Choquet Integral with contamination∗

Paulo C. Coimbra†

EPGE/FGV‡ and FUCAPE§

November 13, 2008

Abstract

Suppose that an economic agent is (1 − ψi) · pi certain that uncertainty she faces on acertain event Ai is characterized by a particular probability which associates pi over thisevent (and similarly for all events), but that she has a fear that, with ψi · pi chance, herconviction over the event Ai is completely wrong and she is left perfectly ignorant about thetrue probability in the present as well in the future. This situation generalizes the so oftencalled “ε-contamination”.

The purpose of this paper is to investigate the general conditions under which a contam-ination of confidence of some probability measure can be represented through the probabilitymeasures in the core of a convex capacity. In the sequel it will be presented two new formulasof the Choquet Integral in a parametric approach (based on the ambiguity level) that gen-eralizes the well known formula of the expected value for the case of ε−contamination andcan be applied to contaminations. A special property of the Choquet Integral (monotonicityof the ambiguity level) is also presented.

JEL Classification: D81.Keywords: Capacity (non-additive probability), ε-contamination, Choquet Integral.

∗This paper is part of my doctoral research at EPGE/FGV. An early version of this paper was circulated withthe title “Integral representation with convex capacities that are squeezes of (additive) probability measures”. Iam deeply indebted to Jose Heleno Faro and to Victor Filipe Martins-da-Rocha for their invaluable commentsand suggestions towards this research and I am also deeply indebted to my advisor Sergio Ribeiro da CostaWerlang, for his guidance, support, collaboration and encouragement through my education and research. I alsogratefully thank Humberto Moreira and Daniel Gottlieb for very useful comments and suggestions as well as theaudiences at the following conferences: XXIX Stochastic Process and Applications and VII Brazilian School onProbability, Angra dos Reis, RJ, Brazil (2003); First Brazilian Conference on Statistical Modeling in Insuranceand Finance, Ubatuba, SP, Brazil (2003); feedbacks from participants of the 2003 Latin American Meeting ofthe Econometric Society, held in Panama City, Panama (2003); The Second Word Congress of the Game TheorySociety, Marseille, France (GAMES 2004) and XXX Meeting of the Brazilian Econometric Society, Salvador,BA, Brazil (EBE 2008) and also at academic seminars: EPGE/FGV (2003); IBMEC-RJ (2003); UFRJ (2004),EPGE/FGV (2006) and FUCAPE (2008). Obviously the usual disclaimer applies. I also gratefully acknowledgepartial financial support for this research provided by CAPES and for Getulio Vargas Foundation Fellowships.†Ph.D. candidate in Economics at EPGE/FGV and Assistant Professor at FUCAPE Business School; e-mail:

pc.coimbra@gmail.com; url’s: http://pc.coimbra.googlepages.com and http://pauloccoimbra.blogspot.com‡Graduate School of Economics at Getulio Vargas Foundation, Praia de Botafogo, 190, sala 1100 - Botafogo,

Rio de Janeiro, RJ, CEP: 22250-900, Brazil; url: http://www.fgv.br/epge§Capixaba Research Foundation at FUCAPE Business School, Av. Fernando Ferrari, 1358 - Goiabeiras,

Vitoria, ES, CEP: 29075-010, Brazil; url: http://www.fucape.br

1

1 Introduction

Suppose that an economic agent is (1 − ψi) · pi certain that uncertainty she faces on a certainevent Ai is characterized by a particular probability which associates pi over this event (andsimilarly for all events), but that she has a fear that, with ψi ·pi chance, her conviction over theevent Ai is completely wrong and she is left perfectly ignorant about the true probability in thepresent as well in the future. This situation generalizes the so often called “ε-contamination”.1

The purpose of this paper is to investigate the general conditions under which a contam-ination of confidence of some probability measure can be represented through the probabilitymeasures in the core of a convex capacity. The point of departure is to impose restrictions onthe ambiguity level, a set-function that has the same properties of the measure of vaguenessassociated to a convex capacity (see Marinacci [9]). As it will be seen in the sequel, these re-strictions on the ambiguity level are necessary to relate contaminations to convex capacities.Then it will be presented two new formulas of the Choquet Integral in a parametric approach(based on the ambiguity level) that generalizes the well known formula of the expected valuefor the case of ε-contamination and can be applied to contaminations. A special property ofthe Choquet Integral (monotonicity of the ambiguity level) is then presented: for any two given(restricted) ambiguity levels, if one of them is greater than the other then the Choquet Integralof a contamination will be so lesser how much bigger it will be the ambiguity level associated.

The other sections of this paper are organized as follows: the next section introduces therequired definitions and basic statements; Section 3 characterizes contaminations; Section 4presents some examples of contaminations and Section 5 presents two new formulas of theChoquet Integral that can be applied for contaminations and also presents a special property ofthe Choquet Integral (monotonicity of the ambiguity level).

2 Set-up and preliminaries

Consider a non-empty set S of states of nature (world), endowed with an algebra Σ of subsetscalled events. We denoted by (S,Σ) a measure space.

Definition 1 (capacity). A real-valued set-function v : Σ → R+ with v(∅) = 0 is called a ca-pacity (also known as non-additive probability measure) on S if it is normalized and monotone:2

i) normalized : v(S) = 1; and

ii) monotone: for all E,F ∈ Σ such that E ⊆ F , v(E) ≤ v(F ).3

Let denote by V the set of capacities on (S,Σ).4

Definition 2 (convex capacity). A real-valued set-function v : Σ→ R+ with v(∅) = 0 is calleda convex capacity on S if it is normalized and convex:

iii) convex : for all E,F ∈ Σ, v(E ∪ F ) + v(E ∩ F ) ≥ v(E) + v(F ).1For further discussions on ε-contamination, see Asano [1], Berger [2], Berger and Berliner [3], Huber [7],

Nishimura and Ozaki´s [11], [12], [13], [14] and Wasserman and Kadane [15]2∅ denotes the empty set.3In this paper the following notation will be used: E ⊆ F means that the set E may not be a proper set of F

(i.e., E = F is possible) and E F means that F is a proper set of F (i.e., E 6= F ).4It is easy to prove that if v ∈ V then v(E) ∈ [0, 1], ∀E ∈ Σ.

2

It is easy to prove that if a real-valued set-function v : Σ→ R+ with v(∅) = 0 is normalizedand convex then it is also monotone (in other words, every convex capacity is a capacity).

Let denote by Λ the set of all convex capacities on (S,Σ), a subset of the set of capacities(i.e., Λ V ):

Λ = {v ∈ V : v(E ∪ F ) + v(E ∩ F ) ≥ v(E) + v(F ), ∀E,F ∈ Σ}.

Definition 3 (probability measure). A real-valued set-function v : Σ → R+ with v(∅) = 0 iscalled a probability measure on S if it is normalized and additive:

iii’) additive: for all E,F ∈ Σ such that E ∩ F = ∅, v(E ∪ F ) = v(E) + v(F ).

It is easy to prove that if a real-valued set-function v : Σ→ R+ with v(∅) = 0 is normalizedand additive then it is also convex (in other words, every probability measure is a convexcapacity).

Let denote by ∆ the set of all probability measures on (S,Σ), a subset of the set of convexcapacities (i.e., ∆ Λ):

∆ = {v ∈ Λ : E ∩ F = ∅ =⇒ v(E ∪ F ) = v(E) + v(F ), ∀E,F ∈ Σ}.

If a convex capacity v ∈ Λ is not a probability measure then there exists at least a pairE,F ∈ Σ such that: v(E ∪ F ) + v(E ∩ F ) > v(E) + v(F ). In particular, if F = S\E thenv(E) + v(S\E) may be less than 1, implying that not all probability mass is allocated to anevent and its complement. This is the basis of the definition of the measure of vagueness.

Definition 4 (measure of vagueness). Let v be a capacity (i.e., v ∈ V ) and consider an eventE (i.e., E ∈ Σ). The measure of vagueness of a capacity v at event E is defined by:5

c(v,E) = 1− v(E)− v(S\E).

Remark 1. Convex capacities are also known as non-additive probabilities reflecting vagueness.

Marinacci [9] shows the properties that are satisfied by the measure of vagueness associatedto a convex capacity. Let v be a convex capacity on (S,Σ) (i.e., v ∈ Λ) and consider eventsE and F (i.e., E, F ∈ Σ). The measure of vagueness associated to a convex capacity shouldsatisfies the following properties:

i) c(v, ∅) = 0 = c(v, S);

ii) for all E ∈ Σ, c(v,E) = c(v, S\E); and

iii) concave: for all E,F ∈ Σ, c(v,E ∪ F ) + c(v,E ∩ F ) ≤ c(v,E) + c(v, F ).

