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Conditional expectation for monotone measures, the discrete case
Journal of Mathematical Economics 37 (2002) 105–121
Conditional expectation for monotone measures, the discrete case Dieter Denneberg FB 3, Universität Bremen, D-28334 Bremen, Germany Received 9 October 2000; received in revised form 16 January 2002; accepted 3 April 2002
Abstract In modelling economic decision processes with risk and uncertainty one has to update the measures and condition the functions given new information. Non-additive measures (also called Choquet capacities) proved to be an important tool for modelling uncertainty. Here, for functions on a finite set Ω, a comprehensive approach to conditioning is made. It builds on conditional expectation for probability measures in representing the monotone measure ν as a min of belief functions below ν and the latter as a max of additive measures in the core of the respective belief function. The same method is applied, too, for defining the product of monotone measures. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Monotone measure; Choquet integral; Conditional expectation; Product measure
1. Introduction For modelling economic decisions under risk and uncertainty non-additive monotone measures (also called capacities or fuzzy measures) proved more and more to be an appropriate tool as they did already for other areas like information processing, multicriteria decision, insurance, etc. If one regards processes in time, one has to update the measure if new information arrives. This procedure is well understood for stochastic processes with σ -additive measures. Thus, it is important to generalise conditional probability, conditional expectation, products and independence to the case of monotone measures. Many approaches have been made but a closed comprehensive theory is still lacking. Most approaches can be divided in two classes. First, one tries to reduce the problem to the additive case in applying the core of the monotone measure ν (perceived as a cooperative game, the core being an issue of game theory). The approaches of the second class apply the Möbius transform of the monotone measure which can be perceived as an additive E-mail address: [email protected] (D. Denneberg). 0304-4068/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 ( 0 2 ) 0 0 0 1 1 - 3
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(or σ -additive) signed measure on a larger set. Both types of approaches have their limitations. In the first case, there exist monotone measures with an empty core, in the second case one has difficulties to translate back from the level of the Möbius transforms to the level of the original monotone measure. A further approach to conditional expectation for the non-additive case should be mentioned. Lehrer (1996) tries to extend the characterisation of conditional expectation as a projection of L2 -spaces to the non-additive case. But there L2 is no longer a Hilbert space, so Lehrer (1996) allows for non-uniqueness of conditional expectation, or conditional expectation of a function is a family of functions. The present paper completes, for the discrete case, the approaches of the first class in introducing the totally monotone core, consisting of the totally monotone measures (or belief functions) below ν, analogously to the usual core, consisting of the additive measures above ν. Using both cores we can represent an arbitrary monotone measure as a max–min of additive measures, generalising the well-known representation of a totally monotone measure (even a convex monotone measure) as a min of additive measures. This max–min additive representation is valid also for the corresponding integrals and we use it for defining conditional expectation going back to the additive case. Many properties of conditional expectation for the additive case are sustained in the general, but still discrete, case. The max–min representation is also a useful tool for defining the product of monotone measures on the basis of the product of additive measures. It turns out that this product coincides with the product defined for totally monotone measures via the Möbius transform of the monotone measures involved. The min–max additive representation has been generalised to the non-discrete case in Brüning and Denneberg (2002) and, with a compact metric space Ω, a min–max σ -additive representation is achieved. These results will be applied to products in Brüning (2002). The author thanks A. Chateauneuf and G.A. Koshevoy for helpful discussions.
2. Monotone measures and the Choquet integral In this section we fix the terminology and recall some well-known facts about monotone measures and their integral. For simplicity we suppose that Ω is a non-empty finite set. Throughout, a set function ν on a set system S ⊂ 2Ω with ∅, Ω ∈ S is a real valued function ν : S → [0, 1] with ν(∅) = 0. Furthermore, if not otherwise stated, we suppose that it is normalised, i.e. ν(Ω) = 1. A set function ν is called a monotone measure if A ⊂ B implies ν(A) ≤ ν(B). A set function ν is called k-monotone, k ≥ 2, if for A1 , . . . , Ak ⊂ Ω k |I | (1) ν Ai + (−1) ν Ai ≥ 0. i=1
I ⊂{1,...,k} I =∅
i∈I
2-monotonicity is also called supermodularity or convexity. Submodularity is the corresponding property with the reversed inequality sign. ν is totally monotone or a belief function if it is monotone and k-monotone for any k ≥ 2.
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Next we adopt some notions and results from cooperative game theory. A unanimity game uK for ‘coalition’ K ∈ 2Ω , K = ∅, is the monotone measure defined by uK (A) = 1 iff A ⊃ K and = 0 else. The additive measures among the {0, 1}-valued monotone measures are the Dirac measures u{ω} , ω ∈ Ω, and we get uK = u{ω} . (2) ω∈K
Here and in the sequel the lattice operator ∨ denotes the minimum or infimum and ∧ the maximum or supremum. Every {0, 1}-valued monotone measure η can be written as a maximum of unanimity games η= uKj (3) j
where Kj are the minimal sets K with η(K) = 1. In cooperative game theory or combinatorics it is well known that every monotone measure ν can uniquely be represented as linear combination of unanimity games (see e.g. Denneberg , 2000a), the coefficient of uK being denoted µν (K), ν= µν (K)uK . (4) K⊂Ω K =∅
µν is called the Möbius representation of ν. In some sense it can be perceived as a derivative of ν since (4) reformulates to ν(A) = K⊂A µν (K) which corresponds to the fundamental theorem of calculus. If the coefficients in (4) are all non-negative, µν ≥ 0, then ν is totally monotone like the uK , and the converse is also true. Thus, total monotonicity corresponds to ordinary monotonicity of differentiable functions which is characterised by non-negative derivative. A monotone measure ν is additive iff µν (K) = 0 for |K| > 1, i.e. µν lives on the singletons of 2Ω , which form a antichain, i.e. there is no inclusion relation between any pair of singletons. A class of monotone measures based on chains will play an important rˆole in the sequel, namely the necessity measures. A monotone measure ν is a necessity measure iff there is a chain1 K ⊂ 2Ω and a monotone set function κ : K → [0, 1] such that ν is the inner set function κ∗ (A) := κ(K), A ∈ 2Ω . K∈K K⊂A
of κ.2 The necessity measures among the {0, 1}-valued monotone measures are the unanimity games uK . This is easily seen regarding a chain containing K. Proposition 2.1. A necessity measure is totally monotone. A chain in 2Ω is a nested sequence of subsets and we always suppose that a chain contains ∅, Ω. cf. Example 6.3 in Denneberg and Grabisch (1999) to see that the present definition coincides with the original one in Shafer (1976), Zadeh (1978). 1
2
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This fact is well known Shafer (1976). We present a proof since it shows that the Möbius representation of ν lives on the chain K, a fact needed later on. Proof (see Proposition 3.1 in (Denneberg , 2000b)). Let ν = κ∗ be the necessity measure as above and denote the elements of the chain K by ∅ = K0 ⊂ · · · ⊂ Kn = Ω. We show that µ(Ki ) := κ(Ki ) − κ(Ki−1 ),
i = 1, . . . , n,
for A ∈ 2Ω \ K
µ(A) := 0
is the Möbius representation of ν. For A ∈ 2Ω let iA be the maximal index i such that Ki ⊂ A. Then ν(A) =
κ(K) = κ(KiA ) =
K∈K K⊂A
iA i=1
µ(Ki ) =
µ(B),
B⊂A
and this is Eq. (4). Finally, µν = µ ≥ 0 implies that ν is totally monotone.
We recall the definition of the conjugate ν¯ of a set function ν, ν¯ (A) := 1 − ν(Ω \ A). The conjugate of a submodular monotone measure is supermodular and vice versa. The conjugate of a necessity measure is usually called a possibility measure and the conjugate of a belief function is a plausibility function. In applications to imprecise probabilities the pair (ν, ν¯ ) describes the imprecision or uncertainty, especially if ν ≤ ν¯ . The difference ν¯ − ν is called ambiguity or vagueness (see e.g. Denneberg , 2000a, Section 9). We do not repeat here the definition and elementary properties of the (Choquet) integral X dν of a function X : Ω → R w.r.t. a monotone measure ν, refer e.g. to Denneberg (1994a) or Denneberg (2000a). Since here Ω is a finite set and ν is finite the integral always exists and is a real number. We also will have to integrate functions with an extended range, which will be described first. In Section 4, we shall need an element in the range of our functions, which behaves neutral w.r.t. the minimum and the maximum simultaneously, i.e. like +∞ w.r.t. the min and like −∞ w.r.t. the max. So we join to R one infinite far point, denoted again ∞ as it is common use for the projective line, with the following properties x ∧ ∞ := x,
x ∨ ∞ := x,
x ± ∞ := ∞,
x · ∞ := ∞
for x ∈ R∞ := R ∪ {∞}. In accordance with these properties we use the following convention for arbitrary S ⊂ R∞ if S ∩ R = ∅, ∞ x := x else, x∈S x∈S∩R
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and similarly with . Furthermore we extend the order relation ≤ on R to a relation, again denoted ≤, on R∞ by x ≤ ∞,
x≥∞
for all x ∈ R∞ .
Be aware that ≤ is not an order relation on R∞ since antisymmetry and transitivity fail. With this extension an inequality between R∞ -valued functions is true if it holds for those points of the domain for which both functions assume finite values, i.e. values in R. For integrating R∞ -valued functions Z : Ω → R∞ w.r.t. a monotone measure ν the appropriate definition seems to be with ΩZ := {ω ∈ Ω|Z(ω) = ∞}. Z dν := Z|ΩZ d(ν|2ΩZ ) Notice that ν|2ΩZ is, in general, not normalised any more like ν. Of course, for real valued Z, i.e. ΩZ = Ω, this definition coincides with the ordinary Choquet integral.
