Calibrated Uncertainty
† Faruk Gul and Wolfgang Pesendorfer Princeton University July 2014 AbstractWe define a binary relation (qualitative uncertainty assessment (QUA)) to describe a decision-maker’s subjective assessment of likelihoods on a collection of events. To model ambiguity, we permit incompleteness. Our axioms yield a representation according to whichA is more likely than Bif an only if a capacity, calleduncertainty measure (UM) as-signs a higher value toAthan toBand a higher valueA-complement thanB-complement. We prove uniqueness, show that UMs are belief functions and describe how QUAs extended to utility functions on acts with a sophistication axiom.
1.
Introduction
A binary relation on a collection of events describes the the likelihood assessments of a group of decision makers. These decision makers may perceive ambiguity; that is, they may be indifferent between betting on an A orAc := Ω\A, indifferent between betting on Band onBc, but may nevertheless prefer betting onAto betting onB. As a result of such perceived ambiguity, the binary relation may be incomplete; that is,it may be impossible to rank certain bets without specifying a decision maker’s attitude to ambiguity. Our main result provides a cardinal representation of this binary relation.
Savage (1972) provides an analogous result for a qualitative probability (QP); that is, when decision makers perceive no ambiguity. It yields a cardinal measure of likelihood, a probability, that can be used in conjunction with either expected utility theory or non-expected utility alternatives. A statement such as “voter i assigns probability .3 to the event that that party X will win the election,” is meaningful even without knowing this decision maker’s attitude toward gambles and even if we don’t agree with her assessment. In particular, if this voter also assigns probability .3 to party Y winning the election and probability .4 to party Z winning, then virtually every theory of choice among gambles would conclude that she is indifferent between betting on a party X victory and a party Y victory, she prefers betting on a party Z victory to either of the previous two bets, and she prefers betting on a partyZ loss (i.e., victory by either partyX or partyY) to betting on a party Z victory.
Thus, a cardinal measure of uncertainty provides a separation between the individual’s
perception of uncertainty andattitude towards uncertainty. Two decision makers may per-ceive the same uncertainty in the prospect f, they may also perceive the same uncertainty in the prospect g; that is, they may assign the same probability distribution over prizes to the act f and the same probability distribution over prizes to act g, but nevertheless one may prefer f while the other prefers act g possibly because they have different levels of risk aversion.
In many economic models, the implicit or explicit assumption is that agents share a common uncertainty perception, that is, have the same priors and their current probability assessments differ only because they have received different information. To extend this
methodology to settings with perceived ambiguity requires a device for uncertainty mea-surement analogous to a probability measure. Such a device is not available in the presence of ambiguity. A nonadditive probability, that is, a capacity, plays a central role in Choquet expected utility theory (Schmeidler (1989)). However, it plays a role both in describing the decision maker’s perception of uncertainty and her attitude towards it. An alternative to a capacity is a set of probabilities, as used in Maxmin Expected utility theory (Gilboa and Schmeidler (1989)). Indeed Ghirardato, Maccheroni and Marinacci (2004) and Ghirardato and Siniscalchi (2013) derive a set of “perceived priors” for a variety of models. As the authors note (see, for example, GMM page 137), this measure may confound ambiguity attitude and perception and, more importantly, is derived under the assumption that all deviations from expected utility theory are due to ambiguity.1
To circumvent these problems, we propose an alternative approach, analogous to that of Gilboa, Maccheroni, Marinacci and Schmeidler (2010). To illustrate the central idea, let E1 andE2 be two unambiguous events and letAbe an ambiguous event. We interpret≽as
the common qualitative uncertainty assessment of a group of individuals andE1 ≽A ≽E2 means that even the most ambiguity averse person in this group prefers a bet on A to a bet onE2 while even the most ambiguity loving member of the group prefersE1 to A. By identifying a “tight” window of unambiguous events of this form, we determine the range of probabilities that A might have. Ambiguity renders the binary relation incomplete; there are events E and A such that the most ambiguity averse members of the group prefer a bet on E over a bet on A while the least ambiguity averse members have the reverse preference.
Our main theorem (Theorem 1) is a representation theorem for qualitative uncertainty assessment; it provides axioms such that a special type of capacity represents it. We call such a capacity anuncertainty measuresand introduce them in section 2 below. Section 3 introduces the axioms for qualitative uncertainty assessments and states the representation theorem. Section 4 analyzes utility functions over monetary acts that are compatible with an uncertainty measure.
2.
Cardinal Measurement of Uncertainty
2.1 PreliminariesThroughout this paper, Ω is the state space and Σ is aσ-algebra of events. Henceforth, all sets considered are elements of Σ. Aset functionq is a mapping from any sub-σ-algebra of Σ to [0,1] such that q(∅) = 0, and q(A) ≤ q(B) whenever A ⊂ B. A set function q is
additive if q(A ∪B) +q(A∩B) = q(A) +q(B); it is continuous if An+1 ⊂ An for all n
implies q(∩An) = limq(An). It is not difficult to verify that an additive set function is continuous if and only if it is countably additive.
When we wish to be explicit about the domain of q, we write (q,A). Given the set function (q,A), the set A ∈ A is null if q(Ac) = 1 and whole if B ⊂ A implies B ∈ A. A set function is complete if every null set is whole. A complete, continuous, and additive set function q is aprobability measureif q(Ω) = 1 and a subprobability measureif q(Ω)<1. A probability measure (µ,A) is nonatomic if A ∈ A, µ(A) > 0 implies there is B ⊂ A, B∈ A such that 0< µ(B)< µ(A).
A set functionq is a capacity ifq(Ω) = 1. LetN ={1, . . . , n}. A capacity q is a belief function if q(∪i∈N Ai) ≥ ∑I⊂N,I≠ ∅(−1)|I|+1q(∩I Ai) for all n and for any collection of sets A1, . . . , An,
2.2 Uncertainty Measures
In this section, we will define a class of capacities, uncertainty measures, that rep-resent the uncertainty perception of agents under the axioms provided in section 3. An uncertainty measure has two components: a probability measure (µ,E) that quantifies the uncertainty of unambiguous events (we refer to (µ,E) as the risk measure) and a sub-probability measure (η,Σ) that quantifies the uncertainty of ambiguous events (we refer to (η,Σ) as the ambiguity measure). When an event E is µ-whole it means that E and every subset of E is unambiguous. In that case, ambiguity plays no role when quantifying the uncertainty of subsets of E. When E is not µ−whole then some subsets of E are not E-measurable. That is, there are A, B ⊂ E such that A∪B = E but neither A nor B are unambiguous. In that case, the ambiguity measure (η,Σ) places bounds on the probabilities of A andB.
A probability measure (µ,E) and a subprobability measure (η,Σ) are compatible if µ(E) > η(E) for every µ-nonnull E ∈ E, and η(E) = 0 for every µ-whole E ∈ E. A capacity π is an uncertainty measure if there exists a non-atomic probability measure (µ,E) and a compatible, non-atomic subprobability measure (η,Σ) such that
π(A) = max
E∈[A]{µ(E) +η(A\E)} (1)
where [A] = {E ∈ E, E ⊂ A}. Thus, the uncertainty measure of a set A is calculated by finding the maximal unambiguous subset of each event and adding to it the ambiguity measure of the residual. Since (µ,E) and (η,Σ) are compatible, it follows that (1) is equivalent to (2) below:
π(A) = max
E∈[A]µ(E) + minE∈[A]η(A\E) (2)
If π, µ, η satisfy the conditions above, we say that µ is a risk measure for π and η is an ambiguity measure for π.
