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Kbenhavn, Danmark, 2013 Aggregation functions for social decision making Vicen c Torra torsdag den 17. oktober 2013 Institut dInvestigaci o en Intel lig` encia Artificial (IIIA-CSIC), Bellaterra Outline Outline 1. Introduction

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SLIDE 1

København, Danmark, 2013

Aggregation functions for social decision making Vicen¸ c Torra torsdag den 17. oktober 2013

Institut d’Investigaci´

  • en Intel·lig`

encia Artificial (IIIA-CSIC), Bellaterra

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Outline

Outline

  • 1. Introduction
  • 2. Aggregation Functions
  • 3. Non-additive measures and integrals
  • 4. Application (a paradox)
  • 5. Distorted Probabilities
  • 6. End (p. 33)

København, Danmark, 2013 1 / 58

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Outline

Introduction

København, Danmark, 2013 2 / 58

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Motivation > What Outline

Introduction

Topic: Aggregation functions

  • They are used in decision problems

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 3 / 58

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Motivation > What Outline

Introduction

Topic: Aggregation functions

  • They are used in decision problems
  • To help establishing preferences for different pareto optimal situations

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 3 / 58

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Motivation > Where Outline

Introduction

MADM/MCDM: Select the best alternative from a set of alternatives

  • Usually a finite set of alternatives (otherwise MODM)
  • Each alternative evaluated in terms of a set of attributes (utilities)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 4 / 58

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Motivation > Where Outline

Introduction

MADM/MCDM: Select the best alternative from a set of alternatives

  • Usually a finite set of alternatives (otherwise MODM)
  • Each alternative evaluated in terms of a set of attributes (utilities)

Attributes (points of view/criteria) are in contradiction

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 4 / 58

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Motivation > Where Outline

Introduction

MADM/MCDM: Select the best alternative from a set of alternatives

  • Usually a finite set of alternatives (otherwise MODM)
  • Each alternative evaluated in terms of a set of attributes (utilities)

Attributes (points of view/criteria) are in contradiction Example: Decision making

  • Criteria to order our car preferences: price, quality, and confort

assign to each car ci ∈ Cars utility values up(ci), uq(ci), uc(ci) assign importances to each criteria (or subset of criteria)

(and combine values w.r.t. importances to find a global value (and order))

  • Contradictory attributes: price vs. quality and confort

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 4 / 58

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Motivation > Where Outline

Introduction

Example: Decision making

  • Criteria to order our car preferences: price, quality, and confort

assign to each car ci ∈ Cars utility values up(ci), uq(ci), uc(ci) assign importances to each criteria (or subset of criteria)

(and combine values w.r.t. importances to find a global value (and order))

Example: Ford T, Peugeot 308, Audi A4 up uq uc C Ford T 0.3 0.7 0.2 C(0.3, 0.7, 0.2) Peugeot 308 0.7 0.5 0.6 C(0.7, 0.5, 0.6) Audi A4 0.6 0.8 0.5 C(0.6, 0.8, 0.5)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 5 / 58

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Motivation > Where Outline

Introduction

MADM/MCDM: Select the best alternative from a set of alternatives

  • Select alternatives with large utilities

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 6 / 58

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Motivation > Where Outline

Introduction

MADM/MCDM: Select the best alternative from a set of alternatives

  • Select alternatives with large utilities

The larger the utility, the better

  • However, in some cases, not possible to improve one criteria without

worsening another

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 6 / 58

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Motivation > Where Outline

Introduction

MADM/MCDM: Select the best alternative from a set of alternatives

  • Select alternatives with large utilities

The larger the utility, the better

  • However, in some cases, not possible to improve one criteria without

worsening another

  • Such solutions define the Pareto set

x1 f1(x2) f1(x1) f1 f2 f2(x2) f2(x1) x2

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 6 / 58

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Motivation > What? Outline

Introduction

Topic: Aggregation functions permit to order different pareto optimal solutions

  • Different aggregation functions lead to different orderings
  • Some functions

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 7 / 58

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Motivation > What? Outline

Introduction

Topic: Aggregation functions permit to order different pareto optimal solutions

  • Different aggregation functions lead to different orderings
  • Some functions
  • Arithmetic mean
  • Weighted mean
  • Ordered Weighting Averaging Operator
  • Choquet integral (integral for non-additive measures)
  • Example: C

up uq uc C Ford T 0.3 0.7 0.2 C(0.3, 0.7, 0.2) Peugeot 308 0.7 0.5 0.6 C(0.7, 0.5, 0.6) Audi A4 0.6 0.8 0.5 C(0.6, 0.8, 0.5)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 7 / 58

