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k -intolerant capacities and Choquet integrals Jean-Luc Marichal - - PowerPoint PPT Presentation

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k -intolerant capacities and Choquet integrals Jean-Luc Marichal marichal@cu.lu University of Luxembourg k -intolerant capacities and Choquet integrals p.1/18 Aggregation in multicriteria decision aid Alternatives A = { a, b, c, . . . }

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SLIDE 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

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SLIDE 2

Aggregation in multicriteria decision aid

  • Alternatives A = {a, b, c, . . .}

k-intolerant capacities and Choquet integrals – p.2/18

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SLIDE 3

Aggregation in multicriteria decision aid

  • Alternatives A = {a, b, c, . . .}
  • Criteria N = {1, 2, . . . , n}

k-intolerant capacities and Choquet integrals – p.2/18

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SLIDE 4

Aggregation in multicriteria decision aid

  • Alternatives A = {a, b, c, . . .}
  • Criteria N = {1, 2, . . . , n}
  • Profile

a ∈ A − → (xa

1, . . . , xa n) ∈ [0, 1]n

k-intolerant capacities and Choquet integrals – p.2/18

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SLIDE 5

Aggregation in multicriteria decision aid

  • Alternatives A = {a, b, c, . . .}
  • Criteria N = {1, 2, . . . , n}
  • Profile

a ∈ A − → (xa

1, . . . , xa n) ∈ [0, 1]n

  • Aggregation function

F : [0, 1]n → [0, 1] (x1, . . . , xn) → F(x1, . . . , xn)

k-intolerant capacities and Choquet integrals – p.2/18

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SLIDE 6

Tolerant and intolerant character of F

k-intolerant capacities and Choquet integrals – p.3/18

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SLIDE 7

Tolerant and intolerant character of F

F(x) = mini xi →

intolerant behavior

k-intolerant capacities and Choquet integrals – p.3/18

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SLIDE 8

Tolerant and intolerant character of F

F(x) = mini xi →

intolerant behavior

F(x) = maxi xi →

tolerant behavior

k-intolerant capacities and Choquet integrals – p.3/18

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SLIDE 9

Tolerant and intolerant character of F

F(x) = mini xi →

intolerant behavior

F(x) = maxi xi →

tolerant behavior

F(x) = x(k) →

intermediate behavior

k-intolerant capacities and Choquet integrals – p.3/18

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SLIDE 10

Tolerant and intolerant character of F

F(x) = mini xi →

intolerant behavior

F(x) = maxi xi →

tolerant behavior

F(x) = x(k) →

intermediate behavior

F(x) =

n

  • i=1

xi

1/n

→ ?

k-intolerant capacities and Choquet integrals – p.3/18

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SLIDE 11

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 12

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx ∈ [0, 1]

1

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 13

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx ∈ [0, 1]

1

Examples

  • E(min) =

1 n+1

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 14

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx ∈ [0, 1]

1

Examples

  • E(min) =

1 n+1

  • E(max) =

n n+1

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 15

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx ∈ [0, 1]

1

Examples

  • E(min) =

1 n+1

  • E(max) =

n n+1

  • E(OSk) =

k n+1

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 16

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx ∈ [0, 1]

1

Examples

  • E(min) =

1 n+1

  • E(max) =

n n+1

  • E(OSk) =

k n+1

  • E(WAMω) = E(median) = 1

2

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 17

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx ∈ [0, 1]

1

Examples

  • E(min) =

1 n+1

(most intolerant)

  • E(max) =

n n+1

  • E(OSk) =

k n+1

  • E(WAMω) = E(median) = 1

2

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 18

Tolerant and intolerant character of F

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

E(F) :=

  • [0,1]n F(x) dx ∈ [0, 1]

1

Examples

  • E(min) =

1 n+1

(most intolerant)

  • E(max) =

n n+1

(most tolerant)

  • E(OSk) =

k n+1

  • E(WAMω) = E(median) = 1

2

k-intolerant capacities and Choquet integrals – p.4/18

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SLIDE 19

Tolerant and intolerant character of F

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

k-intolerant capacities and Choquet integrals – p.5/18

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SLIDE 20

Tolerant and intolerant character of F

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

1 1 E(F)

