4.2 Behavioral Imprecise Probabilities
4.2.2 Rationality Assumptions
For the purposes of defining Behavioral Imprecise Probability Models we do not make restrictions on the knowledge operators used, except to require that the lower probabil- ity is no greater than the upper probability. Nevertheless, it may be useful to consider various restrictions on the operatorsK, and what effect these restrictions have on the resulting Behavioral Imprecise Probability Models. Many restrictions on K can be thought of as assumptions about the rationality of the agent. Definition 4.5 gave the non-dissonance, or Axiom D, restrictionKE ∩K¬E = ∅for all eventsE ∈ 2Ω. This
is a rationality assumption as it requires that the agent cannot know that an event has occurred, and that the complement of that event has also occurred.
This section considers a range of rationality assumptions of varying strengths, and the implications of these assumptions to the resulting Behavioral Imprecise Probability Models. Such rationality assumptions range from the extremely strict requirement of being fully informed, where KE = E for all eventsE, ending with the fairly weak assumption of monotonicity, whereE ⊂F impliesKE ⊂KF for all eventsE,F. We start with the most restrictive assumption, and work down to the least restrictive. Definition 4.6. Let(Ω, K)be an information structure. The operatorK is fully informed if
KE =Efor all eventsE ∈2Ω.
If the knowledge operator is fully informed, all ambiguity is eliminated. There is no imperfect information, and the imprecision in the Behavioral Imprecise Probability entirely disappears. In particular, for any probability measureµ, thenP(K, µ)is essen- tially just the original distributionµ. This justifies the general interpretation in which
µis the agent’s subjective underlying probability, but their limited information means they have access only to the less informative imprecise probabilityP.
Proposition 4.5. Let(Ω, K)be an information structure such thatK is fully informed. Then, for everyµ ∈ ∆Ω, there exists a unique Behavioral Imprecise Probability model(Ω, K, µ, P).
The imprecise probabilityP is given byP∗ =P∗ =µ.
A much more general, though still very restrictive, assumption is that the knowl- edge operator admits a representation as a partition of the state space. Partitions of the state space are the most common way to model imperfect information in economics. A partitionπof a state spaceΩis a set of subsets ofΩ, that isπ⊂2Ω, such that eachω∈Ω
is in exactly one set inπ. For ease of notation we useπ(ω)to denote the set inπwhich containsω. This means ω1 ∈π(ω1), and eitherπ(ω1) = π(ω2)orπ(ω1)∩π(ω2) = ∅for
all statesω1, ω2 ∈Ω.
If the partition π is used to represent the information available to an agent, then the usual interpretation is that the agent can distinguish states which are in different partition elements, but cannot distinguish states in the same partition element. This allows for a measure of uncertainty, as the agent is uncertain between any states in the same partition element. The uncertainty is, however, quite restrictive as, for example,
if the true state isω0 and the agent cannot rule outω1, then it must be that when the
true state isω1, then the agent cannot rule outω0.
Partitions are a useful tool for investigating Behavioral Imprecise Probability Mod- els. One part of Proposition 4.6 is that any partitional knowledge operator necessarily generates a Behavioral Imprecise Probability Model for any measureµ∈∆Ω.
A closely related concept is that ofσ-algebras. For a state spaceΩ, aσ-algebra, or just algebra,2F is a collection of subsets ofΩwhich is closed under unions, intersections,
and complements, and which contains the empty set. As Ωis finite, any partitionπ
generates a unique algebra by taking the set of all unions of partition elements in π. Similarly, an algebra F generates a unique partition by taking the set of non-empty minimal elements ofF. Algebras are useful as they allow discussion of the inner and outer measures associated with a algebra F and standard probability µ ∈ ∆Ω. The
inner measure of an eventE is the measure of the highest-probability event which is both a subset ofEand belongs to the algebra. Similarly, the outer measure ofE is the measure of the lowest-probability event which is both a superset ofEand belongs to the algebra. Inner and outer measures have a natural interpretation as the upper and lower probabilities of an imprecise probability.
Definition 4.7. Let(Ω, K)be an information structure. The operatorK is partitional if there exists a partitionπK of the state spaceΩsuch that
ω ∈KE ⇐⇒ πK(ω)⊂E (4.2)
Definition 4.8. LetΩbe a state space,F aσ-algebra onΩ, andµ∈∆Ωa probability measure.
The inner and outer measure induced by(F, µ)are given by, respectively,
µ∗(E) = sup{µ(F)|F ⊂E, F ∈ F }
µ∗(E) = inf{µ(F)|E ⊂F, F ∈ F }
A pair of measures(µ∗,µ∗)are an inner and outer measure pair if there exists aσ-algebraF
and measureµsuch thatµ∗ andµ∗are the inner and outer measures induced by(F, µ).
Proposition 4.6 shows that given a partitional knowledge operatorK, thenP(K, µ) is a Behavioral Imprecise Probability for every measure µ. Also, for P = P(K, µ),
the pair of upper and lower probabilities(P∗, P∗)is an inner and outer measure pair.
Moreover, any inner and outer measure pair can be generated as the lower and upper probability of a Behavioral Imprecise Probability Model with a partitional knowledge operator.
