Integral representation with convex capacities that are squeeze of (additive) probability measures

· PIEGPE

SA

F U N D A

ç

à O

GETULIO VARGAS

Ｌｾ＠

FGV

EPGE

,

SEMINARIOS DE ALMOÇO

DA EPGE

Integral representation with convex

capacities that are squeeze

of (additive) probability measures

.J

PAULO CÉSAR COIMBRA LISBÔA

(EPGE/FGV)

Data: 24/10/2003 (Sexta-feira)

Horário: 12h 30min

Local:

Praia de Botafogo, 190 - 11

0

_andar

Auditório nO 2

Coordenação:

Integral Representation with Convex Capacities

that are Squeeze of (Additive) Probability

Measures.*

Paulo César Coimbra-Lisboa

t

EPGE/FGV+

October 03, 2003

Abstract

In this paper I will investigate the conditions under which a convex ca-pacity (or a non-additive probability which exhibts uncertainty aversion) can be represented as a squeeze of a(n) (additive) probability measure associate to an uncertainty aversion function. Then I will present two alternatives forrnulations of the Choquet integral (and I will extend these forrnulations to the Choquet expected utility) in a parametric approach that will enable me to do comparative static exercises over the uncertainty aversion function in an easy way.

JEL Classification Numbers: D8l.

Keywords and Phrases: Ellsberg paradox, Knightian uncertainty, capacity (non-additive probability), uncertainty aversion, Choquet inte-gral, Choquet expected utility.

1 INTRODUCTION

The expected utility theory, based in the works of Bernoulli (1739), Ramsey

(1931), de Finetti (1937) and von-Neumann and Morgenstern (1944), is the

*This paper is part of my doctoral research at EPGEjFGV. I grateful thank my supervisor, Sérgio Ribeiro da Costa Werlang, for very useful comments and sugestions and for encouraging me to write this paper. I also thank Hugo Pedro Boff who is carefully revisipg the proofs and audiences at XXIX Stochastic Process and Applications and VII Brazilian School on Probability, Angra dos Reis, RJ, Brazil (2003) and First Brazilian Conference on Statistical Modelling in Insurance and Finance, Ubatuba, SP, Brazil (2003) for comments and discussions and feedbacks from participants of the 2003 Latin American Meeting of the Econometric Society, held in Panama City, Panama (2003). The usual disclaimer applies.

tphD Student in Economics at EPGEjFGV.

+Graduate School of Economics at Getulio Vargas Foundation, Praia de Botafogo, n0190, sala 1100, Rio de Janeiro, RJ, CEPo 22250-900, Brazil; e-mail: [email protected]; url: http://www.fgv.br/users/aluno/coimbra/

basis of the modem theory of decision making under risk. After Knight (1928)' s distintion between risk and uncertainty (risk is when the probability distribution is known by the decision maker and uncertainty is when it is unknown), Savage (1954) proposed a set of 7 axioms under wich choice under uncertaity reduces to choice under risk, subjectivelIy perceived by the decision maker, which is the basis of the subjective expected utility theoryl. However, examples such as ElIsberg paradox (ElIsberg (1961)), and an increasing body of experimental and empirical evidence show that something was wrong with subjective expected utility theory. By the 80-'s, Gilboa and Schmeidler (see references) presented axiomatizations of the use of the Choquet integral on wich the decision making under uncertainty entails the mathematical expectation of the utility function with respect to capacities (or non-additive probabilities). This generalization of subjective expected utility has been calIed Choquet expected utility.

First I will investigate the condictions under which a convex capacity (or a non-additive probability which exhibts uncertainty aversion) can be represented as a squeeze of a(n) (additive) probability measure associate to an uncertainty aversion function. Let ｾ＠ be the power set of a finite state space D; p : ｾ＠ ---t [O, 1)

be a(n) (additive) probability measure and 'ljJ : ｾ＠ ---t [O,lJ be an uncertainty

aversion function. I will define a convex capacity v that is a squeeze of a(n) (additive) probability measure in the folIowing way: v(A) = (1-'ljJ(A»p(A) for alI A E ｾＬ＠ except for the case of A being the whole set (in which p and vare

1). Then I will be able to present two alternatives formulations of the Choquet integral (and I will extend these formulations to the Choquet expected utility) in a parametric approach that will enable me to do comparative static exercises over the uncertainty aversion function in an easy way. In the end of this section I will present a brief discussion on Ellsberg paradoxo The other sections of this paper are organized as follows: the next section introduces the required defi-nitions and basic statements of Knightian uncertainty' s decision theory, I will also discuss about the support of a capacity, the uncertainty aversion measure and the uncertainty aversion function; in section 3, I will introduce formalIy the special class of convex capacity that are squeeze of (additive) probability mea-sures; in section 4, I will discuss Choquet integral and present two new formulas with- a parametric approach and also present a new property that is satisfied by the Choquet integral; in section 5, I will extend the results to Choquet expected utility and; finally, section 6 concludes. The proofs are presented in the end of the paper.

1.1

Ellsberg Paradox

In Ellsberg (1961) 's paper there exists a mind experiment that consist ofmaking a choice over lotteries to show that decisions under uncertainty (in the Knight sense) are not consistent with Savage's paradigm.

Suppose that there exist an um that contains 90 balls on 3 different colors: red, black and yellow. Besides this is also know that there exists 30 red balls and that all that is known about the other 60 balls is that they are black or yellow.

Imagine, first, the two following lotteries: "win $100 if a red ball is drawn from the um and win nothing otherwise" and " win $100 if a black ball is drawn from the um and win nothing otherwise" . Of course is expected that most of participants of this experiment will prefer to bet on " red ball" lottery, beca use it is a known fact that 1/3 of the balls are with certainty reds. However, if in each of these lotteries the option " yellow ball" is included in the following way: " win $ 100 if a red ball or a yellow ball is drawn from the um and win nothing otherwise" and " win $ 100 if a black ball or a yellow ball is drawn from the um and win nothing otherwise" then is expected that most of the participants will now prefer to bet on " black ball - yellow ball" lottery. This occur because it is known with certainty that there exists exactly 60 balls that are black or yellow. So decisions under uncertainty are different from that ones when the probability distributions are objectively known.

2 SET-UP AND PRELIMINARIES

The decision setting that I will use in the paper is developed in a Savage-style (see Savage (1954)). I assume that the uncertainty a decision maker faces can be described by a non-empty and finite set of states n2 . Associate with the set of states is the set of events taken to be an algebra of subsets of n,denoted by

I:3 . I assume that for each w E n, {w} E I:. Let X be a non-empty and finite set of outcomes. Let ｾ＠ be the class of alI simple acts. A simple act is a finite valued function

f

:

n -; X which is measurable with respect to I:4_{. For x E X I}

will define x E ｾ＠ to be the constant act such that x(w)

= x

for all w E n. So, with slight abuse of notation, I shall let X also denote the subclass of constant

acts in ｾＮ＠

A set-function v : I: -; ｾ＠ with v(0) = O is called a capacity (also called a non-additive probability) on (n, I:) if it is normalized and monotone, that is:

21n this paper,

n

== {1,2, ... ,n}.

31n this paper ｾ＠ == 2° satisfying the following properties: i)

n

E E;

ii) For ali A, B E E: A U B E E; iii) For ali A E E: (n\A) E E.

Then, it is easy to prove that if ｾ＠ == 2° satisfy (i) to (iii) above then it is also true that: iv) 0 E E;

v) For ali A, B E E: A n B E E.