3. The max–min additive representation of the Choquet integral It can easily be seen (Denneberg , 2000b) that any monotone measure ν is a maximum of linear combinations of unanimity games with positive coefficients, hence a maximum of totally monotone measures. On the other hand, any totally monotone measure is a minimum of additive measures. Similarly, the Choquet integral w.r.t. an arbitrary monotone measure is a max–min of additive integrals. The tool for the min representation is the well-known core of a cooperative game. The (additive) core of a set function ν on 2Ω is C+ (ν) := {α|α additive on 2Ω , ν ≤ α} (recall that, according to our general assumptions, α(Ω) = 1 = ν(Ω)). Example 3.1. α is a member of the core C+ (uK ) of a unanimity game uK iff α is an additive measure on 2Ω with α(K) = 1 or iff α = ω∈K α(ω)u{ω} . The last formula is the Möbius representation of α, which is a convex combination of Dirac measures u{ω} . Proposition 3.1. If ν is supermodular and monotone on 2Ω then C+ (ν) = ∅ and X dα, α, X dν = ν= α∈C+ (ν)
(5)
α∈C+ (ν)
where X : Ω → R. This well-known fact (Shapley , 1953) proves with the Hahn-Banach theorem (see e.g. Denneberg , 1994a, Proposition 10.3). We present a short proof for totally monotone ν since we apply only this case in the sequel and details of proof are essential to prove Corollary 3.3.
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Proof. For totally monotone ν. Let K, ∅ = K0 ⊂ · · · ⊂ Kn = Ω, be a maximal chain containing the upper level sets {X > x}, x ∈ R, of X. Enumerate the elements of Ω so that Ki \ Ki−1 = {ωi }, i = 1, . . . , n. Now we construct an α ∈ C+ (ν) in shifting the mass of µν on 2Ki \ 2Ki−1 to ωi , α(ωi ) = µα (ωi ) := µν (A), i = 1, . . . , n. A∈2Ki \2Ki−1
Since µν ≥ 0 by assumption, these settings define an additive measure α ≥ 0. By induction on i one checks, using the Möbius representation (4), that α(Ki ) = ν(Ki ). Then the decreasing distribution functions of X w.r.t. α and w.r.t. ν are identical and so are the respective integrals. It remains to show that ν ≤ α.First observe that 2Ki \ 2Ki−1 consists A Ki Ki−1 ) for any A ∈ 2Ω . of all subsets of Ki containing ωi , whence 2 ⊂ i:ωi ∈A (2 \ 2 This implies ν(A) = B⊂A µν (B) ≤ ω∈A α(ω) = α(A). Analogously to C+ (ν), but with the dual view replacing ≤ with ≥, we define the totally monotone core as C (ν) := {β|β totally monotone on 2Ω , β ≤ ν}. As noted already at the beginning of this section, C (ν) = ∅, ν = β
(6)
β∈C (ν)
for arbitrary monotone ν. We can say something more: Proposition 3.2. Given a monotone measure ν on 2Ω and a chain K ⊂ 2Ω , then the necessity measure β := (ν|K)∗ belongs to C (ν). Proof. Since β is totally monotone by Proposition 2.1 we have only to show that β ≤ ν. But this is plain since the inner set function β is the smallest monotone measure extending ν|K monotonically to the whole powerset 2Ω , and ν is such an extension. Combining Propositions 3.2 and 3.1 we get the result, which is basic for the rest of the paper. Corollary 3.3. For a monotone measure ν on 2Ω and a function X : Ω → R there exists a necessity measure βX ∈ C (ν) \ it and an additive measure αX ∈ C+ (βX ) such that X dν = X dβX = X dαX . Proof. Like in proof of Proposition 3.1 let K be a maximal chain containing the upper level sets {X > x}, x ∈ R, of X. Set βX := (ν|K)∗ and αX (ωi ) := µβ (Ki ), i = 1, . . . , n, where ∅ = K0 ⊂ · · · ⊂ Kn = Ω are the elements of the chain K and Ki \ Ki−1 = {ωi }. By construction ν, βX and αX (see proof of Proposition 2.1) coincide on K, hence the decreasing distribution functions of X w.r.t. the three monotone measures are identical and
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so are the respective integrals. Finally βX ∈ C (ν) was shown in Proposition 3.2 and αX ∈ C+ (βX ) was shown within proof of Proposition 3.1. Since the integral behaves monotone w.r.t. monotone measures the corollary implies ν= X dα. (7) α, X dν = β∈C (ν) α∈C+ (β)
β∈C (ν) α∈C+ (β)
We call these formulas the max–min additive representation of ν and of the Choquet integral w.r.t. ν, respectively. For {0, 1}-valued monotone measures η the max–min additive representation is very natural (cf. (3), (2)) and well known (cf. e.g. Ghirardato and Le Breton , 2000) η= u{ω} , X dη = X(ω). (8) K⊂Ω ω∈K uK ≤η
K⊂Ω ω∈K uK ≤η
In the max–min additive representation one could also replace the totally monotone core by the necessity core Cnec (ν) := {γ |γ a necessity measure on 2Ω , γ ≤ ν}. It is contained in the totally monotone core (Proposition 2.1), Cnec (ν) ⊂ C (ν). It would be interesting to know if the totally monotone core is the convex hull of the necessity core. 4. Conditional expectation The present approach to conditional expectation starts with conditional expectation for additive measures and generalises it by means of the max–min additive representation of monotone measures. Things become more complicated than in the last section since there is no analogue to Corollary 3.3. Let be given a function X : Ω → R and a subalgebra B of 2Ω . First we recall the classical issue of conditional expectation Eα (X|B) of X w.r.t. an additive measure α on 2Ω , i.e. a probability measure. Eα (X|B) is characterised as the unique (up to addition of α-nullfunctions) B-measurable function Y with the property X − Y dα = 0 for all B ∈ B. (9) B
Since we are concerned with different monotone measures simultaneously, we produce uniqueness of Y not in the usual way in building classes of functions differing only by α-nullfunctions, but in the following way which works at least in the discrete case. We extend the real line R by one infinite far point ∞ to R∞ as desribed at the end of Section 2 and for the B-measurable function Y in (9) we require Y (ω) := ∞ iff α(Bω ) = 0,
(10)
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where Bω denotes the atom of B containing ω. As a result, conditional expectations Eα (X|B) remain ordinary functions, but with extended range R∞ . With this setting we recall the explicit formula for Eα (X|B). Since we are in the discrete case, B-measurability means that the function is constant on the atoms of the algebra B. Hence (9) implies 1 X dα for α(Bω ) = 0, Eα (X|B)(ω) = α(Bω ) Bω (11) ∞ else, i.e. on an atom A of B with α(A) = 0 the value of Eα (X|B) is the average of X on A. Example 4.1. For the Dirac measure δ = u{ω1 } at point ω1 ∈ Ω we get Eδ (X|B)(ω) = since
Bω1
X(ω1 ) if Bω ω1 ∞ else
X dδ = X(ω1 ) and δ(Bω ) = 1 iff Bω ω1 .
Now we define analogously to (7) the conditional expectation of X : Ω → R w.r.t. a monotone measure ν on 2Ω given the algebra B ⊂ 2Ω as Xν\B := Eα (X|B). (12) β∈C (ν) α∈C+ (β)
According to our convention about the use of ∞, the value Xν\B (ω) is finite iff there is some β ∈ C (ν) and some α ∈ C+ (β) such that Eα (X|B)(ω) is finite. Proposition 4.1. Given a monotone measure ν on 2Ω , an algebra B ⊂ 2Ω and a function X : Ω → R. Then the conditional expectationXν\B is a B-measurable function. If Xν\B (ω) = ∞ then ν(Bω ) = 0. The converse of the last implication does not hold as Example 4.2 will show. Proof. Xν\B is a B-measurable function since the Eα (X|B) are B-measurable and the infimum and the supremeum of measurable functions is again measurable for finite Ω. According to our rules for the infinite point ∞ we get Xν\B (ω) = ∞ iff α∈C+ (β) Eα (X|B)(ω) = ∞ for all β ∈ C (ν). This, again, is equivalent to Eα (X|B)(ω) = ∞ for all α ∈ C+ (β) and all β ∈ C (ν). Then, according to (10), α(Bω ) = 0 for all α ≥ β and all β ≤ ν as above. Then the max–min additive representation (7) of ν implies ν(Bω ) = 0. Our first example generalises Example 4.1. Example 4.2. The conditional expectation of X w.r.t. a unanimity game uK is XuK \B (ω) = X(ω1 ). ω1 ∈K∩Bω
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Like above Bω denotes the atom of B containing ω. Proof relies on the fact uK = ω1 ∈K u{ω1 } (2) and property (ii) in Proposition 4.2 below. Since u{ω1 } ∈ C+ (uK ), ω1 ∈ K, the assertion follows taking the min over α ∈ C+ (uK ) of the following relation, Eu{ω1 } (X|B)(ω) = X(ω1 ) ≤ Eα (X|B)(ω), α ∈ C+ (uK ). ω1 ∈K
ω1 ∈K∩Bω
For verifying the last inequality we employ (11) and the fact that α vanishes outside K since α(K) = 1. If K ∩ Bω = ∅ and Bω /K then we have an example as postulated in the remark after Proposition 4.1, uK (Bω ) = 0 but XuK \B (ω) = ∞. The next example shows that the present issue of conditional expectation is related to the generalisation of conditional probability as developed by Fagin and Halpern (1991) and Jaffray (1992) (see also Denneberg , 1994b). Example 4.3. The conditional expectation of an indicator function X = 1A , A ⊂ Ω, w.r.t. a totally monotone β given the algebra B := {∅, B, B c , Ω} with β(B) > 0 is (1A )α\B (ω) (1A )β\B (ω) = α∈C+ (β)
=
α∈C+ (β)
=
α(A ∩ B) α(B)
for ω ∈ B
β(A ∩ B) . ¯ c ∩ B) β(A ∩ B) + β(A
(13)
For the first equation we applied (ii) in Proposition 4.2 and for the last equation we applied Denneberg (1994b), Theorem 2.4. The second equation is well known in the additive theory. The assumption β(B) > 0 guarantees that the denominator does not vanish. The fraction (13) is called “generalised Bayes update” in Walley (1991) or “full Bayesian update” in Cohen et al. (2000). If β is additive or a unanimity game, (13) becomes the ordinary Bayes update β(A ∩ B)/β(B). Dually to the max–min additive representation (7) there is a min–max additive representation X dα. (14) α, X dν = ν= β∈C (¯ν ) α∈C+ (β)
β∈C (¯ν ) α∈C+ (β)
The duality may be more apparent if one reformulates the conditions on α and β in (14), β ∈ C (¯ν ) iff γ := β¯ is a plausibility function and γ ≥ ν, ¯ α ∈ C+ (β) iff α is additive and α ≤ γ = β. Then, also our max–min conditional expectation (12) allows a dual version. The min–max conditional expectation of X w.r.t. a monotone measure ν on 2Ω given the algebra B ⊂ 2Ω is
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Xν/B :=
Eα (X|B).