Proposition 1, below, shows that for any uncertainty measure π there is a unique decomposition into risk and ambiguity measures. For any capacity π, the set A is π -unambiguous if π(A) +π(Ac) = 1 and E
π is the collection of all π-unambiguous sets.
Proposition 1: (i) If(µ,E)is a risk measure for the uncertainty measureπ, thenE =Eπ;
(ii) every uncertainty measure has a unique risk measure and a unique ambiguity measure.
Not all capacities can be uncertainty measures:
Proposition 2: Every uncertainty measure is a belief function.
In the following section, we consider a binary relation≽on Σ. We will impose axioms on this binary relation that yield the following representation:
A ≽B if and only if {
π(A) ≥π(B) and π(Bc)≥π(Ac) where π is an uncertainty measure.
3.
Qualitative Uncertainty Assessments
The primitive is a binary relation ≽on Σ. We interpret A∈Σ as a bet; one that the decision-maker wins if and only if a state in A occurs and interpret A ≽ B to mean that the decision maker prefers the bet A to the bet B regardless of her ambiguity attitude. Thus, the binary relation ≽ may be incomplete. This incompleteness is resolved once the decision maker’s ambiguity attitude is taken into account.
As a benchmark, we state a continuous/countably additive version of Savage’s QP representation theorem: call a binary relation ≽ on Σ a QP if it is complete, transitive,
nondegenerate (i.e., ∅ ̸≽Ω) and satisfies the following additivity axiom: Axiom A: (A∪B)∩C =∅ implies A ≽B if and only if A∪C ≽B∪C.
Let ≻ denote the strict part of ≽. A finite collection of sets C1, . . . , Cn is a partition if the sets are pairwise disjoint and ∪Ck = Ω. Call a binary relation ≽ fine if it satisfies the following axiom:
Axiom F: A≻B implies there is a partition C1. . . Cn such that A≻B∪Cn for all n.
Finally, call a binary relation ≽ continuous if it satisfies the following axiom. Axiom C: An ≽B and An+1 ⊂An for all n implies
∩
An ≽B.
Then, we say that a binary relation has a (nonatomic) probability representation if there exists a (nonatomic) probability measure (µ,Σ) such that A ≽ B if and only if µ(A) ≥µ(B). Savage provides a finitely additive probability representation of a QP that satisfies Axiom F. It is straightforward to see that with Axiom C, we get a non-atomic probability representation. Formally, we have:
Savage’s Theorem: A binary relation has a nonatomic probability representation if and only if it is a QP that satisfies AxiomsC andF.
Our goal is to provide a version of Savage’s Theorem for decision makers who perceive ambiguity. We will use the following stylized version of an Ellsberg-type thought experi-ment to illustrate our model and the axioms. An urn has n balls, numbered from 1 to n. For each integer i ∈ N := {1, . . . , n}, there is exactly one ball with number i. Each ball has one ofk colors, that is, each ballihas some colork ∈K :={c1, . . . , ck} and, therefore,
the state space is Ω = N ×K. Let Aj ⊂ Ω denote the event that a ball of color cj is drawn and Ei denote the event that ball number i is drawn. Nothing is known about the composition of colors in the urn or about the color of any particular ball i.
Suppose for a moment that n = k = 10. Let Aj = ∪jt=1At and Ei = ∪it=1Et. Thus, Ei denotes the event that one of the first i balls will be drawn while Aj denotes the event that the drawn ball will have one of the colors in {c1, . . . , cj}. How would be
expect experimental subjects to rank Aj and Ei? If the subject is ambiguity averse, she might prefer betting on E5 to betting on A5. On the other hand, an ambiguity loving subject may prefer betting on A5 to betting onE5. Even when comparing two ambiguity
averse subjects, it might be that one prefers E5 to A5 but would rather bet on A5 than on E4 while the other is more ambiguity averse; that is even prefers E4 to A5. Thus, simple comparisons of rankings bets by a single individual do not enable us to distinguish ambiguity perception from attitude towards ambiguity.
So, we propose an alternative approach analogous to that of Gilboa, Maccheroni, Marinacci and Schmeidler (2010). We interpret ≽ as the common qualitative uncertainty assessment of a group of individuals. Thus, E7 ≽ A5 ≽ E3 means that even the most ambiguity averse person in this group prefersA5toE3while even the most ambiguity loving
member of the group prefers E7 to A5. By identifying a “tight” window of unambiguous events of this form, we determine the range of probabilities thatA5might have. Specifically, assume that E7 ≽ A5 ≽ E3 and A5 ̸≽ E4 and E6 ̸≽ A5. Thus, we conclude that the range of probabilities for A5 is [.3, .7]. Hence, the comparison between A5 and E5 is indeterminate; i.e., A5 ̸≽ E5 and E5 ̸≽ A5. This indeterminacy does not reflect the
incompleteness of a single agent’s preferences; rather is reflects the fact that the group’s common uncertainty perception by itself is not sufficient to rank the two bets E5 and A5. Our main axiom, calibration encapsulates this notion of a common uncertainty per-ception. Let W(A) ={E|A≽E}. The binary relation ≽ is calibrated if
A≽B if and only if {
W(B)⊂W(A) and W(Ac)⊂W(Bc)
In our Ellsberg experiment, as in Ellberg’s original thought experiments, the wording of the choice problem makes it clear which class of events are unambiguous and which ones are not. Events that correspond to outcomes of a symmetric game of chance (draws from an urn with a known composition) are unambiguous while events that correspond the decisions of the experimenter are ambiguous. In our model, we do not assume that the distinction between ambiguous events and unambiguous events is self-evident or exogenous. Instead, we provide a criterion for identifying unambiguous events and calibrate the ambiguous events with them.
Implicitly or explicitly, there are at least two different notion of an unambigous event in the literature. Some formal models and empirical/experimental work accept what we might call a “relative” notion of unambiguous events. In this approach, unambiguousness is a property of a collection of sets for which the more likely relationship is preserved whenever two unambiguous sets A, B are combined with a third disjoint unambiguous set C; that is, A≽B if and only ifA∪C ≽B∪C wheneverA∩C =B∩C =∅.2 Other formulations
assume (or imply) what we might call an “absolute” notion of unambiguousness. Here, unambiguousness is a property of a single set and a set E is deemed unambiguous if the more likely relationship is preserved whenanyA, B such that (A∪B)∩E =∅is combined with E.3 We take this second approach.
Definition: An event A is null if B ≽ B∪A for all B; E is unambiguous if A ≽ B if and only if A∪E ≽B∪E whenever (A∪B)∩E =∅.
Let E denote the set of all unambiguous events. Henceforth, we use A, B, C, D to denote generic elements of Σ and E, F, G to denote generic elements of E. The binary relation is monotone if A ⊂B implies B ≽A and B ≻A if, in addition, B\A is non-null. The binary relation ≽ is non-degenerate if Ω is non-null.
Axiom 1: The binary relation ≽ is monotone, non-degenerate and calibrated.
It is easy to verify that a calibrated ≽ must be transitive. However, we permit incompleteness to allow for the decision maker’s ranking of bets to depend on her ambiguity
2 This view is implicit in the notion of a source as in defined in Fox and Tversky (1995) and studied Gul and Pesendorfer (2014).