4 A real-valued function, bounded on

n,

a :

n

-> iR is said to be E-measurable if, for ali

open set O

ç

iR, a-1(O) E ｾＬ＠ where a-1(O) == {w E

n :

a(w) E O}. I will denote by

B(n, E) the c1ass of ali real-valued function, bounded on n, that are E-measurable. Note that

ｾＨｃ＠ B(n, E)).

..

i) normalized: v(O) = 1; ii) monotone: For all A, B E I; such that A

ç

B5 :

v(A) :::; v(B). I will denote by V(O, I;) the class of ali capacities on (O, I;)6. A capacity is convex if, besides (i) and (ii) it also satisfies the following property:

(iii) For ali A, BEL;: v(A U B)

+

v(A

n

B) ｾ＠ v(A)

+

v(B). In fact is easy to prove that if a set-function v : I; -+

R

with v(0)

=

O

satisfy the property (iii) then the property (ii) is also satisfied, Le., v is monotone. I will denote by

A(c V(O, I;» the class of all convex capacities on (O, I;).

A capacity is (finitely) additive (also called a(n) (additive) probability mea-sure) if, besides properties (i) and (ii) it also satisfies the following property: (iii') For ali A, B E I; such that A

n

B

= 0:

v(A U B)

=

v(A)

+

v(B). I will denote by Ó( C A) the class of ali (additive) probability measures on (O, I;).

2.1 Support of a Capacity

The notion of support of a capacity is the first step necessary to study the conditions under which a convex capacity can be understood as a squeeze of a(n) (additive) probability measure.

Definition 1 Support of a Capacity Let v E V(O, I;) and A, B E I;

The support o/ the capacity v is an event B such that:

i) v(O\B) = O;

ii) For all A, BEL;, A C B: v(O\A)

>

O.

The Ellsberg's um, presented at the introduction of this paper, offer an example of a capacity that does not have an unique support.

Example 1 Ellsberg's Um

Let Er be the event "a red ball is drawn from the urn". The events Eb and Ey are similarly defined.

Let (O, I;, v) be a capacity space that refiects the Ellsberg 's um, i.e.:

°

=

{Er,Eb,Ey};

I;=2fl;

v(0) = O; v(O) = 1; v({Er}) ］ｾＬ＠ v({Eb}) = v({Ey}) = k; v( {Er, Eb})

=

v( {Er , Ey})

=

ｾ＠

+

k; v( { Eb, Ey})

=

ｾ［＠

for all k E [O, ｾ｝Ｎ＠

I/ k

E [O, 1]

then v is a convex capacity.

I/ k E (O, \] then the (uni que) support is the event {Er,Eb, Ey}. I/ k

=

O then the supports are the events {Er , Eb} and {Er , Ey}.

5In this paper it will be used the following notation: A

ç

B means that the set A is not a proper subset of B (i.e., A = B is possible) and A C B means that A is a proper set of B (i.e., A "# B always).

The following example, due to Eichberger and Kelsey (2000), shows that the supports may all get a zero mass :

Example 2 (Eichberger and Kelsey (2000)) Let (Sl,:E, v) be a capacity space defined by:

Sl

=

{Wl,W2,W3};

:E -_-

-

')0.

_,

v(0)

=

O; V({Wi})

=

O, for i E {1,2,3}; V({Wl,W2})

=

1;

V({Wl,W3})

=

V({W2,W3})

=

O e v(Sl)

=

l.

The supports are the events {wr} and {W2}, both with zero mass: v({wIl) = V({W2}) = O

The following proposition, due to Marinacci (2000), present the condi-tions under which a capacity has a unique support.

Proposition 1 (Marinacci (2000)) Let v E V(Sl, :E).

The following two conditions are equivalent: i) supp v is unique;

ii) For all A E :E and all w E Sl such that v(A)

=

v( {w}) = O:

v(A

u

{w})

=

v(A)

+

v({w}).

This suggest the following definition:

Definition 2 Convex Capacities with a Unique Support

A convex capacity has a unique support if, besides (i) to (iii) it also satisfy the following property:

iv) For all A E :E and all w E Sl such that v(A) = v({w}) = O:

v(A

u

{w})

=

v(A)

+

v({w}).

Example 3 Ellsberg Um (following example 1)

lf

k = O then (iv) fails:

V({Eb} U {Ey})

=

V({Eb,Ey})

=

t

=I-

O

=

V({Eb})

+

v({Ey})

Example 4 (Eichberger and Kelsey (2000)) (following example 2) (iv) fails:

V({W2,W3} U

{wIl)

=

v(Sl)

=

1 =I-O

=

V({W2,W3})

+

v({wr})

2.2 Uncertainty Aversion Measure

If v E Ais not a(n) (additive) probability measure then there exists at least a pair A, B E ｾ＠ such that: v(A

u

B)

+

v(A

n

B)

>

v(A)

+

v(B). In particular, if B = (Çl\A) then v(A)

+

v(Çl\A) may be less than 1, implying that not ali probability mass is allocated to an event and its complemento Dow and Werlang (1992) proposed an uncertainty aversion measure of a capacity v at event A:

Definition 3 (Dow and Werlang (1992))

Let v E ｖＨￇｬＬｾＩ＠ and A E ｾＮ＠

The uncertainty aversion measure of v at event A, is defined by: c(v,A)

= 1 -

v(A) - v(Çl\Af.

It is easy to prove that if a capacity is convex then for ali A E ｾＺ＠ c(v, A) E [0,1]. Convex capacities are also know as non-additive probabilities reflecting uncertainty aversion. Throughout this paper I will restrict attention to convex capacities.

The following proposition present the properties that are satisfied by the uncertainty aversion measure of v at event A if v is a convex capacity:

Proposition 2 Let v E A and A, B E ｾ｟＠

The uncertainty aversion measure of v at event A satisfy the following prop-erties:

i) c(v, 0)

=

c(v, Çl)

=

0,-ii) For ali A E ｾＺ＠ c( v, A)

=

c(v,

(Çl\A)),-iii) For ali A, B E ｾＺ＠ c(v, (A

u

B))

+

c(v, (A

n

B))

:s:

c(v, A)

+

c(v, B).

2.3 Uncertainty Aversion Function

In the last subsection, I discussed the properties satisfied by an uncertainty aversion measure associate to a convex capacity. Now I will present a set-function with the same properties:

Definition 4 Uncertainty Aversion Function

The set-function 1jJ : ｾ＠ -> [0,1] is an uncertainty aversion function if it satisfy the following properties:

i) 1jJ(0)

=

1jJ(Çl)

=

0,-ii) For ali A E ｾＺ＠ 1jJ(A)

=

1jJ(Çl\A),-iii) For ali A, B E ｾＺ＠ 1jJ(A U B)

+

1jJ(A

n

B)

:s:

1jJ(A)

+

1jJ(B).

..

I will denote by ｉｊＡＨｮＬｾＩ＠ the class of alI uncertainty aversion function on

ＨｮＬｾＩＮ＠

ExampIe 5 Let ＨｮＬｾＩ＠ be defined by:

n

= {Er,Eb,Ey};

ｾ＠

=

20.;

Consider an uncertainty aversion function, 7j; E ｉｊＡＨｮＬｾＩ＠ such that 7j;(0)

=

7j;(n)

= O and for all A

E ｾＬ＠ 0

f=

A

f=

n is defined by:

7j;({Er })

=

7j;({Eb, Ey})

=

O;

7j;({Eb})

= 7j;{{EnEy})

=

ｾ＠

-

2k; 7j;({Ey })

= 7j;({En E b})

=

3 - 2k.

ft is easy to check that, for all k E

[O,

ｾｬ＠ the properties of an uncertainty aversion function are satisfied.