(15)
β∈C (¯ν ) α∈C+ (β)
Notationally, the min–max conditional expectation is characterised by the forward slash, whereas the max–min conditional expectation by the backward slash. Xν/B is needed to ex tend asymmetry of the Choquet integral, −X dν = − X dν¯ , to conditional expectation, which is property (vii) below. Many properties of conditional expectation w.r.t. a probability measure remain valid for monotone measures except, of course, additivity. We list, with the exception of (ii) and (vii), only the properties for the max–min conditional. The corresponding properties of the min– max conditional prove similarly or can be derived by means of the asymmetry property (vii). Proposition 4.2. Let λ, ν be monotone measures on 2Ω , A, B subalgebras of 2Ω and X, Y real valued functions on Ω. Then (i) for a probability measure α the present definitions coincide with the classical one, Xα\B = Eα (X|B) ; (ii) for a totally monotone measure β and its conjugate β¯ Xβ\B = Eα (X|B), Xβ\ Eα (X|B) ¯ B = α∈C+ (β)
α∈C+ (β)
and the max–min and min–max conditional expectations coincide, Xβ\B = Xβ/B ,
Xβ\ ¯ B = Xβ/ ¯ B.
For arbitrary monotone ν we have only the inequality
(iii) (iv) (v) (vi) (vii) (viii) (ix)
Xν\B ≤ Xν/B ; Xν\B = X dν if B := {∅, Ω}; Xν\B = X if X is B-measurable, especially if B = 2Ω ; (XY)ν\B = X(Yν\B ) if X ≥ 0 is B-measurable, in particular (X constant) conditional expectation is positively homogenous; (X + Y )ν\B = Xν\B + Y if Y is B-measurable; (−X)ν\B = −Xν¯ /B ; X ≤ Y implies Xν\B ≤ Yν\B ; λ ≤ ν implies Xλ\B ≤ Xν\B .
We have arranged the properties in such a way that first the properties with a fixed monotone measure and a fixed hypothesis are listed, whereas the last ones are concerned with several hypotheses or several measures simultaneously. Nevertheless, property (ix) is important for proving some of the preceding properties. Proof. Most properties are derived from the corresponding results for additive measures, which are supposed to be known (see e.g. Dudley , 1989). (i) follows from the first equation in (ii) since C+ (α) = {α}.
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(ii) For β ∈ C (β) we have C+ (β ) ⊃ C+ (β) so that α∈C+ (β ) Eα (X|B) ≤ α∈C+ (β) Eα (X|B) . Since β ∈ C (β) the first assertion follows. Using this result we get Xβ\ Xβ \B ≥ Xα\B = Xα\B . ¯ B = ¯ β ∈C (β)
¯ α∈C (β) α additive
α∈C+ (β)
Here, the inequality is plain since the sup on the right hand side is taken with a smaller set than on the left hand side. The last equality is clear since α ≤ β¯ iff α ≥ β. Finally, the reversed inequality follows from Lemma 4.3 using the monotonicity property (ix). The corresponding property for the min–max conditional shows that the max–min and min–max conditionals coincide for totally monotone β and its conjugate. Finally, for an arbitrary ν we derive from (ix) for β ≤ ν ≤ γ , γ a plausibility function, Xβ\B ≤ Xγ \B = Xγ /B so that
Xν\B =
Xβ\B ≤
β∈C (ν)
Xγ /B = Xν/B .
γ¯ ∈C (¯ν )
(iii) Xν\B = β∈C (ν) α∈C+ (β) Eα (X|B) = β∈C (ν) α∈C+ (β) X dα = X dν, where, for the last equation we applied (7). (iv) proves like (iii) X is B-measurable now. Eα (X|B) = X since but with (v) (XY)ν\B = Eα (XY|B) = X Eα (Y |B) β∈C (ν) α∈C+ (β)
=X
β∈C (ν) α∈C+ (β)
Eα (Y |B) = X Yν\B .
β∈C (ν) α∈C+ (β)
(vi) (X + Y )ν\B =
β∈C (ν) α∈C+ (β)
=
Eα (X + Y |B) =
(Eα (X|B) + Y )
β∈C (ν) α∈C+ (β)
Eα (X|B) + Y = Xν\B + Y.
β∈C (ν) α∈C+ (β)
(vii) First for a totally monotone β we derive from equations (ii) Eα (−X|B) = − Eα (X|B) = −Xβ/ (−X)β\B = ¯ B. α∈C+ (β)
α∈C+ (β)
Then replacing in (12) and (15) the expressions from (ii) we get (−X)β\B = (−Xβ/ Xβ/ (−X)ν\B = ¯ B) = − ¯ B = −Xν¯ /B . β∈C (ν)
β∈C (ν)
β∈C (ν)
For the last equation observe that ν¯¯ = ν. (viii) easily derives from the corresponding property for additive measures.
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(ix) By λ ≤ ν we get C (λ) ⊂ C (ν) so that, in the definition of conditional expectation, the maximum is taken over a larger set in case of ν than in case of λ from which the assertion follows. For proving (ii) we used the following version of the sandwich theorem, which is interesting for its own right (see Chateauneuf , 1994). Lemma 4.3. Let β, β be supermodular monotone measures on 2Ω and suppose ¯ β ≤ β. Then there exists an additive monotone measure α ∈ C+ (β) ∩ C+ (β ), i.e. ¯ β ≤ α ≤ β. Proof. On the space, say, of bounded functions X : Ω → R the Choquet integral Γβ X := X dβ is a superadditive and positively homogenous (short superlinear) monotone functional. Similarly Γβ¯ X := X dβ¯ is sublinear and Γβ ≤ Γβ¯ . By the sandwich theorem in Fuchssteiner and Lusky (1981) there exists a monotone linear functional Λ such that Γβ ≤ Λ ≤ Γβ¯ . Then α(A) := Λ1A has the desired properties. Which should be the appropriate form of (9) in the non-additive case? Lehrer (1996) perceives (X − Xν\B )ν\B = 0 as a generalisation of (9) for monotone measures. In our context this equation3 follows from property (vi) since −Xν\B is B-measurable (Proposition 4.1). A partial result with the original form of (9) can already be found at the end of Denneberg (1994b). Proposition 4.4. Let β be totally monotone on 2Ω , B ⊂ 2Ω an algebra and X a real function on Ω. Then X − Xβ\B dβ ≥ 0 for B ∈ B with β(A) > 0 for all atoms A ⊂ B of B B
and equality holds if B is an atom of B itself with β(B) > 0. Here, we understand that B Z dν := 1B Z dν for R∞ -valued functions Z (see end of Section 2). In Denneberg (1994b) this equality for atoms B had been used to define conditional expectation for sub- and supermodular monotone measures. Proof. The integrand Z := 1B (X − Xβ\B ) may assume the value ∞, so we have X − Xβ\B dβ = Z|ΩZ d(β|2ΩZ ). B
3 In order to restrict technicalities, we have defined conditional expectation only for real valued functions, so—strictly speaking—one has to suppose here that Xν\B is real valued.
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Like in proof of Proposition 3.1 we can find an α ∈ C+ (β) which coincides with β on the chain of upper level sets of Z|ΩZ in 2ΩZ ⊂ 2Ω . Then ΩZ ΩZ Z|ΩZ d(β|2 ) = Z|ΩZ d(α|2 ) ≥ 1B (X − Xα\B )|ΩZ d(α|2ΩZ ) = 0, where the inequality holds since Xβ\B ≤ Xα\B and Xα\B is finite on ΩZ by the assumption on B. That the last integral equals zero is just (9). Now suppose that B is an atom. Then Xβ\B is constant and finite on B (Proposition 4.1) and by property (ii) for any ε > 0 there exists an α ∈ C+ (β) such that Xβ\B (ω) ≥ Xα\B (ω) − ε
for ω ∈ B.
This inequality implies X − Xβ\B dβ ≤ X − Xα\B + ε dβ ≤ X − Xα\B dα + εα(B) = εα(B). B
B
B
Again we applied (9) and, since ε > 0 was arbitrary, we are done.
As pointed out at the end of Section 3, the max–min additive representation of the Choquet integral could have been performed also with the necessity core in place of the totally monotone core. Why did we not apply this representation for defining conditional expectation? Denoting it with Xν\\B for the moment, and the dual one with Xν//B , we had got Xν\\B ≤ Xν\B ≤ Xν/B ≤ Xν//B . We decided for the alternative for which the pair of dual conditionals is closer to each other.