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SLIDE 15

Outline

Aggregation Functions

København, Danmark, 2013 8 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

Topic: Aggregation functions permit to order different pareto optimal solutions

  • Different aggregation functions lead to different orderings
  • Some functions

København, Danmark, 2013 9 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

Topic: Aggregation functions permit to order different pareto optimal solutions

  • Different aggregation functions lead to different orderings
  • Some functions
  • Arithmetic mean
  • Weighted mean
  • Ordered Weighting Averaging Operator
  • Choquet integral (integral for non-additive measures)

København, Danmark, 2013 9 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

Aggregation functions

  • Weighting vector (dimension N): v = (v1...vN) iff

vi ∈ [0, 1] and

i vi = 1

  • Arithmetic mean (AM :RN → R): AM(a1, ..., aN) = (1/N) N

i=1 ai

  • Weighted mean (WM: RN → R): WMp(a1, ..., aN) = N

i=1 piai

(p a weighting vector of dimension N)

  • Ordered Weighting Averaging operator (OWA: RN → R):

OWAw(a1, ..., aN) =

N

  • i=1

wiaσ(i), where {σ(1), ..., σ(N)} is a permutation of {1, ..., N} s. t. aσ(i−1) ≥ aσ(i), and w a weighting vector.

København, Danmark, 2013 10 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

Aggregation functions: Arithmetic Mean (AM)

  • Level curbes for Pareto Optimal solutions

København, Danmark, 2013 11 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

Aggregation functions: Weighted Mean (WM)

  • Level curbes for Pareto Optimal solutions

København, Danmark, 2013 12 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

Aggregation functions Interpretation of weights Weights in WM and OWA: p and w

  • Multicriteria Decision Making.

p: importance of criteria, w: degree of compensation

  • Fuzzy Constraint Satisfaction Problems.

p: importance of the constraints, w: degree of compensation

  • Robot Sensing (all data, same time instant).

p: reliability of each sensor, w: importance of small values/outliers

  • Robot Sensing (all data, different time instants).

p: more importance to recent data than old one, w: importance of small values/outliers

København, Danmark, 2013 13 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

Weighted Ordered Weighted Averaging WOWA operator

(WOWA :RN → R):

WOWAp,w(a1, ..., aN) = N

i=1 ωiaσ(i)

where ωi = w∗(

j≤i pσ(j)) − w∗( j<i pσ(j)),

with σ a permutation of {1, ..., N} s. t. aσ(i−1) ≥ aσ(i), and w∗ a nondecreasing function that interpolates the points {(i/N,

j≤i wj)}i=1,...,N ∪ {(0, 0)}.

w∗ is required to be a straight line when the points can be interpolated in this way.

København, Danmark, 2013 14 / 58

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Aggregation Operators > Options Outline

Aggregation Functions

WOWA operator The shape of the function w∗ gives importance

  • (a) to large values
  • (b) to medium values
  • (c) to small values
  • (d) equal importance to all values

(a) (b) (c) (d)

København, Danmark, 2013 15 / 58

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SLIDE 24

Outline

Non-additive measures and integrals (Choquet integral)

København, Danmark, 2013 16 / 58

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Introduction > Definition Outline

Definitions

Additive measures: (X, A) a measurable space; then, a set function µ is an additive measure if it satisfies (i) µ(A) ≥ 0 for all A ∈ A, (ii) µ(X) ≤ ∞ (iii) µ(∪∞

i=1Ai) = ∞ i=1 µ(Ai) for every countable sequence Ai (i ≥ 1)

  • f A that is pairwise disjoint (i.e,. Ai ∩ Aj = ∅ when i = j).

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 17 / 58

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Introduction > Definition Outline

Definitions

Additive measures: (X, A) a measurable space; then, a set function µ is an additive measure if it satisfies (i) µ(A) ≥ 0 for all A ∈ A, (ii) µ(X) ≤ ∞ (iii) µ(∪∞

i=1Ai) = ∞ i=1 µ(Ai) for every countable sequence Ai (i ≥ 1)

  • f A that is pairwise disjoint (i.e,. Ai ∩ Aj = ∅ when i = j).

