Adecuación de un mercado eficiente para el agua

The previous subsection considered the link between Dempster-Shafer Belief Models and Behavioral Imprecise Probability Models when the knowledge operator was a cor- respondence operator. This section investigates what happens for knowledge operators which are Monotonic, but are not necessarily correspondence operators. While such operators retain Monotonicity, they may fail to haveKΩ = Ω, orKE∩KF ⊂K(E∩F)

for some eventsE, F ∈ 2Ω_{. Remark 4.1 has a possible interpretation for agents with}

Monotonic knowledge operators. Such an agent may have multiple ‘selves’, each of which provides correspondence-type information, but which are unable to communi- cate with each other.

Knowledge operatorsK satisfying Monotonicity can be represented using a new formulationbγ : Ω → 2

2Ω

which we call the multi-correspondence associated withK. The multi-correspondenceγbmaps each stateω ∈Ωto the set of events which are both

(a) known atω, and (b) minimal in the sense that no subset of the event is also known atω. Lemma 4.3 guarantees that such a representation exists and is unique.

Lemma 4.3. LetK : 2Ω →2Ωbe a monotonic knowledge operator. Then, there exists a unique functionbγ : Ω→2

2Ω

such that

(a) ω∈KE if and only if there existsF ∈_bγ(ω)such thatF ⊂E.

We callbγthe multi-correspondence associated withK.

Remark 4.1. The concept of multiple-selves has been developed to attempt to explain some of the inconsistencies arising in individual choice theory between the predictions of standard choice theory and empirical observations, for example in Levine and Fudenberg [2006]. In the multiple-self theories, agents are modeled as having a number of different ‘selves’, each of which is rational in some sense, but facing different motivations. This is of particular relevance for time-inconsistent preferences.

Multi-correspondences provide a reasonable framework for considering a multiple-self model for knowledge. This may be the case when each self has at least a correspondence-type level of rationality, but these selves cannot fully integrate their information. Each selfj contributes a correspondenceγj, and the multi-correspondence of the person as a whole is simply the union

b

γ(ω) = [

j∈J

{γj(ω)} (4.7)

In the language of Chapter 2, suppose a collectionJ of selves has correspondence operators

Kj : 2Ω → 2Ω, and associated functionsγj : Ω → 2Ω in the style of Equation 4.3. Then an

agent with multi-correspondences given by the union of the correspondences, as in Equation 4.7, has a knowledge operatorK which is the ‘somebody knows’ operator of Chapter 2.

A reasonable conjecture is that ifKis monotonic and does not satisfy conjunction thenP =_P(K, µ)cannot be represented by a Dempster-Shafer Belief Model. Unfortu- nately, such a conjecture fails, as demonstrated in Example 4.9.

Example 4.9. Let(Ω, K, µ, P)be a Behavioral Imprecise Probability Model whereK is mono- tonic. Let bγ : Ω → 2

2Ω

be the multi-correspondence in Table 4.3. From Lemma 4.3, this generates the knowledge operator, lower probability and implied mass function also shown in Table 4.3.

As long asµ(3) ≥µ(1), thenm is a valid mass function. Moreover,K satisfiesKΩ = Ω

and Axiom D. However, ifµ(3) < µ(1), then the imprecise probabilityP cannot be represented by any Dempster-Shafer Belief Model.

The ideas presented in Example 4.9 are generalized in Definition 4.15 and Proposi- tion 4.14 leading to an algorithm to determine whether a particular Behavioral Impre- cise Probability Model can be represented as a Dempster-Shafer Belief Model. Defini- tion 4.15 defines the alternating count. This is a purely technical object which allows

ω _bγ(ω) 1 {{1,2},{1,3}} 2 {{2}} 3 {{1,2,3}} E KE P∗E =µ(KE) m(E) ∅ ∅ 0 0 {1} ∅ 0 0 {2} {2} µ(2) µ(2) {3} ∅ 0 0 {1,2} {1,2} µ(1) +µ(2) µ(1) {1,3} {1} µ(1) µ(1) {2,3} {2} µ(2) 0 Ω Ω 1 µ(3)−µ(1)

Table 4.3:Example Multi-Correspondence and Knowledge Operator

the construction of a Dempster-Shafer Belief Model from a Behavioral Imprecise Prob- ability Model when this is possible. Proposition 4.14 describes the construction, and the terms under which such a construction are possible.