Associate with each uncertainty aversion function 7j; E ｉｊＡＨｮＬｾＩ＠ there exists a subclass of convex capacities v E A with the property that, for all A E ｾＬ＠ the uncertain aversion measure is such that: c(v, A)

=

7j;(A).

Definition 5 Convex Capacity Associate to an Uncertainty Aversion

Function Let 7j; E IJ!(n, ｾＩＮ＠

f say that v E A is associate to 7j; if, for all A E ｾＬ＠ 0

f=

A

f=

n:

c(v,A) = 7j;(A).

For each 7j; E ｉｊＡＨｮＬｾＩＬ＠ I will denote by A(7j;) (C A) the class of all convex capacities on ＨｮＬｾＩ＠ that are associate to the uncertainty aversion function 7j;.

Proposition 3 Let 7j; E IJ!(n, ｾＩＮ＠

v E A(7j;) if and only ifv E Ais such that, for all A ｅｾＬ＠ 0

f=

A

f=

n:

c(v,A)

= 7j;(A)

Example 6 Ellsberg's Um (following example 1)

For each k E [O, ｾ＠

1

the uncertainty aversion measure associate to the Ells-berg's um has the following characteristic:

i) c(v, {Er}) = c(v, {Eb,Ey}) = O;

ii) c(v, {Eb}) = c(v, {Er, Ey})

=

% - 2k; iii) c(v, {Ey})

=

c(v, {Er, Eb})

= % -

2k.

So, for each k E [O, ｾＩＬ＠ f can define an uncertainty aversion function 7j; E ｉｊＡＨｮＬｾＩ＠ in the following way:

For ali A E ｾＺ＠ 7j;(A)

=

c( v, A), where v is the convex capacity associate with the Ellsberg um. That is:

..

i) 1jJ({Er})

=

1jJ({Eb,Ey})

=

Oi ii) ＱｪｊＨｻｅ｢ｽＩ］ＱｪｊＨｻｅｲＬｅｹｽＩ］ｾＭＲｫ［＠

iii) 1jJ({Ey}) = 1jJ({Er, Eb}) =

i -

2k.

Of course if v is the convex capacity associate to the Ellsberg 's urn then, for each k E ｛ｏＬｾ｝Ｌ＠ v E A(1jJ,k), where'lj; E \[I(!1,E) is the above uncertainty aversion function.

The following example shows that there exist at least a k E [O, ｾｬ＠ such that

A(1jJ, k) contains more than one element:

Example 7 Ellsberg's Um (following example 1)

ff k = ｾ＠ then A(1jJ,k ］ｾＩ＠ is a Non-Empty, Non-Unitary Class Let (!1, E) be defined by:

!1 = {Er , Eb, Ey}i

E = 2°;

Let 'Ij; E 1l1(!1, E) be defined as in the last example.

ft is easy to see that if k = ｾ＠ then, for all A E E: 1jJ(A) = O.

Let p E

t.

be a(n) (additive) probability measure on (!1, E). Then, for any p the unceriainty aversion measure is such that, for all A E E: c(v, A)

=

O. 80,

11.( 1jJ, k

=

ｾＩ＠ defined as in the last example is a non-empty, non-unitary class:

11.( 1jJ, k

=

ｾＩ＠

=

{p E !:li

t.

is the set of probability measures on (!1, E)}

3

SQUEEZE

DF

PRDBABILITY MEASURES

I will define the conditions under which a convex capacity can be represented as a squeeze of a(n) (additive) probability measure associate to an uncertainty aversion function in the following way:

Definition 6 Squeeze of a(n) (Additive) Probability Measure

Asso-ciate to an Uncertainty Aversion Function

Let p E

t.

and 1jJ E \[I(!1, E).

f say that v E V(!1, E) is a squeeze of p associate to 1jJ, if: (A) = { (1 -1jJ(A»p(A) if A

=I

!1

v 1 if A =!1

..

Definition 7 Properties of a Convex Capacity that are Squeeze

Let v E A, with

0

i=

D

=

supp v

ç

D, and C,e:,D,E E L:.

l say that v is a squeeze of a(n) (additive) probability measure p E Do asso-ciate to some 1/J E w(D, L:) if it also satisfy the following properties:

v) For all 0

i=

C, C C D, with C

u

C C D and C

n

C

=

0: v(C U C)

=

v(C)

+

v(C)

vi) For all0

i=

C C D and all E

ç

(D\D): v(C U E)

=

v(C)

Proposition 4 Let 1/J E w(fl, L:).

lf v E A( 1/J) satisfy the properties (v) and (vi) (of definition 7) then v is a squeeze of some p E Do associate to some 1/J.

The following examples shows convex capacities on which (v) or (vi) fails:

Example 8 Convex capacity that not satisfy (v)

Let (fl, L:, v) be a capacity space defined by:

fl

=

{Wl,W2,W3};

L:

=

2°.

v(0) = O; v(fl) = 1; v( {wd) = 0,4; v( {W2})

=

0,3; v( {W3})

=

0,1;

V({Wl,W2})

=

0,8; V({Wl,W3})

=

0,5; V({W2,W3})

=

0,4.

lt is easy to check that v E A and not satisfy (v)

Consider an uncertainty aversion junction 'lj; E w(fl, L:) defined by: 1/J({wd) = 1/J({W2,W3})

=

0,2;

1/J({W2})

= 1/J({Wl,W3})

=

0,2;

1/J({W3})

= 1/J({Wl,W2})

=

0, l.

lt is easy to check that 1/J has the same values of the uncertainty aversion measure associate to v, defined such that, for all A E 2;:

c(v, A)

=

1 - v(A) - v(D\A) So v E A(1/J).

To show that v is not a squeeze of any (additive) probability measure p E Do,

l will define a set function q : L: --- [0,1] by: q(A) = {

ｬｾｾｾｾｽＧ＠

if A

i=

D

1, if A

= D

Note that q f/. Do, because:

({ }) v({w,}) 1

q Wl

=

l-,p({w,})

=

2"

({ }) _ V({W2}) _ 3

q W2 - 1-,p({W2}) - 8

({ }) _ V({Wl,W2}) _ 8

q Wl,W2 - 1 ,p({Wl,W,}) - 9

So, q({wd)

+

q({W2})

<

q({Wl,W2})

Example 9 Convex capacity that not satisfy (vi) Let (D, L:, v) be a capacity space defined by:

D

=

{Wl,W2,W3};

L:

=

2°.

v(O)

=

O; v(D)

=

1; v({wd)

=

0,4; V({W2})

=

0,3; V({W3})

=

O;

V({Wl,W2})

=

0,8; V({Wl,W3})

=

0,5; V({W2,W3})

=

0,3.

It is easy to check that v E A and not satisfy (vi)

Consider an uncerlainty aversion function 1/; E \[I (D, I:) defined by:

1/;({wd) = 1/;({W2,W3}) = 0,3;

1/;({W2})

=

1/;({Wl,W3})

=

0,2;

1/;({W3})

=

1/;({wl,wd)

=

0,2.

ft is easy to check that 1/; has the same values of the uncertainty aversion

measure associate to v, defined such that, for all A E I::

c(v, A)

=

1 - v(A) - v(D\A)

So v E A(1/;).

