k -intolerant capacities and Choquet integrals

(1)

k -intolerant capacities and Choquet integrals

Jean-Luc Marichal

marichal@cu.lu

University of Luxembourg

k-intolerant capacities and Choquet integrals – p.1/18

(2)

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

(3)

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

• Criteria N = {1, 2, . . . , n}

(4)

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

• Criteria N = {1, 2, . . . , n}

• Profile a ∈ A −→ (x ₁ ^a , . . . , x ^a _n ) ∈ [0, 1] ⁿ

(5)

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

• Criteria N = {1, 2, . . . , n}

• Profile a ∈ A −→ (x ₁ ^a , . . . , x ^a _n ) ∈ [0, 1] ⁿ

• Aggregation function

F : [0, 1] ⁿ → [0, 1]

(x ₁ , . . . , x _n ) 7→ F (x ₁ , . . . , x _n )

(6)

Tolerant and intolerant character of F

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Tolerant and intolerant character of F

F (x) = min _i x _i → intolerant behavior

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Tolerant and intolerant character of F

F (x) = min _i x _i → intolerant behavior F (x) = max _i x _i → tolerant behavior

(9)

Tolerant and intolerant character of F

F (x) = min _i x _i → intolerant behavior F (x) = max _i x _i → tolerant behavior

F (x) = x _(k) → intermediate behavior

(10)

Tolerant and intolerant character of F

F (x) = min _i x _i → intolerant behavior F (x) = max _i x _i → tolerant behavior

F (x) = x _(k) → intermediate behavior

F (x) =

µ n

Y

i=1

x _i

¶ _1/n

→ ?

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Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx

(12)

Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx ∈ [0, 1]

0 1

(13)

Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = _n+1 ¹

(14)

Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = _n+1 ¹

• E (max) = _n+1 ⁿ

(15)

Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = _n+1 ¹

• E (max) = _n+1 ⁿ

• E (OS _k ) = _n+1 ^k

(16)

Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = _n+1 ¹

• E (max) = _n+1 ⁿ

• E (OS _k ) = _n+1 ^k

• E (WAM _ω ) = E (median) = ¹ ₂

(17)

Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = _n+1 ¹ (most intolerant)

• E (max) = _n+1 ⁿ

• E (OS _k ) = _n+1 ^k

• E (WAM _ω ) = E (median) = ¹ ₂

(18)

Tolerant and intolerant character of F

Average value of F over [0, 1] ⁿ :

E (F ) :=

Z

[0,1] ⁿ F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = _n+1 ¹ (most intolerant)

• E (max) = _n+1 ⁿ (most tolerant)

• E (OS _k ) = _n+1 ^k

• E (WAM _ω ) = E (median) = ¹ ₂

(19)

Tolerant and intolerant character of F

Position of E (F ) within the interval [E (min), E (max)]

1

(20)

Tolerant and intolerant character of F

Position of E (F ) within the interval [E (min), E (max)]

0 1

E(F)

orness(F)

orness(F ) := E (F ) − E (min) E (max) − E (min)

(Dujmovi´c, 1974)

(21)

Tolerant and intolerant character of F

Position of E (F ) within the interval [E (min), E (max)]

0 1

E(F)

orness(F)

orness(F ) := E (F ) − E (min) E (max) − E (min) andness(F ) := E (max) − E (F )

E (max) − E (min)

(Dujmovi´c, 1974)

(22)

Tolerant and intolerant character of F

Position of E (F ) within the interval [E (min), E (max)]

0 1

E(F)

orness(F)

orness(F ) := E (F ) − E (min) E (max) − E (min) andness(F ) := E (max) − E (F )

E (max) − E (min)

(Dujmovi´c, 1974) andness(F ) + orness(F ) = 1

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Intolerant behavior : application

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Intolerant behavior : application

Selection of candidates for a university permanent position

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Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

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Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae

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Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae 2. Teaching effectiveness

(28)

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae 2. Teaching effectiveness

3. Ability to supervise staff and work in a team environment

(29)

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae 2. Teaching effectiveness

3. Ability to supervise staff and work in a team environment 4. Ability to communicate easily in English

(30)

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae 2. Teaching effectiveness

3. Ability to supervise staff and work in a team environment 4. Ability to communicate easily in English

5. Work experience in the industry

(31)

