboa and Schmeidler (1989) has been widely used in Bayesian decision theory. Its purpose is to model vagueness in beliefs, a critical feature of decision making which is not possible to treat in the classic analysis of de Finetti (1931) and (1937) and Savage (1954). The basic idea of the multiple priors model is simple and appealing: since the decision maker has not enough information to form a meaningful single prior, he uses a set of priors, consisting of all priors compatible with his limited information. Using this set C, the acts f :! X are then ranked via the criterion min P 2C u (f (s)) dp, in which the use of the minimum RS re ects a cautious attitude of the decision maker. 1 The derivation of Gilboa and Schmeidler (1989) rests on a functional analytic result. It plays a critical role both mathematically and decisiontheoretically since it is the place where sets of priors pop up in their derivation. Unfortunately, Gilboa and Schmeidler (1989) o er a quite involved I wish to thank Paolo Ghirardato and Peter Klibano for some very useful discussions. 1 Notation: S is a state space, X a prize space, u : X! R a utility function and P : 2 S! [0; 1] an additive prior. 1

2 proof of this key result. Chateauneuf (1991) proved a similar result, but his proof is based on a powerful inequality result of Fan (1956) and the resulting argument is indirect and its intuition not that transparent. In contrast, this note shows that a very simple and transparent proof of this basic result can be given, a proof directly based on the classic Hahn- Banach Theorem, due to Banach (1932). The advantage of our proof is twofold: it is very simple and it is a direct consequence of a basic result in functional analysis. In this way we improve the understanding of the multiple priors model and, in particular, of how sets of priors pop up in its derivation. 2 The Result Let be a state space and an algebra of subsets of. Let B 0 be the set of simple functions measurable on and B the closure of B 0 with respect to the supnorm kk, i.e., f 2 B if there exists a sequence ff n g n1 B 0 such that kf f n k! 0 as n! 1. We endow B with its natural order structure: we write f g if f(!) g(!) for all! 2. In particular, 0 and 1 denote, respectively, the constant functions f such that f(!) = 0 for all! 2 and f(!) = 1 for all! 2. Finally, P denotes the set of all nitely additive probability measures P :! [0; 1]. The derivation of the multiple prior model rests on the following key result: Theorem 1 Let I : B! R be a functional satisfying: 1. (a) I(0) = 0 and I(1) = 1; (b) I(f) I(g) if f g for all f; g 2 B; (c) I(f + g) I(f) + I(g) for all f; g 2 B; (d) I(f + 1) = I(f) + for all 0 and 2 R. Then there exists a unique convex and w * -compact set C P such that Z I(f) = min fdp for all f 2 B. P 2C As mentioned in the introduction, besides its mathematical interest, Theorem 1 has also a critical conceptual importance in Gilboa and Schmeidler 2

3 (1989) because it is the place in which sets of priors come up in their derivation. And it plays a similar role in the derivations of the multiple priors model of Chateuneuf (1991) and Casadesus-Masanell, Klibano, and Ozdenoren (1998). The uniqueness part is a simple consequence of standard separation results. As to existence, in the next section we provide a direct and very simple proof. Since the argument is identical, we prefer to prove the result in a more abstract setting which highlights its functional analytic features. 2.1 Abstract Set-up Let (X; ) be an ordered real vector space with zero vector 0 and let I be a functional I : X! R such that (i) I(0) = 0 and I(x) I(y) if x y and x; y 2 X; (ii) I(x + y) I(x) + I(y) for all x; y 2 X: Notice that I (x) I( x) for all x 2 X because I(x) + I( x) I(x x) = I(0) = 0: In Theorem 1 the functionals were additive with respect to constants. In this abstract setting we just pick an element x 2 X and assume additivity with respect to all elements in the linear subspace spanned by x. (iii) x-additivity: For a x 2 X we have for all 0, 2 R, and x 2 X. I(x + x) = I(x) + I(x) Theorem 2 Let I a functional I : X! R that satis es (i)-(iii) above. Then there exists a unique convex and w*-compact set C of positive linear functionals L on X such that L(x) = I (x) and I(x) = min L(x) for all x 2 X: L2C 3