Marinacci [9] defined ambiguity level as a set-function ψ : Σ→ [0, 1] that satisfies the sameproperties of the measure of vagueness associated to a convex capacity.

Definition 5 (ambiguity level). A set-function ψ : Σ→ [0, 1] is an ambiguity level if it satisfies:

i) ψ(∅) = 0 = ψ(S);

ii) for all E ∈ Σ: ψ(E) = ψ(S\E); and5It is easy to prove that: i) if v ∈ Λ then, for all E ∈ Σ: c(v,E) ∈ [0, 1]; and ii) if v ∈ ∆ then, for all

E ∈ Σ: c(v,E) = 0.

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iii) concave: for all E,F ∈ Σ: ψ(E ∪ F ) + ψ(E ∩ F ) ≤ ψ(E) + ψ(F ).

Let denote by Ψ the set of ambiguity levels on (S,Σ). For each ψ ∈ Ψ, let denote by Λ(ψ)the subset of convex capacities on (S,Σ) that are associated to the ambiguity level ψ; i.e., foreach ψ ∈ Ψ:

Λ(ψ) = {v ∈ Λ : c(v,E) = ψ(E), ∀E ∈ Σ}.6

Note that:Λ =

⋃ψ∈Ψ

Λ(ψ).

It is important to note that there exist non-convex capacities associated to some ψ ∈ Ψ asin the following example.

Example 1. Let S = {s1, s2, s3} and let ψ be the ambiguity level (i.e., ψ ∈ Ψ) defined by:

ψ({s1}) =12

= ψ({s2, s3}) and ψ({si}) =14

= ψ({s1, sj}), i, j = 2, 3 and j 6= i.

Let v be the capacity (i.e., v ∈ V ) defined by:

v({s1}) =14

= v({s2, s3}); v({si}) =316

and v({s1, si}) =916

, i = 2, 3.

It is immediate to note that v is a capacity associated to this ambiguity level ψ, i.e.,

v ∈ V (ψ) = {w ∈ V : c(w,E) = ψ(E), ∀E ∈ Σ}.7

However, v is not a convex capacity associated to this ambiguity level ψ (i.e., v /∈ Λ(ψ)). Tosee this, note that:

v({s2}) + v({s3}) =316

+316

>14

= v({s2, s3})

Actually, for any ambiguity level ψ ∈ Ψ, there exists a convex capacity v ∈ Λ such that itsmeasure of vagueness c(v, �) coincides with ψ(�). It is easy to prove that:

Ψ = {c(v, ·) : v ∈ Λ}.

In some economic applications, for instance the Dow-Werlang’s proof of the existence the-orem of Nash equilibrium under uncertainty (see Dow and Werlang [6]), it is interesting thata convex capacity can be described in a parametric way, based on an ambiguity level and aprobability measure. In their proof they used a special subset of convex capacities, know as“ε-contaminations” (also known as simple capacities or uniform squeezes).

Definition 6 ((preliminary) ε-contamination of confidence of some probability measure). Letp be a probability measure (i.e., p ∈ ∆) and let ε ∈ (0, 1). The ε-contamination of confidenceof p, denoted by ε-cont(p), is a subset of the set of probability measures defined by:

ε− cont(p) = {q ∈ ∆ : (1− ε) · p+ ε · q}

where q is a “contamination” of p and ε reflects the amount of error in p that is deemed possible.6The set Λ(ψ) is convex and throughout this paper we will only consider ψ ∈ Ψ such that Λ(ψ) 6= ∅.7Note that: ⋃

ψ∈Ψ

V (ψ) V

4

This definition of ε-contamination of some probability measure can be rewritten based on aconvex capacity. To attend this purpose we need the definition of the core of a capacity.

Definition 7 (core of a capacity). Let v be a capacity (i.e., v ∈ Λ). The core of a capacity v,denoted by core(v), is defined by:

core(v) = {p ∈ ∆ : p(E) ≥ v(E), ∀E ∈ Σ}.

Remark 2. Shapley [16] shows that if a capacity is convex, then the core is a non-empty andconvex set.

Now we can rewriting the definition of ε-contamination of some probability measure in termsof a convex capacity:

Definition 8 (ε-contamination of confidence of some probability measure). Let ψε be an am-biguity level (i.e., ψε ∈ Ψ), defined by ψε(E) = ε ∈ (0, 1) if E 6∈ {∅, S} and ψε(E) = 0 ifE ∈ {∅, S}; and let p be a probability measure (i.e., p ∈ ∆).

Let v : Σ→ R+ be a real-valued set-function defined by:8

v(E) ≡{

(1− ε) · p(E) if E 6= S1 if E = S

The ε-contamination of confidence of p is the core of the above convex capacity: 9

core(v) = {q ∈ ∆ : (1− ε) · p+ ε · q}.

Example 2 (ε-contamination). Let S = {s1, s2, s3}, let ψε be the ambiguity level (i.e., ψε ∈ Ψ)defined by ψε(E) = ε ∈ (0, 1) if E /∈ {∅, S} and ψε(E) = 0 if E ∈ {∅, S}; and let p be theprobability measure defined by:

p({si}) =13

, i = 1, 2, 3.

If the real-value set-function v : Σ→ R+ is defined by v ≡ (1− ψε) · p, i.e.:

v({si}) =(1− ε)

3, i = 1, 2, 3 and v({si, sj}) =

2 · (1− ε)3

, i, j = 1, 2, 3 6= i, ε ∈ (0, 1)

then v is a convex capacity associated to this ambiguity level ψε (i.e., v ∈ Λ(ψε)).The ε-contamination of confidence of p is defined as the core of v:

core(v) = {q ∈ ∆ : (1− ε) · p+ ε · q}.

An interesting property of the ε-contamination is that if an ambiguity level ψε (i.e., ψε ∈ Ψ)is defined by ψε(E) = ε ∈ (0, 1) if E 6∈ {∅, S} and ψε(E) = 0 if E ∈ {∅, S} then there mayexist other convex capacities associated to ψε (i.e., w ∈ Λ(ψε)) such that if q : Σ → R+ is areal-valued set-function defined by:

q(E) ≡

1

(1− ε)· w(E) if E 6= S

1 if E = S

then q is a probability measure (i.e., q ∈ ∆) that belongs to the ε-contamination of confidenceof p.

8It is easy to prove that v is a convex capacity (i.e., v ∈ Λ).9Kelley [8] shows this result.

5

3 Contaminations

In this section it will be investigated the conditions under which it is possible to relate thecontamination of confidence of some probability measure to the probability measures in the coreof a convex capacity. Our start point is to show through two examples that if ψ is an ambiguitylevel (i.e., ψ ∈ Ψ) and p is a probability measure (i.e., p ∈ ∆) then the real-valued set-functionv : Σ→ R+ defined by v ≡ ((1− ψ) · p) is not always a convex capacity.

Example 3. Let S = {s1, s2, s3}, let ψ be the ambiguity level defined by:

ψ({s1}) = 1 = ψ({s2, s3}) and ψ({si}) =12

= ψ({s1, sj}), i, j = 2, 3 and i 6= j

and let p be the probability measure defined by :

p({s1}) = 0, p({si}) =12

, i = 2, 3.

If the real-valued set-function v : Σ→ R+ is defined by v ≡ (1− ψ) · p, i.e.:

v({s1}) = 0 = v({s2, s3}) and v({si}) =14

= v({s1, si}), i = 2, 3

then v is not a capacity associated to this ambiguity level ψ (i.e., v /∈ V (ψ)). To see this notethat:

v({s2}) = v({s3}) =14> 0 = v({s2, s3}).

Note the counter-intuitive fact on which associates the maximum ambiguity to the event s1 andstill agree with the “miracle” that this event happens (i.e., v({s1}) = 0).

Example 4 (back to Example 1). Let S = {s1, s2, s3}, let ψ be the ambiguity level defined by:

ψ({s1}) =12

= ψ({s2, s3}) and ψ({si}) =14

= ψ({s1, sj}), i, j = 2, 3 and i 6= j

and let p be the probability measure defined by:

p({s1}) =12

and p({si}) =14

, i = 2, 3.

If the real-valued set-function v : Σ→ R+ is defined by v ≡ (1− ψ) · p, i.e.:

v({s1}) =14

= v({s2, s3}); v({si}) =316

and v({s1, si}) =916

, i = 2, 3

then, as already noted in Example 1, v is a capacity associated to ψ (i.e., v ∈ V (ψ)), but v isnot a convex capacity associated to ψ (i.e., v /∈ Λ(ψ)).

The notion of support of a capacity is the first necessary step to understand the conditionsunder which the contamination of confidence of some probability measure can be representedfrom the point of view of a convex capacity. The notion of support that it will be used here isthe same notion used in Dow and Werlang [6].