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae 2. Teaching effectiveness

3. Ability to supervise staff and work in a team environment 4. Ability to communicate easily in English

5. Work experience in the industry

6. Recommendations by faculty and other individuals

(32)

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae 2. Teaching effectiveness

3. Ability to supervise staff and work in a team environment 4. Ability to communicate easily in English

5. Work experience in the industry

6. Recommendations by faculty and other individuals

Example of procedure rules

The complete failure of any two of these criteria results in automatic rejection of the applicant

(33)

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

1. Scientific value of curriculum vitae 2. Teaching effectiveness

3. Ability to supervise staff and work in a team environment 4. Ability to communicate easily in English

5. Work experience in the industry

6. Recommendations by faculty and other individuals

Example of procedure rules

The complete failure of any two of these criteria results in automatic rejection of the applicant

x _i = 0 for any two i ∈ N ⇒ F (x) = 0

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k -intolerant aggregation functions

For any fixed k ∈ {1, . . . , n} , consider the condition

x _i = 0 for any k criteria i ∈ N ⇒ F (x) = 0

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k -intolerant aggregation functions

For any fixed k ∈ {1, . . . , n} , consider the condition

x _i = 0 for any k criteria i ∈ N ⇒ F (x) = 0

This is equivalent to

x _(k) = 0 ⇒ F (x) = 0

(36)

k -intolerant aggregation functions

For any fixed k ∈ {1, . . . , n} , consider the condition

x _i = 0 for any k criteria i ∈ N ⇒ F (x) = 0

This is equivalent to

x _(k) = 0 ⇒ F (x) = 0

When F ≡ C _v is the Choquet integral then this condition is equivalent to

F (x) 6 _x _(k) _(x _∈ _[0, _1] ⁿ ₎

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Choquet integral

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Choquet integral

Capacity on N

v : 2 ^N → [0 , 1] , monotone, v ( ∅ ) = 0 , and v ( N ) = 1 F ⁿ := { capacities on N }

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Choquet integral

Capacity on N

v : 2 ^N → [0 , 1] , monotone, v ( ∅ ) = 0 , and v ( N ) = 1 F ⁿ := { capacities on N }

Choquet integral of x ∈ [0 , 1] ⁿ w.r.t. v C ^v ( x ) :=

n

X

i =1

x ₍ i )

h v ^£ ( i ) , . . . , ( n ) ^¤ − v ^£ ( i + 1) , . . . , ( n ) ^¤ ⁱ

with the convention that x ₍₁₎ 6 · · · 6 x ₍ n ) .

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Choquet integral

Capacity on N

v : 2 ^N → [0 , 1] , monotone, v ( ∅ ) = 0 , and v ( N ) = 1 F ⁿ := { capacities on N }

Choquet integral of x ∈ [0 , 1] ⁿ w.r.t. v C ^v ( x ) :=

n

X

i =1

x ₍ i )

h v ^£ ( i ) , . . . , ( n ) ^¤ − v ^£ ( i + 1) , . . . , ( n ) ^¤ ⁱ

with the convention that x ₍₁₎ 6 · · · 6 x ₍ n ) .

Example

If x ₃ 6 x ₁ 6 x ₂ , we have

C ^v ( x ₁ , x ₂ , x ₃ ) = x ₃ [ v (3 , 1 , 2) − v (1 , 2)] + x ₁ [ v (1 , 2) − v (2)] + x ₂ v (2)

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k -intolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-intolerant if F 6 OS ^k and F OS ^k ₋₁ .

(42)

k -intolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-intolerant if F 6 OS ^k and F OS ^k ₋₁ .

Proposition

Let k ∈ {1 , . . . , n } and v ∈ F ⁿ .

Then the following assertions are equivalent:

i ) C ^v ( x ) 6 x ₍ k ) ∀ x ∈ [0 , 1] ⁿ ,

ii ) ∀ x ∈ [0 , 1] ⁿ : x ₍ k ) = 0 ⇒ C ^v ( x ) = 0 ,

(43)

k -intolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-intolerant if F 6 OS ^k and F OS ^k ₋₁ .

Proposition

Let k ∈ {1 , . . . , n } and v ∈ F ⁿ .