3 See for example, Epstein and Zhang’s (2001) notion of an unambiguous event and Gul and Pesendor-fer’s (2014) notion of an ideal event.
attitude. For any eventA, let⟨A⟩denote the sets of unambiguous events thatA preserves; that is,
⟨A⟩={E|E∩A ∈ E, E∩Ac ∈ E}
Suppose the first color in our urn example is red and the second color is blue. Then, A = A1∩Ej is the event that ball j is red whereas B =A2∩Ej is the event that ball j
is green. These two events are similar because they both preserve the same unambiguous events; that is,
⟨A⟩=⟨B⟩={E|E∩Ej =∅}
That is, ⟨A⟩ and ⟨B⟩ consist of all unambiguous events that do not containEj. Definition: A, B are similar (A∼=B) if ⟨A⟩=⟨B⟩.
When two events are similar, we expect the decision maker to be able to compare them solely based on her perception of ambiguity. To put it differently, when two sets are similar, they are equally ambiguous and hence the decision maker’s attitude towards ambiguity plays no role in determining her betting preference over these events. In the example, we might expect decision makers to findA andB to be equally likely. Moreover, should they perceive one event to be more likely than the other, then this preference is unaffected by their ambiguity attitude. This is the first part of Axiom 2 below.
The second part of Axiom 2 requires unambiguous events to be similar. The two parts together ensure that the restriction of ≽ to the set of unambiguous events is complete. A second consequence of Axiom 2 is that the set of unambiguous events is closed under unions and complements; that is, it is an algebra.
Axiom 2: (i) A ∼=B implies A≽B or B≽A. (ii)E ∼=F.
The familiar additivity axiom of qualitative probability requires that A ≽ B if and only if A∪C ≽B∪C wheneverA, C and B, C are both disjoint. Axiom 3 below weakens the additivity requirement of qualitative probability to permit ambiguity.
To see why we need a weaker form of additivity consider again our Ellsberg experiment. Consider a decision maker who finds the events A = A1 ∩E1 (“ball 1 will be drawn and
Consider the event C = E1\A1 (“ball 1 will be drawn and it will not be red.”) Then
A∪C =E1 (“ball one is drawn”) is an unambiguous event whileB∪C is not. Therefore,
ambiguity averse agent might preferA∪C toB∪C while an ambiguity loving agent might have the opposite preference.
Such a situation arises when A∪C creates a new unambiguous event while B∪C does not or B∪C destroys new unambiguous events while A∪C does not. For any two collections of unambiguous sets A,B ⊂ E, let A − B ={E ∈ A |E∩F ∼ ∅ for all F ∈ B} be the unambiguous sets in the collection A that B does not intersect. Note that in the above example:
⟨A∪C⟩ − ⟨A⟩={E1,∅}
and, therefore, adding C to A creates the new unambiguous set E1. Conversely,
⟨B⟩ − ⟨B∪C⟩={E1,∅}
and, therefore, adding C to B destroys the unambiguous set E1. Finally, adding C to B
creates no new unambiguous sets (the first line of the display equation below) and adding C toA destroys no new unambiguous sets (the second line of the display equation below):
⟨B∪C⟩ − ⟨B⟩={∅}
⟨A⟩ − ⟨A∪C⟩={∅}
We require additivity to hold only when adding C to A creates the same new unam-biguous sets as adding C to B and adding C to A destroys the same unambiguous sets as adding C to B. Formally,
Definition: A, C and B, C conform if (A∪B)∩C =∅, ⟨A∪C⟩ − ⟨A⟩=⟨B∪C⟩ − ⟨B⟩
and ⟨A⟩ − ⟨A∪C⟩=⟨B⟩ − ⟨B∪C⟩.
When A, C and B, C conform, we are in a situation such that if A was more likely than B for all decision makers with a given perception of uncertainty, then Ellberg-style thought experiments do not suggest that additivity should be violated. For example, suppose A = (E1 ∪ E2) ∩ A1 (“ball 1 or 2 will be drawn and it will be red”) while
two events are not similar and, therefore, they need not be ranked. However, if there is agreement and, for example, all decision makers agree thatA andBare equally likely then we require that they consider A∪C andB∪C equally likely where C =E3∩A2 (“ball 3
will be drawn and it is green.”) To see that A, C and B, C conform, note that
⟨A∪C⟩ − ⟨A⟩={∅}
⟨B∪C⟩ − ⟨B⟩={∅}
⟨A⟩ − ⟨A∪C⟩={E3,∅}
⟨B⟩ − ⟨B∪C⟩={E3,∅}
Axiom 3: If A, C and B, C conform, then A ≽B if and only if A∪C ≽B∪C.
Our axioms allow but do not require that the decision-maker perceives ambiguity. However, we do impose the existence of a rich set of unambiguous events. This assumption is implicit in any uncertainty model stated in the Anscombe-Aumann framework and in Savage’s formulation of subjective expected utility. We also assume that every event, unambiguous or not, can be divided into finer, similar events. The following version of Axiom F ensures that this is so.
Axiom 4: A ≻ B implies there is a partition C1, . . . , Ck such that Cn\B ∼= B and
A≻B∪Cn for all n.
The following axiom is the counterpart of the continuity axiom (Axiom C) from the Savage framework. It ensures that unambiguous events are closed not countable (not just finite) unions. It also ensures that the risk measure and ambiguity measure of the representing uncertainty measure are indeed countably additive.
Axiom 5: An+1 ⊂ An, An ≽ B for all n and either B ∈ E or An\An+1 ∈ E for all n,
implies ∩An ≽B.
We call a binary relation that satisfies the five axioms above aqualitative uncertainty assessment (QUA). Let ≽be any binary relation on Σ. A capacity π represents≽ if
A≽B if and only if {
π(A)≥π(B) π(Bc)≥π(Ac)
Theorem 1: A binary relation≽ onΣ can be represented by an uncertainty measure if and only if it is a qualitative uncertainty assessment.
Theorem 1 formalizes our notion of separating ambiguity attitude from ambiguity perception. It establishes that every QUA can be represented by a nonatomic uncertainty assessment.
4.
Preferences over Acts and Sophistication
Let X = [w, z] be a nondegenerate compact interval. Let F be the set of all simple, Σ-measurable functions on Ω; that is,
F ={f : Ω→X|f−1(x)∈Σ andf(Ω) finite}
For any f ∈ F, x ∈ X and A ∈ Σ, let f Ag denote h ∈ F such that h(s) = f(s) for all s ∈A and h(s) =g(s) for all s /∈A. We identify constant act that always yields x with x and hence write x∈ F and xAh.
Let L be the set of cumulative distributions with a finite support in X. Given a non-atomic uncertainty assessment π, and h∈ F, define (F, G)∈ L × Lsuch that
Fh(x) = ∑ y≤x π(h−1(y)) Gh(x) = 1−∑ y>x π(h−1(y))
We say that f dominates g if Ff(x) ≤ Fg(x), Gf(x) ≤ Gg(x) for all x and Ff ̸= Fg and Gf ̸=Gg.
A complete and transitive binary relation≽∗ on F is an act preference. We say that
≽∗ is monotone with respect to the the non-atomic uncertainty assessment π if f ≻∗ g
wheneverf dominatesg. The act preference is continuous with respect toπif g≽∗ fn ≽h and Ffn and Gfn converge in distribution to Ff and Gf imply g ≽ f ≽ h. An act
preference is sophisticated if it is monotone and continuous with respect to some some nonatomic uncertainty assessment π.
Given an uncertainty measure π, let ϕ:F → L × L be the function ϕ(f) = (Ff, Gf) and let Φ = ϕ(F) be the range of ϕ. We say that ≽∗ has a double-lottery representation
if there is an uncertainty measure π and a continuous increasing function V : Φ → IR such that f ≽∗ g if and only if V(Ff, Gf) ≥ V(Fg, Gg). If (π, V) is a double-lottery representation for≽∗, we callπ its uncertainty measure andV its utility function. We call a preference that has a double-lottery representation a double-lottery preference.