5. Products There are many approaches for a product of monotone measures (e.g. Walley and Fine , 1982; Gilboa and Schmeidler , 1989; Hendon et al. , 1996; Koshevoy , 1998; Denneberg , 2000b). Here, we define a product again by means of the max–min additive representation of monotone measures going back to the product of additive measures. For supermodular set functions this method is known (Walley and Fine , 1982; Gilboa and Schmeidler , 1989). Surprisingly, for totally monotone measures this product coincides with the Möbius product of Hendon et al. (1996). Let Ω1 and Ω2 be finite sets and Ω := Ω1 ×Ω2 their cartesian product. For two monotone measures ν1 on 2Ω1 and ν2 on 2Ω2 we define their (max–min) product by ν1 ⊗ ν2 := α1 ⊗ α 2 , (16) βi ∈C (νi ) αi ∈C+ (βi ) i=1,2 i=1,2
where α1 ⊗ α2 denotes the usual product on 2Ω of the additive measures αi on 2Ωi , i = 1, 2. Obviously, the product is a monotone set function. Also, the product is a monotone operator, λi ≤ νi , i = 1, 2, implies λ1 ⊗ λ2 ≤ ν1 ⊗ ν2
(17)
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for monotone λ1 , ν1 on 2Ω1 , λ2 , ν2 on 2Ω2 . For totally monotone βi on 2Ωi , i = 1, 2, β1 ⊗ β2 := α1 ⊗ α 2 . (18) αi ∈C+ (βi ) i=1,2
For unanimity games uKi on 2Ωi , Ki ⊂ Ωi , i = 1, 2, one easily shows (Example 3.1, Proposition 3.1) uK1 ⊗ uK2 = uK1 ×K2 . Proposition 5.1. For totally monotone βi on 2Ωi , i = 1, 2, β1 ⊗ β2 = µβ1 (K1 )µβ2 (K2 )uK1 ×K2 ,
(19)
(20)
K1 ,K2
where Ki runs through all non-empty subsets of Ωi , i = 1,2. The sum on the right hand side of (20) is the Möbius product of Hendon et al. (1996) (see also Denneberg , 2000b). Since the Möbius product for arbitrary monotone measures is not monotone in general, an alternative product had been proposed in Section 6 of Denneberg (2000b) which, by Proposition 5.1, coincides with the present definition of a product. Proposition 5.1 implies that the product behaves associative at least for totally monotone set functions. See Denneberg (2000b) for further properties and examples. Before proving Proposition 5.1 we recall a part of Corollary 4 in Danilov and Koshevoy (2000), which is already implicit in Shapley (1953). Lemma 5.2. Let β be a totally monotone measure with Möbius representation β= µβ (K)uK . K⊂Ω K =∅
Then for any α ∈ C+ (β) there are αK ∈ C+ (uK ), K ⊂ Ω, K = ∅, such that α= µβ (K)αK . K⊂Ω K =∅
Conversely any sum like on the right hand side is an element of C+ (β). Hence, C+ (β) is a convex combination of the C+ (uK ) with coefficients µβ (K), K ⊂ Ω, K = ∅, C+ (β) = µβ (K)C+ (uK ). K⊂Ω K =∅
Proof of Proposition 5.1. Applying Lemma 5.2 to β1 on 2Ω1 and β2 on 2Ω2 we get
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β1 ⊗ β 2 =
α1 ⊗ α 2
α1 ∈C+ (β1 ) α2 ∈C+ (β2 )
=
(αL1 )L1 (αL2 )L2 αL1 ∈C+ (uL1 ) αL2 ∈C+ (uL2 )
=
µβ1 (K1 )αK1 ⊗
µβ2 (K2 )αK2
K1
µβ1 (K1 )µβ2 (K2 )
K1 ,K2
=
K2
µβ1 (K1 )µβ2 (K2 )αK1 ⊗ αK2
K1 ,K2 (αL1 )L1 (αL2 )L2 αL1 ∈C+ (uL1 ) αL2 ∈C+ (uL2 )
=
119
αK1 ⊗ αK2
αK1 ∈C+ (uK1 ) αK2 ∈C+ (uK2 )
µβ1 (K1 )µβ2 (K2 ) uK1 ⊗ uK2 .
K1 ,K2
Here, Ki , Li run over all non-empty subsets of Ωi and (αLi )Li denotes the collection of the αLi , Li ∈ 2Ωi \ {∅}. For the third equation we used that the product of additive measures is a bilinear operator. For the second and the fourth equation µβi ≥ 0 is crucial, i.e. βi totally monotone. Fianlly, the last equation follows from (18) applied with uK1 , uK2 . Like for conditional expectation there is a dual approach to the product (16) by means of the min–max additive representation (14). The min–max product of ν1 and ν2 is defined as ¯ ν2 := α1 ⊗ α 2 , (21) ν1 ⊗ βi ∈C (νi ) αi ∈C+ (βi ) i=1,2 i=1,2
The min–max product coincides with the max–min product for belief functions and plausibility functions and for the general case an inequality remains true. Proposition 5.3. Let βi be totally monotone on 2Ωi , i = 1, 2, then ¯ β2 , β1 ⊗ β2 = β1 ⊗
¯ β2 . β1 ⊗ β2 = β1 ⊗
For arbitrary monotone measures νi on 2Ωi , i = 1, 2, ¯ ν2 = ν1 ⊗ ν2 , ν1 ⊗ ¯ ν2 . ν1 ⊗ ν2 ≤ ν1 ⊗ Proof. Proof uses Lemma 4.3, analogously to the reasoning for Proposition 4.2 (ii), β1 ⊗ β2 = β1 ⊗ β2 ≥ α1 ⊗ α 2 βi ∈C (βi ) i=1,2
=
αi ∈C+ (βi ) i=1,2
αi ∈C (βi ) additive i=1,2
¯ β2 . α1 ⊗ α 2 = β 1 ⊗
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Here, the inequality is plain since the sup on the right hand side is taken with a smaller set than on the left hand side. The next equality is clear since αi ≤ βi iff αi ≥ βi . The last ¯ Finally, the reversed inequality follows from Lemma equality is the analogue of (18) for ⊗. 4.3 using monotonicity property (17). Proof for belief functions runs similarly. For monotone ν1 , ν2 we get ν1 ⊗ ν2 (A) = 1 − β1 ⊗ β2 (Ac ) = (1 − β1 ⊗ β2 (Ac )) βi ∈C (νi ) i=1,2
=
βi ∈C (νi ) i=1,2
β1 ⊗ β2 (A) =
βi ∈C (νi ) i=1,2
¯ β2 (A) = ν1 ⊗ν ¯ 2 (A). β1 ⊗
βi ∈C (νi ) i=1,2
¯ β2 ¯ and, for the fourth equation, β1 ⊗ β2 = β1 ⊗ Again we used the analogue of (18) for ⊗ which proves with similar arguments. The last assertion proves like the inequality in Proposition 4.2 (ii).
6. Conclusion For the discrete case with finite Ω we have seen that the max–min additive representation of the Choquet integral provides a unifying tool to generalise conditional expectation and products for arbitrary monotone measures. Remaining questions are: how to treat iterated conditioning? How stochastic independence should be incorporated in the present approach? What about the non-discrete case? Some answers can be found in Brüning (2002), Brüning and Denneberg (2002). But the main question is if the concepts developed here can become useful in applications like information processing, economic decision processes, finance, insurance etc. For this purpose at least an inequality between iterated and one step conditioning is desirable.
References Brüning, M., 2002. Products of Non-additive Measures, Universität Bremen (preprint). Brüning, M., Denneberg, D., 2002. Max–min σ -additive representation of monotone measures. Statistical Papers 34, 23–35. Chateauneuf, A., 1994. Combination of compatible belief functions and relation of specifity. In: Fedrizzi, M., Kaprczyk, J., Yager, R.R. (Eds.), Advances in the Dempster–Shafer theory of evidence. Wiley, New York, pp 97–114. Cohen, M., Gilboa, I., Jaffray, J.Y., Schmeidler, D., 2000. An experimental study of updating ambiguous beliefs. Risk Decision and Policy 5, 123–133. Danilov, V.I., Koshevoy, G.A., 2000. Cores of cooperative games, superdiffereentials of functions and the Minkowski difference of sets. Journal of Mathematical Analysis and Applications 247, 1–14. Denneberg, D., 1994a. Non-additive Measure and Integral, Theory and Decision Library, 2nd Edition, Vol. 27, Series B. Kluwer Academic Publishers, Dordrecht. Denneberg, D., 1994b. Conditioning (updating) non-additive measures. Annals of Operations Research 52, 21–42.