Then the following assertions are equivalent:

i ) C ^v ( x ) 6 x ₍ k ) ∀ x ∈ [0 , 1] ⁿ ,

ii ) ∀ x ∈ [0 , 1] ⁿ : x ₍ k ) = 0 ⇒ C ^v ( x ) = 0 ,

Recruiting problem

The global evaluation is bounded above by x ₍₂₎

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k -intolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-intolerant if F 6 OS ^k and F OS ^k ₋₁ .

Proposition

Let k ∈ {1 , . . . , n } and v ∈ F ⁿ .

Then the following assertions are equivalent:

i ) C ^v ( x ) 6 x ₍ k ) ∀ x ∈ [0 , 1] ⁿ ,

ii ) ∀ x ∈ [0 , 1] ⁿ : x ₍ k ) = 0 ⇒ C ^v ( x ) = 0 , iii ) C ^v ( x ) is independent of x ₍ k +1) , . . . , x ₍ n ) ,

(45)

k -intolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-intolerant if F 6 OS ^k and F OS ^k ₋₁ .

Proposition

Let k ∈ {1 , . . . , n } and v ∈ F ⁿ .

Then the following assertions are equivalent:

i ) C ^v ( x ) 6 x ₍ k ) ∀ x ∈ [0 , 1] ⁿ ,

ii ) ∀ x ∈ [0 , 1] ⁿ : x ₍ k ) = 0 ⇒ C ^v ( x ) = 0 , iii ) C ^v ( x ) is independent of x ₍ k +1) , . . . , x ₍ n ) ,

Recruiting problem

The global evaluation depends only on x ₍₁₎ and x ₍₂₎

(46)

k -intolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-intolerant if F 6 OS ^k and F OS ^k ₋₁ .

Proposition

Let k ∈ {1 , . . . , n } and v ∈ F ⁿ .

Then the following assertions are equivalent:

i ) C ^v ( x ) 6 x ₍ k ) ∀ x ∈ [0 , 1] ⁿ ,

ii ) ∀ x ∈ [0 , 1] ⁿ : x ₍ k ) = 0 ⇒ C ^v ( x ) = 0 , iii ) C ^v ( x ) is independent of x ₍ k +1) , . . . , x ₍ n ) ,

iv ) v ( T ) = 0 ∀ T ⊆ N such that | T | 6 n − k

(47)

k -intolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-intolerant if F 6 OS ^k and F OS ^k ₋₁ .

Proposition

Let k ∈ {1 , . . . , n } and v ∈ F ⁿ .

Then the following assertions are equivalent:

i ) C ^v ( x ) 6 x ₍ k ) ∀ x ∈ [0 , 1] ⁿ ,

ii ) ∀ x ∈ [0 , 1] ⁿ : x ₍ k ) = 0 ⇒ C ^v ( x ) = 0 , iii ) C ^v ( x ) is independent of x ₍ k +1) , . . . , x ₍ n ) ,

iv ) v ( T ) = 0 ∀ T ⊆ N such that | T | 6 n − k

Definition

v ∈ F ⁿ is k-intolerant if iv ) holds for k and not for k − 1 .

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Tolerant behavior : application

(49)

Tolerant behavior : application

Parents want to buy a house

(50)

Tolerant behavior : application

Parents want to buy a house House buying criteria

(51)

Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

(52)

Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

2. With parks for their children to play in

(53)

Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

2. With parks for their children to play in

3. With safe neighborhood for children to grow up in

(54)

Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

2. With parks for their children to play in

3. With safe neighborhood for children to grow up in 4. At least 100 meters from the closest major road

(55)

Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

2. With parks for their children to play in

3. With safe neighborhood for children to grow up in 4. At least 100 meters from the closest major road 5. At a fair distance from the nearest shopping mall

(56)

Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

2. With parks for their children to play in

3. With safe neighborhood for children to grow up in 4. At least 100 meters from the closest major road 5. At a fair distance from the nearest shopping mall 6. Within reasonable distance of the airport

(57)

Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

2. With parks for their children to play in

3. With safe neighborhood for children to grow up in 4. At least 100 meters from the closest major road 5. At a fair distance from the nearest shopping mall 6. Within reasonable distance of the airport

To be realistic

The parents are ready to consider a house that would fully succeed any five over the six criteria

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Tolerant behavior : application