  • rness(F)
  • rness(F)

:= E(F) − E(min) E(max) − E(min)

(Dujmovi´ c, 1974)

k-intolerant capacities and Choquet integrals – p.5/18

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SLIDE 21

Tolerant and intolerant character of F

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

1 1 E(F)

  • rness(F)
  • rness(F)

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

(Dujmovi´ c, 1974)

k-intolerant capacities and Choquet integrals – p.5/18

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SLIDE 22

Tolerant and intolerant character of F

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

1 1 E(F)

  • rness(F)
  • rness(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

k-intolerant capacities and Choquet integrals – p.5/18

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SLIDE 23

Intolerant behavior : application

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 24

Intolerant behavior : application

Selection of candidates for a university permanent position

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 25

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 26

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

  • 1. Scientific value of curriculum vitae

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 27

Intolerant behavior : application

Selection of candidates for a university permanent position Academic selection criteria

  • 1. Scientific value of curriculum vitae
  • 2. Teaching effectiveness

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 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

xi = 0 for any two i ∈ N ⇒ F(x) = 0

k-intolerant capacities and Choquet integrals – p.6/18

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SLIDE 34

k-intolerant aggregation functions

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

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

k-intolerant capacities and Choquet integrals – p.7/18

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SLIDE 35

k-intolerant aggregation functions

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

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

This is equivalent to

x(k) = 0 ⇒ F(x) = 0

k-intolerant capacities and Choquet integrals – p.7/18

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SLIDE 36

k-intolerant aggregation functions

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

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

This is equivalent to

x(k) = 0 ⇒ F(x) = 0

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

F(x) x(k) (x ∈ [0, 1]n)

k-intolerant capacities and Choquet integrals – p.7/18

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SLIDE 37

Choquet integral

k-intolerant capacities and Choquet integrals – p.8/18

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Choquet integral

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

k-intolerant capacities and Choquet integrals – p.8/18

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SLIDE 39

Choquet integral

Capacity on N v : 2N → [0, 1], monotone, v(∅) = 0, and v(N) = 1 Fn := {capacities on N} Choquet integral of x ∈ [0, 1]n w.r.t. v Cv(x) :=

n

  • i=1

x(i)

  • v

(i), . . . , (n) − v (i + 1), . . . , (n)

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

k-intolerant capacities and Choquet integrals – p.8/18

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SLIDE 40

Choquet integral

Capacity on N v : 2N → [0, 1], monotone, v(∅) = 0, and v(N) = 1 Fn := {capacities on N} Choquet integral of x ∈ [0, 1]n w.r.t. v Cv(x) :=

n

  • i=1

x(i)

  • v

(i), . . . , (n) − v (i + 1), . . . , (n)

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

Example

If x3 x1 x2, we have Cv(x1, x2, x3) = x3[v(3, 1, 2) − v(1, 2)] + x1[v(1, 2) − v(2)] + x2v(2)

k-intolerant capacities and Choquet integrals – p.8/18

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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 OSk and F OSk−1.

k-intolerant capacities and Choquet integrals – p.9/18

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SLIDE 42

k-intolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-intolerant if F OSk and F OSk−1. Proposition Let k ∈ {1, . . . , n} and v ∈ Fn. Then the following assertions are equivalent: i) Cv(x) x(k) ∀x ∈ [0, 1]n, ii) ∀x ∈ [0, 1]n : x(k) = 0 ⇒ Cv(x) = 0,

k-intolerant capacities and Choquet integrals – p.9/18

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SLIDE 43

k-intolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-intolerant if F OSk and F OSk−1. Proposition Let k ∈ {1, . . . , n} and v ∈ Fn. Then the following assertions are equivalent: i) Cv(x) x(k) ∀x ∈ [0, 1]n, ii) ∀x ∈ [0, 1]n : x(k) = 0 ⇒ Cv(x) = 0, Recruiting problem The global evaluation is bounded above by x(2)