Definition 9 (support of a capacity). Let v be a capacity (i.e., v ∈ V ). The support of acapacity v, denoted by supp(v), is a set of events, such that, for every F ∈ supp(v):

i) v(S\F ) = 0; and

6

ii) for all E F : v(S\E) > 0.

In other words, we have

supp(v) = {F ∈ Σ : v(S\F ) = 0 and ∀E F ⇒ v(S\E) > 0}

The following example recovers Marinacci´s example (see Marinacci [9], pp.205), by an“exotic” combination of a particular ambiguity level ψ ∈ Ψ and a particular probability measurep ∈ ∆ (Dirac´s delta) and illustrates that, according to this definition of support, there mayexists the possibility that some capacities have more than one element in the support (and more,in the example all elements of the support have zero mass).

Example 5. Let S = {s1, s2, s3}, let ψ be the ambiguity level defined by:

ψ({s1}) = 1 = ψ({s2, s3}) and ψ({si}) =12

= ψ({s1, sj}), i, j = 2, 3 and i 6= j

and let p be the probability measure, defined by δ{s1} (Dirac’s delta):

p({s1}) = 1 and p({si}) = 0, i = 2, 3.

Note that supp(p) = {s1}.If the real-valued set-function v : Σ→ R+ is defined by v ≡ (1− ψ) · p, i.e.:

v({si}) = 0 = v({s2, s3}), i = 1, 2, 3 and v({s1, si}) =12

, i = 2, 3

then v is a convex capacity associated to this ambiguity level ψ (i.e., v ∈ Λ(ψ)) and supp(v) ={{s1}, {s2, s3}}. Note also that v({s1}) = v({s2, s3}) = 0.

The next example illustrates the extreme case for which ψ(E) = 1 for all E 6∈ {∅, S}.Example 6. Let S = {s1, s2, s3}, and let ψ be the ambiguity level defined by:

ψ(E) = 1 if E 6∈ {∅, S} and ψ(E) = 0 if E ∈ {∅, S}.

and consider any probability measure p ∈ ∆.If the real-valued set-function v : Σ→ R+ is defined by v ≡ (1− ψ) · p, i.e.:

v(A) = 0 if A 6= S and v(S) = 1 if A = S

then v is a convex capacity associated to this ambiguity level ψ (i.e., v ∈ Λ(ψ)) and supp(v) ={{s1}, {s2}, {s3}}.

Marinacci (see Marinacci [9], pp. 204) shows the conditions that should be satisfied for acapacity such that supp(v) has only one element: for all s ∈ S and for all E ∈ Σ such thatv(s) = v(E) = 0 it is required that v(s ∪ E) = v(s) + v(E).10

Note that, if we want to represent the contamination of confidence of some probabilitymeasure p through the probability measures in the core of a convex capacity then it is necessaryto know the conditions under which every convex capacity associated to each ambiguity levelψ and to the probability measure p, described as v ≡ (1 − ψ) · p, has only one element in thesupport. The following lemma present these necessary conditions.

10The interest reader should see the proof of this proposition in Marinacci (see Marinacci [9], pp.217).

7

Lemma 1. Let ψ be an ambiguity level satisfying ψ(E) < 1 for all E ∈ Σ and let p be aprobability measure.

If v : Σ → R+ is a real-valued set-function defined by v ≡ (1 − ψ) · p then v has the samesupport of p (i.e., supp((1− ψ) · p) = {supp(p)}).11.

Proof.

supp(v) = {F ∈ Σ : v(S\F ) = 0 and ∀E F ⇒ v(S\E) > 0}= {F ∈ Σ : (1− ψ(S\F )) · p(S\F ) = 0 and ∀E F ⇒ (1− ψ(S\E)) · p(S\E) > 0}

But, because ψ(E) < 1, ∀E ∈ Σ, then (1− ψ(E)) > 0, which implies:

supp(v) = {F ∈ Σ : p(S\F ) = 0 and ∀E F ⇒ p(S\E) > 0} = {supp(p)}

So, with slight abuse of notation, supp(v) = supp(p).

Lemma 1 suggests us to consider the following subset of Ψ:

Ψ = {ψ ∈ Ψ : ψ(E) < 1, ∀E ∈ Σ}.

One may consider the set Λ(Ψ) of convex capacities associated to measure of vagueness in Ψ,i.e.,

Λ(Ψ) ≡{v ∈ Λ : c(v, ·) ∈ Ψ

}.

Observe thatΛ(Ψ) =

⋃ψ∈Ψ

Λ(ψ).

In the following example we propose to slightly modify example 5 by posing ψ({s1}) =1− α = ψ({s2, s3}), where α ∈ (0, 1

2).

Example 7 (modification of Example 5). Let S = {s1, s2, s3}, let ψ be the ambiguity leveldefined by:

ψ({s1}) = 1−α = ψ({s2, s3}), α ∈ (0,12

) and ψ({si}) =12

= ψ({s1, sj}), i, j = 2, 3 and i 6= j

and let p be the probability measure defined by δ{s1} (Dirac’s delta):

p({s1}) = 1 and p({si}) = 0, i = 2, 3.

Note that supp(p) = {s1}.If the real-valued set-function v : Σ→ R+ is defined by v ≡ (1− ψ) · p, i.e.:

v({s1}) = α, α ∈ (0,12

); v({si}) = 0 = v({s2, s3}) and v({s1, si}) =12

, i = 2, 3

then v is a convex capacity, with supp(v) = {s1}, asssociated to this ambiguity level ψ (i.e.,v ∈ Λ(ψ)) .

11Let with slight abuse of notation: supp(v) = {supp(p)} ≡ supp(p) = {s ∈ S : p(s) > 0}

8

The following definition is the basis of the characterization of the subsets of convex capacitiesthat can be represented as contaminations.

Definition 10 (ambiguity level in Ψ restricted to D). Fix a non-empty event D (i.e., ∅ 6= D ∈Σ). An ambiguity level ψ Ψ is said to be restricted to D if it satisfies:

i) ψ(∅) = ψ(S) = 0;

ii) for all E ∈ Σ, 0 ≤ ψ(E) = ψ(S\E) < 1;

iii) concave: for all E,F ∈ Σ, ψ(E ∪ F ) + ψ(E ∩ F ) ≤ ψ(E) + ψ(F ).

iv) uniformity in proper subsets of D: for all ∅ 6= C,C ′ D, with C ∪C ′ 6= D and C ∩C ′ = ∅,ψ(C) = f(C ′) = ψ(C ∩ C ′) ≥ ψ(D);

v) nullity in the complement of D: for all ∅ 6= C D and for all E ⊆ (S\D), ψ(C∪E) = ψ(C);and

vi) monotone decreasing in D: for all E1 ⊆ E2 ⊆ · · · ⊆ Ej = S\D, ψ(D) ≥ ψ(D ∪ E1) ≥ψ(D ∪ E2) ≥ · · · ≥ ψ(D ∪ Ej) = ψ(S) = 0.

Fix a non-empty event D and let denote by Ψ(D) the set of ambiguity levels in Ψ restrictedto D.

Contaminations are defined as follows:

Definition 11 (contamination of confidence of some probability measure). Let ψ be an ambi-guity level in Ψ restricted to D (i.e., ψ ∈ Ψ(D)) and let p be a probability measure (i.e., p ∈ ∆)with supp(p) = D (∅ 6= D ⊆ S).

Let the real-valued set-function v : Σ→ R+ be defined by: 12

v ≡ (1− ψ) · p

The contamination of confidence of p, denoted by cont(p), is the core of the above real-valuedset-function.

cont(p) = {q ∈ ∆: (1−ψ(E))·p(E)+ψ(E)·q(E), ∀E ∈ Σ, where ψ ∈ Ψ(D), for D ≡ supp(p)}

The following theorem shows that contaminations are represented through convex capacities.

Theorem 1. Let ψ be an ambiguity level in Ψ restricted to D (i.e., ψ ∈ Ψ(D)) and let pbe a probability measure (i.e., p ∈ ∆) with supp(p) = D. Furthermore define a real-valuedset-function v : Σ→ R+ by:13

v ≡ (1− ψ) · p

then ((1− ψ) · p) is a convex capacity (i.e., ((1− ψ) · p) ∈ Λ).

Proof. Note, firstly, that (1− ψ(∅)) · p(∅) = 0 because p(∅) = 0.

i) normalized : (1− ψ(S)) · p(S) = 1;

So, it remains to verify convexity :12Note that, because ψ(E) < 1 for all E ∈ Σ the support of v is the same of p (i.e., supp((1 − ψ) · p) =

supp(p))=D.13Note that supp((1− ψ) · p) = {supp(p)}.

9

iii) convex : for all E,F ∈ Σ, (1 − ψ(E ∪ F )) · p(E ∪ F ) + (1 − ψ(E ∩ F )) · p(E ∩ F ) ≥(1− ψ(E)) · p(E) + (1− ψ(F )) · p(F ).