4 We only prove existence as uniqueness follows from standard separation results. Proof. Let L be the set of all linear functionals on X, and set C = fl 2 L : L(x) I(x) for all x 2 X and L (x) = I (x)g : By (i), C consists of positive linear functionals. Let bx 2 X and X bx = fx : x = bx + x for some ; 2 Rg : Let L bx be the linear functional on the linear subspace X bx de ned by L bx (bx + x) = I (bx) + I (x) for ; 2 R. By (iii), L bx (bx + x) = I (bx + x) if 0 and 2 R; if < 0, L bx (bx + x) I (bx + x) because L bx ( bx x) = I ( bx x) I (bx + x) : Thus, L bx (x) I(x) for all x 2 X bx. By the Hahn-Banach Theorem there exists a linear functional b L on X such that b L(x) = L bx (x) for all x 2 X bx, and b L(x) I(x) for all x 2 X. Therefore, b L 2 C, and I(bx) = b L(bx) = min L2C L(bx). This is the abstract formulation of the result on which rests the derivation of the multiple prior model. In fact, since B is a vector space, Theorem 1 is a straightforward version of Theorem 2 in which the constant function 1 plays the role of x and the normalization I(x) = 1 is adopted. More generally, however, in place of the constant 1 we could have picked any other function f 2 B and then require the functional I to be additive with respect to the linear subspace spanned by f. In this case, instead of probability measures, we should look at all the positive nitely additive measures such that R fd = I f. We close this section by observing that it is possible to obtain a version of Theorem 2 in which the monotonicity property (ii) is not required. In such a case, the linear functionals in C are no longer necessarily positive and the corresponding multiple prior model would feature signed priors. 4

5 3 An Extension In Theorem 2 we have considered the additivity of I with respect to the elements of a subspace spanned by a single element x 2 X. A natural extension is to consider the case in which the additivity of I is with respect to the subspace generated by a collection of elements of X. (iv) X-ADDITIVITY: e For a collection X e X we have I x + X! i ex i = I (x) + X i I (ex i ) for all 0, x 2 X, fexg e X, f i g R, and all nite index sets I such that jij jxj. Notice that there are no cardinality restrictions on e X and that e X-additivity implies that I is additive on the linear subspace spanned by e X. In particular, when e X spans the whole space X we get back to standard additivity, while when e X = fxg we get back to x-additivity. In other words, the linear subspace spanned by e X is the domain of additivity of I. We now extend Theorem 2 to this more general setting. Theorem 3 Let I be a functional I : X! R that satis es (i), (ii), and ex-additivity for a collection e X X. Then there exists a unique convex and w*-compact set C of positive linear functionals L on X such that L(ex) = I(ex) for all ex 2 e X and I(x) = min L(x) for all x 2 X. L2C Proof. Let L be the set of all linear functionals on X, and set n C = L 2 L : L(x) I(x) for all x 2 X and L (ex) = I (ex) for all ex 2 X e o : By (i), C consists of positive linear functionals. Let bx 2 X and ( X bx = x : x = bx + X ) i ex i : 5

6 where 2 R, fexg X, e f i g R, and I is a nite index set. Let L bx be the linear functional on X bx de ned by L bx bx + X! i ex i = I (bx) + X i I (ex i ) : By X-additivity, e L bx bx + P iex i = I bx + P iex i if 0; if < 0, L bx bx + P iex i I bx + P iex i because!! X X L bx bx i ex i = I bx i ex i I bx + X! i ex i : Thus, L bx (x) I(x) for all x 2 X bx. By applying the Hahn-Banach Theorem as in the proof of Theorem 2 we get the desired conclusion. References [1] S. Banach, Théorie des opérations linéaires, Warsaw, [2] R. Casadesus-Masanell, P. Klibano, and E. Ozdenoren, Maxmin expected utility over Savage acts with a set of priors, CMSEMS DP 1218, Northwestern University, [3] A. Chateauneuf, On the use of capacities in modeling uncertainty aversion and risk aversion, J. Math. Econ. 20 (1991), [4] B. de Finetti, Sul Signi cato Soggettivo della Probabilità, Fund. Math. 17 (1931), (English translation in B. de Finetti, Probabilità ed Induzione, CLUEB, Bologna, 1993). [5] B. de Finetti, La Prevision: Ses Lois Logiques, Ses Sources Subjectives, Ann. Inst. Poincaré 7 (1937), (English translation in H. E. Kyburg and H. E. Smokler (eds.), Studies in Subjective Probability, Wiley, New York, 1964). [6] I. Gilboa and D. Schmeidler, Maxmin expected utility with a non-unique prior, J. Math. Economics 18 (1989), [7] K. Fan, On systems of linear inequalities, in Linear inequalities and related systems, Annals of Mathematical Studies 38,

7 [8] L.J. Savage, The Foundations of Statistics, New York, Wiley,