k-intolerant capacities and Choquet integrals – p.9/18

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SLIDE 44

k-intolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-intolerant if F OSk and F OSk−1. Proposition Let k ∈ {1, . . . , n} and v ∈ Fn. Then the following assertions are equivalent: i) Cv(x) x(k) ∀x ∈ [0, 1]n, ii) ∀x ∈ [0, 1]n : x(k) = 0 ⇒ Cv(x) = 0, iii) Cv(x) is independent of x(k+1), . . . , x(n),

k-intolerant capacities and Choquet integrals – p.9/18

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SLIDE 45

k-intolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-intolerant if F OSk and F OSk−1. Proposition Let k ∈ {1, . . . , n} and v ∈ Fn. Then the following assertions are equivalent: i) Cv(x) x(k) ∀x ∈ [0, 1]n, ii) ∀x ∈ [0, 1]n : x(k) = 0 ⇒ Cv(x) = 0, iii) Cv(x) is independent of x(k+1), . . . , x(n), Recruiting problem The global evaluation depends only on x(1) and x(2)

k-intolerant capacities and Choquet integrals – p.9/18

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SLIDE 46

k-intolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-intolerant if F OSk and F OSk−1. Proposition Let k ∈ {1, . . . , n} and v ∈ Fn. Then the following assertions are equivalent: i) Cv(x) x(k) ∀x ∈ [0, 1]n, ii) ∀x ∈ [0, 1]n : x(k) = 0 ⇒ Cv(x) = 0, iii) Cv(x) is independent of x(k+1), . . . , x(n), iv) v(T) = 0 ∀T ⊆ N such that |T| n − k

k-intolerant capacities and Choquet integrals – p.9/18

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SLIDE 47

k-intolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-intolerant if F OSk and F OSk−1. Proposition Let k ∈ {1, . . . , n} and v ∈ Fn. Then the following assertions are equivalent: i) Cv(x) x(k) ∀x ∈ [0, 1]n, ii) ∀x ∈ [0, 1]n : x(k) = 0 ⇒ Cv(x) = 0, iii) Cv(x) is independent of x(k+1), . . . , x(n), iv) v(T) = 0 ∀T ⊆ N such that |T| n − k Definition v ∈ Fn is k-intolerant if iv) holds for k and not for k − 1.

k-intolerant capacities and Choquet integrals – p.9/18

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SLIDE 48

Tolerant behavior : application

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 49

Tolerant behavior : application

Parents want to buy a house

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 50

Tolerant behavior : application

Parents want to buy a house House buying criteria

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 51

Tolerant behavior : application

Parents want to buy a house House buying criteria

  • 1. Close to a school

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 58

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

xi = 1 for any five i ∈ N ⇒ F(x) = 1

k-intolerant capacities and Choquet integrals – p.10/18

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SLIDE 59

k-tolerant aggregation functions

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

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

k-intolerant capacities and Choquet integrals – p.11/18

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SLIDE 60

k-tolerant aggregation functions

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

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

This is equivalent to

x(n−k+1) = 1 ⇒ F(x) = 1

k-intolerant capacities and Choquet integrals – p.11/18

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SLIDE 61

k-tolerant aggregation functions

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

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

This is equivalent to

x(n−k+1) = 1 ⇒ F(x) = 1

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

F(x) x(n−k+1) (x ∈ [0, 1]n)

k-intolerant capacities and Choquet integrals – p.11/18

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SLIDE 62

k-tolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-tolerant if F OSn−k+1 and F OSn−k+2.

k-intolerant capacities and Choquet integrals – p.12/18

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SLIDE 63

k-tolerant capacities and Choquet integrals

Definition Let k ∈ {1, . . . , n}. F : [0, 1]n → [0, 1] is k-tolerant if F OSn−k+1 and F OSn−k+2.

