Choquet integral

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Tolerant or intolerant character of interacting criteria in aggregation by the

Choquet integral

Jean-Luc Marichal HCP’2003

Slide 1

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Aggregation in multicriteria decision aid

• Alternatives A = {a, b, c, . . .}

• Criteria N = {1, 2, . . . , n}

• Profile a ∈ A −→ (x

, . . . , x

| {z }

same interval scale

) ∈ R

or [0, 1]

• Aggregation function

M : R

→ R M : [0, 1]

→ [0, 1]

(x

, . . . , x

) 7→ M (x

, . . . , x

)

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Choquet integral

Capacity on N :

v : 2

→ [0, 1], monotone, v (∅) = 0, and v(N ) = 1 F

:= {capacities on N }

Choquet integral of x ∈ R

w.r.t. v:

C

(x) :=

X

i=1

x

h

v £

(i), . . . , (n) ¤

− v £

(i + 1), . . . , (n) ¤i with the convention that x

6 · · · 6 x

.

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Examples

If x

6 x

6 x

, we have

C

(x

, x

, x

) = x

[v(3, 1, 2) − v(1, 2)]

+ x

[v (1, 2) − v(2)]

+ x

v(2) If x

6 x

6 x

, we have

C

(x

, x

, x

) = x

[v(2, 3, 1) − v(3, 1)]

+ x

[v (3, 1) − v(1)]

+ x

v(1)

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Properties of the Choquet integral

• Linearity w.r.t. the capacity

∃ f

: R

→ R (T ⊆ N ) s.t.

C

= X

T⊆N

v(T )f

(v ∈ F

)

• Monotonicity

For any x, x

∈ R

and any v ∈ F

x 6 x

⇒ C

(x) 6 C

(x

)

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• Affine invariance

For any x ∈ R

, v ∈ F

, r > 0 and s ∈ R,

C

(rx

+ s, . . . , rx

+ s) = r C

(x

, . . . , x

) + s

⇒ The partial scores may be embedded in [0, 1]

• Meaningfulness of weights

v(S ) = C

(1

) (S ⊆ N, v ∈ F

) Example : N = {1, 2, 3, 4}

v (2, 3, 4) = C

(0, 1, 1, 1)

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Axiomatization

Consider a family of n-variable aggregation functions {M

: R

→ R | v ∈ F

}

This family satisfies the four properties above if and only if M

= C

(v ∈ F

)

(Marichal, 2000)

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Evaluation of students

Student St Pr Al

a 19 15 18

b 19 18 15

c 11 15 18

d 11 18 15

←→

St Pr Al

0.95 0.75 0.90 0.95 0.90 0.75 0.55 0.75 0.90 0.55 0.90 0.75

on [0, 20] on [0, 1]

x

= rx

+ s

r = 1/20, s = 0

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Assumptions

• St and Pr more important than Al

• St and Pr are somewhat substitutive

• a  b and d  c (⇒ nonadditive Choquet integral)

v (∅) = 0 v (St, Pr) = 0.50 v (St) = 0.35 v (St, Al) = 0.80 v (Pr) = 0.35 v (Pr, Al) = 0.80 v (Al) = 0.30 v (St, Pr, Al) = 1

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Results

Student St Pr Al Global evaluation

a 19 15 18 17.75

b 19 18 15 16.85

c 11 15 18 15.10

d 11 18 15 15.25

Ranking

¤

£

¡

a  b  d  c ¢

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Special types of interaction:

veto and favor effects

A criterion k ∈ N is

• a veto for C

if

C

(x) 6 x

(x ∈ [0, 1]

)

• a favor for C

if

C

(x) > x

(x ∈ [0, 1]

) (Grabisch, 1997)

If k is both a veto and a favor then k is a dictator

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Surprising behavior of C

Example of rule

“ x

6 18/20 ⇒ C

(x) 6 18/20 ”

⇓ k is a veto Proposition

1) k is a veto for C

if and only if ∃ λ ∈ [0, 1[ s.t. ∀x ∈ [0, 1]

x

6 λ ⇒ C

(x) 6 λ

2) k is a favor for C

iff ∃ λ ∈ ]0, 1] s.t. ∀x ∈ [0, 1]

x

> λ ⇒ C

(x) > λ

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More surprising...