Parents want to buy a house House buying criteria

1. Close to a school

2. With parks for their children to play in

3. With safe neighborhood for children to grow up in 4. At least 100 meters from the closest major road 5. At a fair distance from the nearest shopping mall 6. Within reasonable distance of the airport

To be realistic

The parents are ready to consider a house that would fully succeed any five over the six criteria

x _i = 1 for any five i ∈ N ⇒ F (x) = 1

(59)

k -tolerant aggregation functions

For any fixed k ∈ {1, . . . , n} , consider the condition

x _i = 1 for any k criteria i ∈ N ⇒ F (x) = 1

(60)

k -tolerant aggregation functions

For any fixed k ∈ {1, . . . , n} , consider the condition

x _i = 1 for any k criteria i ∈ N ⇒ F (x) = 1

This is equivalent to

x _(n−k+1) = 1 ⇒ F (x) = 1

(61)

k -tolerant aggregation functions

For any fixed k ∈ {1, . . . , n} , consider the condition

x _i = 1 for any k criteria i ∈ N ⇒ F (x) = 1

This is equivalent to

x _(n−k+1) = 1 ⇒ F (x) = 1

When F ≡ C _v is the Choquet integral then this condition is equivalent to

F (x) > _x _(n−k+1) _(x _∈ _[0, _1] ⁿ ₎

(62)

k -tolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-tolerant if F > OS ⁿ ₋ ^k ₊₁ and F ® OS ⁿ ₋ ^k ₊₂ .

(63)

k -tolerant capacities and Choquet integrals

Definition

Let k ∈ {1 , . . . , n } .

F : [0 , 1] ⁿ → [0 , 1] is k-tolerant if F > OS ⁿ ₋ ^k ₊₁ and F ® OS ⁿ ₋ ^k ₊₂ .

When F ≡ C ^v

we have a similar proposition as for intolerance...

(64)

Intolerance and tolerance indices

(65)

Intolerance and tolerance indices

Given a Choquet integral C _v

andness(C _v ) measures the degree to which C _v is intolerant

(66)

Intolerance and tolerance indices

Given a Choquet integral C _v

How can we measure the degree to which the inequality C _v 6 _OS _k holds ?

(67)

Intolerance and tolerance indices

Given a Choquet integral C _v

How can we measure the degree to which the inequality C _v 6 _OS _k holds ?

Recall that:

C _v 6 _OS _k _⇐⇒ ^h _x _(k) _{= 0} _{⇒ C} _v _{(x) = 0} ⁱ

(68)

Intolerance and tolerance indices

Given a Choquet integral C _v

How can we measure the degree to which the inequality C _v 6 _OS _k holds ?

Recall that:

C _v 6 _OS _k _⇐⇒ ^h _x _(k) _{= 0} _{⇒ C} _v _{(x) = 0} ⁱ

⇐⇒ ^h x _(k) = 0 ⇒ C _v (x) = min _i x _i ⁱ

(69)

Intolerance and tolerance indices

Given a Choquet integral C _v

How can we measure the degree to which the inequality C _v 6 _OS _k holds ?

Recall that:

C _v 6 _OS _k _⇐⇒ ^h _x _(k) _{= 0} _{⇒ C} _v _{(x) = 0} ⁱ

⇐⇒ ^h x _(k) = 0 ⇒ C _v (x) = min _i x _i ⁱ Definition

For any k ∈ {1, . . . , n − 1} and any v ∈ F _n , we define intol _k (C _v ) := andness(C _v | x _(k) = 0)

(70)

Intolerance and tolerance indices

Given a Choquet integral C _v

How can we measure the degree to which the inequality C _v 6 _OS _k holds ?

Recall that:

C _v 6 _OS _k _⇐⇒ ^h _x _(k) _{= 0} _{⇒ C} _v _{(x) = 0} ⁱ

⇐⇒ ^h x _(k) = 0 ⇒ C _v (x) = min _i x _i ⁱ Definition

For any k ∈ {1, . . . , n − 1} and any v ∈ F _n , we define intol _k (C _v ) := andness(C _v | x _(k) = 0)

Idea : defined from the conditional expectation E (C ^v | x ₍ k ) = 0)

(71)

Intolerance and tolerance indices

intol _k (C _v ) := andness(C _v | x _(k) = 0)

(72)