When F ≡ Cv we have a similar proposition as for intolerance...

k-intolerant capacities and Choquet integrals – p.12/18

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SLIDE 64

Intolerance and tolerance indices

k-intolerant capacities and Choquet integrals – p.13/18

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SLIDE 65

Intolerance and tolerance indices

Given a Choquet integral Cv

andness(Cv) measures the degree to which Cv is intolerant

k-intolerant capacities and Choquet integrals – p.13/18

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SLIDE 66

Intolerance and tolerance indices

Given a Choquet integral Cv How can we measure the degree to which the inequality

Cv OSk holds ?

k-intolerant capacities and Choquet integrals – p.13/18

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SLIDE 67

Intolerance and tolerance indices

Given a Choquet integral Cv How can we measure the degree to which the inequality

Cv OSk holds ?

Recall that:

Cv OSk ⇐ ⇒

  • x(k) = 0

⇒ Cv(x) = 0

  • k-intolerant capacities and Choquet integrals – p.13/18
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SLIDE 68

Intolerance and tolerance indices

Given a Choquet integral Cv How can we measure the degree to which the inequality

Cv OSk holds ?

Recall that:

Cv OSk ⇐ ⇒

  • x(k) = 0

⇒ Cv(x) = 0

  • ⇐

⇒

  • x(k) = 0

⇒ Cv(x) = minixi

  • k-intolerant capacities and Choquet integrals – p.13/18
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SLIDE 69

Intolerance and tolerance indices

Given a Choquet integral Cv How can we measure the degree to which the inequality

Cv OSk holds ?

Recall that:

Cv OSk ⇐ ⇒

  • x(k) = 0

⇒ Cv(x) = 0

  • ⇐

⇒

  • x(k) = 0

⇒ Cv(x) = minixi

  • Definition

For any k ∈ {1, . . . , n − 1} and any v ∈ Fn, we define

intolk(Cv) := andness(Cv | x(k) = 0)

k-intolerant capacities and Choquet integrals – p.13/18

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SLIDE 70

Intolerance and tolerance indices

Given a Choquet integral Cv How can we measure the degree to which the inequality

Cv OSk holds ?

Recall that:

Cv OSk ⇐ ⇒

  • x(k) = 0

⇒ Cv(x) = 0

  • ⇐

⇒

  • x(k) = 0

⇒ Cv(x) = minixi

  • Definition

For any k ∈ {1, . . . , n − 1} and any v ∈ Fn, we define

intolk(Cv) := andness(Cv | x(k) = 0)

Idea : defined from the conditional expectation E(Cv | x(k) = 0)

k-intolerant capacities and Choquet integrals – p.13/18

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SLIDE 71

Intolerance and tolerance indices

intolk(Cv) := andness(Cv | x(k) = 0)

k-intolerant capacities and Choquet integrals – p.14/18

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SLIDE 72

Intolerance and tolerance indices

intolk(Cv) := andness(Cv | x(k) = 0)

In terms of v, this index reads

intolk(Cv) = 1 − 1 n − k

n−k

  • t=0

1

n

t

  • T⊆N

|T|=t

v(T)

k-intolerant capacities and Choquet integrals – p.14/18

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SLIDE 73

Intolerance and tolerance indices

intolk(Cv) := andness(Cv | x(k) = 0)

Some properties

  • 1. intolk(Cv) = 1 if and only if Cv OSk

k-intolerant capacities and Choquet integrals – p.14/18

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SLIDE 74

Intolerance and tolerance indices

intolk(Cv) := andness(Cv | x(k) = 0)

Some properties

  • 1. intolk(Cv) = 1 if and only if Cv OSk
  • 2. intolk(Cv) is nondecreasing as k increases

k-intolerant capacities and Choquet integrals – p.14/18

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SLIDE 75

Intolerance and tolerance indices

intolk(Cv) := andness(Cv | x(k) = 0)

Some properties

  • 1. intolk(Cv) = 1 if and only if Cv OSk
  • 2. intolk(Cv) is nondecreasing as k increases
  • 3. Graph of intolk(OSj) for fixed k:

1 j 1 intol (OS )

<k

n k

j

k-intolerant capacities and Choquet integrals – p.14/18

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SLIDE 76

Intolerance and tolerance indices

intolk(Cv) := andness(Cv | x(k) = 0)

Some properties

  • 1. intolk(Cv) = 1 if and only if Cv OSk
  • 2. intolk(Cv) is nondecreasing as k increases
  • 3. Graph of intolk(OSj) for fixed k:

1 j 1 intol (OS )

<k

n k

j

OSj OSk ⇐ ⇒ j k

k-intolerant capacities and Choquet integrals – p.14/18

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SLIDE 77

Intolerance indices : characterization

k-intolerant capacities and Choquet integrals – p.15/18

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SLIDE 78

Intolerance indices : characterization

Theorem Let k ∈ {1, . . . , n − 1} and consider a family of real numbers {ψk(Cv) | v ∈ Fn} These numbers are

k-intolerant capacities and Choquet integrals – p.15/18

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SLIDE 79

Intolerance indices : characterization

Theorem Let k ∈ {1, . . . , n − 1} and consider a family of real numbers {ψk(Cv) | v ∈ Fn} These numbers are

  • 1. linear with respect to the capacity

k-intolerant capacities and Choquet integrals – p.15/18

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SLIDE 80

Intolerance indices : characterization

Theorem Let k ∈ {1, . . . , n − 1} and consider a family of real numbers {ψk(Cv) | v ∈ Fn} These numbers are

  • 1. linear with respect to the capacity
  • 2. independent of the numbering of criteria (symmetry)

k-intolerant capacities and Choquet integrals – p.15/18

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SLIDE 81

Intolerance indices : characterization

Theorem Let k ∈ {1, . . . , n − 1} and consider a family of real numbers {ψk(Cv) | v ∈ Fn} These numbers are

  • 1. linear with respect to the capacity
  • 2. independent of the numbering of criteria (symmetry)
  • 3. such that intolk(OSj) has the graph showed above

k-intolerant capacities and Choquet integrals – p.15/18

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SLIDE 82

Intolerance indices : characterization

Theorem Let k ∈ {1, . . . , n − 1} and consider a family of real numbers {ψk(Cv) | v ∈ Fn} These numbers are

  • 1. linear with respect to the capacity
  • 2. independent of the numbering of criteria (symmetry)
  • 3. such that intolk(OSj) has the graph showed above

if and only if ψk(Cv) = intolk(Cv) for all v ∈ Fn.

k-intolerant capacities and Choquet integrals – p.15/18

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SLIDE 83

The recruiting problem

k-intolerant capacities and Choquet integrals – p.16/18

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SLIDE 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

k-intolerant capacities and Choquet integrals – p.16/18

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SLIDE 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 intolk(Cv) for k = 1, . . . , 5

k-intolerant capacities and Choquet integrals – p.16/18

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SLIDE 86

Intolerance and tolerance indices

Similarly, we can define k-tolerant indices

k-intolerant capacities and Choquet integrals – p.17/18

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SLIDE 87

Intolerance and tolerance indices

Similarly, we can define k-tolerant indices

tolk(Cv) := orness(Cv | x(n−k+1) = 1)

k-intolerant capacities and Choquet integrals – p.17/18

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SLIDE 88

Intolerance and tolerance indices

Similarly, we can define k-tolerant indices

tolk(Cv) := orness(Cv | x(n−k+1) = 1)

...with similar motivation, characterization, properties.

k-intolerant capacities and Choquet integrals – p.17/18

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SLIDE 89

Conclusion

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

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SLIDE 90

Conclusion

  • We have defined
  • k-intolerant and k-tolerant capacities and Choquet

integrals

  • k-intolerance and k-tolerance indices

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

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SLIDE 91

Conclusion

  • We have defined
  • k-intolerant and k-tolerant capacities and Choquet

integrals

  • k-intolerance and k-tolerance indices
  • Behavioral parameters :
  • importance
  • interaction
  • dispersion
  • tolerance (veto, favor, andness, orness, intol, tol...)

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

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SLIDE 92

Conclusion

  • We have defined
  • k-intolerant and k-tolerant capacities and Choquet

integrals

  • k-intolerance and k-tolerance indices
  • Behavioral parameters :
  • importance
  • interaction
  • dispersion
  • tolerance (veto, favor, andness, orness, intol, tol...)
  • Identification of capacities :
  • by optimization
  • learning data
  • constraints on behavioral parameters...

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