There exist two non-trivial cases (in what follows, let ∅ 6= X ⊆ (E ∩ (S\F )), ∅ 6= Y ⊆ (E ∩ F )and ∅ 6= Z ⊆ (F ∩ (S\E))):

1st case: E ∪ F = S and E ∩ F 6= ∅.

i) If D ⊆ E, with D ∩ (S\F ) = X and D ∩ (E ∩ F ) = Y then according to the property ofnullity in the complement of D : (1− ψ(Y )) = (1− ψ(E ∩ F )) = (1− ψ(F ));

ii) If D ⊆ S\(E ∩ F ), with D ∩ (S\F ) = X and D ∩ (S\E) = Z then according to theproperty of nullity in the complement of D : (1−ψ(X)) = (1−ψ(S\F )) = (1−ψ(E)) and(1− ψ(Z)) = (1− ψ(S\E)) = (1− ψ(F ));

iii) If D ⊆ S, with D∩ (S\F ) = X, D∩ (E∩F ) = Y and D∩ (S\E) = Z then according to theproperty of uniformity in proper subsets of D : (1− ψ(X)) = (1− ψ(Y )) = (1− ψ(Z)) =(1−ψ(X∪Y )) = (1−ψ(X∪Z)) = (1−ψ(Y ∪Z)) and according to the property of nullityin the complement of D : (1− ψ(X ∪ Y )) = (1− ψ(E)); (1− ψ(Y )) = (1− ψ(E ∩ F )) and(1− ψ(Y ∪ Z)) = (1− ψ(F )).

2nd case: E ∪ F 6= S and E ∩ F 6= ∅.

i) If D ⊆ ((E ∩ (S\F )) ∪ (S\(E ∪ F ))), with D ∩ (E ∩ (S\F )) = X then according to theproperty of nullity in the complement of D : (1− ψ(X)) = (1− ψ(E)) = (1− ψ(E ∪ F ));

ii) If D ⊆ (E∩(S\F )) , with D∩(E∩(S\F )) = X then according to the property of monotonedecreasing in D: (1− ψ(E)) ≤ (1− ψ(E ∪ F ));

iii) If D ⊆ (E ∪ (S\(E ∪F ))), with D∩ (E ∩ (S\F )) = X and D∩ (E ∩F ) = Y then accordingto the property of nullity in the complement of D : (1−ψ(X∪Y )) = (1−ψ((E∩(S\F )))∪(E ∩ F )) = (1−ψ(E)) = (1−ψ(E ∪ F )) and (1−ψ(Y )) = (1−ψ(E ∩ F )) = (1−ψ(F ));

iv) If D ⊆ E, with D∩(E∩(S\F )) = X and D∩(E∩F ) = Y then according to the property ofnullity in the complement of D : (1−ψ(Y )) = (1−ψ(E ∩F )) = (1−ψ(F )) and accordingto the property of monotone decreasing in D: (1− ψ(E)) ≤ (1− ψ(E ∪ F ));

v) If D ⊆ ((S\(E∩F )), with D∩ (E∩ (S\F )) = X and D∩ (F ∩ (S\E)) = Z then according tothe property of uniformity in proper subsets of D : (1−ψ(X)) = (1−ψ(Z)) = (1−ψ(X∪Z))and according to the property of nullity in the complement of D : (1−ψ(X)) = (1−ψ(E)),(1− ψ(Z)) = (1− ψ(F )) and (1− ψ(X ∪ Z)) = (1− ψ(E ∪ F ));

vi) If D ⊆ ((E ∩ (S\F ))∪ (F ∩ (S\E))), with D ∩ (E ∩ (S\F )) = X and D ∩ (F ∩ (S\E)) = Zthen according to the property of nullity in the complement of D : (1−ψ(X)) = (1−ψ(E))and (1−ψ(Y )) = (1−ψ(F )) and according to the property of monotone decreasing in D:(1− ψ(E)) ≤ (1− ψ(E ∪ F )) and (1− ψ(F )) ≤ (1− ψ(E ∪ F ));

vii) If D ⊆ S, D∩(E∩(S\F )) = X; D∩(E∩F ) = Y and D∩((S\E)∩F ) = Z then according tothe property of uniformity in proper subsets of D : (1−ψ(X)) = (1−ψ(Y )) = (1−ψ(Z)) =(1−ψ(X ∪Y )) = (1−ψ(X ∪Z)) = (1−ψ(Y ∪Z)) = (1−ψ(X ∪Y ∪Z)) and according tothe property of nullity in the complement of D : (1−ψ(X ∪Y )) = (1−ψ(E)); (1−ψ(Y ) =(1− ψ(E ∩ F )); (1− ψ(Y ∪ Z)) = (1− ψ(F )) and (1− ψ(X ∪ Y ∪ Z)) = (1− ψ(F ∪E)).

10

viii) If D ⊆ (E ∪ F ), D ∩ (E ∩ (S\F )) = X; D ∩ (E ∩ F ) = Y and D ∩ ((S\E) ∩ F ) = Z thenaccording to the property of uniformity in proper subsets of D : (1−ψ(X)) = (1−ψ(Y )) =(1−ψ(Z)) = (1−ψ(X∪Y )) = (1−ψ(X∪Z)) = (1−ψ(Y ∪Z)) and according to the propertyof nullity in the complement of D : (1−ψ(X∪Y )) = (1−ψ(E)); (1−ψ(Y )) = (1−ψ(E∩F ))and (1− ψ(Y ∪ Z)) = (1− ψ(F )).

The convex capacity (1−ψ) · p has other properties, presented in the following proposition.

Proposition 1 (special properties of the convex capacity ((1−ψ) · p)). Let ψ be an ambiguitylevel in Ψ restricted to D (i.e., ψ ∈ Ψ(D)) and let p be a probability measure (i.e., p ∈ ∆) withsupp(p) = D.

If the convex capacity is defined by (1− ψ) · p then it also satisfies the following properties:

iv) additivity in proper subsets of D : for all ∅ 6= C,C ′ D, with C ∪ C ′ 6= D and C ∩ C ′ = ∅,((1− ψ) · p)(C ∪ C ′) = ((1− ψ) · p)(C) + ((1− ψ) · p)(C ′); and

v) nullity in the complement of D : for all ∅ 6= C D and all E ⊆ S\D, ((1− ψ) · p)(C ∪E) =((1− ψ) · p)(C).

Proof. Verifying the property of additivity in the support:

((1−ψ)·p)(C∪C ′) = (1−ψ)(C∪C ′)·p(C∪C ′) = (1−ψ)(C)(p(C)+p(C ′)) = ((1−ψ)·p)(C)+((1−ψ)·p)(C ′)

Verifying the property of nullity in the complement of the support:

((1− ψ) · p)(C ∪ E) = (1− ψ)(C ∪ E) · p(C ∪ E) = (1− ψ)(C) · p(C) = ((1− ψ) · p)(C)

One interesting question that emerges is: taking the reverse procedure in Theorem 1 isalways possible to recover a probability measure? Unhappily the answer is no.

Let ψ be an ambiguity level in Ψ restricted to D (i.e., ψ ∈ Ψ(D)) and let w be a convexcapacity, with supp(w) = D, associated to this ambiguity level (i.e., w ∈ Λ(ψ)). Furthermore,define a real-valued set-function ( w

(1−ψ)) : Σ→ R+ such that, for all E ∈ Σ:

(w

(1− ψ))(E) ≡ 1

(1− ψ(E))· w(E)

then ( w(1−ψ))(E) is not always a probability measure, as the following example shows.

Example 8. Let S = {s1, s2, s3}. Fix D = {s1, s2} and let ψ be the ambiguity level in Ψrestricted to {s1, s2} (i.e., ψ ∈ Ψ({s1, s2})) defined by:

ψ({si}) = 0, 10 = ψ({sj , s3}), for i, j = 1, 2 and j 6= i; and ψ({s3}) = 0, 05 = ψ({s1, s2})

and let w be the convex capacity, with supp(w) = {s1, s2}, associated to this ambiguity level ψ(i.e., w ∈ Λ(ψ)) defined by:

w({s1}) = 0, 40 = w({s2}); w({s3}) = 0; w({s1, s2}) = 0, 95 and w({s1, s3}) = 0, 50 = w({s2, s3}).

11

If the real-valued set-function ( w(1−ψ)) : Σ→ R+ is defined by:

(w

(1− ψ))(E) ≡ 1

(1− ψ(E))· w(E)

that is:

(w

(1− ψ))({s1}) =

0, 400, 90

= (w

(1− ψ))({s2}); (

w

(1− ψ))({s3}) = 0; (

w

(1− ψ))({s1, s2}) = 1 and

(w

(1− ψ))({s1, s3}) =

0, 500, 90

= (w

(1− ψ))({s2, s3})

then ( w(1−ψ)) is not a probability measure (i.e., ( w

(1−ψ)) /∈ ∆).