Definition 4.15. LetK : 2Ω →2Ωbe monotonic, and represented by the multi-correspondence

b

γ as in Lemma 4.3. The alternating count ofK is the functionη: Ω×2Ω _→

Zdefined by η(ω, E) = X E1,...,En∈bγ(ω) s.t.E1∪···∪En=E andEi6=Ej (−1)n+1 (4.8)

The alternating count at (ω, E) counts up the number of ways E can be written as the union of an odd number of elements ofbγ(ω), and counts down the number of

even ways. In the limiting case where K is representable by the correspondence γ, thenη(ω, E) = 1ifγ(ω) =Eandη(ω, E) = 0otherwise. The benefit of the alternating count is that it can be determined directly for each pair (ω, E). For example, for the multi-correspondence in Example 4.9, we findη(1,{1,2,3}) =−1because{1,2,3}can be written as the union of elements ofbγ(1)as{1,2,3}={1,2} ∪ {2,3}, and in no other

way. As this was a union of an even number of elements, the alternating count counts down, to -1.

The concept of the alternating count is useful because it allows us to determine whether a Behavioral Imprecise Probability is representable as a Dempster-Shafer Im- precise Probability. Moreover, the alternating count allows a direct construction of the Dempster-Shafer mass function if this is the case.

Lemma 4.4. Let(Ω, K, µ, P)be a Behavioral Imprecise Probability Model whereK is mono- tonic. Letη: Ω×2Ω _→

Zbe the alternating count ofK. Define the functionm : 2Ω →Rat each eventE ∈2Ωby

m(E) =X

ω∈Ω

η(ω, E)µ(ω) (4.9)

Then P∗E = P_F∈2Ω_|F⊂Em(F) and m is the only function 2Ω → R with this property. Moreover, ifKΩ = Ω, then P E =   X F∈2Ω_|F⊂E m(F), X F∈2Ω_|F∩E6=∅ m(F)  

Proposition 4.14. Let (Ω, K, µ, P)be a Behavioral Imprecise Probability Model whereK is monotonic. Letη : Ω×2Ω _→

Zbe the alternating count ofK. Letmbe defined as in Equation 4.9.

The imprecise probabilityP is a Dempster-Shafer Imprecise Probability if and only ifmis a Dempster-Shafer mass function.

One consequence of Proposition 4.14 is that we can identify circumstances in which monotonic information structures can be consistent, or inconsistent, with Dempster- Shafer Belief Models. As shown in Corollary 4.3, every information structure where the knowledge operator is monotonic, but not conjunctive, will be inconsistent with Dempster-Shafer for at least some probability distributionsµ. IfK is monotonic, but not conjunctive, then there are distributionsµsuch thatP(K, µ)is not Dempster-Shafer. Corollary 4.4 goes further and establishes a technical condition for K under which P(K, µ)is not consistent with Dempster-Shafer for all distributionsµ. Specifically, if the alternating countηis non-positive over allω ∈Ω, for some eventF ∈2Ω_{, and strictly}

negative somewhere, thenP(K, µ)is not a Dempster-Shafer Imprecise Probability for any probabilityµ.

Corollary 4.3. Let (Ω, K) be an information structure such that K is monotonic and does not satisfy conjunction. Then, there exists a distributionµ ∈ ∆Ω _{and Behavioral Imprecise}

Probability Model(Ω, K, µ, P)such thatP is not a Dempster-Shafer Imprecise Probability.

Corollary 4.4. Let(Ω, K)be an information structure such thatKis monotonic andKΩ = Ω. Let η be the alternating count of K. Suppose that there exists an event F ∈ 2Ω _{such that}

any Behavioral Imprecise Probability Model(Ω, K, µ, P), the imprecise probabilityP is not a Dempster-Shafer Imprecise Probability.