To show that v is noi a squeeze of any (additive) probability measure p E ll.,

f will define a set function q : L: ->

[0,1]

by:

q(A) = {

ｬｾｾｾｾＩＧ＠

if A

# D

1, if A

=

D

Note that q rJ. ll., because:

({ }) _ v({wIk) _ 4

q Wl - 1 ,p({w,j) - "7

({ }) V({W3})

°

q W3 = 1-,p({W3})

=

({ }) _ V({Wl.W3}) _ 5

q Wl,W3 - 1 ,p({W"W3}) - 8

So, q({wd)

+

q({W3}) < Q({Wl,W3})

Let ll. be the class of (additive) probability measures and \[I(D, I:) be the class of uncertainty aversion function. I wilI denote by 8(ll., \[I) the class of convex capacities that are squeeze of (additive) probability measures associate to some 1/;.

Proposition 5 Let v E 8(ll., \[I), with

O

#

D = supp v

ç

D and C, C, D and E E L:.

The uncertainty aversion measure of valso satisfy the following properties:

iv) For all

O

#

C, C C D, with C

u

C C D e C

n

C

=

O:

c(v,C)

=

c(v,C);::: c(v,D);

v) For allO

#

C C D and all E

ç

(D\D):

c(v,(CUE»

=

c(v,C).

Example 10 Let (D, L:, v) be a capacity space defined as in example 8.

For each A E I:, the uncerlaíniy aversion measure is:

c( v, ( {W2})

=

c( v, ( {Wl, W3})

=

0,2;

c(v, ({W3})

=

c(v, ({Wl,W2})

=

O, L

Because supp p v

=

rl: c(v, ({wd)

=

c(v, ({wd)

=

c(v, ({W3})' SO, lhe property (iv) of lhe uncertainly aversion measure fails.

Example 11 Let ＨｲｬＬｾＬ＠ v) be a capacity space defined as in example 9. For each A E ｾＬ＠ the uncertainty aversion measure is:

c(v, ({ wd)

=

c( v, ({ W2, W3})

=

0,3;

c(V,({W2})

=

c(V,({Wl,W3})

=

0,2;

c(v, ({ W3})

=

c( v, ({ Wl, W2})

=

0,2.

Because supp p v

=

{Wl,W2}: 1,I>({Wl,W3})

=

1,I>({Wl})'

So, lhe property (v) of the uncertainty aversion measure fails.

Proposition 6 Let 1,1> E w(rl, ｾＩＬ＠ p E ｾＬ＠ with

0

i=

D

=

supp p

ç

rl and C,C:,D,E ｅｾＮ＠

lf 1,1> also satisfy the following properties:

iv) For all 0

i=

C,

c:

C D, with C U

c:

C D and C

n c: =

0:

1,I>(C) = 1,I>(C)

2:

1,I>(D);

v) For all0

i=

C C D and all E

ç

(rl\D): 1,I>(CUE) = 1,I>(C).

And ifv E ｖＨｲｬＬｾＩ＠ is defined by: (A)

= {

(1 -1,I>(A))p(A) if A

i=

rl

v 1 if A = rl

Then v E 11.(1,1».

Proposition 7 Let 1,1> E ｷＨｲｬＬｾＩＬ＠ p E ｾＬ＠ with

0

i=

D = supp p

ç

rl and C,C:,D,E ｅｾＮ＠

lf 1,1> E ｷＨｲｬＬｾＩ＠ also satisfy the following properties:

iv) For all 0

i=

C,

c:

C D, with C

u c:

C D and C

n c:

=

0:

1,I>(C)

=

1,I>(C)

2:

1,I>(D);

v) For all0

i=

C C D and all E

ç

(n\D): 1,I>(CUE)

=

1,I>(C).

And ifv E 11.(1,1» is defined by:

(A) = { (1 -1,I>(A))p(A) if A

i=

rl

v 1 ifA=rl

Then valso satisfy the following properties:

v) For all0

i=

C,

c:

C D, with C U

c:

C D e C

n c: =

0:

v(C U C)

=

v(C)

+

v(C)

vi) For all0

i=

C C D and all E

ç

(rl\D): v(C U E)

=

v(C).

EXaII1ple 12 Convex Capacity with a Unique 8upport which has 1 El-ement.

Let (1l,I:,v) be a capacity space defined by:

11

=

{Wl,W2,W3}i

I:

=

2!1.

v(0)

=

Oi v(ll)

=

li v({wd)

=

Oi V({W2})

=

Oi V({W3})

=

ai V({Wl,W2})

=

Oi V({Wl,W3})

=

a

+

bi V({W2,W3}) = a

+

c.

lt is easy to check that if a

+

b

+

c :::: 1 then v E A and also satisfy properlies (v) and (vi).

The (unique) support is the event {W3}

Consider an uncertainty aversion function 'Ij; E 1li(1l, I:), defined by: 'Ij;({wd)

= 'Ij;({W2,W3})

=

1 - a - ci

'Ij;({W2})

= 'Ij;({Wl,W3})

= 1 -

a - bi 'Ij;({wd)

=

'Ij;({Wl,W2})

=

1 - a.

lt is easy to check that 'Ij; satisfy properlies (i) to (v) of an uncertainty aver-sion function and, for each A E I:, has the same values that the uncertainty aversion measure associate to v, i. e.:

c( v, A) = 1 - v(A) - v(Il\A)

80 v E A('Ij;).

lf 1 will define the set-function q : I: - 7

[0,1],

by:

q(A)

= {

ｬｾｾｾｾＩＧ＠

if A =1= 11 1, if A

=

11

lt is easy to check that q E Êl.

80, v is a squeeze of q associate to 'Ij;, i.e., v E 8(Êl, Ili).

EXaII1ple 13 Convex Capacity with a Unique 8upport which has 2 El-ements

Let (11, I:, v) be a capacity space defined by:

11

=

{Wl,W2,W3}i

I:

=

2!1.

v(0) = Oi v(ll) = li v({wd) = ai V({W2})

= bi V({W3})

= Oi v ( { Wl , w2 }) = a

+

b

+

c i v ( { Wl , W3}) = a i v ( { W2, w3 }) = b.

lt is easy to check that if a

+

b

+

c :::: 1 then v E

A

and also satisfy properlies (v) and (vi).

The (unique) supporl is the event {W2, W3}'

Consider an uncerlainty aversion function 'Ij; E 1li(1l, I:), defined by: 'Ij;({wd)

=

'Ij;({W2,W3})

=

1 - a - bi

'Ij;({W2})

=

'Ij;({Wl,W3})

=

1 - a - bi

'Ij; ( { W3})

=

'Ij; ( { Wl , W2 })

=

1 - a - b - c.

Jt is easy to check that 'Ij; satisfy properties (i) to (v) of an uncerlainty aver-sion function and, for each A E I:, has the same values that the uncerlainty aversion measure associate to v, i. e.:

c(v, A)

=

1 - v(A) - v(Il\A)

"

lf 1 define the set-function q : L: - t

[O, 1],

by:

q(A) = {

ＱｾｌＱｾＩＬ＠

if A -1=

n

1, if A

=

n

It is easy to check that q E .6..

So, v is a squeeze of q associate to 1/;, i.e., v E 8(.6., 1lJ).