Proposition 3: An act preference is sophisticated if and only if it is a double-lottery preference.
It is easy to verify that each double-lottery preference has a unique uncertainty mea-sure and its utility is unique up to a monotone transformation. Double-lottery preferences are analogous to the general class of probabilistically sophisticated preferences in Machina and Schmeidler (1992). These are preferences consistent with some qualitative uncertainty assessment and the monotonicity and continuity requirements described above.
Example: Let τ : [0,1] → [0,1] be convex, strictly increasing and onto and suppose the agent’s utility is V(F, G) = ∫ xdGτ where Gτ(x) = 1−τ(1−G(x)). In that case, the agent is a rank dependent expected utility maximizer (with a linear utility index) when prospects are unambiguous. For general uncertain prospects, f ∈ F the agent disregards Ff and calculates the expected value of the transformed cdf Gf. Notice that the following functional form is an equivalent representation of this agent
U(f) = min ν∈CEµ[f]
whereEν[f] is the expectation of the random variablef ∈ F under the probability measure (ν,Σ) andCis the core of the convex capacityκ=τ◦π.4 Thus, the maxmin representation
of this agent yields a set of priors that is larger than the set of priors implied by the uncertainty measure π because it captures both the uncertainty perception (κ) and the non-linearity of the preference.
Next, we provide conditions under which ambiguity is the only source of deviations from expected utility theory. In that case, the utility function is α−maxmin expected utility with a set of priors equal to the core of the uncertainty measure κ. To formalize
the idea that ambiguity is the only source of deviations from expected utility theory, we must add two of Savage’s axioms.
Fix an uncertainty measure π and let (µ,E) be its risk measure. An act preference
≽∗ satisfies the sure-thing principle with respect to π if, for all E ∈ E,
f Eh ≽∗ gEh impliesf Eh′ ≽∗ gEh′ (ST P)
Thus, STP imposes the sure thing principle for π−unambiguous events only. An act preference ≽∗ satisfies Savage’s axiom P4 if for all x, y, x′, y′ such thatx > y, x′ > y′ and all A, B ∈Σ,
xAy ≽xBy implies x′Ay′ ≽x′By′ (P4)
A function V : L × L → IR is a Double Expected Utility (DEU) function if there exist continuous utility indicesu1 :X →IR, u2 :X →IRsuch thatu1 andu2 are non-decreasing
and u1+u2 is strictly increasing and
V(F, G) = ∫ X u1dF + ∫ X u2dG (DEU)
We say that ≽∗ has a DEU preference if it can be represented by a DEU. In the special case where u1 and u2 are affine transformations of one another we refer to the utility as
a symmetric double expected utility. A function V : L × L → IR is a Symmetric Double Expected Utility(SDEU) function if there exists continuous, strictly increasing utility index u:X →IR, α∈(0,1) such that
V(F, G) = (1−α) ∫ X udF +α ∫ X vdG (SDEU)
We say that ≽∗ is a SDEU preference if it can be represented by a SDEU.
Proposition 4: (i) An act preference is sophisticated and satisfies STP if and only if it is a DEU preference; (ii) a DEU preference satisfies P4 if and only if it is an SDEU preference.
An SDEU preference with parameters u, α, π is an α−maxim expected utility. Let C be the core of the capacityπ. It is easy to show that theα−maxmin representation of this preference is
U(f) =αmin
ν∈CEν[f ◦u] + (1−α) maxEν[f ◦u]
where C is the core of the capacity π. In this case, the set of priors used in the represen-tation coincides with the set of priors that characterize π, the uncertainty measure.
5.
Appendix A: Preliminaries
For any σ-algebra E and A∈Σ, let [A]E ={E ∈ E |E ⊂ E}. When the choice of E is clear, we suppress it and write [A]. For any σ-continuous probability (µ,E), let
µ∗(A) = sup E∈[A]
µ(E)
We call µ∗ the inner probability of µ. Also, let E(A) = {E ∈ E |µ(E) =µ∗(A)}. Lemma A1, below, ensures that E(A) is nonempty for allA and hence we can replace the sup with max in the definition of µ∗.
For any n≥ 1, let N = {1, . . . , n} and let N denote the set of all nonempty subsets of N. Then, let (µ,E) be any σ-continuous probability measure and A ={Ai|i∈ N} be any partition of Ω. The partition{Eκ| ∅ ̸=κ ⊂N}is a µ-split ofA if Eκ ∈ E for allκand
∪ κ∈K Eκ ⊂ ∪ i∈K Ai µ ( ∪ κ∈K Eκ ) =µ∗ ( ∪ i∈K Ai ) (1)
for all ∅ ̸=K ⊂N and K={κ∈ N |κ ⊂K}.
If A1 =A and A2 =Ac, we say that and E1, E2, E12 is a µ-split of (A, Ac).
Lemma A1: Let(µ,E) be aσ-continuous probability. Then, every partition Ahas a µ -split. Moreover, if {Eκ| ∅ ̸=κ ⊂N} is aµ-split of A, then the partition {Fκ| ∅ ̸=κ ⊂N}
is also a µ-split of A if and only if µ(Eκ\Fκ) = 0for all κ.
Proof: Suppose (µ,E), N = {1, . . . , n} and A = {Ai|i ∈ N} satisfy the hypotheses of the Lemma. The partition{Eκ| ∅ ̸=κ⊂N} is a µ-split of A if Eκ ∈ E.
First, we will show that for Ai, E(Ai) ̸= ∅. Let Ei ∈ [A] be a sequence such that limµ(Ei) = µ∗(A). By definition such a sequence exists. Then, since µ is a probability measure, E =∪Ei is the desired set.
Let E∅ =∅ and choose Eκ ∈ C(∪i∈κAi )
for all κ ∈and let
Eκ =Eκ\ ∪ κ̸=ˆκ⊂κ
Eˆκ
for all κ ∈ N. It is straightforward but somewhat tedious to verify that {Eκ|κ ∈ N } is the desired partition.
Now, if{Eκ} ⊂ E and{Fκ} ⊂ E are two partitions andµ(Eκ\Fκ)>0, thenµ∗(Aκ)≥ µ(Eκ∪Fκ)> µ(Eκ) =µ∗(Aκ), a contradiction.
If (E1, E2, E12) is a µ-split of (A, Ac), then by Lemma A1, E1, E12 is unique up to a
set of measure 0. Therefore, we call E(A) = E1 the core of A and F(A) the boundary of
A. Note that F(A) =F(Ac) and (E(A), E(Ac), F(A)) is a µ-split of (A, Ac). Lemma A2: If (µ,E) is a probability measure, then µ∗ is a belief function.
Proof: Let A = {A1, . . . , An} be any partition of Ω. Assume without loss of generality that ∅ ∈ A/ . Let ˆA be the finite collection of sets that can be expressed as the unions of elements in A. Let
γ(A) =µ(Eκ)
for all A =∪i∈κ∈N Ai. Clearly, γ(A)≥0 for all A∈Aˆ. Lemma A1 ensures that
µ∗(A) = ∑
B∈Aˆ B⊂A
γ(A)
for all A ∈ A. Dempster (1967) shows that a capacity on the algebra of sets generated by some partition A is a belief function if and only if there exists a γ ≥ 0 satisfying the above display equation. Hence, the restriction of µ∗ to ˆA ∪ {∅} is a belief function. Since the partition A was arbitrary, this proves that µ∗ is a belief function. . Proof of Proposition 1
Suppose, π is an uncertainty measure let (µ, η) and (η,Σ) satisfy the appropriate properties. In particular,
π(A) = max
E∈[A]µ(E) + minE∈[A]η(A\E)
for all A.
