A Dutch book coherence condition for conditional completely alternating Choquet expectations

Stemming from de Finetti’s coherence for finitely additive (conditional) probabilities, the paradigm of coherence has been extended to other uncertainty calculi. We study the notion of coherence for conditional completely alternating Choquet expectations, providing an avoiding Dutch book like condition.


Introduction
In the second half of the past century, decision theory and artificial intelligence saw the introduction of different non-additive measures and integrals, generalizing probability and expectation. That was motivated by the need to manage knowledge acquisition and decision processes when the available information is incomplete, imprecise or expressed in vague terms. In this context, the Dempster-Shafer theory of evidence [7,17,18] is the most known framework, relying on two non-additive dual set functions Bel and Pl, said belief and plausibility functions. Belief and plausibility functions are connected to finitely additive (f.a. for short) probabilities, since every f.a. probability P on an algebra A determines a belief and a Belief and plausibility functions have been introduced in [7,17,18]. We stress that, in this paper no form of continuity is required for Bel and Pl.
Every plausibility function Pl and its dual belief function Bel on A are (see Theorem A in [11]) in bijection with a finitely additive probability measure μ defined on an algebra A possibly strictly contained in P(U), where U = A 0 . The algebra A is built as follows: for every A ∈ A, define A L , A U ∈ P(U), as A L = {B ∈ U : B ⊆ A} and A U = {B ∈ U : B ∩ A = ∅} and let A be the algebra generated by {A L : A ∈ U} (or, equivalently, by {A U : A ∈ U}). The finitely additive probability μ on A allows to provide an integral expression of Bel and Pl obtained, for every A ∈ A, as where the integrals are of Stieltjes type [1]. The finitely additive probability μ on A is called the Möbius inverse of Bel. This paper adopts the conditioning rule expressed by the axiomatic definition of conditional plausibility or belief functions given in [5], which generalizes the Dempster rule [7], allowing for conditioning to events of null plausibility.
Given a conditional plausibility function, the dual conditional belief function Bel(·|·) is defined, for every E|H ∈ A × H, as Bel(E|H ) = 1 − Pl(E c |H ).
In case the additive class H is finite, then a conditional plausibility Pl(·|·) is completely characterized by a linearly ordered class {Pl 0 , . . . , Pl k } of plausibility functions on A, said minimal agreeing class [5]. Given Pl(·|·) set: This shows, in case of a finite H, the presence of a one-to-one correspondence between the class of conditional plausibility functions on A ×H and the class of minimal agreeing classes on A (see also [5,15]).
multiplication by a constant, as well as ordering comparisons, are always assumed pointwise on Ω. Let L(Ω) be the set of all bounded gambles which is a topological linear space under the topology of uniform convergence induced by the supremum norm X = sup ω∈Ω |X (ω)|. In particular, denote by L(Ω) + = {X ∈ L(Ω) : X ≥ 0} the convex cone of non-negative bounded gambles. We further denote by L(Ω) * the topological dual space of L(Ω). If we set U = P(Ω) 0 , then L(U), L(U) + and L(U) * have analogous meaning. Let Bel and Pl be a pair of dual belief and plausibility functions on P(Ω), then denote by C Bel and C Pl the functionals on L(Ω), determined by the Choquet integral with respect to Bel and Pl, respectively: for every X ∈ L(Ω), For every k ≥ 2, for every X 1 , . . . , X k ∈ L(Ω), it holds that where ∧ and ∨ denote the pointwise minimum and maximum of functions. C Bel satisfies an analogous property, obtained by inverting the inequality and by switching minima and maxima. Then, we refer to C Bel and C Pl as completely monotone and completely alternating Choquet expectations. Notice that the two functionals are dual, that is, for every X ∈ L(Ω), it holds C Bel (X ) = −C Pl (−X ). Moreover, they can be interpreted as lower and upper expectation functionals, since, for every X ∈ L(Ω), it holds For X ∈ L(Ω), the corresponding lower and upper generalized gambles X L , X U ∈ L(U) are defined, for every B ∈ U, as If μ is the Möbius inverse of Bel defined on the algebra A generated by {A L : A ∈ U}, then Corollary 2 in [14] implies The following proposition will be useful in the rest of the paper.

Proposition 3.1 Let U = P(Ω) 0 and μ be the Möbius inverse on A of the belief function
Bel, the latter defined on P(Ω) and with dual Pl. Then, for every X ∈ L(Ω) and every H ∈ P(Ω) 0 , it holds that Proof Since X ∈ L(Ω), there exists a sequence {X n } n∈N of simple functions in L(Ω) converging uniformly to X . We first show that the statement holds for every X n . Suppose By the definition of the Choquet integral for simple functions [8,12], we have that where the second equality follows since Pl( converges to X uniformly, for every > 0, there exists n 2 ∈ N such that for every n ≥ n 2 it holds that sup ω∈Ω |X n (ω) − X (ω)| < 2 . For every n ≥ n 2 and every B ∈ U such that B ∩ H = ∅, by condition (x) of Theorem 4.13 in [19], we have that Therefore, since X U Finally, by the continuity of both the Choquet integral and the Stieltjes integral with respect to the topology of uniform convergence [19], and the fact that limits keep equalities, we have that Notice that all finitely additive probability measures on P(U) extending μ give rise to the same Bel and Pl on P(Ω) through 1 and satisfy the equation in Proposition 3.1.