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Denneberg, D., 2000a. Non-additive Measure and Integral, Basic Concepts and their Rˆole for Applications. In: Grabisch, M., Murofushi, T., Sugeno, M. (Eds.), Fuzzy Measures and Integrals—Theory and Applications, Studies in Fuzzyness and Soft Computing, Vol. 40. Physica-Verlag, Heidelberg, pp. 42–69. Denneberg, D., 2000b. Totally monotone core and products of monotone measures. International Journal of Approximate Reasoning 24, 273–281. Denneberg, D., Grabisch, M., 1999. Interaction transform of set functions over a finite set. Information Sciences 121, 149–170. Dudley, R.M., 1989. Real Analysis and Probability. Wadsworth, Pacific Grove. Fagin, R., Halpern, J.Y., 1991. A new approach to updating beliefs. Uncertainty in Artificial Intelligence 6, 347– 374. Fuchssteiner, B., Lusky, W., 1981. Convex Cones, North-Holland Mathematics Studies, Vol. 56. North-Holland, Amsterdam. Ghirardato, P., Le Breton, M., 2000. Choquet rationality. Journal of Economic Theory 90, 277–285. Gilboa, I., Schmeidler, D., 1989. Max–min expected utility with non-unique prior. Journal of Mathematical Economics 18, 141–153. Hendon, E., Jacobsen, H.J., Sloth, B., Tranæs, T., 1996. The product of capacities and belief functions. Mathematical Social Sciences 32, 95–108. Jaffray, J.-Y., 1992. Bayesian updating and belief functions. IEEE Transactions Systems Man Cybernetics 22, 1144–1152. Koshevoy, G.A., 1998. Distributive lattices and products of capacities. Journal of Mathematical Analysis and Applications 219, 427–441. Lehrer, E., 1996. An approach to the conditional expectation with non-additive probabilities. Mimeo. Tel Aviv University. Shafer, G., 1976. A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ. Shapley, L.S., 1953. A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (Eds.), Contributions to the Theory of Games. Annals of Mathematics Studies 28, Vol. II. Princeton University Press, Princeton, NJ, pp. 307–317. Walley, P., 1991. Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, London. Walley, P., Fine, T.L., 1982. Towards a frequentist theory of upper and lower probability. Annals of Statistics 10, 741–761. Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets & Systems 1, 3–28.
Conditional expectation for monotone measures, the discrete case Dieter Denneberg FB 3, Universität Bremen, D-28334 Bremen, Germany Received 9 October 2000; received in revised form 16 January 2002; accepted 3 April 2002
Abstract In modelling economic decision processes with risk and uncertainty one has to update the measures and condition the functions given new information. Non-additive measures (also called Choquet capacities) proved to be an important tool for modelling uncertainty. Here, for functions on a finite set Ω, a comprehensive approach to conditioning is made. It builds on conditional expectation for probability measures in representing the monotone measure ν as a min of belief functions below ν and the latter as a max of additive measures in the core of the respective belief function. The same method is applied, too, for defining the product of monotone measures. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Monotone measure; Choquet integral; Conditional expectation; Product measure
1. Introduction For modelling economic decisions under risk and uncertainty non-additive monotone measures (also called capacities or fuzzy measures) proved more and more to be an appropriate tool as they did already for other areas like information processing, multicriteria decision, insurance, etc. If one regards processes in time, one has to update the measure if new information arrives. This procedure is well understood for stochastic processes with σ -additive measures. Thus, it is important to generalise conditional probability, conditional expectation, products and independence to the case of monotone measures. Many approaches have been made but a closed comprehensive theory is still lacking. Most approaches can be divided in two classes. First, one tries to reduce the problem to the additive case in applying the core of the monotone measure ν (perceived as a cooperative game, the core being an issue of game theory). The approaches of the second class apply the Möbius transform of the monotone measure which can be perceived as an additive E-mail address: [email protected] (D. Denneberg). 0304-4068/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 ( 0 2 ) 0 0 0 1 1 - 3
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(or σ -additive) signed measure on a larger set. Both types of approaches have their limitations. In the first case, there exist monotone measures with an empty core, in the second case one has difficulties to translate back from the level of the Möbius transforms to the level of the original monotone measure. A further approach to conditional expectation for the non-additive case should be mentioned. Lehrer (1996) tries to extend the characterisation of conditional expectation as a projection of L2 -spaces to the non-additive case. But there L2 is no longer a Hilbert space, so Lehrer (1996) allows for non-uniqueness of conditional expectation, or conditional expectation of a function is a family of functions. The present paper completes, for the discrete case, the approaches of the first class in introducing the totally monotone core, consisting of the totally monotone measures (or belief functions) below ν, analogously to the usual core, consisting of the additive measures above ν. Using both cores we can represent an arbitrary monotone measure as a max–min of additive measures, generalising the well-known representation of a totally monotone measure (even a convex monotone measure) as a min of additive measures. This max–min additive representation is valid also for the corresponding integrals and we use it for defining conditional expectation going back to the additive case. Many properties of conditional expectation for the additive case are sustained in the general, but still discrete, case. The max–min representation is also a useful tool for defining the product of monotone measures on the basis of the product of additive measures. It turns out that this product coincides with the product defined for totally monotone measures via the Möbius transform of the monotone measures involved. The min–max additive representation has been generalised to the non-discrete case in Brüning and Denneberg (2002) and, with a compact metric space Ω, a min–max σ -additive representation is achieved. These results will be applied to products in Brüning (2002). The author thanks A. Chateauneuf and G.A. Koshevoy for helpful discussions.
2. Monotone measures and the Choquet integral In this section we fix the terminology and recall some well-known facts about monotone measures and their integral. For simplicity we suppose that Ω is a non-empty finite set. Throughout, a set function ν on a set system S ⊂ 2Ω with ∅, Ω ∈ S is a real valued function ν : S → [0, 1] with ν(∅) = 0. Furthermore, if not otherwise stated, we suppose that it is normalised, i.e. ν(Ω) = 1. A set function ν is called a monotone measure if A ⊂ B implies ν(A) ≤ ν(B). A set function ν is called k-monotone, k ≥ 2, if for A1 , . . . , Ak ⊂ Ω k |I | (1) ν Ai + (−1) ν Ai ≥ 0. i=1
I ⊂{1,...,k} I =∅
i∈I
2-monotonicity is also called supermodularity or convexity. Submodularity is the corresponding property with the reversed inequality sign. ν is totally monotone or a belief function if it is monotone and k-monotone for any k ≥ 2.
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Next we adopt some notions and results from cooperative game theory. A unanimity game uK for ‘coalition’ K ∈ 2Ω , K = ∅, is the monotone measure defined by uK (A) = 1 iff A ⊃ K and = 0 else. The additive measures among the {0, 1}-valued monotone measures are the Dirac measures u{ω} , ω ∈ Ω, and we get uK = u{ω} . (2) ω∈K
Here and in the sequel the lattice operator ∨ denotes the minimum or infimum and ∧ the maximum or supremum. Every {0, 1}-valued monotone measure η can be written as a maximum of unanimity games η= uKj (3) j
where Kj are the minimal sets K with η(K) = 1. In cooperative game theory or combinatorics it is well known that every monotone measure ν can uniquely be represented as linear combination of unanimity games (see e.g. Denneberg , 2000a), the coefficient of uK being denoted µν (K), ν= µν (K)uK . (4) K⊂Ω K =∅
µν is called the Möbius representation of ν. In some sense it can be perceived as a derivative of ν since (4) reformulates to ν(A) = K⊂A µν (K) which corresponds to the fundamental theorem of calculus. If the coefficients in (4) are all non-negative, µν ≥ 0, then ν is totally monotone like the uK , and the converse is also true. Thus, total monotonicity corresponds to ordinary monotonicity of differentiable functions which is characterised by non-negative derivative. A monotone measure ν is additive iff µν (K) = 0 for |K| > 1, i.e. µν lives on the singletons of 2Ω , which form a antichain, i.e. there is no inclusion relation between any pair of singletons. A class of monotone measures based on chains will play an important rˆole in the sequel, namely the necessity measures. A monotone measure ν is a necessity measure iff there is a chain1 K ⊂ 2Ω and a monotone set function κ : K → [0, 1] such that ν is the inner set function κ∗ (A) := κ(K), A ∈ 2Ω . K∈K K⊂A
of κ.2 The necessity measures among the {0, 1}-valued monotone measures are the unanimity games uK . This is easily seen regarding a chain containing K. Proposition 2.1. A necessity measure is totally monotone. A chain in 2Ω is a nested sequence of subsets and we always suppose that a chain contains ∅, Ω. cf. Example 6.3 in Denneberg and Grabisch (1999) to see that the present definition coincides with the original one in Shafer (1976), Zadeh (1978). 1
2
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This fact is well known Shafer (1976). We present a proof since it shows that the Möbius representation of ν lives on the chain K, a fact needed later on. Proof (see Proposition 3.1 in (Denneberg , 2000b)). Let ν = κ∗ be the necessity measure as above and denote the elements of the chain K by ∅ = K0 ⊂ · · · ⊂ Kn = Ω. We show that µ(Ki ) := κ(Ki ) − κ(Ki−1 ),
i = 1, . . . , n,
for A ∈ 2Ω \ K
µ(A) := 0
is the Möbius representation of ν. For A ∈ 2Ω let iA be the maximal index i such that Ki ⊂ A. Then ν(A) =
κ(K) = κ(KiA ) =
K∈K K⊂A
iA i=1
µ(Ki ) =
µ(B),
B⊂A
and this is Eq. (4). Finally, µν = µ ≥ 0 implies that ν is totally monotone.
We recall the definition of the conjugate ν¯ of a set function ν, ν¯ (A) := 1 − ν(Ω \ A). The conjugate of a submodular monotone measure is supermodular and vice versa. The conjugate of a necessity measure is usually called a possibility measure and the conjugate of a belief function is a plausibility function. In applications to imprecise probabilities the pair (ν, ν¯ ) describes the imprecision or uncertainty, especially if ν ≤ ν¯ . The difference ν¯ − ν is called ambiguity or vagueness (see e.g. Denneberg , 2000a, Section 9). We do not repeat here the definition and elementary properties of the (Choquet) integral X dν of a function X : Ω → R w.r.t. a monotone measure ν, refer e.g. to Denneberg (1994a) or Denneberg (2000a). Since here Ω is a finite set and ν is finite the integral always exists and is a real number. We also will have to integrate functions with an extended range, which will be described first. In Section 4, we shall need an element in the range of our functions, which behaves neutral w.r.t. the minimum and the maximum simultaneously, i.e. like +∞ w.r.t. the min and like −∞ w.r.t. the max. So we join to R one infinite far point, denoted again ∞ as it is common use for the projective line, with the following properties x ∧ ∞ := x,
x ∨ ∞ := x,
x ± ∞ := ∞,
x · ∞ := ∞
for x ∈ R∞ := R ∪ {∞}. In accordance with these properties we use the following convention for arbitrary S ⊂ R∞ if S ∩ R = ∅, ∞ x := x else, x∈S x∈S∩R
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and similarly with . Furthermore we extend the order relation ≤ on R to a relation, again denoted ≤, on R∞ by x ≤ ∞,
x≥∞
for all x ∈ R∞ .