Theorem 2. Let ψ be an ambiguity level in Ψ restricted to D (i.e., ψ ∈ Ψ(D)) and let w bea convex capacity, with supp(w) = D, associated this ambiguity level ψ (i.e., w ∈ Λ(ψ)) whichalso satisfies the extra properties of a convex capacity in Proposition 1 (i.e., additivity in theproper subsets of D and nullity in the complement of D). Furthermore, define a real-valuedset-function q : Σ→ R+ by:

q ≡ 1(1− ψ)

· w

then q is a probability measure.

Proof. Because w is a convex capacity, with supp(w) = D, associated to the ambiguity levelψ in Ψ restricted to D (i.e., ψ ∈ Ψ(D)), which also satisfies the extra properties of a convexcapacity in Proposition 1, it follows that, for all E ∈ Σ:

ψ(E) ≡ c(w,E) = 1− w(E)− w(S\E),

which implies that:(1− ψ(E)) = w(E) + w(S\E).

So, the real-valued set-function q : Σ→ R+ can be re-written as:

q(E) ≡ 1(w(E) + w(S\E))

· w(E)

Note firstly, that 1(w(∅)+w(S\∅)) · w(∅) = 0;

i) normalized : 1(w(S)+w(S\S)) · w(S) = 1;

So, it remains to verify additivity :

iii’) additive: for all E,F ∈ Σ such that E ∩ F = ∅,

1(w(E) + w(S\E))

· w(E ∪ F ) =1

(w(E) + w(S\E))· w(E) +

1(w(E) + v(S\E))

· w(F )

To check it note that if S = E ∪ F then the condition is always satisfied. Otherwise, whenS 6= E ∪F , there exist three non trivial cases (in what follows, let ∅ 6= X ⊆ E and ∅ 6= Y ⊆ F ):

1st case: If D ⊆ (S\F ), with D ∩ E = X then w(F ) = 0 and according to the property ofnullity in the complement of D (in Proposition 1): w(X) = w(E) = w(E ∪ F ) and w(S\(E ∪F )) = w(S\E);

12

2nd case: If D ⊆ (E ∪ F ), with D ∩ E = X and D ∩ F = Y then v(S\(E ∪ F )) = 0 andaccording to the property of nullity in the complement of D (in Proposition 1): w(X) = w(E) =w(S\F ) and w(Y ) = w(F ) = w(S\E);

3rd case: If D ⊆ S, with D∩E = X; D∩F = Y and D∩(S\(E∪F )) = Z then, according tothe property of additivity in proper subsets of D (in Proposition 1): w(X ∪Y ) = w(X) +w(Y ),w(X∪Z) = w(X)+w(Z) and w(Y ∪Z) = w(Y )+w(Z) and according to the property of nullityin the complement of D (in Proposition 1): w(X) = w(E), w(Y ) = w(F ); w(Z) = w(S\(E∪F ));w(X∪Y ) = w(E∪F ); w(X∪Z) = w(E∪ (S\(E∪F )); and w(Y ∪Z) = w(F ∪ (S\(E∪F )).

In Example 8 the convex capacity w, with supp(w) = {s1, s2}, is associated to the ambiguitylevel in Ψ restricted to {s1, s2} (i.e., ψ ∈ Ψ({s1, s2})), but w ∈ Λ(ψ) does not satisfy the propertyof nullity in D:

w({si}) = 0, 40 6= 0, 50 = w({si, s3}), for i = 1, 2.

We propose to slightly modify this example by posing:

w({si}) = 0, 45 = w({si, s3}), for i = 1, 2.

Example 9. Let S = {s1, s2, s3}. Fix D = {s1, s2} and let ψ be the ambiguity level restrictedto {s1, s2} (i.e., ψ ∈ Ψ({s1, s2})) defined by:

ψ({si}) = 0, 10 = ψ({sj , s3}), for i, j = 1, 2 and j 6= i; and ψ({s3}) = 0, 05 = ψ({s1, s2})

and now let w be the convex capacity, with supp(w) = {s1, s2}, associated to this ambiguitylevel ψ (i.e., w ∈ Λ(ψ)) which also satisfies the extra properties in Proposition 1, defined by:

w({si}) = 0, 45 = w({si, s3}), for i = 1, 2; w({s3}) = 0 and w({s1, s2}) = 0, 95.

If the real-valued set-function ( w(1−ψ)) : Σ→ R+ is defined by:

(w

(1− ψ))(E) ≡ 1

(1− ψ(E))· w(E)

that is:

(w

(1− ψ))({si}) =

0, 450, 90

= (w

(1− ψ))({si, s3}), for i = 1, 2; (

w

(1− ψ))({s3}) = 0 and

(w

(1− ψ))({s1, s2}) = 1

then ( w(1−ψ)) is a probability measure (i.e., ( w

(1−ψ)) ∈ ∆).

To complete the characterization, it remains to show that if v is a convex capacity, withsupp(v) = D, satisfying the special properties in the Theorem 1, then the measure of vaguenessof this capacity will satisfy the same properties of the ambiguity level ψ in Ψ restricted to D.

Theorem 3. If v is a convex capacity (i.e., v ∈ Λ), with supp(v) = D (∅ 6= D ⊆ S), thatalso satisfy the special properties in Proposition 1 ( additivity in the proper subset of D andnullity in the complement of D) then the associated measure of vagueness will satisfy the sameproperties of the ambiguity level is ψ restricted to D.

13

Proof. Verifying the property of uniformity in proper subsets of D: If supp(v) ≡ D(∅ 6= D ⊆ S) and there exists ∅ 6= C,C ′ D with C ∪C ′ 6= D and C ∩C ′ = ∅ then there exists∅ 6= B S such that D = C∪C ′∪B. According to the property of additivity in proper subsets ofD : v(C∪C ′) = v(C)+v(C ′), v(C∪B) = v(C)+v(B) and v(C ′∪B) = v(C ′)+v(B) and accordingto the property of nullity in the complement of D : v(S\(C ∪ C ′)) = v(B ∪ (S\D)) = v(B),v(S\C) = v((C ′ ∪B) ∪ (S\D)) = v(C ′ ∪B) and v(S\C ′) = v((C ∪B) ∪ (S\D)) = v(C ∪B).

Verifying the property of nullity in the complement of D: If supp(v) ≡ S (∅ 6=D =⊆ S) and there exists ∅ 6= C D then there exists ∅ 6= B′ S such that D = C ∪ B′.According to the property of nullity in the complement of D : v(C ∪E) = v(C), v(S\(C ∪E)) =v(B′ ∪ ((S\E)\D)) = v(B′) and v(S\C) = v(B′ ∪ (S\D)) = v(B′).

Let, with a slight abuse of notation, denote by Λ(ψ,D) the subset of convex capacitieson (S,Σ), with supp(v) = D, associated to the ambiguity level ψ in Ψ restricted to D (i.e.,ψ ∈ Ψ(D)) and that also satisfy the extra properties in Proposition 1.

4 Examples of contaminations

Example 10. (back to Example 2 (ε-contamination)) Let S = {s1, s2, s3}. Fix D ≡ S andlet ψε be the ambiguity level defined by ψε(E) = ε ∈ (0, 1) if E /∈ {∅, S} and ψε(E) =0 if E ∈ {∅, S} and note that according to the property of uniformity in proper subsets of S,ψε(E) = ε ∈ (0, 1) if E /∈ {∅, S} and so ψε is an ambiguity level in Ψ restricted to S (i.e.,ψ ∈ Ψ(S)).

Let p be the probability measure defined by:

p({si}) =13

, i = 1, 2, 3

and note that supp(p) = S.If the real-value set-function v : Σ→ R+ is defined by v ≡ (1− ψε) · p, i.e.:

v({si}) =(1− ε)

3, i = 1, 2, 3 and v({si, sj}) =

2 · (1− ε)3

, i, j = 1, 2, 3 and j 6= i; ε ∈ (0, 1)

then v is a convex capacity, with supp(v) = S, associated to this above ambiguity level ψε inΨ restricted to S and according to the property of additivity in proper subsets of S for all {si}and {sj}, i,j= 1,2,3 and j 6= i, v({si} ∪ {sj}) = v({si, sj}) = v({si}) + v({sj}) and so v alsosatisfies the extra properties in Proposition 1 (i.e., v ∈ Λ(ψε, S)).