Intolerance and tolerance indices

intol _k (C _v ) := andness(C _v | x _(k) = 0)

In terms of v , this index reads

intol _k (C _v ) = 1 − 1 n − k

n−k

X

t=0

1 ³ _n

t

´

X

T ⊆N

|T |=t

v(T )

(73)

Intolerance and tolerance indices

intol _k (C _v ) := andness(C _v | x _(k) = 0)

Some properties

1. intol _k (C _v ) = 1 if and only if C _v 6 _OS _k

(74)

Intolerance and tolerance indices

intol _k (C _v ) := andness(C _v | x _(k) = 0)

Some properties

1. intol _k (C _v ) = 1 if and only if C _v 6 _OS _k

2. intol _k (C _v ) is nondecreasing as k increases

(75)

Intolerance and tolerance indices

intol _k (C _v ) := andness(C _v | x _(k) = 0)

Some properties

1. intol _k (C _v ) = 1 if and only if C _v 6 _OS _k

2. intol _k (C _v ) is nondecreasing as k increases 3. Graph of intol _k (OS _j ) for fixed k :

1 j

1 intol (OS )

<k

k n

j

(76)

Intolerance and tolerance indices

intol _k (C _v ) := andness(C _v | x _(k) = 0)

Some properties

1. intol _k (C _v ) = 1 if and only if C _v 6 _OS _k

2. intol _k (C _v ) is nondecreasing as k increases 3. Graph of intol _k (OS _j ) for fixed k :

1 j

1 intol (OS )

<k

k n

j

OS _j 6 _OS _k _⇐⇒ _j 6 _k

(77)

Intolerance indices : characterization

(78)

Intolerance indices : characterization

Theorem

Let k ∈ {1, . . . , n − 1}

and consider a family of real numbers {ψ _k (C _v ) | v ∈ F _n } These numbers are

(79)

Intolerance indices : characterization

Theorem

Let k ∈ {1, . . . , n − 1}

and consider a family of real numbers {ψ _k (C _v ) | v ∈ F _n } These numbers are

1. linear with respect to the capacity

(80)

Intolerance indices : characterization

Theorem

Let k ∈ {1, . . . , n − 1}

and consider a family of real numbers {ψ _k (C _v ) | v ∈ F _n } These numbers are

1. linear with respect to the capacity

2. independent of the numbering of criteria (symmetry)

(81)

Intolerance indices : characterization

Theorem

Let k ∈ {1, . . . , n − 1}

and consider a family of real numbers {ψ _k (C _v ) | v ∈ F _n } These numbers are

1. linear with respect to the capacity

2. independent of the numbering of criteria (symmetry) 3. such that intol _k (OS _j ) has the graph showed above

(82)

Intolerance indices : characterization

Theorem

Let k ∈ {1, . . . , n − 1}

and consider a family of real numbers {ψ _k (C _v ) | v ∈ F _n } These numbers are

1. linear with respect to the capacity

2. independent of the numbering of criteria (symmetry) 3. such that intol _k (OS _j ) has the graph showed above if and only if ψ _k (C _v ) = intol _k (C _v ) for all v ∈ F _n .

(83)

The recruiting problem

(84)

The recruiting problem

3-intolerant solution learnt from prototypic applicants : v(T ) = 0 for all T ⊆ {1, . . . , 6} except

v({1, 2, 4, 5}) = v({1, 2, 3, 4, 5}) = v({1, 3, 4, 5, 6}) = 1/3 v({1, 2, 3, 4, 6}) = 2/3

v({1, 2, 4, 5, 6}) = v({1, 2, 3, 4, 5, 6}) = 1

(85)

The recruiting problem

3-intolerant solution learnt from prototypic applicants : v(T ) = 0 for all T ⊆ {1, . . . , 6} except

v({1, 2, 4, 5}) = v({1, 2, 3, 4, 5}) = v({1, 3, 4, 5, 6}) = 1/3 v({1, 2, 3, 4, 6}) = 2/3

v({1, 2, 4, 5, 6}) = v({1, 2, 3, 4, 5, 6}) = 1 Sequence intol _k (C _v ) for k = 1, . . . , 5

k -intolerant capacities and Choquet integrals

k -intolerant capacities and Choquet integrals

Jean-Luc Marichal

marichal@cu.lu

University of Luxembourg

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

• Criteria N = {1, 2, . . . , n}

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

• Criteria N = {1, 2, . . . , n}

• Profile a ∈ A −→ (x 1 a , . . . , x a n ) ∈ [0, 1] n

Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

• Criteria N = {1, 2, . . . , n}

• Profile a ∈ A −→ (x 1 a , . . . , x a n ) ∈ [0, 1] n

• Aggregation function

F : [0, 1] n → [0, 1]