Since µis complete, every µ-null set isµ-whole. Hence, η(E) = 0 whenever µ(E) = 0 and therefore µ(E) ≥ η(E) for all E ∈ E. Next, we claim that for all E ∈ [A], µ(E) = maxF∈[A]µ(F) impliesπ(A) =µ(E) +η(A\E).
To see this, choose any E ∈ [A] such that µ(E) = maxF∈[A]µ(F). Suppose there
is ˆE ∈ [A] such that η(A\E)ˆ < η(A\E). Hence, η(A\( ˆE ∪E)) < η(A\E) and therefore η( ˆE∪E)> η(E) which means η( ˆE\E)>0 and henceµ( ˆE\E)>0, contradicting the fact that µ(E) = maxF∈[A]µ(F).
Next, we will show that π is superadditive; that is,π(A∪B)≥π(A) +π(B) whenever A∩B = ∅. To see this, suppose π(A) =µ(E) +η(A\E) and π(B) =µ(E) +η(A\F) for some E ∈[A] andF ∈[B]. Then, sinceE ∪F ∈[A∪B], we have
π(A∪B)≥µ(E∪F)+η((A∪B)\(E∪F)) =µ(E)+µ(F)+η(A\F)+η(B\F) =π(A)+π(B)
as desired.
Next, we claim that E =Eπ. To see this, suppose A∈ Eπ, then
1 =π(A) +π(Ac) =µ(E) +η(A\E) +µ(F) +η(Ac\F)
for some E ∈ [A] and F ∈ [Ac]. Hence, 1 = µ( ˆE) + η( ˆEc) where ˆE = (E ∪ F). If η( ˆEc) > 0, we have µ( ˆEc) > η( ˆEc) and hence µ( ˆE) +µ( ˆEc) > 1 contradicting the fact that µis a probability. So, we must have µ( ˆEc) = 0. Then, the completeness ofµ ensures that A\E ∈ E and hence A ∈ E.
Suppose E ∈ E. Then, π(E) ≥ µ(E) and π(Ec) ≥ µ(Ec). Since π is superadditive, 1 =π(E∪Ec)≥π(E) +π(Ec)≥µ(E) +µ(Ec) = 1. Hence, E ∈ Eπ proving that E =Eπ. It follows that µ(E) = π(E) for all E ∈ Eπ proving that the probability measure of any uncertainty measure is unique. Let (F, Fc) be any essential partition and choose
B ⊂Fc such that µ
∗(B) =µ∗(Fc\B) = 0. Hence, for any E ⊂ Fc, π(E∩B) =η(E ∩B)
and π(E\B) = η(E ∩B). Hence, η(E) = π(E ∩B) +π(E\B). Then, for any E ∈ E, η(E) = η(E ∩F) + η(E ∩ Fc) = η(E ∩ Fc) = π(E ∩B) + π(E\B). Hence, any two ambiguity measures for π must agree on E.
Take any A and E ∈ [E] such that µ(E) = µ∗(E) and note that π(A) = µ(E) + η(A\E) = µ(E) +η(A)−η(E). Hence, η(A) = π(A)−η(E). Since any two ambiguity measures for π must agree on E, it follows that they must agree everywhere. This proves Proposition 1.
5.1 Proof of Proposition 2
Let π be an uncertainty measure with probability measure µ and ambiguity measure η. If η(Ω) = 0, then π =µ∗ and hence, Lemma A2 ensures that π is a belief function. If η(Ω)>0, then leta = 1−η(Ω). Since, µ dominates η, η(Ω)< µ(Ω) and hence a∈(0,1). Also, µ(E) = 0 implies η(E) = 0. To see this, note that if µ(E) = 0 and η(E) > 0 for some E, then π(Ω)> µ(Ec) +η(E)>1, contradicting the fact that π is a capacity.
Let ˆη(A) =η(A)/(1−a) for all E ∈ E. Clearly, Since µdominate η, theσ-continuity of µ ensures the σ-continuity of the restriction of η to E and hence of the σ-continuity of the restriction of ˆηtoE. Hence, ˆµ= (µ−η)/aˆ is a probability measure on E and therefore, by Lemma A2, its inner probability, ˆµ∗, is a belief function.
Clearly µ(E\F) = 0 for all E, F ∈ E(A) and hence η(E\F) = 0 for all E, F ∈ E(A) and therefore,
π(A) =µ(E) +η(A\E) =µ∗(A) +η(A)−η(E) for all E ∈ E(A). Then,
π =aµˆ∗+ (1−a)η
Clearly a convex combination of a belief function and a probability is a belief function. This completes the proof of Proposition 2.
6.
Appendix B: Proof of Theorem 1
Calibration implies transitivity (A ≽ B and B ≽ C implies A ≽ C) and consistency
(A ≽ B implies Bc ≽ Ac). For any two sets X, Y, let X∆Y := (X ∪Y)\(X ∩Y). For any two collection of events A, B, let A ⊔ B = {A ∪B|A ∈ A, B ∈ B}. Finally, let [A] ={E ⊂A}.
Lemma B1:
(i) ⟨A⟩= [A]⊔[Ac].
(ii) E is an algebra.
(iii) A∈ E if and only if A ∼= Ω. (iv) A∩E =∅ implies A∪E ∼=A.
(v) E, F ∈ E and E ̸≽F implies F ≻E.
(vi) If(A∪B)∩C =∅,A∼=BandA∪C ∼=B∪C, thenA ≽Bif and only ifA∪C ≽B∪C. (vii) If B≽An, An ⊂An+1, An+1\An∈ E for all n, then B ≽
∪ An.
Proof: (i) Suppose E ∈ ⟨A⟩. Then, E ∩A ∈ [A] and E ∩Ac ∈ [Ac] and hence E = (E ∩A) ∪(E ∩Ac) ∈ [A] ⊔[Ac]. Next, assume that F
1 ∈ [A] and F2 ∈ [Ac] and let
B=F1∪F2. Then, B is the union of two disjoint elements ofE. Therefore, it is enough to
show that ifE∩F =∅, thenE∪F ∈ E. SupposeA≽Band (A∪B)∩(E∪F) =E∩F =∅. Then, A∪E ≽B∪E and hence (A∪E)∪F ≽(B∪E)∪F as desired.
(ii) Clearly, Ω∈ E and∅ ∈ E. Then, Axiom 2(ii) and part (i) imply [E]⊔[Ec] = [Ω]⊔[∅] and hence Ω∈[E]⊔[Ec] which yieldsEc ∈ E. To complete the proof that E is an algebra, we need to show that E ∩F ∈ E. By Axiom 2(ii) and part (i), [E]⊔[Ec] = [F]⊔[Fc]. Clearly, E ∈[E]∪[Ec] and therefore E ∈[F]∩[Fc] which implies E∩F ∈[F] and hence E∩F ∈ E.
(iii) Since Ω is unambiguous, one direction follows from Axiom 2(ii) and part (i). For the other direction, assume A ∼= Ω and hence Ω∈ [A]⊔[Ac] by part (i); that is, A ∈ [A] and therefore A ∈ E.