Example 14 Convex Capacity with a Unique Support which has 3 El-ements

Let

(n,

L:, v) be a capacity space defined by:

n

=

{Wl,W2,W3,W4}; L:

=

2°.

v(0)

=

O; v(n)

=

1;

v({wJ})

=

O; V({W2})

= a;

V({W3})

= b;

V({W4})

= c;

V({Wl,W2})

= a;

V({Wl,W3})

= b;

V({Wl,W4})

= c;

v( {W2, W3})

=

a + b; v( {W2, W4})

=

a + c; v( {W3, W4})

=

b

+

c; v( {W1,W2,W3}) = a

+

b; v( {Wl,W2,W4}) = a

+

c;

v( {W1, W3, W4})

= b

+

c; v( {W2, W3, W4})

=

a

+

b

+

c

+

d.

lt is easy to check that if a

+

b

+

c

+

d ::;: 1 then v E A and also satisfy properties (v) and (vi).

The (unique) support is the event {Wl, W3, W4}.

Consider an uncertainty aversion function 1/; E llJ(n, L:), defined by:

1/;({wd) = 1/;({W2,W3,W4}) = 1 - a - b - c - d; 1/;({W2}) = 1/;({Wl,W3,W4}) = 1-a - b - c;

1/;({W3}) = 1/;({W1,W2,W4}) = 1-a - b - c;

1/;({W4}) = 1/;({Wl,W2,W3}) = 1-a - b - c;

1/;({Wl,W2})

=

1/;({W3,W4})

=

1 - a - b - c;

1/;({W1,W3})

=

1/;({W2,W4})

=

1-a - b - c;

1/;({W1,W4}) = 1/;({W2,W3}) = 1-a - b - c.

lt is easy to check that 1/; satisfy properties (i) to (v) of an uncertainty aver-sion function and, for each A E L:, has the same values that the uncertainty aversion measure associate to v, i.e.:

c(v,A) = 1 - v(A) - v(n\A)

So v E A(1/;).

lf 1 define the set-function q : L: - t

[O, 1],

by:

q(A)

= {

ＱｾｾｾｾＩＧ＠

if A -1=

n

1, if A

=

n

lt is easy to check that q E .6..

So, v is a squeeze of q associate to 1/;, i.e., v E 8(.6., 1lJ).

"

4 CHOQUETINTEGRAL

Since that capacities can be a non-additive measure I can't use an integral in the sense of Lebesgue. The appropriate notion of integral is due to Choquet (1953). For any given real-valued function, bounded on

n,

a E B(n, ｾＩＬ＠ the Choquet integral of a with respect to a capacity v E ｖＨｮＬｾＩ＠ is defined as follows:

o 00

J adv == J [v(w E

n :

a(w)

2:

0:) - l]do:

+

J v(w E

n :

a(w)

2:

o:)do: 8

-00 o

where the right hand side is a well defined integral in the sense of Riemann (because a is bounded and v is monotone)9.

All the real-valued function considered in this paper will be sim pIe acts. Besides this, every simple act f E

çs

will be defined such that: f(wI) ::; f(W2) ::; ... ::; f(wn ). I will be particularly interested in a special subclass of the class of

simple acts, which are constituted by the co-monotonic acts. Two acts,

f,

9 E

çs

are said to be co-monotonic if, for all ,w, wl E

n:

(f(w)- f(WI))(g(w)-g(wI))

2:

o.

I will denote by F (c

çs)

the class of co-monotonic acts which is constituted by sim pIe acts that are pairwise co-monotonics.

For all

f,

9 E

çs

and v, VI E V(n, ｾＩＬ＠ the following proposition present some properties of Choquet integral (proved in Donnensberg (1994); Simonsen and Werlang (1991) proved in the case of v, VI E A):

Proposition 8 For all f, 9 E

çs

and v, VI E V(n, ｾＩＮ＠

a)Positive homogeneous.J cfdv

=

c J fdv , for c E R+;

b)Monotonic: f

2:

9 =} J fdv

2:

J gdv ; c) J(I

+

c)dv = J fdv

+

c, for c E R+;

d) Co-monotonic additive: If f, 9 E F are non-negative then: J(I

+

g)dv

=

J fdv

+

J gdv ;

e) Monotonic in the capacity: Ifv

2:

VI thenJ fdv

2:

J fdv' .

For all

f,

9 E

çs

and v, VI E A, the following proposition present other properties of Choquet integral (proved in Simonsen and Werlang (1991)):

8From now on we will use the following simplification: v(a ｾ＠ a) = v({w E

n:

a(w) ｾ＠ a}). so the Choquet integral can be re-writer as:

o

J

adv ==

J

(v (a ｾ＠ a) - l)da +

J

v(a ｾ＠ a)da

-oc o

Proposition 9 For all f, 9 E ｾ＠ and v, VI E A.

f) Jfdv

+

J(-f).dv :::; O ;

g) Jensen Inequality: For ali concave functions, u: R --> R :

J u(f)dv :::; u

(J

fdv) .

4.1 Choquet Integral with Convex Capacities that are Squeeze

of (additive) Probability Measures

Theorem 1 Let

f

E F, v E 8(6., \]i). The foliowing statements are equivalent:10

o 00

i) J fdv

==

J (v(f

2':

ex) - l)dex

+

J v(f

2':

ex)dex;

-00 o

ii) J fdv

==

"I,b,f({w,})

+

(1-'!jJ,)J fdp+

n n

+ 2:=

("l,b1 -'!jJj, ... ,n)(2:= p( {Wi} ))(f(Wj) - f(Wj-1»

j=3 i=j

n

iii) J fdv

==

f(w,)[P({wd)

+

'!jJ,

2:=

p({Wj})]+ j=2

n - l n

+ 2:=

f(Wi)[(I-'!jJ1....,i-dp({Wi})+('!jJ1, ... ,i-"l,b1, ... ,i-1)

2:

p({Wj})]+

i=2 j=i+1

+ f(wn )[(1 -'!jJ1, ... ,n-dp( {wn })]

where: p: ｾ＠ --> [0,1] is a(n) (additive) probability measure on

(n,

ｾＩＬ＠

and

J fdp is its expected value.

Remark 1 Note that if for all A E ｾＬ＠

0

#-

A

#-

ri, '!jJ(A)

=

c E [0,1] then I will have the uniform squeeze formula:

J fdv

==

"l,bf(w,)

+

(1 -'!jJ) J fdp

Example 15 Gonvex Gapacity with a Unique Support which has 1 El-ement (following example 12)

Let p E 6. be defined by:

p({wd)

=

O; P({W2})

=

O; P({W3})

=

1.