Be aware that ≤ is not an order relation on R∞ since antisymmetry and transitivity fail. With this extension an inequality between R∞ -valued functions is true if it holds for those points of the domain for which both functions assume finite values, i.e. values in R. For integrating R∞ -valued functions Z : Ω → R∞ w.r.t. a monotone measure ν the appropriate definition seems to be with ΩZ := {ω ∈ Ω|Z(ω) = ∞}. Z dν := Z|ΩZ d(ν|2ΩZ ) Notice that ν|2ΩZ is, in general, not normalised any more like ν. Of course, for real valued Z, i.e. ΩZ = Ω, this definition coincides with the ordinary Choquet integral.
3. The max–min additive representation of the Choquet integral It can easily be seen (Denneberg , 2000b) that any monotone measure ν is a maximum of linear combinations of unanimity games with positive coefficients, hence a maximum of totally monotone measures. On the other hand, any totally monotone measure is a minimum of additive measures. Similarly, the Choquet integral w.r.t. an arbitrary monotone measure is a max–min of additive integrals. The tool for the min representation is the well-known core of a cooperative game. The (additive) core of a set function ν on 2Ω is C+ (ν) := {α|α additive on 2Ω , ν ≤ α} (recall that, according to our general assumptions, α(Ω) = 1 = ν(Ω)). Example 3.1. α is a member of the core C+ (uK ) of a unanimity game uK iff α is an additive measure on 2Ω with α(K) = 1 or iff α = ω∈K α(ω)u{ω} . The last formula is the Möbius representation of α, which is a convex combination of Dirac measures u{ω} . Proposition 3.1. If ν is supermodular and monotone on 2Ω then C+ (ν) = ∅ and X dα, α, X dν = ν= α∈C+ (ν)
(5)
α∈C+ (ν)
where X : Ω → R. This well-known fact (Shapley , 1953) proves with the Hahn-Banach theorem (see e.g. Denneberg , 1994a, Proposition 10.3). We present a short proof for totally monotone ν since we apply only this case in the sequel and details of proof are essential to prove Corollary 3.3.
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Proof. For totally monotone ν. Let K, ∅ = K0 ⊂ · · · ⊂ Kn = Ω, be a maximal chain containing the upper level sets {X > x}, x ∈ R, of X. Enumerate the elements of Ω so that Ki \ Ki−1 = {ωi }, i = 1, . . . , n. Now we construct an α ∈ C+ (ν) in shifting the mass of µν on 2Ki \ 2Ki−1 to ωi , α(ωi ) = µα (ωi ) := µν (A), i = 1, . . . , n. A∈2Ki \2Ki−1
Since µν ≥ 0 by assumption, these settings define an additive measure α ≥ 0. By induction on i one checks, using the Möbius representation (4), that α(Ki ) = ν(Ki ). Then the decreasing distribution functions of X w.r.t. α and w.r.t. ν are identical and so are the respective integrals. It remains to show that ν ≤ α.First observe that 2Ki \ 2Ki−1 consists A Ki Ki−1 ) for any A ∈ 2Ω . of all subsets of Ki containing ωi , whence 2 ⊂ i:ωi ∈A (2 \ 2 This implies ν(A) = B⊂A µν (B) ≤ ω∈A α(ω) = α(A). Analogously to C+ (ν), but with the dual view replacing ≤ with ≥, we define the totally monotone core as C (ν) := {β|β totally monotone on 2Ω , β ≤ ν}. As noted already at the beginning of this section, C (ν) = ∅, ν = β
(6)
β∈C (ν)
for arbitrary monotone ν. We can say something more: Proposition 3.2. Given a monotone measure ν on 2Ω and a chain K ⊂ 2Ω , then the necessity measure β := (ν|K)∗ belongs to C (ν). Proof. Since β is totally monotone by Proposition 2.1 we have only to show that β ≤ ν. But this is plain since the inner set function β is the smallest monotone measure extending ν|K monotonically to the whole powerset 2Ω , and ν is such an extension. Combining Propositions 3.2 and 3.1 we get the result, which is basic for the rest of the paper. Corollary 3.3. For a monotone measure ν on 2Ω and a function X : Ω → R there exists a necessity measure βX ∈ C (ν) \ it and an additive measure αX ∈ C+ (βX ) such that X dν = X dβX = X dαX . Proof. Like in proof of Proposition 3.1 let K be a maximal chain containing the upper level sets {X > x}, x ∈ R, of X. Set βX := (ν|K)∗ and αX (ωi ) := µβ (Ki ), i = 1, . . . , n, where ∅ = K0 ⊂ · · · ⊂ Kn = Ω are the elements of the chain K and Ki \ Ki−1 = {ωi }. By construction ν, βX and αX (see proof of Proposition 2.1) coincide on K, hence the decreasing distribution functions of X w.r.t. the three monotone measures are identical and
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so are the respective integrals. Finally βX ∈ C (ν) was shown in Proposition 3.2 and αX ∈ C+ (βX ) was shown within proof of Proposition 3.1. Since the integral behaves monotone w.r.t. monotone measures the corollary implies ν= X dα. (7) α, X dν = β∈C (ν) α∈C+ (β)
β∈C (ν) α∈C+ (β)
We call these formulas the max–min additive representation of ν and of the Choquet integral w.r.t. ν, respectively. For {0, 1}-valued monotone measures η the max–min additive representation is very natural (cf. (3), (2)) and well known (cf. e.g. Ghirardato and Le Breton , 2000) η= u{ω} , X dη = X(ω). (8) K⊂Ω ω∈K uK ≤η
K⊂Ω ω∈K uK ≤η
In the max–min additive representation one could also replace the totally monotone core by the necessity core Cnec (ν) := {γ |γ a necessity measure on 2Ω , γ ≤ ν}. It is contained in the totally monotone core (Proposition 2.1), Cnec (ν) ⊂ C (ν). It would be interesting to know if the totally monotone core is the convex hull of the necessity core. 4. Conditional expectation The present approach to conditional expectation starts with conditional expectation for additive measures and generalises it by means of the max–min additive representation of monotone measures. Things become more complicated than in the last section since there is no analogue to Corollary 3.3. Let be given a function X : Ω → R and a subalgebra B of 2Ω . First we recall the classical issue of conditional expectation Eα (X|B) of X w.r.t. an additive measure α on 2Ω , i.e. a probability measure. Eα (X|B) is characterised as the unique (up to addition of α-nullfunctions) B-measurable function Y with the property X − Y dα = 0 for all B ∈ B. (9) B
Since we are concerned with different monotone measures simultaneously, we produce uniqueness of Y not in the usual way in building classes of functions differing only by α-nullfunctions, but in the following way which works at least in the discrete case. We extend the real line R by one infinite far point ∞ to R∞ as desribed at the end of Section 2 and for the B-measurable function Y in (9) we require Y (ω) := ∞ iff α(Bω ) = 0,
(10)
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where Bω denotes the atom of B containing ω. As a result, conditional expectations Eα (X|B) remain ordinary functions, but with extended range R∞ . With this setting we recall the explicit formula for Eα (X|B). Since we are in the discrete case, B-measurability means that the function is constant on the atoms of the algebra B. Hence (9) implies 1 X dα for α(Bω ) = 0, Eα (X|B)(ω) = α(Bω ) Bω (11) ∞ else, i.e. on an atom A of B with α(A) = 0 the value of Eα (X|B) is the average of X on A. Example 4.1. For the Dirac measure δ = u{ω1 } at point ω1 ∈ Ω we get Eδ (X|B)(ω) = since
Bω1
X(ω1 ) if Bω ω1 ∞ else
X dδ = X(ω1 ) and δ(Bω ) = 1 iff Bω ω1 .