(iv) We will show that when A∩E =∅, [A]⊔[Ac] = [A∪E]⊔[Ac ∩Ec] and appeal to part (i). Suppose F1 ∈ [A] and F2 ∈ [Ac]. Then, F1 ⊂ A ∪E, F2 = F3∪F4 where
F3 := F2 ∩E and F4 = F2 ∩Ec. Since E is an algebra (by part (ii)), F1∪F3 ∈ [A∪E]
For the converse, take F1 ∈ [A∪E] and F2 ∈ [Ac ∩Ec]. Then, let F3 = F1 ∩ A
and F4 = F1 ∩E. Since E is an algebra, F4 ∈ E and therefore, F3 = F1\F4 ∈ E. But
F1∩A = F3. Hence, F3 ∈ [A] and F2, F4 ∈ [Ac]. Hence, by part (ii), F2∪F4 ∈ [Ac] and
therefore F1∪F2 =F3∪F4∪F2 ∈[A]⊔[Ac].
(v) By Axiom 2, we have F ≽ E and, by calibration, we have W(E) ⊂ W(F) and W(Fc)⊂W(Ec). Moreover, sinceE ̸≽F ∈W(F), the first inclusion is strict. It remains to show that the second inclusion is strict as well. Since, by consistency, Ec ≽Fc implies F ≽E it follows that Ec ̸∈W(Fc), as desired.
(vi) IfA ∼=B, A∪C ∼=B∪C and (A∪B)∩C =∅, then A, C andB, C conform and, therefore, the desired result follows from Axiom 3.
(vii) Suppose B ≽An,An ⊂An+1 andAn+1\An ∈ E for all n. Then, by consistency,
Ac
n ≽Bc and also Acn+1 ⊂Acn, An\Acn+1 ∈ E for all n. Hence, Acn converges to ∩
Ac n and by Axiom 5 ∩Ac
n ≽Bc. Hence, again by consistency,B ≽ ∪
An.
Lemma B2: E is a σ-algebra and there exists a nonatomic probability measureµ on E such that E ≽F if and only if µ(E)≥µ(F).
Proof: First, we show thatE is aσ-algebra. LetE =∪Fi. DefineEn= ∪n
i=1Fi and note thatE =∪Ei. By Lemma B1(ii),En∈ E for alln. SupposeA≽BandA∩E =B∩E =∅. Then, A∪En ≽ B ∪En for all n and, by monotonicity, A∪E ≽ A∪En ≽ B∪En for all n. Hence, transitivity implies A∪E ≽B∪En for all n. Then, Lemma B1 (vii) yields A∪E ≽B∪E.
For the converse, assume A ∪E ≽ B∪ E and (A ∪B)∩E = ∅. By consistency, Bc∩Ec ≽Ac∩Ec. Hence, (Bc∩Ec)∪En ≽(Ac∩Ec)∪En. Monotonicity and transitivity yield (Bc ∩Ec)∪E ≽ (Ac ∩Ec)∪En. Then Lemma B1 (vii) implies (Bc ∩Ec)∪E ≽ (Ac ∩Ec)∪E; that is, Bc ≽ Ac and hence by consistency A ≽B as desired. This proves that E is a σ-algebra.
Let ≽E be the restriction of ≽ to E. Lemma B1(v) ensures that ≽E is complete and transitive and also that ≻ has the usual interpretation. By definition, E ≽F if and only if E∪F′ ≽E′∪F′ whenever (E∪F)∩F′ =∅. Hence, ≽ restricted to≽E is a qualitative probability. By Lemma B1(iii), Axiom 4 restricted to ≽E yields Savage’s small event continuity axiom. Repeating Savage’s proof for theσ-algebraE instead of the σ-algebra of
all subsets of Ω yields a finitely additive, convex valued probability µ that represents ≽E. Then, the restriction of Axiom 5 to sets in E yields establishes that µ is in fact countably additive.
Next, we will show that µ is complete; that is, C ⊂ E and µ(E) = 0 implies C ∈ E. Assume C ⊂E andµ(E) = 0. First, we will show that E is null. Then, sinceµ represents
≽E,E ∼ ∅. By Lemma B1(iv),A\E ∼=A∪Eand therefore, by Lemma B1(vi),A∪E ∼A\E
and hence by monotonicity,A ≽A∪E, proving thatE is null. Monotonicity ensures that any subset of a null set is also null and hence C is null. Then, for any A, B such that A≽B, we have A∪C ∼A≽B ∼B∪C and hence C ∈ E.
Finally, we will show thatµis nonatomic. Supposeµ(E)>0. Then, sinceµrepresents
≽E, E ≻ ∅. Then, by Axiom 4, there exists a partition E1, . . . , En of Ω such that Ei ∼=E for all i and E ≻ Ei. By Lemma B1(iii), Ei ∈ E and hence by Lemma B1(ii), Ei∩E ∈ E for all i. If Ei ∩E were null for all i, then we would get ∅ ≽ E, contradicting the fact that µ(E) > 0 and µ represents ≽E. Hence, E ∩Ei ∈ E is nonnull for some i, say i = 1 µ(E∪E1)>0 and therefore, µ(E)> µ(E∩E1)>0 as desired.
For the remainder of this proof we fix the non-atomic probability measure (µ,E). We let N(A) denote the set of all null subsets of A, Eo(A) ={E ⊂ A|µ(E) = 0}. Let E(A) be the core of A,F(A) the boundary ofA and let (E(A), F(A), E(Ac)) be aµ−split of A. Definition: (i) E is blank if there exists A ⊂ E such that µ∗(A) = µ∗(E\A) = 0; (ii)
(E, Ec) is an essential partition forµ if E ∈ E and is whole andEc is blank.
Lemma B3:
(i) N(A) =Eo(A)
(ii) ⟨A⟩={E|µ(E∩F(A)) = 0}.
(iii) If A ⊂ E and B ⊂ Ec, then F(A ∪B) = F(A) ∪ F(B); ⟨A ∪B⟩ = ⟨A⟩ ⊔ ⟨B⟩;
⟨A∪B⟩ − ⟨A⟩={∅} and ⟨A⟩ − ⟨A∪B⟩= [F(B)]∪ N(Ω). (iv) Every probability measure has an essential partition.
(v) If (E, Ec) and(F, Fc) are two essential partitions forµ, then µ(E∆F) = 0.
Proof: (i) While proving that µ is complete (Lemma B2), we have shown that µ(E) = 0 implies E is null. It follows that Eo(A) ⊂ N(A). For the converse, assume C is null.
Then, A′∪C ∼ A′ for allA′ and hence A′ ≽ B′ implies A′∪C ∼A′ ≽ B′ ∼B′ ∪C and by transitivity A′ ∪C ≽ B′∪C proving that C ∈ E. But if C ∈ E is null then clearly, µ(C) = 0. It follows that N(A)⊂ Eo(A).
(ii) If E = E1 ∪E2 and E1 ∈ [A] and E2 ∈ [Ac], then clearly µ(E ∩F(A)) = 0.
Conversely, if µ(E∩F(A)) = 0, then E = E3∪E4 for someE3 ⊂[A]∪[Ac] and E4 such
that µ(E4) = 0. Then, since µ is complete, E4 ∩A ⊂ [A], E4 ∩Ac ∈ [Ac] and hence
E ∈[A]⊔[Ac] =⟨A⟩.