80 v E 8(6., \]i) can be defined as: (A)

= {

(1 -'!jJ(A»p(A) if A

#-

n

v 1

ifA=n

10_{1 will use the following simplification:}

,p, =,p({w,});

,pl. .... i =,p({w" ... ,wi });

,pj .... n = ,p({Wj, ... ,Wn})

Let f :

n

-+ R be a real-valued function defined by:

f(Wi)

=

i, for i E {1,2,3}.

lt is easy to check that the "usual" Choquet integral associate is 1

+

2a

+

c

Using (ii):

J

fdv

==

lPJ( {w,})

+

(1 -11',)

J

fdp+ +(11'1 -1P3)P({W3})(J(W3) - f(W2» =

=

(1 - a - c)

+

3(1 - (1 - 1 - c»

+

((1 - 1 - c) - (1 - 1)(3 - 2)

=

=

1

+

2a

+

c

Using (iii):

J

fdv

==

f(w,)[P( {wd) + lP,p( {W2.3} )]+

+ f(w2)[(1 -lPdp( {W2}) + (11'1.2 - 1P1)P( {W3} )]+

+f(W3)[(1-1P1.2)P({W3})]

=

=

[0+(1-a-c)]+2[(1-(1-a-c»0+((1-a)-(1-a-c»]+3[(1-(1-a}]

=

=

1

+

2a

+

c

Example 16 Convex Capacity with a Unique Support which has 2

El-ements (following example 13)

Let p E 6 be defined by:

p({wd)

=

O; P({W2})

=

｡ｾ｢［＠ P({W3})

=

a!b·

80 v E 8(6, 'lt) can be defined as:

(A)

= {

(1 -lP(A»p(A) if A

i=

n

v 1 ijA=n

Let f :

n

-+ R be a real-valued function defined by:

f(Wi)

=

i, fori E {l,2,3}.

lt is easy to check that the "usual" Choquet integral associate is 1

+

a

+

2b

+

c

Using (ii):

J

fdv

==

lP,f({w,})

+

(1 -11',)

J

fdp+

+( 11'1 -1P3)P( {W3} )(J(W3) - f(W2» =

= (1 - a - b - c)

+

(1 - (1 - a - b - c»

ＨＲｾＡｾ｢Ｉ＠

+

+((1 - a - b - c) - (1 - a - b»

(a!b)

(3 - 2)

=

=

1

+

a

+

2b+ c

Using (iii):

J

fdv

==

f(w,)[P({wd)

+

lP,P({W2.3})]+

+ f(w2)[(1 -lPdp( {wd)

+

(11'1.2 -lPdp( {W3} )]+

+f(W3)[(1-lPl,2)P({W3})]

=

=

[O

+

(1 - a - b - c)]

+

2[(1 - (1 - a - b - c»

Ｈ｡ｾ｢Ｉ＠

+

((1 - a - b)--(1-a - b-c»

(a!b)]

+

3[(1- (1 - a - b)

(a!b)]

=

Example 17 Convex Capacity with a Unique Support which has 3 El-ements (following example 14)

Let p E ｾ＠ be defined by:

p({wd) = O; P({W2}) = ｡ＫｾＫ｣［＠ p({W3}) = ｡ＫｾＫ｣［＠ P({W4}) = ｡ＫｾＫ｣Ｇ＠

80 v E ･ＨｾＬ＠ \li) can be defined as: (A) = { (1 -1/I(A)p(A) if A

1=

n

v 1

ifA=n

Let f :

n ...

R be a real-valued functíon defined by: f(Wi)

=

i, for í E {1, 2, 3}.

It ís easy to check that the "usual" Choquet integral associate is equal to:

1 +

a

+

2b

+ 3c +

d

Using (ii):

J

fdv

==

1/1,/(

{w,})

+

(1

-1/1,)

J

fdp+

4 4

+ 2:= (1/11 -1/Ij, ...

.4)(

2:=

p( {Wi} »(f(Wj) - f(Wj-1»

=

j=3 i=j

= (1 -

a - b - c - d)

+

(1 - (1 - a - b - c - d))

･｡｡ｾＳ｢｢Ｚ｣Ｔｃ＠

)

+

+( (1 - a - b - c - d) - (1 - a - b - c»

Ｈ｡ｾｴｾ｣Ｉ＠

(3 - 2)+ +( (1 - a - b - c - d) - (1 - a - b - c»

Ｈ｡ＫｾＫ｣Ｉ＠

(4 - 3) =

=

1

+

a

+

2b

+ 3c + d

Using (iií):

4

J

fdv == f(w,)[P({wd)

+1/1,2:=

p({Wj})]+ j=2

3 4

+ 2:=

f(Wi)

[(1-1/I1, ... ,i-1)P({W;})+(1/I1, ... ,i-1/IL. ..

,i-l)

2:=

p({Wj})]+

i=2 j=i+1

+

f(w4)[(1 -1/I1, ... ,3)P( {W4})]

=

= [0+ (1-a - b-c-d)] + 2[(1- (1-a - b-c-d»

Ｈ｡ＫｾＫ｣Ｉ＠

+

+( (1 - a - b - c) - (1 - a - b - c - d»

Ｈ｡Ａｴｾ｣Ｉ｝Ｋ＠

+3[(1 -

(1 - a - b - c»

Ｈ｡ＫｾＫ｣Ｉ＠

+

+ ((

1 - a - b - c) - (1 - a - b - c - d»

Ｈ｡ＫｾＫ｣Ｉ＠

+

+4[(1- (1 -

a - b - c)

Ｈ｡ＫｾＫ｣Ｉｬ＠

=

=

1 +

a

+

2b

+ 3c +

d

4.2

An Interesting Property

The following lemma is helpful to understand the next proposition which estab-lish a new property of Choquet integral which values for a convex capacity that are squeeze of a(n) (additive) probability measure:

Lemma 1 Let v E 8(.6.,111). For all0

i=

C C D = supp v and ali Ei

ç

(D\D)

(i

=

1, ... m) such that El

ç

E2

ç .... ç

Em

=

(D\D), the uncertainty aversion

measure associate to vare summaT"Ízed by:

1::::: c(v,C)::::: c(v,D)::::: c(v,D U

Ed:::::

c(v,DU

E

2 )::::: ..• ::::: c(v""D) = O

Now I can present a new property of Choquet Integral which is satisfied by convex capacities that are squeeze of (additive) probability measures.

Proposition 10 For all J E ｾ＠ and p E .6. , let 'Ij;, 'lj;1 E ｉＱＱＨｄＬｾＩ＠ satisfying (iv) and (v)

Let v, VI E 8(.6.,111) be respectively defined by: (A) = { (1 - 'Ij;(A))p(A) if A

i=

D

v 1 if A

=

D

and

I(A)

= {

(1 - 'lj;/(A))p(A) if A

i=

D

v 1 if A

=

D

lf'lj;

s:

'lj;1 then

J

Jdv :::::

J

Jdv'

5

INTEGRAL REPRESENTATION WITH

CON-VEX CAPACITIES THAT ARE SQUEEZE

OF (ADDITIVE) PROBABILITY MEASURES

In subjective expected utility theory choice under uncertainty is perceived as the maximization of the mathematical expectation of a utility function with respect to the subjective probability that represent the individual' s subjective assess-ment of the reI ative likelihood of events (see Savage (1954) for the defini tive statement and Anscombe and Aumann (1963) for an alternative and simpler treatment). Motivated by examples as the Ellsberg Paradox (discussed in the beginning of this paper) and an increasing body of experimental and empirical evidence, decision theorists have developed and analyzed mo deIs for decision making under uncertainty that entail the mathematical expectation of a utility function with respect to capacities (non-additive probabilities). This general-ization of subjective expected utility has been called Choquet expected utility as it utilizes the Choquet integral for non-additive measuresl l_.

A utility function u : ｾ＠ -> R defined on the class of simple acts is said to be

affine if for any pair of simple acts

J,

9 E ｾＬ＠ and any Q E (0,1) :

11 See Gilboa (1987)'s axiomatization of the use of the Choquet integral as a choice criterion,

u(af

+

(1 - a)g)

=

au(f)

+

(1 - a)u(g).

Definition 8 Fix an affine utility function u : ';5 --> 1R, a capacity v E V(n, I;) and any simple act f E ';5. The Choquet expected utility of the act f with respect to u and v is defined to be:

o 00

J

u(f)dv ==

J

(v(u(f) :2:: a) - l)da

+

J

v(u(f) :2:: a)da12

-00 o

The following theorem extends the results of theorem 1 to Choquet expected utility:

Theorem 2 Corollary of Theorem 1 Let f E F, u(f) bounded and v E e(.6., 1lJ).