One can show that

k is a veto ⇔ ∃ x < y : C

(y, . . . , y,

x , y . . . , y) = x k is a favor ⇔ ∃ x > y : C

(y, . . . , y,

x , y . . . , y) = x Example

C

(10, 11, 11) = 10 ⇒ k = 1 is a veto

⇒ C

(10, 20, 20) = 10 !!

C

(19, 18, 18) = 19 ⇒ k = 1 is a favor

⇒ C

(19, 0, 0) = 19 !!

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Analysis

• Veto and favor criteria are rarely encountered

• Some criteria might behave nearly like a veto or a favor

• How can we measure the degree of veto (or favor)

of criteria ?

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Average value of C

over [0, 1]

: E(C

) :=

Z

[0,1]ⁿ

C

(x) dx ∈ [0, 1]

Examples

• E(min) =

(most intolerant)

• E(max) =

(most tolerant)

• E(OS

) =

• E(WAM

) = E(median) =

E(min) 6 E(C

) 6 E(max)

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Conjunction and disjunction degrees

Position of E(C

) within the interval [E(min), E (max)]:

orness(C

) := E(C

) − E(min) E(max) − E(min) andness(C

) := E(max) − E(C

)

E(max) − E(min) (Dujmovi´c, 1974)

andness(C

) + orness(C

) = 1

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Examples

• orness(min) = 0 (most intolerant)

• orness(max) = 1 (most tolerant)

• orness(WAM

) = orness(median) =

k is a veto for C

⇒ orness(C

) 6 1/2 (not tolerant) Proof:

C

(x) 6 x

⇓ E(C

) =

Z

[0,1]ⁿ

C

(x) dx 6 Z

[0,1]ⁿ

x

dx = 1/2

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Veto and favor degrees

veto(C

, k) := 1 −

P

T⊆N\{k}

(n−t−1)!t!

(n−1)!

v(T ) favor(C

, k) :=

P

T⊆N\{k}

(n−t−1)!t!

(n−1)!

v(T ∪ {k}) −

Why these ugly definitions ?

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Properties of veto(C

, k)

• Linearity w.r.t. the capacity

∃ p

∈ R (T ⊆ N ) s.t.

veto(C

, k) = X

T⊆N

v(T )p

(k ∈ N, v ∈ F

)

• Symmetry

For any permutation π on N

veto(C

, k) = veto(C

, π(k)) (k ∈ N, v ∈ F

) where πv(π(S)) = v(S ) for all S ⊆ N .

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• Boundary conditions For any ∅ 6= T ⊆ N we have

veto(min

T

, k) = 1 ∀k ∈ T (min

x

6 x

∀k ∈ T )

• Normalization Given v ∈ F

veto(C

, 1) = veto(C

, 2) = · · · = veto(C

, n)

⇓

veto(C

, i) = andness(C

)

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Axiomatization

Consider a family of real numbers

{ψ(C

, k) ∈ R | k ∈ N, v ∈ F

}

This family satisfies the four properties above if and only if ψ(C

, k) = veto(C

, k) (k ∈ N, v ∈ F

)

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Evaluation of students

orness(C

) = 0.517

St Pr Al

veto(C

, k) 0.437 0.437 0.575 favor(C

, k) 0.500 0.500 0.550 φ

(v ) 0.292 0.292 0.417 Shapley coefficients:

φ

(v) := X

T⊆N\{k}

(n − t − 1)! t!

n! [v (T ∪ {k}) − v(T )]

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1 n

X

k=1

veto(C

, k) = andness(C

) 1

n X

k=1

favor(C

, k) = orness(C

) 1

n X

k=1

φ

(v) = 1 n

St Pr Al

veto(C

, k)/andness(C

) 0.905 0.905 1.190 favor(C

, k)/orness(C

) 0.968 0.968 1.064 n φ

(v) 0.875 0.875 1.250

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Conclusion

• We have introduced

– global tolerance indices: orness, andness – local tolerance indices: veto, favor

• Behavioral parameters: importance, interaction, dispersion, tolerance...

• Identification of capacities: by optimization, learn-

ing data, constraints on behavioral parameters...

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Paper presented

J.-L. Marichal, Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral, European Journal of Operational Research, to appear.

Preprint available at: math.byu.edu/∼marichal