(iii) For the first assertion, first note that [A∪B] = [A]∪ [B]. This follows since E′ ∈[A∪B] impliesE∩E′ ∈[A],Ec∩E ∈[B] and, therefore,E′ ∈[A]∪[B]. The converse is obvious. Similarly, since Ac ∩Bc = ((E\A)∪Ec)∩((Ec\B)∪E) = (E\A)∪(Ec\B) it follows that [Ac ∩Bc] = [E\A]∪ [Ec\B] by the preceding argument. The last two observations imply
E(A∪B) =E(A)∪E(B) E(Ac∩Bc) =E(Ac)∩E(Bc)
It follows thatF(A∪B) = (E(A∪B)∪E(Ac∩Bc))c = (E(A)∪E(Ac))c∪(E(B)∪E(Bc))c = F(A) ∪ F(B). Part (ii) and the first assertion yield the second assertion from which
⟨A∪B⟩ − ⟨A⟩={∅} follows as does the fact that⟨A⟩ − ⟨A∪B⟩=⟨A⟩ −(⟨A∩ ⟨B⟨). Then, by the part (ii), ⟨A⟩ − ⟨A∪B⟩ = {E|µ(E ∩F(A)) = 0 and µ(E∩F(B)c) = 0}. By the first assertion, F(A) ⊂ F(B)c and therefore ⟨A⟩ − ⟨A∪B⟩ = {E|µ(E∩F(B)c) = 0} = [F(B)]∪ N(Ω) as desired.
(iv) Let B = {E ∈ E |E is blank} and b = supE∈Bµ(E). First, we will show that a countable union of blank sets is a blank set. Suppose E = ∪En and define F1 = E1,
Fn = En\ ∪
i<nEn. Hence, the sets Fn are pairwise disjoint and ∪
Fn = F. Choose An ⊂ Fn for all n so that µ∗(An) = µ∗(Fn\An) = 0. Suppose there exists E′ ⊂ F such thatµ(E′)>0 and (i) E′∩(∪An) =∅) or (ii)E′∩(E\(
∪
An)) =∅. Then, the countable additivity (i.e., σ-continuity) ofµ ensures that we have µ(E′∩Fn)>0 for some n. Then, (i) yields µ∗(Fn\An)>0 while (ii) yields µ∗(An)>0 both are contradictions.
Let En ∈ B be a sequence such that limµ(En) = b. Then, µ(∪En) = b and by the argument above, ∪En ∈ B. Set G = (
∪
En)c. Assume there exists A ⊂ G such that A /∈ E. Since µ is complete, part (i) implies that µ(F(A))>0. Thus, F(A) is a blank set
and, by the above argument, so isF(A)∪Gc. But, µ(F(A))∪Gc)> b, which contradicts the definition of b and proves that F is whole. So, (G, Gc) is the desired partition.
(v) The proof is straightforward and omitted.
If E = Σ. Then, set π =µ and η(A) = 0 for all A ∈ E and note that π is the desired uncertainty measure. So, from now on, we will assume E ̸= Σ. Let (G, Gc) be an essential partition for µ. Since µ is complete and E ̸= Σ, µ(Gc) >0. For any subset E of Gc such that µ(E)>0 and let
CE ={A⊂E|µ∗(A) =µ∗(E\A) = 0}
Since (G, Gc) is an essential partition, there is B⊂Gc such that µ
∗(B) =µ∗(Gc\B) = 0.
Setting A1 =B∩F and verifying thatA1 ∈ CE establishes that CE ̸=∅. Define,
ηE(A) = sup{µ(E′)|A≽E′}
for allA ∈ CE. Call (CE, ηE) alocal ambiguity measurement (LAM). If µ(E)≤1/3 we call (CE, ηE) asmall LAM (SLAM).
Lemma B4: Let A, B∈ CE and F, F′ ⊂Ec. Then,
(i) A is non-null.
(ii) A∪F ∼=C if and only if C =B∪E′ for some B ∈ CE and some E′ ⊂Ec;
(iii) A⊂C ⊂E,B∩C =∅ implies B∈ CE;
(iv) A∪F ≻ B∪F′ implies there is C ∈ CE such that B∩C =∅, A∪F ≻ B∪F′ ∪C
and B∪C ∈ CE.
(v) A∪F ≻ B∪F′, µ(E) ≤ 1/3, and either F = ∅ or F′ = ∅ implies there is E′ ⊂ Ec
such that E′∩(F ∪F′) =∅, µ(E′)>0 and A∪F ≻B∪F′∪E′. (v) If E1. . . , En are pairwise disjoint andE =
∪n
i=1Ei, then, CE =CE1 ⊔. . .⊔ CEn.
Proof: (i) If A is null then A ∈ E by Lemma B3(i). Then, Ac ∩ E1 ∈ E and hence
µ(Ac ∩E1) =µ(E1)>0 contradicting the fact that A∈ CE.
(ii) First we show that A ∼= B if B ∈ CE. To so so, we will show that [A]⊔[Ac] = [B]⊔[Bc] and appeal to Lemma B1(i). Recall that (E(A), E(Ac), F(A)) is a µ-split of
(A, Ac). We begin by showing that µ(Ec∆F(Ac)) = 0. LetF∗ =Ec\E(Ac). Ifµ(F∗)>0 then, since A⊂E, Ec ⊂Ac, we haveµ(F∗∪E(Ac))> µ(E(Ac)) andF∗ ∪E(Ac)⊂[Ac], contradicting the fact thatE(A), E(Ac), F(A) is aµ-split for (A, Ac). Ifµ(E(Ac)\Ec)>0, then we have µ(E(Ac)∩ E) > 0. Since E(Ac) ∩ E ⊂ E\A, we have µ∗(E\A) > 0, contradicting the fact that A ∈ CE and proving the assertion.
Then, by symmetry, µ(Ec∆E(Bc)) = 0. Now, suppose F1 ∈ [A] and F2 ∈ [Ac]. Let
F3 = F1∩B, F4 =F1∩Bc, F5 = F2 ∩E∩B, F6 = F2 ∩E∩Bc, F7 =F2∩Ec∩B and
F8 =F2∩Ec ∩Bc. Clearly,
F1 ∪F2 =F3 ∪F4∪F5 ∪F6∪F7 ∪F8
Since µ is complete and µ∗(A) = µ∗(B) = 0, we have F3, F4, F5, F7 ∈ E. Since F2 ∈[Ac],
µ(F2\E(Ac)) = 0. Then, since µ(Ec∆E(Ac)) = 0, we have µ(F2\Ec) = 0 and therefore
F6 ∈ E. Also, sinceµ(Ec∆E(Bc)) = 0, we can write F = (E(Bc)\F9)∪F10 for F9, F10 ∈ E
such that µ(F9 ∪ F10) = 0. Then, F ∩Bc = (E(Bc)∩ F9c ∩Bc)∪(F10 ∩Bc). Since
E(Bc) ⊂ Bc, we have Ec ∩Bc = (E(Bc)∩Fc
9)∪(F10∩Bc). Clearly, E(Bc)∩F9c ∈ E.
Since µ is complete, we have (F10 ∩ Bc) ∈ E and therefore, Ec ∩ Bc ∈ E and hence
F8 ∈ E. It follows thatF1∪F3∪F5∪F7 ∈[B] and F2∪F4∪F6∪F8 ∈[Bc] and therefore
F1∪F2 ∈[B]⊔[Bc] proving that A∼=B.