Finite case: µ(A ∪ B) = µ(A) + µ(B) for disjoint A, B

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 17 / 58

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Introduction > Definition Outline

Definitions

Non-additive measures: (X, A) a measurable space, a non-additive (fuzzy) measure µ on (X, A) is a set function µ : A → [0, 1] satisfying the following axioms: (i) µ(∅) = 0, µ(X) = 1 (boundary conditions) (ii) A ⊆ B implies µ(A) ≤ µ(B) (monotonicity)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 18 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • In additive measures: µ(A) =

x∈A px

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 19 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • In additive measures: µ(A) =

x∈A px

  • In non-additive measures: additivity no longer a constraint

→ three cases possible

  • µ(A) =

x∈A px

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 19 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • In additive measures: µ(A) =

x∈A px

  • In non-additive measures: additivity no longer a constraint

→ three cases possible

  • µ(A) =

x∈A px

  • µ(A) <

x∈A px

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 19 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • In additive measures: µ(A) =

x∈A px

  • In non-additive measures: additivity no longer a constraint

→ three cases possible

  • µ(A) =

x∈A px

  • µ(A) <

x∈A px

  • µ(A) >

x∈A px

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 19 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 20 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 20 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

  • Why?

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 20 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

  • Why?

some cases represent interactions

  • µ(A) =

x∈A px (no interaction)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 20 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

  • Why?

some cases represent interactions

  • µ(A) =

x∈A px (no interaction)

  • µ(A) <

x∈A px (negative interaction)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 20 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

  • Why?

some cases represent interactions

  • µ(A) =

x∈A px (no interaction)

  • µ(A) <

x∈A px (negative interaction)

  • µ(A) >

x∈A px (positive interaction)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 20 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 21 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 21 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

  • In our example:

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 21 / 58

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Introduction > Differences Outline

Differences

Non-additive measures vs. additive measures:

  • Is non-additivity useful ?

Yes

  • In our example:
  • µ({price}), µ({quality}), µ({confort})
  • µ({price, quality}), µ({price, confort}), µ({quality, confort})
  • µ({price, quality, confort})

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 21 / 58

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Introduction > Number of parameters Outline

Number of parameters

Non-additive measures vs. additive measures:

  • How to define an additive measure on X?

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 22 / 58

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Introduction > Number of parameters Outline

Number of parameters

Non-additive measures vs. additive measures:

  • How to define an additive measure on X?

One probability value for each element → |X| values

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 22 / 58

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Introduction > Number of parameters Outline

Number of parameters

Non-additive measures vs. additive measures:

  • How to define an additive measure on X?

One probability value for each element → |X| values

  • How to define a non-additive measure?

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 22 / 58

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Introduction > Number of parameters Outline

Number of parameters

Non-additive measures vs. additive measures:

  • How to define an additive measure on X?

One probability value for each element → |X| values

  • How to define a non-additive measure?

One value for each set → 2|X| values

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 22 / 58

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Introduction > What to do? Outline

What can we do with a measure?

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 23 / 58

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Introduction > What to do? Outline

What can we do with a measure?

Non-additive measures and additive measures:

  • Integrate a function f with respect to a measure:

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 23 / 58

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Introduction > What to do? Outline

What can we do with a measure?

Non-additive measures and additive measures:

  • Integrate a function f with respect to a measure:
  • Integral w.r.t. additive measure p

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 23 / 58

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Introduction > What to do? Outline

What can we do with a measure?

Non-additive measures and additive measures:

  • Integrate a function f with respect to a measure:
  • Integral w.r.t. additive measure p

→ expectation: pxf(x) − → Lebesgue integral (continuous case:

  • fdp)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 23 / 58

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Introduction > What to do? Outline

What can we do with a measure?

Non-additive measures and additive measures:

  • Integrate a function f with respect to a measure:
  • Integral w.r.t. additive measure p

→ expectation: pxf(x) − → Lebesgue integral (continuous case:

  • fdp)
  • Integral w.r.t. non-additive measure µ

→ expectation like

N

  • i=1

f(xσ(i))[µ(Aσ(i)) − µ(Aσ(i−1))]

− → Choquet integral (continuous case: (C)

  • fdµ)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 23 / 58

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SLIDE 51

Introduction > What to do? Outline

What can we do with a measure?