(x 1 , . . . , x n ) 7→ F (x 1 , . . . , x n )

Tolerant and intolerant character of F

Tolerant and intolerant character of F

F (x) = min i x i → intolerant behavior

Tolerant and intolerant character of F

F (x) = min i x i → intolerant behavior F (x) = max i x i → tolerant behavior

Tolerant and intolerant character of F

F (x) = min i x i → intolerant behavior F (x) = max i x i → tolerant behavior

F (x) = x (k) → intermediate behavior

Tolerant and intolerant character of F

F (x) = min i x i → intolerant behavior F (x) = max i x i → tolerant behavior

F (x) = x (k) → intermediate behavior

F (x) =

µ n

Y

i=1

x i

¶ 1/n

→ ?

Tolerant and intolerant character of F

Average value of F over [0, 1] n :

E (F ) :=

Z

[0,1] n F (x) dx

Tolerant and intolerant character of F

Average value of F over [0, 1] n :

E (F ) :=

Z

[0,1] n F (x) dx ∈ [0, 1]

0 1

Tolerant and intolerant character of F

Average value of F over [0, 1] n :

E (F ) :=

Z

[0,1] n F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = n+1 1

Tolerant and intolerant character of F

Average value of F over [0, 1] n :

E (F ) :=

Z

[0,1] n F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = n+1 1

• E (max) = n+1 n

Tolerant and intolerant character of F

Average value of F over [0, 1] n :

E (F ) :=

Z

[0,1] n F (x) dx ∈ [0, 1]

0 1

Examples

• E (min) = n+1 1

• E (max) = n+1 n

• E (OS k ) = n+1 k

Tolerant and intolerant character of F

Average value of F over [0, 1] n :

E (F ) :=

• Profile a ∈ A −→ (x ₁ ^a , . . . , x ^a _n ) ∈ [0, 1] ⁿ

• Profile a ∈ A −→ (x ₁ ^a , . . . , x ^a _n ) ∈ [0, 1] ⁿ

F : [0, 1] ⁿ → [0, 1]

(x ₁ , . . . , x _n ) 7→ F (x ₁ , . . . , x _n )

F (x) = min _i x _i → intolerant behavior

F (x) = min _i x _i → intolerant behavior F (x) = max _i x _i → tolerant behavior

F (x) = min _i x _i → intolerant behavior F (x) = max _i x _i → tolerant behavior

F (x) = x _(k) → intermediate behavior

F (x) = min _i x _i → intolerant behavior F (x) = max _i x _i → tolerant behavior

F (x) = x _(k) → intermediate behavior

x _i

¶ _1/n

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx ∈ [0, 1]

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx ∈ [0, 1]

• E (min) = _n+1 ¹

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx ∈ [0, 1]

• E (min) = _n+1 ¹

• E (max) = _n+1 ⁿ

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx ∈ [0, 1]

• E (min) = _n+1 ¹

• E (max) = _n+1 ⁿ

• E (OS _k ) = _n+1 ^k

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx ∈ [0, 1]

• E (min) = _n+1 ¹

• E (max) = _n+1 ⁿ

• E (OS _k ) = _n+1 ^k

• E (WAM _ω ) = E (median) = ¹ ₂

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx ∈ [0, 1]

• E (min) = _n+1 ¹ (most intolerant)

• E (max) = _n+1 ⁿ

• E (OS _k ) = _n+1 ^k

• E (WAM _ω ) = E (median) = ¹ ₂

Average value of F over [0, 1] ⁿ :

[0,1] ⁿ F (x) dx ∈ [0, 1]

• E (min) = _n+1 ¹ (most intolerant)

• E (max) = _n+1 ⁿ (most tolerant)

• E (OS _k ) = _n+1 ^k

• E (WAM _ω ) = E (median) = ¹ ₂