Next, we show that C ∼=A if and only if C =B∪F for some B∈ CE. First, assume C =B∪F for some F ⊂ Ec. Then, by Lemma B1(iv), B ∼=B∪E and, since A ∼=B, it follows that A ∼= B∪F = C. For the converse, note first that [A]⊔[Ac] = N(Ω)⊔[Ec]. To see, this if F1 ∈ [A] and F2 ∈[Ac], then µ(F1) = 0 and µ(F2 ∩E) = 0. Hence, F1 and
F2∩E are in N(Ω) by Lemma B3(i). Since the union of null sets is null, F1∪(F2∩E)∈
N(Ω). Let F3 = F2\E and note that F2 ∈ [Ec] and therefore, F1 ∪F2 ∈ N(Ω)⊔[Ec].
Similarly, if F1 ∈[Ec] and F2 ∈ N(Ω), then, since µ is complete, F3 = F2∩A ∈ [A] and
F4 = F2 ∩Ac ∈ [Ac] and also F1 ∈[Ac]. Hence, F1∪F2 =F3∪(F1∪F4)∈ [A]⊔[Ac] as
desired.
Now, suppose C ∩E /∈ CE. Then, either there is F ∈ C ∩E such that µ(F) > 0 or F ∈ [Cc ∩E] such that µ(F) > 0. Since, [A]⊔[Ac] = N(Ω)⊔[Ec], either of these establishes that [A]⊔[Ac] ̸= [C]⊔[Cc]. If C ∩Ec ∈ E/ , then, let E1, E2, E12 be a µ-split
of the partition A1, A2 where A1 =C∩Ec and A2 =Ac1. Then, µ(E12\E) = µ(E12)>0.
Hence, E12′ \E ∈[A]⊔[Ac] and E12′ \E /∈[C]∪[Cc].
It remains to show that A∪F ∼= B∪F′ whenever F, F′ ⊂ Ec. By Lemma B1(iv), A∼=A∪E and B=∼B∪F. The result now follows from A ∼=B.
(iii) Note that if C ∈ CE, B ⊂ E and B ∩ C = ∅, then B ⊂ E\C and hence µ∗(B)≤ µ∗(E\C) = 0. Similarly, E\B ⊂ E\A and therefore, µ∗(E\B) ≤ µ∗(E\A) = 0. Hence, B ∈ CE.
(iv) Assume A∪F ≻ B∪F′. By Axiom 4 there exists a partition C1, . . . , Ck of Ω
such that B ∼=Cn\(B∪F) and A∪F ≻ B∪F′∪Cn for all n. By part (ii) this implies that there areBn, Fn such thatBn∪Fn=Cn\(B∪F) forBn ∈ CE andFn∩E =∅. Note that B2∩B1 =∅ and therefore B∪B1 ∈ CE by part (iii). Since B1 ∈ CE, part(i) implies
it is non-null; then, monotonicity implies that B1∪B∪F′ ≻B∪F′ and, therefore, B1 is
the desired set.
(v) Argue as in the proof of part (iv) to getBn∪Fn =Cn\(B∪F) such thatBn ∈ CE, Fn∩E =∅. SinceFn is a partition ofEc andµ(Ec)>0 we must haveµ(Fn)>0 for some n. Then if F′ =∅, E′ =Fn is the desired set. Similarly, ifF =∅, thenA ≻B∪F′ implies µ(F′) ≤µ(E) < µ(Ec). Hence, there must be Fn such that µ(Fn\F′)> 0 and therefore, E′ =Fn\F′ is the desired set.
(vi) We will prove the results for n = 2. Then, the general case follows from an inductive argument. Suppose A ∈ CE. Then, for i = 1,2, µ∗(A ∩Ei) ≤ µ∗(A) = 0 and µ∗(Ei\A) = µ∗((E\A)∩Ei) ≤ µ∗(E\A) = 0 and hence A ∈ CE1 ⊔ CE2. Conversely, if
Ai ∈ CEi for i = 1,2, then for any F ⊂ A1∪A2, F ∩Ei ⊂ Ai and hence µ(F ∩Ei) = 0.
Therefore,µ(F) =µ(F∩E1)+µ(F∩E2) = 0 and similarly,µ(F) = 0 wheneverF ⊂Ac1∪Ac2
and therefore A1∪A2 ∈ CE.
Lemma B5: Let (CE, ηE) be a LAM and A ∈ CE. Then,
(i) E ≻A≻ ∅.
(ii) ηE(A)≥µ(E) if and only if A ≽E;
Proof: (i) Lemma B4(i) implies that A is non-null. Therefore, monotonicity implies that A ≻ ∅. From the definition of CE it follows that E\A ∈ CE and, therefore, by Lemma B4(i) and monotonicity, E ≻A.
(ii) IfηE(A)≥µ(F) we can find a sequenceFn ∈ E such thatA≽Fnand limµ(Fn)≥ µ(F). Since µ is nonatomic, we can choose this sequence so that Fn ⊂ Fn+1. Hence, by
Lemma B1(vii),A≽∪Fn≽F, establishingA≽F. The converse follows from calibration. (iii) By part (i), E ≻ A. Hence, Axiom 4, there is a partition C1, . . . , Cn of Ω such that ˆBn := (Cn∩E)\A ∈ CE, A ∪Bnˆ ∈ CE. Let B1 = ˆB1. Replacing A with A ∪B1
and repeating the argument yields B2 ∈ CE such that (A∪B1)∩B2 = ∅, B2 ∈ CE and A∪B1∪B2 ∈ CE. Continuing in this fashion, we get a sequence of pairwise disjoints setBn such thatBn ∈ CE and Bn ∈E1\A for all n. By part (i) the sequence ηE(Bn) is bounded. If limηE(Bn) = 0, we are done. Otherwise, ηE(Bnj) ≥ ϵ > 0 some subsequence Bnj. Assume, without loss of generality, that this subsequence is Bn and let An =
∪
i≥nBn. Then, by Lemma B4(iii), An ∈ CE. Also, by monotonicity, An ≽ E for any E such that µ(E) ∈(0, ϵ). Hence, by Axiom 5, ∩An ≽ E for some such E. But
∩
An = ∅ and hence we have a contradiction.
Lemma B6: Let (CE, ηE) be a SLAM, A, B ∈ CE and F, F′ ∈Ec. Then,
(i) F, F′ ⊂Ec implies A∪F ≽B∪F′ if and only if ηE(A) +µ(F)≥ηE(B) +µ(F′); (ii) A∪B ∈CE impliesηE(A∪B) =ηE(A) +ηE(B).
Proof: (i) First, we will show that A∪F ≽ A∪F′ if and only if µ(F) ≥ µ(F′). Note that F ∼=F′ and A∪F =∼A∪F′ by Lemma B4(iii). Sinceµ represents ≽ restricted to E it follows that F ≽F′ if and only ifµ(F)≥µ(F′). Lemma B1(vi) implies that F ≽F′ if and only if A∪F ≽A∪F′ and, therefore, proves the assertion.
Next, we show that A∪F ≽ B implies ηE(A) +µ(F) ≥ ηE(B). Choose F∗ ⊂ Ec such that ηE(B) = µ(F∗). Then, B ∼ F∗ by Lemma B5(ii). If µ(F) ≥ µ(F∗), we get ηE(A)+µ(F)≥ηE(B) immediately. Otherwise, sinceµis non-atomic, we may assumeF ⊂ F∗. Since F\F∗ ∈ E, A ≽F∗\F∗∗. By Lemma B5(ii) this implies ηE(A)≥µ(F∗\F∗∗) = ηE(B)−µ(F∗∗) as desired.
Next, we show that A ≽ B∪F′ implies ηE(A) ≥ ηE(B) +µ(F′). By Lemma B5(i) and (ii), it follows that η(A)<1/3. Thereforeµ(F′)<1/3 and, since µis non-atomic, we