Now we define analogously to (7) the conditional expectation of X : Ω → R w.r.t. a monotone measure ν on 2Ω given the algebra B ⊂ 2Ω as Xν\B := Eα (X|B). (12) β∈C (ν) α∈C+ (β)
According to our convention about the use of ∞, the value Xν\B (ω) is finite iff there is some β ∈ C (ν) and some α ∈ C+ (β) such that Eα (X|B)(ω) is finite. Proposition 4.1. Given a monotone measure ν on 2Ω , an algebra B ⊂ 2Ω and a function X : Ω → R. Then the conditional expectationXν\B is a B-measurable function. If Xν\B (ω) = ∞ then ν(Bω ) = 0. The converse of the last implication does not hold as Example 4.2 will show. Proof. Xν\B is a B-measurable function since the Eα (X|B) are B-measurable and the infimum and the supremeum of measurable functions is again measurable for finite Ω. According to our rules for the infinite point ∞ we get Xν\B (ω) = ∞ iff α∈C+ (β) Eα (X|B)(ω) = ∞ for all β ∈ C (ν). This, again, is equivalent to Eα (X|B)(ω) = ∞ for all α ∈ C+ (β) and all β ∈ C (ν). Then, according to (10), α(Bω ) = 0 for all α ≥ β and all β ≤ ν as above. Then the max–min additive representation (7) of ν implies ν(Bω ) = 0. Our first example generalises Example 4.1. Example 4.2. The conditional expectation of X w.r.t. a unanimity game uK is XuK \B (ω) = X(ω1 ). ω1 ∈K∩Bω
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Like above Bω denotes the atom of B containing ω. Proof relies on the fact uK = ω1 ∈K u{ω1 } (2) and property (ii) in Proposition 4.2 below. Since u{ω1 } ∈ C+ (uK ), ω1 ∈ K, the assertion follows taking the min over α ∈ C+ (uK ) of the following relation, Eu{ω1 } (X|B)(ω) = X(ω1 ) ≤ Eα (X|B)(ω), α ∈ C+ (uK ). ω1 ∈K
ω1 ∈K∩Bω
For verifying the last inequality we employ (11) and the fact that α vanishes outside K since α(K) = 1. If K ∩ Bω = ∅ and Bω /K then we have an example as postulated in the remark after Proposition 4.1, uK (Bω ) = 0 but XuK \B (ω) = ∞. The next example shows that the present issue of conditional expectation is related to the generalisation of conditional probability as developed by Fagin and Halpern (1991) and Jaffray (1992) (see also Denneberg , 1994b). Example 4.3. The conditional expectation of an indicator function X = 1A , A ⊂ Ω, w.r.t. a totally monotone β given the algebra B := {∅, B, B c , Ω} with β(B) > 0 is (1A )α\B (ω) (1A )β\B (ω) = α∈C+ (β)
=
α∈C+ (β)
=
α(A ∩ B) α(B)
for ω ∈ B
β(A ∩ B) . ¯ c ∩ B) β(A ∩ B) + β(A
(13)
For the first equation we applied (ii) in Proposition 4.2 and for the last equation we applied Denneberg (1994b), Theorem 2.4. The second equation is well known in the additive theory. The assumption β(B) > 0 guarantees that the denominator does not vanish. The fraction (13) is called “generalised Bayes update” in Walley (1991) or “full Bayesian update” in Cohen et al. (2000). If β is additive or a unanimity game, (13) becomes the ordinary Bayes update β(A ∩ B)/β(B). Dually to the max–min additive representation (7) there is a min–max additive representation X dα. (14) α, X dν = ν= β∈C (¯ν ) α∈C+ (β)
β∈C (¯ν ) α∈C+ (β)
The duality may be more apparent if one reformulates the conditions on α and β in (14), β ∈ C (¯ν ) iff γ := β¯ is a plausibility function and γ ≥ ν, ¯ α ∈ C+ (β) iff α is additive and α ≤ γ = β. Then, also our max–min conditional expectation (12) allows a dual version. The min–max conditional expectation of X w.r.t. a monotone measure ν on 2Ω given the algebra B ⊂ 2Ω is
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Xν/B :=
Eα (X|B).
(15)
β∈C (¯ν ) α∈C+ (β)
Notationally, the min–max conditional expectation is characterised by the forward slash, whereas the max–min conditional expectation by the backward slash. Xν/B is needed to ex tend asymmetry of the Choquet integral, −X dν = − X dν¯ , to conditional expectation, which is property (vii) below. Many properties of conditional expectation w.r.t. a probability measure remain valid for monotone measures except, of course, additivity. We list, with the exception of (ii) and (vii), only the properties for the max–min conditional. The corresponding properties of the min– max conditional prove similarly or can be derived by means of the asymmetry property (vii). Proposition 4.2. Let λ, ν be monotone measures on 2Ω , A, B subalgebras of 2Ω and X, Y real valued functions on Ω. Then (i) for a probability measure α the present definitions coincide with the classical one, Xα\B = Eα (X|B) ; (ii) for a totally monotone measure β and its conjugate β¯ Xβ\B = Eα (X|B), Xβ\ Eα (X|B) ¯ B = α∈C+ (β)
α∈C+ (β)
and the max–min and min–max conditional expectations coincide, Xβ\B = Xβ/B ,
Xβ\ ¯ B = Xβ/ ¯ B.
For arbitrary monotone ν we have only the inequality
(iii) (iv) (v) (vi) (vii) (viii) (ix)
Xν\B ≤ Xν/B ; Xν\B = X dν if B := {∅, Ω}; Xν\B = X if X is B-measurable, especially if B = 2Ω ; (XY)ν\B = X(Yν\B ) if X ≥ 0 is B-measurable, in particular (X constant) conditional expectation is positively homogenous; (X + Y )ν\B = Xν\B + Y if Y is B-measurable; (−X)ν\B = −Xν¯ /B ; X ≤ Y implies Xν\B ≤ Yν\B ; λ ≤ ν implies Xλ\B ≤ Xν\B .
We have arranged the properties in such a way that first the properties with a fixed monotone measure and a fixed hypothesis are listed, whereas the last ones are concerned with several hypotheses or several measures simultaneously. Nevertheless, property (ix) is important for proving some of the preceding properties. Proof. Most properties are derived from the corresponding results for additive measures, which are supposed to be known (see e.g. Dudley , 1989). (i) follows from the first equation in (ii) since C+ (α) = {α}.
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(ii) For β ∈ C (β) we have C+ (β ) ⊃ C+ (β) so that α∈C+ (β ) Eα (X|B) ≤ α∈C+ (β) Eα (X|B) . Since β ∈ C (β) the first assertion follows. Using this result we get Xβ\ Xβ \B ≥ Xα\B = Xα\B . ¯ B = ¯ β ∈C (β)
¯ α∈C (β) α additive
α∈C+ (β)
Here, the inequality is plain since the sup on the right hand side is taken with a smaller set than on the left hand side. The last equality is clear since α ≤ β¯ iff α ≥ β. Finally, the reversed inequality follows from Lemma 4.3 using the monotonicity property (ix). The corresponding property for the min–max conditional shows that the max–min and min–max conditionals coincide for totally monotone β and its conjugate. Finally, for an arbitrary ν we derive from (ix) for β ≤ ν ≤ γ , γ a plausibility function, Xβ\B ≤ Xγ \B = Xγ /B so that
Xν\B =
Xβ\B ≤
β∈C (ν)
Xγ /B = Xν/B .
γ¯ ∈C (¯ν )
(iii) Xν\B = β∈C (ν) α∈C+ (β) Eα (X|B) = β∈C (ν) α∈C+ (β) X dα = X dν, where, for the last equation we applied (7). (iv) proves like (iii) X is B-measurable now. Eα (X|B) = X since but with (v) (XY)ν\B = Eα (XY|B) = X Eα (Y |B) β∈C (ν) α∈C+ (β)
=X
β∈C (ν) α∈C+ (β)
Eα (Y |B) = X Yν\B .
β∈C (ν) α∈C+ (β)
(vi) (X + Y )ν\B =
β∈C (ν) α∈C+ (β)
=
Eα (X + Y |B) =
(Eα (X|B) + Y )
β∈C (ν) α∈C+ (β)
Eα (X|B) + Y = Xν\B + Y.
β∈C (ν) α∈C+ (β)
(vii) First for a totally monotone β we derive from equations (ii) Eα (−X|B) = − Eα (X|B) = −Xβ/ (−X)β\B = ¯ B. α∈C+ (β)
α∈C+ (β)
Then replacing in (12) and (15) the expressions from (ii) we get (−X)β\B = (−Xβ/ Xβ/ (−X)ν\B = ¯ B) = − ¯ B = −Xν¯ /B . β∈C (ν)
β∈C (ν)
β∈C (ν)
For the last equation observe that ν¯¯ = ν. (viii) easily derives from the corresponding property for additive measures.
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(ix) By λ ≤ ν we get C (λ) ⊂ C (ν) so that, in the definition of conditional expectation, the maximum is taken over a larger set in case of ν than in case of λ from which the assertion follows. For proving (ii) we used the following version of the sandwich theorem, which is interesting for its own right (see Chateauneuf , 1994). Lemma 4.3. Let β, β be supermodular monotone measures on 2Ω and suppose ¯ β ≤ β. Then there exists an additive monotone measure α ∈ C+ (β) ∩ C+ (β ), i.e. ¯ β ≤ α ≤ β. Proof. On the space, say, of bounded functions X : Ω → R the Choquet integral Γβ X := X dβ is a superadditive and positively homogenous (short superlinear) monotone functional. Similarly Γβ¯ X := X dβ¯ is sublinear and Γβ ≤ Γβ¯ . By the sandwich theorem in Fuchssteiner and Lusky (1981) there exists a monotone linear functional Λ such that Γβ ≤ Λ ≤ Γβ¯ . Then α(A) := Λ1A has the desired properties. Which should be the appropriate form of (9) in the non-additive case? Lehrer (1996) perceives (X − Xν\B )ν\B = 0 as a generalisation of (9) for monotone measures. In our context this equation3 follows from property (vi) since −Xν\B is B-measurable (Proposition 4.1). A partial result with the original form of (9) can already be found at the end of Denneberg (1994b). Proposition 4.4. Let β be totally monotone on 2Ω , B ⊂ 2Ω an algebra and X a real function on Ω. Then X − Xβ\B dβ ≥ 0 for B ∈ B with β(A) > 0 for all atoms A ⊂ B of B B
and equality holds if B is an atom of B itself with β(B) > 0. Here, we understand that B Z dν := 1B Z dν for R∞ -valued functions Z (see end of Section 2). In Denneberg (1994b) this equality for atoms B had been used to define conditional expectation for sub- and supermodular monotone measures. Proof. The integrand Z := 1B (X − Xβ\B ) may assume the value ∞, so we have X − Xβ\B dβ = Z|ΩZ d(β|2ΩZ ). B
3 In order to restrict technicalities, we have defined conditional expectation only for real valued functions, so—strictly speaking—one has to suppose here that Xν\B is real valued.