For any convex capacity w, with supp(w) = S, associated to this above ambiguity level ψεrestricted to S that also satisfies the extra properties in Proposition 1 (i.e., w ∈ Λ(ψε, S)) if thereal-valued set-function q : Σ→ R+ is defined by:

q ≡ 1(1− ψε)

· w

then q is a probability measure (i.e., q ∈ ∆).For instance, let the real-valued set-function w : Σ→ R+, with w(∅) = 0, be defined by:

w({s1}) =(1− ε)

3− α; w({s2}) =

(1− ε)3

; w({s3}) =(1− ε)

3+ α

w({s1, s2}) =2 · (1− ε)

3− α; w({s1, s3}) =

2 · (1− ε)3

; w({s2, s3}) =2 · (1− ε)

3+ α and w(S) = 1

14

where α ∈ (0, (1−ε)3 ).

Then it is easy to note that w ∈ Λ(ψε, S), where ψε is the above ambiguity level in Ψrestricted to S. So, if the real-valued set-function q : Σ→ R+ is defined by q ≡ 1

(1−ψε) · w, i.e.:

q({s1}) =13− α

(1− ε); q({s2}) =

13

; q({s3}) =13

+α

(1− ε);

q({s1, s2}) =23− α

(1− ε); q({s1, s3}) =

23

; q({s2, s3}) =23

+α

(1− ε)

then q is a probability measure (i.e., q ∈ ∆), with supp(q) = S.

Remark 3. If D = {si}, i ∈ {1, · · · , n}, which requires that the support of the probabilitymeasure is the same singleton (i.e., supp(p) = {si}), then:

i) The extra properties of the ambiguity level in Definition 10 (uniformity in propersubsets of {si} and nullity in the complement of {si}) are innocuous;

ii) The extra properties of the convex capacity in Proposition 1 (i.e., additivity in propersubsets of {si} and nullity in the complement of {si}) are innocuous; and

iii) For each ambiguity level ψ in Ψ restricted to {si} (i.e., ψ ∈ Ψ({si})) and for eachprobability measure p (i.e., p ∈ ∆), with supp(p) = {si},14 there exits only one element inΛ(ψ, {si}: the real-valued set-function v : Σ→ R+ defined by v ≡ (1− ψ) · p (i.e., Λ(ψ, {si}) ={v}).

The next example illustrates this remark

Example 11. (back to Example 7) Let S = {s1, s2, s3}. Fix D ≡ {s1} and let ψ be the ambiguitylevel defined by:

ψ({s1}) = 1− α = ψ({s2, s3}), α ∈ (0,12

) and ψ({si}) =12

= ψ({s1, si}), i = 1, 2.

and note that according to remark 3, item (i), ψ is an ambiguity level in Ψ restricted to {si}(i.e., ψ ∈ Ψ({si}).

Let p be a probability measure, defined by:

p({s1}) = 1 and p({si}) = 0, i = 2, 3.

and note that supp(p) = {s1}.If the real-valued set-function v : Σ→ R+ is defined by v ≡ (1− ψ) · p, i.e.:

v({s1}) = α, α ∈ (0,12

); v({si}) = 0, i = 2, 3; v({s1, si}) =12

, i = 2, 3; and v({s2, s3}) = 0

then v is a convex capacity, with supp(v) = {s1}, associated to this above ambiguity level ψ inΨ restricted to {si} and according to remark 3, item (ii), v also satisfies the extra properties inProposition 1 (i.e., v ∈ Λ(ψ, {si})).

Note that v is the only convex capacity with supp(v) = {s1} that belongs to Λ(ψ, {s1}),where ψ is the above ambiguity level ψ restricted to {si} (as already noted, according to remark3, item (iii)).

14Note that if supp(p) = {si} this means that p is a probability measure defined by δ{si} (Dirac’s delta), i.e.:p({si}) = 1 and p({sj}) = 0 for all j 6= i.

15

Example 12. (back to Example 9) Let S = {s1, s2, s3}. Fix D ≡ {s1, s2} and let ψ be theambiguity level defined by:

ψ({si}) = 0, 10 = ψ({si, s3}), i = 1, 2 and ψ({s3}) = 0, 05 = ψ({s1, s2})

and note that according to the property of nullity in the complement of {s1, s2} for all ∅ 6= C {s1, s2} and for all E ⊆ (S\{s1, s2}) = {s3} it should be verified that ψ(CUE) = ψ(C). In fact:

ψ({si} ∪ {s3}) = ψ({si, s3}) = 0, 10 = ψ({si}), i = 1, 2

an so ψ is an ambiguity level in Ψ restricted to {s1, s2} (i.e., ψ ∈ Ψ({s1, s2})).Let p be the probability measure defined by:

p({s1}) =13

, p({s2}) =23

and p({s3}) = 0

and note that supp(p) = {s1, s2}.If the real-valued set-function v is defined by v ≡ (1− ψ) · p, i.e.:

v({s1}) = 0, 3 = v({s1, s3}); v({s2}) = 0, 6 = v({s2, s3}); v({s3}) = 0 and v({s1, s2}) = 0, 95

then v is a convex capacity, with supp(v) = {s1, s2}, associated to this above ambiguity levelψ in Ψ restricted to {s1, s2} and according to the property of nullity in the complement of{s1, s2} for all ∅ 6= C {s1, s2} and all E ⊆ S\{s1, s2} = {s3} it should be verified thatv(C ∪ E) = v(C). In fact:

v({s1}∪{s3}) = v({s1, s3}) = 0, 30 = v({s1}) and v({s2}∪{s3}) = v({s2, s3}) = 0, 60 = v({s2})

and so v also satisfies the extra properties in Proposition 1 (i.e., v ∈ Λ(ψ, {si, s2})).For any convex capacity w associated to this above restricted ambiguity level ψ with

supp(w) ≡ {s1, s2} (i.e., w ∈ Λ(ψ, {s1, s2})) if the real-valued set-function q : Σ → R+ isdefined by:

q ≡ 1(1− ψ)

· w

then q is a probability measure (i.e., q ∈ ∆).For instance, let the real-valued set-function w : Σ→ R+, with w(∅) = 0, be defined by:

w({si}) = 0, 45 = w({s2, s3}), i = 1, 2, w({s3}) = 0, w({s1, s2}) = 0, 95 and w(S) = 1

then it is easy to note that w ∈ Λ(ψ, {s1, s2}), where ψ is the above restricted ambiguity level.So, if a set-function q : Σ→ R+ is defined by q ≡ 1

(1−ψ) · w, i.e.:

q({si}) =12

= q({si, s3}), i = 1, 2; q({s3}) = 0 and q({s1, s2}) = 1

then q is a probability measure (i.e., q ∈ ∆), with supp(q) = {s1, s2}).The next example illustrates theorem 3, which says that if a convex capacity also satisfy the

properties in theorem 1 then the ambiguity level associated is restricted.

16

Example 13. Let S = {s1, s2, s3}, and let the real-valued set-function v : Σ→ R+, with v(∅) = 0,be defined by:

v({s1}) = 0, 6 = v({s1, s3}); v({s2}) = 0, 3 = v({s2, s3}); v({s3}) = 0; v({s1, s2}) = 0, 95 and v(S) = 1

then v is a convex capacity, with supp(v) = {s1, s2}.Fix D = {s1, s2} and note that v trivially satisfies the property of additivity in proper

subsets of {s1, s2} and according to the property of nullity in the complement of {s1, s2} for all∅ 6= C {s1, s2} and all E ⊆ (S\{s1, s2}) = {s3} it should be verified that v(C ∪ E) = v(C),so, in fact {s1}, {s2} {s1, s2}:

v({s1}∪{s3}) = v({s1, s3}) = 0, 60 = v({s1}) and v({s2}∪{s3}) = v({s2, s3}) = 0, 30 = v({s2})

so, according to theorem 3, it should be checked that the measure of vagueness of v has thesame properties of a restricted ambiguity level.

The measure of vagueness of a capacity v at event A is defined by c(v,A) = 1 − v(A) −v(S\A) = c(v, S\A). So, it follows that:

c(v, ∅) = 0 = c(v, S); c(v, {si}) = 0, 10 = c(v, {si, s3}), i = 1, 2; c(v, {s3}) = 0, 05 = c(v, {s1, s2})

By construction, the ambiguity level associated to a convex capacity is defined by ψ(·) ≡ c(v, ·).Note that this is the same ambiguity level in example 12 and, as already noted, ψ is an ambiguitylevel in Ψ restricted to {s1, s2} (i.e., ψ ∈ Ψ({s1, s2})).

5 Choquet integral with contamination

Since that capacities can be a non-additive measure it is not possible to use an integral in thesense of Lebesgue. The appropriate notion of integral is due to Choquet [4].

