Decomposition-Integral: Unifying - PowerPoint Presentation

Slide1

Decomposition-Integral: Unifying Choquet and the Concave Integrals

Yaarit

Even

Tel-Aviv University

December 2011Slide2

Non-additive integralDecision making under uncertainty

Game theory

Multi-criteria decision aid (MCDA)

Insurance and financial assets pricingSlide3

In this presentationA new definition for integrals w.r.t. capacities.

Defining

Choquet

and the concave integrals by terms of the new integral.

Properties of the new integral.

Desirable properties and the conditions for which the new integral maintains them.Slide4

Definitions

Let

be a

finite set,

.

A

capacity

over is a function satisfying: (i) . (ii) if , then .A random variable (r.v.) over is a function .A subset of will be called an event.

 Slide5

Let

be a random variable.

A

sub-decomposition

of

is a finite summation

such that:

(i) (ii) and for every If there is an equality in (i), then

is a decomposition of .The value of a decomposition w.r.t. is .

Sub-decompositions and decompositions of a random variableSlide6

-sub-decompositions and

-decompositions

Let

be a set of subsets of

,

.

is a -sub-decomposition of if it is a sub-decomposition of and for every

is a

-decomposition of if it is a decomposition of and for every  Slide7

Examples

Suppose

and

.

and

are both

-decompositions of

.Suppose , , and .

has a

-decomposition: , and has only

-sub-decompositions, such as:

.

 Slide8

The decomposition-integral

A vocabulary

is a set of subsets of

.

A sub-decomposition of

is

-allowable if it is a

-sub-decomposition of and .The decomposition-integral w.r.t. is defined: = is -allowable sub-decomposition of X}. The sub-decomposition attaining the maximum is called the

optimal sub-decomposition

of . Slide9

Examples

Suppose

,

,

,

.

and . has an optimal decomposition:

,

and

=

.

has an optimal sub-decomposition:

, and

=

.

 Slide10

The decomposition-integral as a generalization of known integrals

Choquet

integral

The concave integral

Riemann integral

Shilkret

integral

And other plausible integration schemes.Slide11

The concave integral

Definition (Lehrer):

, where the minimum is taken over all concave and homogeneous functions

, such that

for every

.

Lemma (Lehrer):

=

.

 Slide12

The concave integral as a decomposition-integral

Define

.

=

=

is

-allowable sub-decomposition of X}.

Since is monotonic w.r.t. inclusion, we have: =

is

-allowable decomposition of X}.

 Slide13

The concave integral

Since

allows for all decompositions, for every vocabulary

, the following inequality holds:

, for every

.

Concluding, that of all the decomposition-integrals, the concave integral attains the highest value. Slide14

Choquet integral

Definition:

=

=

,

where

is a permutation over

that satisfies

and

.

, where

.

 Slide15

Choquet integral as a decomposition-integral

Definitions:

Any

two subsets

and

of

are nested if either

or .A set is called a chain if any two events are nested. Slide16

Choquet integral as a decomposition-integral

Define

to be the

set

of all chains.

=

= is -allowable sub-decomposition of X}=

is

-allowable decomposition of X}.

 Slide17

Examples

Suppose

,

,

,

and

.

 Slide18

Riemann integral

A partition of

is a set

, such that all

’s are pairwise disjoint and their union is

.

Define

to be the set of all partitions of .The Riemann integral can be defined as . Slide19

Shilkret integral

Define

.

The

Shilkret

integral can be defined as:

=

is -allowable sub-decomposition of X} =

=

.

 Slide20

Properties of the decomposition-integral

Positive homogeneity of degree one

:

for

every

for every

,

and .The decomposition-integral and additive capacities:Proposition: Let be a probability and a vocabulary. Then, for every r.v. iff every has a -decomposition, . Slide21

Properties of the decomposition-integral - continuation

Monotonicity

:

1.

Monotonicity w.r.t.

r.v.’s

: Fix

and and suppose . Then, .2. Monotonicity w.r.t. capacities: Fix . If for every and every , , then for every r.v. , .  Slide22

Properties of the decomposition-integral - continuation

3.

Monotonicity w.r.t. vocabularies: Fix

and suppose

and

are two vocabularies.

Proposition

: iff for every and every minimal set , there is such that .A set is minimal if the variables are algebraically independent. Slide23

Properties of the decomposition-integral - continuation

Additivity

:

Two variables

and

are

comonotone

if for every , .Comonotone additivity means that if X and Y are comonotone, then: .Let . and

are

comonotone iff their optimal decompositions use the same in . Slide24

Properties of the decomposition-integral - continuation

Example:

,

and

. Fix

. Suppose

is small enough so that the optimal D-decompositions of X and Y use

:

,

but for , taking :

 Slide25

Properties of the decomposition-integral - continuation

Fix

and

.

is

leaner

than

if there are optimal decompositions in which employ every indicator that employs:The optimal decomposition of is , , and the optimal decomposition of is , , and .

 Slide26

Properties of the decomposition-integral - continuation

Proposition

: Fix a vocabulary

such that every

has an optimal decomposition for every

. Suppose that for every

, whenever there are two different decompositions of the same variable,

, there is that contains all the ’s with and all the ’s with . Then, for every and every and , where

is leaner than

, . Slide27

Desirable properties

Concavity

Monotonicity w.r.t. stochastic dominance

Translation-invarianceSlide28

Concavity

is

concave

if for every two

r.v

.

and

, and : Theorem 1: The decomposition-integral is concave for every , iff there exists a vocabulary containing only one

such that

. Slide29

A new characterization of the concave integral

Corollary 1

: A decomposition-integral

satisfies (

i

)

for every event

and capacity ; and (ii) is concave, iff . Slide30

Monotonicity w.r.t. stochastic dominance

stochastically dominates

w.r.t.

if for every number

,

. is monotonic w.r.t. stochastic dominance if implies Theorem 2: The decomposition-integral is monotonic w.r.t. stochastic dominance iff is the collection of all chains of the same size

(

). Slide31

Monotonicity w.r.t. stochastic dominance

Example:

,

,

,

and .Obviously, .

:

>

:

<

 Slide32

Translation-invariance

is

translation-invariant

for every

, if for every

,

, when

.Theorem 3: The decomposition-integral is translation-invariant for every iff the vocabulary is (i) composed of chains; and (ii) any is contained in that includes .

 Slide33

Translation-invariance

Example:

,

,

and

. and

. , but -

.

 Slide34

A new characterization of Choquet integral

Corollary 2

: A decomposition-integral

satisfies (

i

)

for every probability

; and (ii) it is monotonic w.r.t. stochastic dominance for every iff .Corollary 3: A decomposition-integral satisfies (i) for every probability ; and (ii) it is translation-invariant for every iff .

 Slide35

For ConclusionA new definition for integrals w.r.t. capacities.

A new characterization of the concave

integral.

Two new characterizations of integral

Choquet

(that

do not use

comonotone additivity).Finding a trade-off between different desirable properties.Slide36

THE END