Non-additive measures and additive measures:

  • Integrate a function f with respect to a measure:
  • Integral w.r.t. additive measure p

→ expectation: pxf(x) − → Lebesgue integral (continuous case:

  • fdp)
  • Integral w.r.t. non-additive measure µ

→ expectation like

N

  • i=1

f(xσ(i))[µ(Aσ(i)) − µ(Aσ(i−1))]

− → Choquet integral (continuous case: (C)

  • fdµ)

The Choquet integral is a Lebesgue integral when the measure is additive

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 23 / 58

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Introduction > What to do? Outline

Choquet integrals

Integral:

  • fdµ =

(for additive measures)

(6.5)

x∈X f(x)µ({x})

(6.6) R

i=1 biµ({x|f(x) = bi})

(6.7) N

i=1(ai − ai−1)µ({x|f(x) ≥ ai})

(6.8) N

i=1(ai − ai−1)

  • 1 − µ({x|f(x) ≤ ai−1})
  • bi

bi−1 ai ai−1 bi bi−1 x1 x1 x1 xN xN x {x|f(x) ≥ ai} {x|f(x) = bi} (a) (b) (c)

Among (6.5), (6.6) and (6.7), only (6.7) satisfies internality.

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 24 / 58

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Introduction > What to do? Outline

Choquet integrals

Aggregation functions: Choquet integral (CI)

  • Level curbes for Pareto Optimal solutions

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 25 / 58

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Outline

Application (a paradox)

København, Danmark, 2013 26 / 58

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Applications > Decision Making Outline

Example

Decision making: Ellsberg paradox (Ellsberg, 1961), an urn, 90 balls ... Color of balls Red Black Yellow Number of balls 30 60 fR $ 100 fB $ 0 $ 100 fRY $ 100 $ 100 fBY $ 0 $ 100 $ 100

  • Usual (most people’s) preferences
  • fB ≺ fR

København, Danmark, 2013 27 / 58

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Applications > Decision Making Outline

Example

Decision making: Ellsberg paradox (Ellsberg, 1961), an urn, 90 balls ... Color of balls Red Black Yellow Number of balls 30 60 fR $ 100 fB $ 0 $ 100 fRY $ 100 $ 100 fBY $ 0 $ 100 $ 100

  • Usual (most people’s) preferences
  • fB ≺ fR
  • fRY ≺ fBY

København, Danmark, 2013 27 / 58

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SLIDE 57

Applications > Decision Making Outline

Example

Decision making: Ellsberg paradox (Ellsberg, 1961), an urn, 90 balls ... Color of balls Red Black Yellow Number of balls 30 60 fR $ 100 fB $ 0 $ 100 fRY $ 100 $ 100 fBY $ 0 $ 100 $ 100

  • Usual (most people’s) preferences
  • fB ≺ fR
  • fRY ≺ fBY
  • No solution exist with additive measures,

but can be solved with non-additive ones

København, Danmark, 2013 27 / 58

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SLIDE 58

Outline

Distorted Probabilities

København, Danmark, 2013 28 / 58

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SLIDE 59

Distorted Probabilities > Introduction Outline

Distorted Probabilities: introduction

An open question:

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 29 / 58

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Distorted Probabilities > Introduction Outline

Distorted Probabilities: introduction

An open question: Non-additive measures vs. additive measures: How to define a non-additive measure? One value for each set → 2|X| values

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 29 / 58

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Distorted Probabilities > Introduction Outline

Distorted Probabilities: introduction

An open question: Non-additive measures vs. additive measures: How to define a non-additive measure? One value for each set → 2|X| values A possible solution:

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 29 / 58

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Distorted Probabilities > Introduction Outline

Distorted Probabilities: introduction

An open question: Non-additive measures vs. additive measures: How to define a non-additive measure? One value for each set → 2|X| values A possible solution: Distorted Probabilities.

  • Compact representation of non-additive measures:

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 29 / 58

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Distorted Probabilities > Introduction Outline

Distorted Probabilities: introduction

An open question: Non-additive measures vs. additive measures: How to define a non-additive measure? One value for each set → 2|X| values A possible solution: Distorted Probabilities.