A real-valued function, bounded on S, a : S → R is said to be Σ-measurable if , for allopen set O ⊆ R, a−1(O) ∈ Σ, where a−1(O) = {s ∈ S : a(s) ∈ O}. Let B(S,Σ) denote theset of all real-valued function, bounded on S, that are Σ-measurable. Choquet [4] defined anintegration operator with respect to a capacity. Let a be a real-valued function, bounded on Sand Σ-measurable and let v be a capacity on (S,Σ), then the integral of a with respect to v is:

∫Sadv ≡

0∫−∞

[v(Lλ(a))− 1]dλ+

∞∫0

v(Lλ(a))dλ

where Lλ(a) = {s ∈ S : a(s) ≥ λ} and the integral on the right hand side is a well definedintegral in the usual sense (Riemann or Lebesgue), because a is bounded and v is monotone.

For the special case on which v is a probability measure, the definition above coincides withthe standard integral.

Let a : S → R be a real-valued function, bounded on S and Σ-measurable. In particular, ifΣ = 2S is finite, then B(S,Σ) = RS . A real-valued function, bounded on S and Σ-measurableis simple if it is a finite-valued function, i.e., if the set {a(s) : s ∈ S} is finite. Each simple

function a admits a unique representation as a finite step function a =k∑i=1

αi1Ai15 with α1 <

15Let s ∈ S, then 1Ai is the characteristic function of the set Ai, which means: i) 1Ai = 1 if s ∈ Ai and ii)1Ai = 0 if s ∈ S\Ai.

17

α2 < · · · < αk, where Ai = a−1({a(si)}) ⊆ S andk⋃i=1

Ai = S.16 Let α0 = 0, the definition of∫S adv for a capacity v and a simple function a is given by

∫Sadv =

k∑i=1

(αi − αi−1) v

k⋃j=i

Aj

(1)

The next proposition presents a couple of equivalent ways of the above Choquet integral(that are valid for finite S) which can be applied to convex capacities that can be representedas contaminations. These formulations generalizes the usual Choquet integral formula withε−contamination and are very useful for comparative static exercises over the ambiguity level.

Theorem 4. Let ψ be an ambiguity level (i.e., ψ ∈ Ψ) and let p be a probability measure (i.e.,p ∈ ∆).

If a is a simple function and if the real-valued set-function v : Σ → R+ is defined byv ≡ (1− ψ) · p then it holds:17

∫Sadv=α1ψ(A1)+ (1− ψ(A1))

∫Sadp+

k∑i=3

(αi−αi−1)

ψ(A1)−ψ

k⋃j=i

Aj

p

k⋃j=i

Aj

(2)

= α1

p(A1) + ψ(A1) p

k⋃j=2

Aj

+ (3)

+k−1∑i=2

αi

1− ψ

i−1⋃j=1

Aj

p(Ai) +

ψ i⋃j=1

Aj

− ψi−1⋃j=1

Aj

p

k⋃j=i+1

Aj

+

+ αk

1− ψ

k−1⋃j=1

Aj

p(Ak)

where: p : Σ→ [0, 1] is a probability measure on S; and∫Sadp is its expected value.

Proof. It is enough to prove the first equality, the other being a simple rearrangement of itsterms. Let a be positive so that α1 > 0. If λ > αk then {s ∈ S : a(s) ≥ λ} = ∅. If λ ∈ (αi−1, αi],

then (recall that α0 = 0) {s ∈ S : a(s) ≥ λ} =k⋃j=i

Aj .

Hence, for each λ ∈ R+

16That is, {Ai}ki=1 is a partition of S.17If, for all A ∈ Σ, ∅ 6= A 6= S, ψ(A) = (1 − ε) ∈ [0, 1) then it is easy to prove that these formulas collapses

with the formula of Choquet integral with ε−contamination, i.e.:∫S

adv ≡ (1− ε) α1 + ε

∫S

adp

18

v(Lλ(a)) =k∑i=1

v

k⋃j=i

Aj

1(αi−1,αi](λ)

Using the definition of v(·), for each λ ∈ R+

v(Lλ(a)) =k∑i=1

1− ψ

k⋃j=i

Aj

p

k⋃j=i

Aj

1(αi−1,αi](λ)

So that:

∫adv =

∞∫0

v(Lλ(a))dλ

where Lλ(a) = {s ∈ S : a(s) ≥ λ}

∫adv=

∞∫0

k∑i=1

1− ψ

k⋃j=i

Aj

p

k⋃j=i

Aj

1(αi−1,αi](λ)dλ

=k∑i=1

1− ψ

k⋃j=i

Aj

p

k⋃j=i

Aj

∞∫0

1(αi−1,αi](λ)dλ

=k∑i=1

(αi−αi−1)

1− ψ

k⋃j=i

Aj

p

k⋃j=i

Aj

=α1ψ(A1)+ (1− ψ(A1))

∫adp+

n∑i=3

(αi−αi−1)

ψ(A1)−ψ

k⋃j=i

Aj

p k⋃j=i

Aj

as desired.

This proof the first equality for a positive simple act a. The case of a general act ais easily obtained using translation invariance (the interest reader should see Marinacci andMontrucchio.[10]).

Corollary 1. Let ψ be an ambiguity level in Ψ restricted to D (i.e., ψ ∈ Ψ(D) and let p be aprobability measure (i.e., p ∈ ∆).

If a is a simple function and if the real-valued set-function v : Σ → R+ is defined byv ≡ (1− ψ) · p then the integral of a with respect to v can be calculated using formulas (2) and(3) of the theorem 4.

Let a be simple function defined by a(si) = i, for i = 1, 2, 3. In the sequel it will be calculatedthe expected value of a for the convex capacities in the preceding section, using the formulas(2) and (3) of the Choquet integral, presented in the theorem 4.

Example 14. (Expected value of the ε-contamination (back to Examples 2 and 10))Let S = {s1, s2, s3}. Fix D ≡ S, let ψε be the ambiguity level, defined by:

ψε(A) = ε if A 6∈ {∅, S} and ψε(A) = 0 if A ∈ {∅, S}

19

and let p be the probability measure (i.e., p ∈ ∆) defined by:

p({si}) =13, i = 1, 2, 3

note that supp(p) = S.If v : Σ→ R+ is the real-valued set-function defined by v ≡ (1− ψε) · p, i.e.:

v({si}) =(1− ε)

3, i = 1, 2, 3; v({si, sj}) =

2 · (1− ε)3

, i, j = 1, 2, 3 6= i.

then v is a convex capacity, with supp(v) ≡ S.It is easy to check that the expected value using the usual Choquet integral is 2− ε.Using formula (2):∫

Sadv ≡ α1ψε(A1)+ (1− ψε(A1))

∫Sadp+ (α3 − α2) (ψε(A1)−ψε (A3)) p (A3) = 2− ε

Using formula (3):

∫Sadv ≡ α1

p(A1) + ψε(A1) p

3⋃j=2

Aj

+α2

(1− ψε (A1)) p(A2)+

ψε 2⋃j=1

Aj

−ψε (A1)

p (A3)

+

+α3

1− ψε

2⋃j=1

Aj

p(A3) = 2− ε

Example 15. (Expected value of the convex capacity in Examples 7 and 11)Let S = {s1, s2, s3}. Fix D ≡ {s1}, let ψ be the ambiguity level defined by:

ψ({s1}) = 1− α = ψ({s2, s3}), α ∈ (0,12

) and ψ({si}) =12

= ψ({s1, si}), i = 2, 3

As showed in example 8, ψ is an ambiguity level restricted to {s1} (i.e., ψ ∈ Ψ′({s1})).and let p be the probability measure defined by:

p({s1}) = 1({si}) = 0, i = 2, 3

note that supp(p) = {s1}.If a set-function v : Σ→ R+ is defined by

v ≡ (1− ψ) p

that is:

v({si}) = α, α ∈ (0,12

); v({si}) = 0, i = 2, 3; v({s2, s3}) = 0 and v({s1, si}) =12

0, i = 2, 3

Then v is a convex capacity and supp(v) ≡ {s1}.It is easy to check that the expected value using the usual Choquet integral is 1.Using formula (2):∫

Sadv ≡ α1ψ(A1)+ (1− ψ(A1))

∫Sadp+ (α3 − α2) (ψ(A1)−ψ (A3)) p (A3) = 1

20

Using formula (3):∫Sadv ≡ α1

p(A1) + ψ(A1)p

3⋃j=2

Aj

+α2

(1− ψ (A1)) p(A2)+

ψ 2⋃j=1

Aj

−ψ (A1)

p (A3)

+

+α3

1− ψ

2⋃j=1

Aj

p(A3) = 1

Example 16. (Expected value of the convex capacities in Examples 9 and 12)Let S = {s1, s2, s3} and let p be a probability measure, defined by:

p({s1}) =13

, p({s2}) =23

and p({s3}) = 0

note that supp(p) = {s1, s2}.Fix D ≡ {s1, s2} and let ψ be an ambiguity level, defined by:

ψ({si}) = 0, 10 = ψ({si, s3}), i = 1, 2 and ψ({s3}) = 0, 05 = ψ({s1, s2})

As showed in example 13, ψ is an ambiguity level restricted to {s1, s2} (i.e., ψ ∈ Ψ({s1, s2})).If a set-function v : Σ→ R+ is defined by:

v ≡ (1− ψ) p

that is:

v({s1}) = 0, 3 = v({s1, s3}); v({s2}) = 0, 6 = v({s2, s3}); v({s3}) = 0 and v({s1, s2}) = 0, 95

Then v is a convex capacity and supp(v) ≡ {s1, s2}. In fact, as already noted in example13, this convex capacity also belongs to the subset of contaminations in example 12 (i.e., v ∈Λ(ψ, {s1, s2}, where ψ is the above restricted ambiguity level).