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Like in proof of Proposition 3.1 we can find an α ∈ C+ (β) which coincides with β on the chain of upper level sets of Z|ΩZ in 2ΩZ ⊂ 2Ω . Then ΩZ ΩZ Z|ΩZ d(β|2 ) = Z|ΩZ d(α|2 ) ≥ 1B (X − Xα\B )|ΩZ d(α|2ΩZ ) = 0, where the inequality holds since Xβ\B ≤ Xα\B and Xα\B is finite on ΩZ by the assumption on B. That the last integral equals zero is just (9). Now suppose that B is an atom. Then Xβ\B is constant and finite on B (Proposition 4.1) and by property (ii) for any ε > 0 there exists an α ∈ C+ (β) such that Xβ\B (ω) ≥ Xα\B (ω) − ε
for ω ∈ B.
This inequality implies X − Xβ\B dβ ≤ X − Xα\B + ε dβ ≤ X − Xα\B dα + εα(B) = εα(B). B
B
B
Again we applied (9) and, since ε > 0 was arbitrary, we are done.
As pointed out at the end of Section 3, the max–min additive representation of the Choquet integral could have been performed also with the necessity core in place of the totally monotone core. Why did we not apply this representation for defining conditional expectation? Denoting it with Xν\\B for the moment, and the dual one with Xν//B , we had got Xν\\B ≤ Xν\B ≤ Xν/B ≤ Xν//B . We decided for the alternative for which the pair of dual conditionals is closer to each other.
5. Products There are many approaches for a product of monotone measures (e.g. Walley and Fine , 1982; Gilboa and Schmeidler , 1989; Hendon et al. , 1996; Koshevoy , 1998; Denneberg , 2000b). Here, we define a product again by means of the max–min additive representation of monotone measures going back to the product of additive measures. For supermodular set functions this method is known (Walley and Fine , 1982; Gilboa and Schmeidler , 1989). Surprisingly, for totally monotone measures this product coincides with the Möbius product of Hendon et al. (1996). Let Ω1 and Ω2 be finite sets and Ω := Ω1 ×Ω2 their cartesian product. For two monotone measures ν1 on 2Ω1 and ν2 on 2Ω2 we define their (max–min) product by ν1 ⊗ ν2 := α1 ⊗ α 2 , (16) βi ∈C (νi ) αi ∈C+ (βi ) i=1,2 i=1,2
where α1 ⊗ α2 denotes the usual product on 2Ω of the additive measures αi on 2Ωi , i = 1, 2. Obviously, the product is a monotone set function. Also, the product is a monotone operator, λi ≤ νi , i = 1, 2, implies λ1 ⊗ λ2 ≤ ν1 ⊗ ν2
(17)
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for monotone λ1 , ν1 on 2Ω1 , λ2 , ν2 on 2Ω2 . For totally monotone βi on 2Ωi , i = 1, 2, β1 ⊗ β2 := α1 ⊗ α 2 . (18) αi ∈C+ (βi ) i=1,2
For unanimity games uKi on 2Ωi , Ki ⊂ Ωi , i = 1, 2, one easily shows (Example 3.1, Proposition 3.1) uK1 ⊗ uK2 = uK1 ×K2 . Proposition 5.1. For totally monotone βi on 2Ωi , i = 1, 2, β1 ⊗ β2 = µβ1 (K1 )µβ2 (K2 )uK1 ×K2 ,
(19)
(20)
K1 ,K2
where Ki runs through all non-empty subsets of Ωi , i = 1,2. The sum on the right hand side of (20) is the Möbius product of Hendon et al. (1996) (see also Denneberg , 2000b). Since the Möbius product for arbitrary monotone measures is not monotone in general, an alternative product had been proposed in Section 6 of Denneberg (2000b) which, by Proposition 5.1, coincides with the present definition of a product. Proposition 5.1 implies that the product behaves associative at least for totally monotone set functions. See Denneberg (2000b) for further properties and examples. Before proving Proposition 5.1 we recall a part of Corollary 4 in Danilov and Koshevoy (2000), which is already implicit in Shapley (1953). Lemma 5.2. Let β be a totally monotone measure with Möbius representation β= µβ (K)uK . K⊂Ω K =∅
Then for any α ∈ C+ (β) there are αK ∈ C+ (uK ), K ⊂ Ω, K = ∅, such that α= µβ (K)αK . K⊂Ω K =∅
Conversely any sum like on the right hand side is an element of C+ (β). Hence, C+ (β) is a convex combination of the C+ (uK ) with coefficients µβ (K), K ⊂ Ω, K = ∅, C+ (β) = µβ (K)C+ (uK ). K⊂Ω K =∅
Proof of Proposition 5.1. Applying Lemma 5.2 to β1 on 2Ω1 and β2 on 2Ω2 we get
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β1 ⊗ β 2 =
α1 ⊗ α 2
α1 ∈C+ (β1 ) α2 ∈C+ (β2 )
=
(αL1 )L1 (αL2 )L2 αL1 ∈C+ (uL1 ) αL2 ∈C+ (uL2 )
=
µβ1 (K1 )αK1 ⊗
µβ2 (K2 )αK2
K1
µβ1 (K1 )µβ2 (K2 )
K1 ,K2
=
K2
µβ1 (K1 )µβ2 (K2 )αK1 ⊗ αK2
K1 ,K2 (αL1 )L1 (αL2 )L2 αL1 ∈C+ (uL1 ) αL2 ∈C+ (uL2 )
=
119
αK1 ⊗ αK2
αK1 ∈C+ (uK1 ) αK2 ∈C+ (uK2 )
µβ1 (K1 )µβ2 (K2 ) uK1 ⊗ uK2 .
K1 ,K2
Here, Ki , Li run over all non-empty subsets of Ωi and (αLi )Li denotes the collection of the αLi , Li ∈ 2Ωi \ {∅}. For the third equation we used that the product of additive measures is a bilinear operator. For the second and the fourth equation µβi ≥ 0 is crucial, i.e. βi totally monotone. Fianlly, the last equation follows from (18) applied with uK1 , uK2 . Like for conditional expectation there is a dual approach to the product (16) by means of the min–max additive representation (14). The min–max product of ν1 and ν2 is defined as ¯ ν2 := α1 ⊗ α 2 , (21) ν1 ⊗ βi ∈C (νi ) αi ∈C+ (βi ) i=1,2 i=1,2
The min–max product coincides with the max–min product for belief functions and plausibility functions and for the general case an inequality remains true. Proposition 5.3. Let βi be totally monotone on 2Ωi , i = 1, 2, then ¯ β2 , β1 ⊗ β2 = β1 ⊗
¯ β2 . β1 ⊗ β2 = β1 ⊗
For arbitrary monotone measures νi on 2Ωi , i = 1, 2, ¯ ν2 = ν1 ⊗ ν2 , ν1 ⊗ ¯ ν2 . ν1 ⊗ ν2 ≤ ν1 ⊗ Proof. Proof uses Lemma 4.3, analogously to the reasoning for Proposition 4.2 (ii), β1 ⊗ β2 = β1 ⊗ β2 ≥ α1 ⊗ α 2 βi ∈C (βi ) i=1,2
=
αi ∈C+ (βi ) i=1,2
αi ∈C (βi ) additive i=1,2
¯ β2 . α1 ⊗ α 2 = β 1 ⊗
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Here, the inequality is plain since the sup on the right hand side is taken with a smaller set than on the left hand side. The next equality is clear since αi ≤ βi iff αi ≥ βi . The last ¯ Finally, the reversed inequality follows from Lemma equality is the analogue of (18) for ⊗. 4.3 using monotonicity property (17). Proof for belief functions runs similarly. For monotone ν1 , ν2 we get ν1 ⊗ ν2 (A) = 1 − β1 ⊗ β2 (Ac ) = (1 − β1 ⊗ β2 (Ac )) βi ∈C (νi ) i=1,2
=
βi ∈C (νi ) i=1,2
β1 ⊗ β2 (A) =
βi ∈C (νi ) i=1,2
¯ β2 (A) = ν1 ⊗ν ¯ 2 (A). β1 ⊗
βi ∈C (νi ) i=1,2
¯ β2 ¯ and, for the fourth equation, β1 ⊗ β2 = β1 ⊗ Again we used the analogue of (18) for ⊗ which proves with similar arguments. The last assertion proves like the inequality in Proposition 4.2 (ii).
6. Conclusion For the discrete case with finite Ω we have seen that the max–min additive representation of the Choquet integral provides a unifying tool to generalise conditional expectation and products for arbitrary monotone measures. Remaining questions are: how to treat iterated conditioning? How stochastic independence should be incorporated in the present approach? What about the non-discrete case? Some answers can be found in Brüning (2002), Brüning and Denneberg (2002). But the main question is if the concepts developed here can become useful in applications like information processing, economic decision processes, finance, insurance etc. For this purpose at least an inequality between iterated and one step conditioning is desirable.
References Brüning, M., 2002. Products of Non-additive Measures, Universität Bremen (preprint). Brüning, M., Denneberg, D., 2002. Max–min σ -additive representation of monotone measures. Statistical Papers 34, 23–35. Chateauneuf, A., 1994. Combination of compatible belief functions and relation of specifity. In: Fedrizzi, M., Kaprczyk, J., Yager, R.R. (Eds.), Advances in the Dempster–Shafer theory of evidence. Wiley, New York, pp 97–114. Cohen, M., Gilboa, I., Jaffray, J.Y., Schmeidler, D., 2000. An experimental study of updating ambiguous beliefs. Risk Decision and Policy 5, 123–133. Danilov, V.I., Koshevoy, G.A., 2000. Cores of cooperative games, superdiffereentials of functions and the Minkowski difference of sets. Journal of Mathematical Analysis and Applications 247, 1–14. Denneberg, D., 1994a. Non-additive Measure and Integral, Theory and Decision Library, 2nd Edition, Vol. 27, Series B. Kluwer Academic Publishers, Dordrecht. Denneberg, D., 1994b. Conditioning (updating) non-additive measures. Annals of Operations Research 52, 21–42.
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