The following statements. are equivalent:

o 00

i)

J

u(f)dv ==

J

(v(u(f) :2:: a) - l)da

+

J

v(u(f) :2:: a)da;

- x o

ii)

J

u(f)dv == 1jJ, u(f)( {w,})

+

(1 - 1jJ,)

J

u(f)dp+

n n

+

I:

(1jJl -1jJj .... n)(

I:

p( {w;}) )(u(f)(Wj) - u(f)(Wj-l))

j=3 i=j

n

iii)

J

u(f)dv == u(f)(w,)[P( {wd)

+

1jJ,

I:

p({Wj})]+

j=2

n - l n

+

I:

u(f)(Wi) [(l-1jJl ... i-dp({w;})+(1jJL.. . .;-1jJl . ..

i-d

I:

p({Wj})]+

i=2 j=i+l

+u(f)(wn)[(l -1jJl, .... n-dp({wn })]

where: p: 2:: --> [0,1] is a(n) (additive) probability measure on

(n,

I;);

and

J

u(f)dp is its subject expected utility.

6

CONCLUSIONS

In this paper I showed that it is necessary to make some restrictions over the class of all convex capacities to a special class on which it is possible to present the convex capacities as a squeeze of a(n) uncertainty aversion function. This parametric approach allows to do comparative static exercises over the uncertainty aversion function in an easy way. Coimbra-Lisboa (2003) used this framework to generalize Dow and Werlang (1994) 's existence theorem of Nash equilibrium under Knightian Uncertainty.

12Note that if the capacity is in fact a(n) (additive) probability measure then the expression on the right-hand side collapses to the standard subject expected utility formulation (see Savage (1954)).

It remains to explore other properties of the Choquet integral (besides lemma 1 and proposition 10) that are valid when the convex capacities are squeeze of (additive) probability measures.

References

[1] Ascombe, F. and R. Aumann (1963j: "A definition of Subjective Probabil-ity", Annals of Mathematical Statistics, 34, 199-205.

[2] Bernoulli, D. (1738): "Specimen Theoriae Novae de Mensure Sortis", Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5, 175-192. Translated by L. Sommer in Econometrica, 22, 23-26.

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2° ed, Robert E. Krieger Publishing Company. Huntington, NY.

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20

[14] Cilboa, I. and D. Schmeidler (1991): "Updating ambiguous beliefs", Jour-nal of Economic Theory, 59, 33-49.

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A

Proofs

Proof. Proposition 1

Marinacci (2000) . •

Proof. Proposition 2

The uncertainty aversion measure is defined by:

c(v, A)

=

1 - v(A) - v(Sl\A)

Satisfy the usual conditions:

(i) c(v, 0)

=

1 - v(0) - v(Sl\0)

=

c(v, Sl)

From the definition of convex capacity: v(0)

=

o

and v(n)

=

l.

80: c(v,0) = c(v,n) = O

(ii) For all A E ｾＺ＠ c(v, A)

=

1 - v(A) - v(n\A)

=

c( v, n\A) 8atisfied by definition.

It remains to check (iii). Note, firstly, that:

c(v, A) = 1 - v(A) - v(n\A) c(v, B)

=

1 - v(B) - v(n\B)

c(v, (A U B» = 1 - v(A U B) - v(n\(A U B»

=

M0:;jan 1 - v(A U B) - v((n\A)

n

(n\B)) c(v, (A

n

B»

=

1 - v(A

n

B) - v(n\(A

n

B»

=

M0:;jan 1 - v(A

n

B) - v((n\A) U (n\B»

From the definition of convex capacity, it is known that for all A, B E ｾＺ＠

v(A)

+

v(B) :S; v(A U B)

+

v(A

n

B)

v(n\A)

+

v(n\B) :S; v((n\A) U (n\B»

+

v((n\A)

n

(n\B» Rewriting:

1 - v(A) - v(B) ｾ＠ 1 - v(A U B) - v(A

n

B)

1 - v(n\A) - v(n\B) ｾ＠ 1 - v((n\A) U (n\B» - v((n\A)

n

(n\B)) 8um up then:

(1 - v(A) - v(B»

+

(1 - v(n\A) - v(n\B» ｾ＠

ｾ＠ (1 - v(A U B) - v(A

n

B))

+

(1 - v((n\A) U (n\B» - v((n\A)

n

(n\B») Rewriting (using Morgan 's law):

(1 - v(A) - v(n\A»

+

(1 - v(B) - v(n\B» ｾ＠

ｾ＠ (1 - v(A U B) - v(n\(A U B»)

+

(1 - v(A

n

B) - v(n\(A·n B») Finally, using the definition of uncertainty aversion measure:

｣ＨｶＬａＩＫ｣ＨｶＬｂＩｾ｣ＨｶＬａｕｂＩＫ｣ＨｶＬａｮｂＩ＠ •

Proof. Proposition 3

.;::

Let 'Ij; E W(n, ｾＩＮ＠

80 'Ij; satisfy the following properties:

i) 'Ij;(0)

=

'Ij;(n)

=

O;

ii) For all A E ｾＺ＠ 'Ij;(A)

=

'Ij;(n\A);

iii) For all A, B E ｾＺ＠ 'Ij;(A U B)

+

'Ij;(A

n

B) :S; 'Ij;(A)

+

'Ij;(B) ..

Consider the capacities V E ｖＨｮＬｾＩ＠ that are associate to 'Ij; E w(n, ｾＩＬ＠ i.e., ca-pacities such that the uncertainty aversion measure, c(v) : ｾ＠ ...

[0,1]

are such that, for all A E ｾＺ＠

c( v, A)

=

'Ij;(A)

The uncertainty aversion measure associate to v must satisfy the same properties that the uncertainty aversion function. 80 it remains to check if v E A.

(i) c(v,0)

=

c(v,n)

=

O

80, 1 - v(0) - v(n)

=

O

By definition v(0) = O and v(n) = 1.

(ii) For all A E ｾＺ＠ c(A) = c(n\A)

80 it is satisfied, by definition.

(iii} For alI A,B E I;: c(v,A U B)

+

c(v,A

n

B) :::: c(v, A)

+

c(v,B) Using (ii):

[1 - v(A) - v(n\A)]

+

[1 - v(B) - v(n\B)] ?:

?: [1 - v(A

u

B) - v(n\(A U B»]

+

[1 - v(A

n

B) - v(n\(A

n

B»]

Rewriting:

[v(A)

+

v(n\A)]

+

[v(B)

+

v(n\B)] ::::

:::: [v(A U B)

+

v(n\(A

u

B»]

+

[v(A

n

B)

+

v(n\(A

n

B»].". :. [v(A)

+

v(B)]

+

[v(n\A)

+

v(n\B)] ::::

:::: [v(A

u

B)

+

v(A

n

B)]

+

[v(n\(A

u

B»

+

v(n\(A

n

B»] :. :. [v(A)

+

v(B) - v(A U B) - v(A

n

B)]+

+[v(n\A)

+

v(n\B) - v(n\(A U B» - v(n\(A

n

B»] :::: 013

If V E A ('Ij;) then the condition is satisfied.