It is easy to check that the expected value using the usual Choquet integral is 0, 9.Using formula (2):∫

Sadv ≡ α1ψ(A1)+ (1− ψ(A1))

∫Sadp+ (α3 − α2) (ψ(A1)−ψ (A3)) p (A3) = 0, 9

Using formula (3):∫Sadv ≡ α1

p(A1) + ψ(A1) p

3⋃j=2

Aj

+α2

(1− ψ (A1)) p(A2)+

ψ 2⋃j=1

Aj

−ψ (A1)

p (A3)

+

+α3

1− ψ

2⋃j=1

Aj

p(A3) = 0, 9

The next proposition present a new property of the Choquet integral that can be appliedfor the subset of convex capacities that can be represented as contaminations (more specifically,this new property can be applied for the formulas (2) and (3) presented in the last proposition)on which for any two restricted ambiguity levels (i.e., ψ,ψ′ ∈ Ψ(D)) such that if one is greaterthan the other then the Choquet integral of a convex capacity that can be represented as acontamination will be so lesser how much bigger it will be ambiguity level associated.18

18The interest reader should check some properties of the Choquet integral in Donnensberg [?] (when v ∈ V )and Simonsen and Werlang [17] (when v ∈ Λ).

21

Proposition 2. Monotonicity of the Choquet integral with contaminationsLet a be a simple function. Fix a non-empty event D (∅ 6= D ⊆ S) and let ψ and ψ′ be

ambiguity levels in Ψ restricted to D (i.e., ψ,ψ′ ∈ Ψ(D)) and let p be a probability measure(i.e., p ∈ ∆), with supp(p) = D. Let v and w contaminations associated, respectively, to ψ andψ′. (i.e.: v ∈ Λ(ψ,D), w ∈ Λ(ψ′, D)), so it follows that:

If ψ′ ≥ ψ 19 then∫Sadw ≤

∫Sadv

Proof. For each p ∈ ∆ if ψ ≤ ψ′ then v ≥ w. So, it follows from the property of monotonicityof the convex capacities (see Simonsen and Werlang [17]) that

∫S adv ≥

∫S adw

The following examples illustrates this proposition and also present some static comparativeexercises .

Example 17. Let S = {s1, s2, s3} and let ψ,ψ′ : Σ→ [0, 1) be ambiguity levels, defined by:

ψ(A) = 0, 49 and ψ′(A) = 0, 51 for all A S.

Let p : Σ→ R+ be a probability measure, defined by:

p({si}) =13

, i = 1, 2, 3.

note that supp(p) = S.Fix D ≡ S and then note that ψ and ψ′ are restricted ambiguity levels associated to D (i.e.,

ψ,ψ′ ∈ Ψ(S)).If v is a set-function v : Σ→ R+ defined by:

v ≡ (1− ψ) p

that is:

v({si}) = (1− 0, 49) · 13

, i = 1, 2, 3; v({si, sj}) = (1− 0, 49) · 23

, i, j = 1, 2, 3 6= i.

Then v is a convex capacity associated to the restricted ambiguity level ψ with supp(v) ≡ S(i.e., v ∈ Λ(ψ, S)).

If w is a set-function w : Σ→ R+ defined by:

w ≡ (1− ψ′) p

that is:

w({si}) = (1− 0, 51) · 13

, i = 1, 2, 3; w({si, sj}) = (1− 0, 51) · 23

, i, j = 1, 2, 3 6= i.

Then w is a convex capacity associated to the restricted ambiguity level ψ′ with supp(w) ≡ S(i.e., v ∈ Λ(ψ′, S)).

Let a : S → R be a real-valued function defined by a(si) = i, for i = 1, 2, 3.It is easy to check that the expect value of a associated to v is 1, 51 and the expected value

of a of w is 1, 49.So, as desired: ∫

Sadw = 1, 49 ≤ 1, 51 =

∫Sadv

22

Let b : S → R be a real-valued function defined by: b(s1) = 0; b(s2) = 3; b(s3) = 6.It is easy to check that the expect value of b associated to v is 1, 53 and the expected value

of b associated to w is 1, 47.So, as desired: ∫

Sbdw = 1, 47 ≤ 1, 53 =

∫Sbdv

Now, suppose that an economic agent evaluates choices according to the expected value.If the uncertainty her faces is represented by the restricted ambiguity level ψ then

∫S adv =

1, 51 < 1, 53 =∫S bdv and so b � a.

By other hand, if the uncertainty her faces is represented by the restricted ambiguity levelψ′ then

∫S adw = 1, 49 > 1, 47 =

∫S bdw and so a � b.

More generally, if ψ ∈ (12 ; 1) then a � b, if ψ = 1

2 then a ∼ b and if ψ ∈ (0; 12) then a ≺ b.

Example 18. Let S = {s1, s2, s3} and let ψ,ψ′ : Σ→ [0, 1) be ambiguity levels, defined by:

ψ({si}) = 0, 1 = ψ({si, s3}), i = 1, 2; ψ({s3}) = 0, 05 = ψ({s1, s2})

and:ψ′({si}) = 0, 20 = ψ′({si, s3}), i = 1, 2; ψ′({s3}) = 0, 05 = ψ′({s1, s2})

Note that ψ′ ≥ ψ.Let p : Σ→ R+ be a probability measure, defined by:

p({s1}) =13

; p({s2}) =23

and p({s3}) = 0

note that supp(p) = {s1, s2}.Fix D ≡ {s1, s2} and then note that ψ and ψ′ are restricted ambiguity levels (i.e., ψ,ψ′ ∈

Ψ({s1, s2})).If v is a set-function v : Σ→ R+ defined by:

v ≡ (1− ψ) p

that is:

v({s1}) = 0, 3 = v({s1, s3}); v({s2}) = 0, 6 = v({s2, s3}); v({s3}) = 0 and v({s1, s2}) = 0, 95

Then v is a convex capacity associated to ψ and supp(v) = {s1, s2} (i.e., v ∈ Λ(ψ, {s1, s2})).If w is a set-function w : Σ→ R+ defined by:

w ≡ (1− ψ′) p

that is:

w({s1}) =14

= w({s1, s3}); w({s2}) =12

= w({s2, s3}); w({s3}) = 0; w({s1, s2}) =1920

and w(S) = 1

Then w is a convex capacity associated to ψ′ and supp(w) = {s1, s2} (i.e., v ∈ Λ(ψ′, {s1, s2})).Let a : S → R be a real-valued function defined by a(si) = i, for i = 1, 2, 3.It is easy to check that the expect value of a associated to v is 1, 60 and the expected value

of a of w is 1, 50.So, as desired: ∫

Sadw = 1, 50 ≤ 1, 60 =

∫Sadv

23

Let b : S → R be a real-valued function defined by: b(s1) = 1, 915; b(s2) = 1, 5; b(s3) = 1.It is easy to check that the expect value of b associated to v is 1, 5995 and the expected

value of b associated to w is 1, 57875.So, as desired: ∫

Sbdw = 1, 57875 ≤ 1, 5995 =

∫Sbdv

Now, suppose that an economic agent evaluates choices according to the expected value.If the uncertainty her faces is represented by the restricted ambiguity level ψ then

∫S adv =

1, 6000 > 1, 5995 =∫S bdv and so a � b.

By other hand, if the uncertainty her faces is represented by the restricted ambiguity levelψ′ then

∫S adw = 1, 50000 < 1, 57875 =

∫S bdw and so b � a.

More generally, if ψi ∈ [0, 1) are such that 0 ≤ ψ3 ≤ ψ1 = ψ2 < 1 and if

0, 0283... > (0, 483...) · ψ1 −12ψ3

then a � b. In particular, note that if ψi = (1− ε) ∈ [0, 1) (i.e., under ”uniform squeeze”) thenthe above inequality is satisfied for any ε ∈ (0, 1] and so, a � b.

By other hand, if

0, 0283... = (0, 483...) · ψ1 −12ψ3

then a ∼ b.And, finally, if

0, 0283... < (0, 483...) · ψ1 −12ψ3

then a ≺ b.

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