Conditioning and coherence
Given a gamble X ∈ L(Ω) and an event H ∈ P(Ω) 0 , a conditional gamble is a pair (X , H ), usually denoted by X |H , which consists in regarding X under the hypothesis H . In particular, a conditional event E|H ∈ P(Ω) × P(Ω) 0 is identified with the conditional gamble 1 E |H , and an unconditional gamble X ∈ L(Ω) is identified with X |Ω.

Also in this case, C Bel (·|·) satisfies an analogous property, obtained by inverting the inequality and by switching minima and maxima.
In what follows the conditional functionals C Bel (·|·) and C Pl (·|·) are said conditional completely monotone and alternating Choquet expectations.
Notice that, C Bel (·|·) and C Pl (·|·) are dual since, for every X ∈ L(Ω) and H ∈ H, it holds that C Bel (X |H ) = −C Pl (−X |H ), therefore we can limit to deal with C Pl (·|·).
We stress that a conditional functional C Pl (·|·) essentially relies on the structure of its domain. If we want to weaken the hypotheses on the domain we need to introduce a suitable notion of coherence.
Let G ⊆ L(Ω) × P(Ω) 0 be an arbitrary subset of conditional bounded gambles. Further, let H G be the additive class generated by {H ∈ P(Ω) 0 : X |H ∈ G}. Here we provide a notion of coherence for an assessment Ψ : G → R, showing its characterization in terms of a form of generalized Dutch book.
Definition 4.2 An assessment Ψ : G → R avoids conditional CPl-Dutch book if, for every n ∈ N, for every F = {X 1 |H 1 , . . . , X n |H n } ⊆ G, and every λ 1 , . . . , λ n ∈ R, the function G F : U → R defined as The function G F can be seen as a random gain under partially resolving uncertainty [13]: we admit bets where the gain can be computed only knowing that an event B = ∅ occurs, even if we are not able to identify the true ω ∈ B. The gain G F is asked not to be uniformly negative over B's contained in H 0 0 .   [11], every such Pl α is in bijection with a finitely additive probability measure μ α defined on A that, by the Hahn-Banach theorem, can be extended, generally not in a unique way, to a continuous positive linear functional with unit norm f α ∈ L(U) * . Notice that the restriction of f α to the indicators of events in P(U) determines a finitely additive probability measure extending μ α , that we continue to denote by μ α . Taking into account Proposition 3.1, the functional f α satisfies, Hence, condition (i) is equivalent to the existence of a linearly ordered class { f 0 , . . . , f k } of elements of L(U) * with the above properties. In turn, this is equivalent to the solvability of the sequence of systems S 0 , . . . , S k defined below.
If f 0 is a solution of S 0 , its restriction to indicators of events in P(U) determines a finitely additive probability measure μ 0 . In turn, μ 0 induces a plausibility function Pl 0 on P(Ω) through 1 satisfying Pl 0 (H 0 0 ) = 1, Pl 0 ((H 0 0 ) c ) = 0 and, by Proposition 3.1, We search for f ∈ L(U) * solving the system S α : Also in this case, system S α has solution if and only if for every λ i ∈ R, i ∈ I α , defining the function G α : U → R as we have sup B⊆H α 0 G α (B) ≥ 0. As before, from a solution f α we derive the finitely additive probability measure μ α on P(U) that induces through 1 the plausibility function Pl α on P(Ω). Notice that the sequence stops at index k such that Pl k (H i ) > 0 for all H i ⊆ H k 0 . Finally, since the same Pl(·|·) determines C Pl (·|·) whose restriction to L(Ω) × H F extends the restriction Ψ |F for every F ⊆ G, repeating the previous construction for such a restriction, we have that taking the supremum of all the resulting functions G α for B ⊆ H α 0 is equivalent to condition (ii). Now we consider an arbitrary G. We only need to prove (ii) ⇒ (i), as the other implication follows by the proof for the finite case. Indeed, if Ψ is CPl-coherent, then also Ψ |F is, for every finite F ⊆ G, and this implies (ii).
For every finite F ⊆ G, let P F ⊆ X |H ∈L(Ω)×H G inf ω∈Ω X (ω), sup ω∈Ω X (ω) , be the set of all inf/sup bounded real-valued functions on L(Ω) × H G whose restriction to L(Ω) × H F is a conditional completely alternating Choquet expectation extending Ψ |F , determined by a conditional plausibility function on P(Ω) × H F . The set P F is not empty by the proof for the finite case and is easily seen to be a closed subset of the compact set for every net {D γ } γ ∈Γ in P F converging pointwise to D, a simple application of properties of limits of real nets and the main Theorem in [16] imply that D ∈ P F . It is also easily seen that the family P = {P F : F = {X 1 |H 1 , . . . , X n |H n } ⊆ G, n ∈ N}, possesses the finite intersection property, thus it holds P = ∅ and so there exists C Pl ∈ P which is a conditional completely alternating Choquet expectation extending Ψ .