Proposition 4.6. Let(Ω, K)be an information structure such thatKis partitional. Then, for everyµ∈∆Ω, there exists a unique Behavioral Imprecise Probability model(Ω, K, µ, P). The
functionsP∗andP∗are inner and outer measures, respectively.
Moreover, let F be a σ-algebra over Ω, µ ∈ ∆Ω, and µ
∗, µ∗ inner and outer measures
induced by (F, µ). Then, there exists a Behavioral Probability Model(Ω, K, µ, P)such that
P∗ =µ∗,P∗ =µ∗, andK is partitional.
Weakening somewhat the idea of partitions, we come to another commonly used model of uncertainty in economics: the correspondence. A correspondence is a func- tionγ : Ω→2Ω\ ∅, with the interpretation thatγ(ω
0)is the set of states which the agent
thinks is possible whenω0is the true state. Definition 4.9 describes when a knowledge
operator can be represented by a correspondence.
Definition 4.9. Let(Ω, K)be an information structure. The operatorKis a correspondence operator if there exists a function γ : Ω → 2Ω \ ∅such that for all statesω ∈ Ωand events
E ∈2Ω
ω∈KE ⇐⇒ γ(ω)⊂E (4.3)
The property of being a correspondence operator is equivalent to a collection of specific behavioral restrictions on the structure of the operatorK. Indeed,K is a cor- respondence operator if and only ifKΩ = Ω, K∅ = ∅, and K(E ∩F) = KE ∩KF
for all eventsE, F ∈2Ω.3 We will reserve most discussion of the Behavioral Imprecise
Probability Models whereK is a correspondence operator to Section 4.3. Here we only note that ifK is correspondence operator, then it satisfies Axiom D. Hence, by Propo- sition 4.4, for all priors µ ∈ ∆Ω, the tuple (Ω, K, µ,
P(K, µ))is a Behavioral Imprecise Probability Model.
3In previous chapters we did not require thatK∅=∅. This additional assumption is without loss of
generality in the context of this chapter as, if(Ω, K, µ, P)is a Behavioral Imprecise Probability Model andKsatisfiesKΩ = Ω, then, by Proposition 4.1.1,K∅=∅.
Corollary 4.1. Let(Ω, K)be an information structure such thatK is a correspondence oper- ator. Then, for everyµ ∈ ∆Ω, there exists a unique Behavioral Imprecise Probability model
(Ω, K, µ, P).
Being a correspondence operator in this setting is slightly more restictive than in, for example, Mukerji [1997]. Mukerji defines information correspondences of the formΓ : S →2Ω\ ∅whereSis an arbitrary signal space. Notably, the domain of the information correspondence may be unrelated to the codomain. Due to this difference, some of the results in Section 4.3.2, including Corollary 4.2, differ from the results in Mukerji [1997]. The broad framework in Mukerji [1997] also does not allow generalization beyond the ‘signal’ approach to knowledge modeling.
The final, and least restrictive, assumption we make on knowledge operators is Monotonicity. An operatorK is monotonic if whenever an eventE is a sub-event of event F, that is E ⊂ F, and the agent knows E has occurred, then the agent must also know F has occurred. Formally, E ⊂ F implies KE ⊂ KF. This is a fairly weak rationality assumption, but is nonetheless violated empirically. Tversky and Kahneman [1983] provides such an empirical violation, and is discussed in Section 4.4.4
Definition 4.10. Let(Ω, K)be an information structure. The operatorK is Monotonic if for all eventsE, F ∈2Ω,
E ⊂F =⇒ KE ⊂KF
A useful, and very general, model for imprecise probabilities is the capacity. A capacityP is an imprecise probability with the only restrictions being that i) the lower probability is monotone in that P∗E ≤P∗F wheneverE ⊂F, ii) the upper and lower
probabilities are consistent in thatP∗E = 1−P∗¬E for allE ∈2Ω, and iii)PΩ = [1,1].
These conditions together also imply that the upper probabilities are monotone in that
P∗E ≤P∗F wheneverE ⊂F. Capacities are normally described only in terms of the lower probability, but within this work it is more helpful to work with capacities as imprecise probabilities. As the lower and upper probabilities must be consistent this is without loss or gain of generality.
Definition 4.11. LetΩbe a state space andP : 2Ω → Ian imprecise probability overΩ. The
eventsE, F ∈2Ωsuch thatE ⊂F.
Capacities form quite a large class of imprecise probabilities. Under the fairly weak rationality assumption thatKis monotonic, any Behavioral Imprecise Probability P(K, µ) will necessarily be a capacity. Moreover, there are capacities which cannot be modeled using a Behavioral Imprecise Probability Model. However, the class of Behavioral Imprecise Probabilities does include imprecise probabilities which are not capacities. This may occur when the knowledge operatorK is not monotonic.
Proposition 4.7. Let(Ω, K, µ, P)be a Behavioral Imprecise Probability Model such thatK is monotonic andKΩ = Ω. Then,P is a capacity overΩ.
The converse does not hold. There exist imprecise probabilitiesP whereP is a capacity over
ΩandP is not a Behavioral Imprecise Probability.