=}

Let v E A('Ij;)

80, according to proposition 1, it is enough to see if v satisfy the following prop-erties:

i) v(0)

=

O;

ii) v(n)

=

1;

iii) For alI A, B E I;: v(A)

+

v(B) :::: v(A U B)

+

v(A n B)

If v E A( 'Ij;) then v E A. It follows that the uncertainty aversion measure satisfy the following properties:

(i) c(v, 0) = c(v, n) = 1 - v(0) - v(n) = O;

(ii) For ali A E I;: c(A) = c(n\A);

(iii) For ali A, B E I;: c(v, A)

+

c(v, B) ?: c(v, (A

n

B»

+

c(v, (A U B». Define the function 9 : I; --> [O, 1] such that, for ali A E I;.

g(A) = c(v,A)

It follows that g: I; --> [0,1] has the same properties that some 'Ij; E lJt(n, I;), so

:J'Ij; E lJt(n, I;) such that 1Jt(·)

=

g(-) •

Proof. Proposition 4

By contraposition. Hypothesis:

:J'Ij; E lJt (n, I;) such that :Jv E A ( 'Ij;) that could not be represented by a squeeze of any p associated to 'Ij;

Thesis:

v E A('Ij;) not satisfy (v) or (vi) 1st case:

Let

C,

Cand DEI;, such that:

0

=f=.

C, C

C D

ç

n,

C U C C D and C

n

C =

0.

Consider 'Ij; E lJt(n, I;) with 'Ij;(C) =f=. 'Ij;(C') and v E A('Ij;), with

0

=f=. D = supp vçn

The uncertainty aversion function is associated to the uncertainty aversion measure of the capacity v, so it is also true that:

'Ij;(C)

=

c(v, C)

=

1 - v(C) - v(n\C) 'Ij;(C') = c(v, C') = 1 - v(C') - v(n\C')

where: c( v, .) is the uncertainty aversion measure of the capacity v

•

So:

So:

1jJ(C)

i=

1jJ(C){:?1 - v(C) - v(O\C)

i=

1 - v(C) - v(O\C){:?v(C)

+

v(O\C)

i=

v(C)

+

v(O\C)

o

i=

c,

c

c

D, C

u

C

c

D and C

n

C

=

0

=>3B

i=

0

such that D

=

C U

Cu

B

(O\C) =

(Cu

B

u

(O\D»

(O\C)

=

(C

u

B

u

(O\D» Then:

v(C)

+

v(Cu B U (O\D))

i=

v(C)

+

v(C U B U (O\D»

If the properties (v )and (vi) are satisfied, then:

(vi) (v)

v(Cu B U (O\D»

=

v(CU B)=v(C)

+

v(B) v(C

u

B

u

Ｈｏ｜ｄＩＩＨｾＩｶＨｃ＠

u

B)0:)v(C)

+

v(B) Inequality requires some condition being violated

Sub case: supp v

=

O (i.e., D

=

O)

1jJ(C)

i=

1jJ(C){:?1 - v(C) - v(O\C)

i=

1 - v(C) - v(O\C){:?v(C)

+

v(O\C)

i=

v(C)

+

v(O\C)

o

i=

C, C C Oe C U C C 0=>3E

i=

0

such that O

=

C

u Cu

E So:

(O\C)

= (CU

B)

(O\C)

=

(C

u

B) Then:

v(C)

+

v(Cu B)

i=

v(C)

+

v(C U B) If property (v) is satisfied, then:

v(Cu B)0:)v(C)

+

v(B)

v(C U B)0:)v(C)

+

v(B)

So, to have an inequality it is necessary that condition (v) fails.14 2nd case:

Let C, D and E E ｾＬ＠ such that, 0

i=

C C D

ç

O, E

ç

(O\D).

Consider 1jJ E ＱｊｲＨＰＬｾＩ＠ with 1jJ(C

u

E)

i=

1jJ(C) and v E A(1jJ), with

0

i=

D =supp

p pv

ç

o.

So:

1jJ(C U E)

=

c(v, C U E)

=

1 - v(C

u

E) - v(O\(C

u

E» 1jJ(C)

=

c(v, C) = 1 - v(C) - v(O\C)

Then:

1jJ(C U E)

i=

1jJ(C){:?l - v(C

u

E) - v(O\(C

u

E})

i=

1 - v(C) - v(O\C){:?

{:? v(C

u

E)

+

v(O\(C

u

E»)

i=

v(C)

+

v(O\C)

o

i=

C C D=>3E

i=

0 such that D

=

C U E So:

(O\(C u E»

=

Eu [(O\D)\E]

(O\C)

=

Eu (O\D) Then:

v(C U E)

+

v(EU [(O\D)\E])

i=

v(C)

+

v(EU (O\D»

14Beside this, if the suport concides with the space of states then for the convex capacity be represented as a squeeze of a(n) (additive) probability measure, it is nec-essary that the uncertainty aversion function associate be uniform.(i.e., 1/;c E wc(n, ｾＩＩ＠

If property (vi) is satisfied, then: v(C

u

ｅＯｾＩｶＨｃＩ＠

v(13'U

｛Ｈｑ｜ｄＩ｜ｅ｝ＩＨｾＩｶＨｂＩ＠

v(13'U Ｈｑ｜ｄＩＩＨｾＩｶＨｂＩ＠

So, to have an inequality it is necessary that condition (vi) fails. •

Proof. Proposition 5

Verifying (iv):

From the definition of uncertainty aversion measure: c(v, C)

=

1 - v(C) - v(Q\C)

c(v, C)

=

1 - v(C) - v(Q\C) c(v, D)

=

1 - v(D) - v(Q\D)

o

=f. D

=

supp v

ç

Q=}:JB =f.

0

such that D

=

C U

Cu

B So:

v(Q\C) = v((Cu B) U (Q\D)) v(Q\C)

=

v((C

u

B) U (Q\D)) v(Q\D)

=

O (because D

=

supp v) By definition: (C

u

B),

(Cu

B)cD So from property (vi) of 9(6., 'li):

v(Q\C)

=

v((Cu B) U (Q\D)

=

v(CU B) v(Q\C)

=

v((C U B) U (Q\D))

=

v(C U B) According to property (v) of 8(6., 'li):

v(Q\C)

=

v(CU B)

=

v(C)

+

v(B) v(Q\C)

=

v(C U B)

=

v(C)

+

v(B) Substituting in the definitions:

So:

c(v, C)

=

1 - v(C) - [v(C)

+

v(B)] c(v, C)

=

1 - v(C) - [v(C)

+

v(B)] c(v, D)

=

1 - v(D)

c( v, C)

=

c( v, C)

Finally, D = C U CU B so, from property (iii) of 8(6., 'li):

v(D)

=

v(C U

Cu

B) ;::: v(C U C)

+

v(B)

=

v(C)

+

v(C)

+

v(B) Then:

c(v,C)

=

c(v, C) ;::: c(v,D) Verifying (v):

From the definition of uncertainty aversion function: c(v, (C U E))

=

1 - v(C U E) - v(Q\(C U E» c(v,C)

=

1-v(C) - v(Q\C)

If C C D and E

ç

(Q\D) then from property (vi)of 9(6., 'li): v(C U E)

=

v(C)

o

=f. D

=

supp v

ç

Q, C C D=}:J13'=f.

0

such that D

=

Cu 13' So, v(Q\(C U E))

=

v(13'U ((Q\E)\D»)

From property (vi) of 8(6., 'li), follows that (using ((Q\E)\D) ç (Q\D)): v(Q\(C U E)) = v(13'U ((Q\E)\D) = v(B)

It is true that v(Q\C) = v(13'U (Q\D»)