Existence and Uniqueness

Questions of existence and uniqueness of Behavioral Imprecise Probability Models exhibit a surprising amount of nuance. For existence, there is the easy question of whether, for a tuple (Ω, K, µ), there exists a Behavioral Imprecise Probability Model

(Ω, K, µ, P). This simple problem was addressed previously, and is simply to check whetherµ(KE) +µ(K¬E)≤1for all eventsE.

The more interesting question is whether an information structure(Ω, K)admits a Behavioral Imprecise Probability Model (Ω, K, µ, P). Some knowledge operators have a structure which guarantees that (Ω, K, µ,_P(K, µ)) is a Behavioral Imprecise Probability Model for any measure µ ∈ ∆Ω_{. This is examined in Definition 4.5 and}

Proposition 4.4.

The uniqueness question has less subtlety in this framework, in that there is very little uniqueness. Let(Ω, K, µ, P) be a Behavioral Imprecise Probability Model. It is certainly the case that ifΩ,Kandµare fixed, thenP =_P(K, µ), and is unique.

However, we do not otherwise have uniqueness. IfP =_P(K, µ), there is, in general, no special reason to thinkP 6= _P(K0, µ), or P 6= _P(K, µ0)whereK0 6= K, andµ0 6= µ. That is, multiple different knowledge operatorsK, and measuresµcan be associated with the same Behavioral Imprecise ProbabilityP. Propositions 4.2 and 4.3 explore this issue further.

Proposition 4.2. Let (Ω, K, µ, P), (Ω0, K0, µ0, P0) be two Behavioral Imprecise Probability Models.

1. IfΩ6= Ω0, thenK 6=K0,µ6=µ0, andP 6=P0. 2. IfΩ = Ω0,K =K0, andµ=µ0, thenP =P0.

3. IfΩ = Ω0,K =K0, andP =P0, then it need not be the case thatµ=µ0. 4. IfΩ = Ω0,µ=µ0, andP =P0, then it need not be the case thatK =K0.

Proposition 4.2.3 can be strengthened to give a condition where, for any Behavioral Imprecise Probability Model(Ω, K, µ, P), there exists another distributionµ6=µ_bover

Ωsuch thatP(K, µ) = P(K,µ)b . Following the topology naming conventions, we say

a knowledge operatorK is Kolmogorov if it allows the agent to distinguish between all states in the sense that for any pair of states, there is some eventEwhere the agent knowsE at one state, but does not knowEat the other.

Definition 4.4. Let(Ω, K)be an information structure. Distinct statesω1, ω2 ∈Ωare distin-

guishable byK if there exists an event E ∈ 2Ω _{such that exactly one ofω}

1 andω2 belongs to

KE. Formally

ω1 ∈KE ⇐⇒ ω2 ∈/ KE

OperatorKis Kolmogorov if all pairs of distinct states are distinguishable byK.

Proposition 4.3 shows that if an operator is not Kolmogorov, then for any distribu- tionµwe can find a different distributionµb6=µwhere nonetheless,P(K, µ) =P(K,µ)b ,

That is, if K is not Kolmogorov, then multiple different distributions can certainly construct the same Behavioral Imprecise Probability.

Proposition 4.3. Let(Ω, K, µ, P)be a Behavioral Imprecise Probability Model.

IfKis not Kolmogorov, then there exists a Behavioral Imprecise Probability Model(Ω, K,µ, P_b )

whereµb6=µ.

Example 4.1 illustrates the method of the proof of Proposition 4.3 by giving a knowl- edge operatorK which is not Kolmogorov, and provides multiple distributions which lead to the same Behavioral Imprecise Probability.

Example 4.1. LetΩ ={a, b, c}and knowledge operatorK : 2Ω _→2Ω_be

KE =

(

E , if{a, b} ⊂E

∅ , otherwise

The statesaandbare indistinguishable by operatorK, as for every eventE, we havea∈KE

if and only ifb ∈ KE. Therefore,K is not Kolmogorov. Consider the distributionsµandµb

where

µ(a) = 1/3, µ(b) = 1/3, µ(c) = 1/3

b

This gives values for µ◦K and µb◦K as outlined in Table 4.1. From Lemma 4.1 it is

sufficient to check that the lower probabilities are equal to prove that the imprecise probabilities are equal. Event E ∅ {a} {b} {c} {a, b} {a, c} {b, c} Ω KE ∅ ∅ ∅ ∅ {a, b} ∅ ∅ Ω µ(KE) 0 0 0 0 2/3 0 0 1 b µ(KE) 0 0 0 0 2/3 0 0 1

Table 4.1:Lower Probabilities for a non-Kolmogorov Operator

Indeed, so long asµ(a) +e µ(b) = 2/3e , thenµe◦K =µ◦K. This is why multiple different

distributions over Ωcan lead to the same imprecise probabilities even when paired with the same knowledge operator.

Proposition 4.3 cannot be strengthened into an equivalence result. Example 4.2 provides an information structure(Ω, K)whereK is Kolmogorov, and yet for every distributionµ∈∆Ω_{, we can find another distribution}

b

µsuch thatP(K, µ) = P(K,µ)b .

Example 4.2. LetΩ ={a, b, c, d}and knowledge operatorK : 2Ω _→2Ω_be

KE =

(

E , ifE ={a, b}, {a, c}, orΩ

∅ , otherwise

OperatorKis Kolmogorov as each pair of states is distinguishable. For example,a∈K{a, c}, whileb /∈ K{a, c}, and similarly for all other pairs of states. Letµ∈∆Ω be any distribution over Ω, and ε > 0some very small number. In particular, make sure ε < minω∈Ωµ(ω)and

ε <minω∈Ω(1−µ(ω)). Define a new distributionµbsuch that b

µ(a) = µ(a)−ε, µ(b) =_b µ(b) +ε, _bµ(c) =µ(c) +ε, µ(d) =_b µ(d)−ε

This is a distribution asεis sufficiently small. Then

µ({a, b}) = µ(a) +µ(b) =µ(a)−ε+µ(b) +ε=µ(a) +_{b b}µ(b) = µ(_b {a, b}) µ({a, c}) = µ(a) +µ(c) =µ(a)−ε+µ(c) +ε=µ(a) +_{b b}µ(c) =µ(_b {a, c})

In addition,µ(b ∅) =µ(∅)andbµ(Ω) =µ(Ω). As{a, b},{a, c},∅, andΩare all of the sets in the

image ofK, this gives

µ(KE) =µ(KE)_b for all eventsE ∈2Ω

As previously noted, it is not the case that for every pair (K, µ) of operator K ∈ K and distribution µ ∈ ∆Ω_{, that}

P(K, µ) is a Behavioral Imprecise Probability. It is possible thatµ(KE) > µ(¬K¬E); the lower probability of the eventEwould exceed the upper probability of the eventE. We can guarantee that this is avoided by making the assumption thatKE ∩K¬E = ∅for all eventsE ∈ 2Ω. This is a non-dissonance assumption. It requires that the agent does not simultaneously know that an event has occurred, and know that it has not occurred. In the Modal Logic literature this assumption is called Axiom D.

Definition 4.5. Let(Ω, K)be an information structure. The operatorK satisfies Axiom D if

KE∩K¬E =∅for all eventsE ∈2Ω_.

Axiom D is the minimal requirement on K which guarantees the existence of a Behavioral Imprecise Probability for all priorsµ. Specifically,µ(KE)is guaranteed to be smaller thanµ(¬K¬E)exactly whenKsatisfies Axiom D. This does not mean that just because(Ω, K, µ, P)is a Behavioral Imprecise Probability Model thenKsatisfies Axiom D. It is possible thatK does not satisfy Axiom D, and neverthelessµ(KE)≤µ(¬K¬E)

for all events E ∈ 2Ω_{. Axiom D merely allows us to be certain that (Ω, K, µ, P) is a}

Behavioral Imprecise Probability Modelfor any distributionµ.

Proposition 4.4. Let(Ω, K)be an information structure such thatKsatisfies Axiom D. Then, for everyµ∈∆Ω, there exists a unique Behavioral Imprecise Probability model(Ω, K, µ, P).

Moreover, ifK does not satisfy Axiom D, then there existsµ∈∆ΩandE ∈2Ω such that

µ(KE) > µ(¬K¬E). Consequently, there does not exist a Behavioral Imprecise Probability Model(Ω, K, µ, P).