  • Compact representation of non-additive measures:
  • Only |X| values (a probability) and a function (distorting function)

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 29 / 58

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Distorted Probabilities > Definition Outline

Distorted Probabilities: Definition

  • Representation of a fuzzy measure:
  • f and P represent a fuzzy measure µ, iff

µ(A) = f(P(A)) for all A ∈ 2X f a real-valued function, P a probability measure on (X, 2X)

  • f is strictly increasing w.r.t.

a probability measure P iff P(A) < P(B) implies f(P(A)) < f(P(B))

  • f is nondecreasing w.r.t. a probability measure P iff P(A) < P(B)

implies f(P(A)) ≤ f(P(B))

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 30 / 58

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Distorted Probabilities > Definition Outline

Distorted Probabilities: Definition

  • Representation of a fuzzy measure: distorted probability
  • f and P represent a fuzzy measure µ, iff

µ(A) = f(P(A)) for all A ∈ 2X f a real-valued function, P a probability measure on (X, 2X)

  • f is strictly increasing w.r.t.

a probability measure P iff P(A) < P(B) implies f(P(A)) < f(P(B))

  • f is nondecreasing w.r.t. a probability measure P iff P(A) < P(B)

implies f(P(A)) ≤ f(P(B))

  • µ is a distorted probability if µ is represented by a probability

distribution P and a function f nondecreasing w.r.t. a probability P.

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 31 / 58

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Distorted Probabilities > Definition Outline

Distorted Probabilities: Definition

  • Representation of a fuzzy measure: distorted probability
  • µ is a distorted probability if µ is represented by a probability

distribution P and a function f nondecreasing w.r.t. a probability P.

  • So, for a given reference set X we need:
  • Probability distribution on X: p(x) for all x ∈ X
  • Distortion function f on the probability measure: f(P(A))

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 32 / 58

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Outline

The End

København, Danmark, 2013 33 / 58

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SLIDE 68

Outline

m-dimensional Distorted Probabilities

København, Danmark, 2013 34 / 58

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m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities

Justification: Why any extension of distorted probabilities?

København, Danmark, 2013 35 / 58

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m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities

Justification: Why any extension of distorted probabilities?

  • The number of distorted probabilities.

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m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities

Justification: Why any extension of distorted probabilities?

  • The number of distorted probabilities.

Observe the following

  • For X = {1, 2, 3}, 2/8 of distorted probabilities.
  • For larger sets X ...

... the proportion of distorted probabilities decreases rapidly

  • For µ({1}) ≤ µ({2}) ≤ . . .

|X| Number of possible orderings for Number of possible orderings for Distorted Probabilities Fuzzy Measures 1 1 1 2 1 1 3 2 8 4 14 70016 5 546 O(1012) 6 215470 –

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m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities

Justification: Why any extension of distorted probabilities? The number of distorted probabilities. Goal:

  • To cover a larger region of the space of fuzzy measures

DP Unconstrained fuzzy measures

→ (similar to the property of k-additive fuzzy measures) DP1,X ⊂ DP2,X ⊂ DP3,X · · · ⊂ DP|X|,X

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SLIDE 73

m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities

  • In distorted probabilities:
  • One probability distribution
  • One function f to distort the probabilities
  • Extension to:
  • m probability distributions
  • One function f to distort/combine the probabilities

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SLIDE 74

m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities

  • In distorted probabilities:
  • One probability distribution
  • One function f to distort the probabilities
  • Extension to:
  • m probability distributions Pi

⋆ Each Pi defined on Xi ⋆ Each Xi is a partition element of X (a dimension)

  • One function f to distort/combine the probabilities

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SLIDE 75

m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities: Example

  • Running example:
  • A fuzzy measure that is not a distorted probability:

µ(∅) = 0 µ({M, L}) = 0.9 µ({M}) = 0.45 µ({P, L}) = 0.9 µ({P}) = 0.45 µ({M, P}) = 0.5 µ({L}) = 0.3 µ({M, P, L}) = 1

  • Partition on X:

⋆ X1 = {L} (Literary subjects) ⋆ X2 = {M, P} (Scientific Subjects)

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m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities: Definition

  • m-dimensional distorted probabilities.
  • µ is an at most m dimensional distorted probability if

µ(A) = f(P1(A ∩ X1), P2(A ∩ X2), · · · , Pm(A ∩ Xm)) where, {X1, X2, · · · , Xm} is a partition of X, Pi are probabilities on (Xi, 2Xi), f is a function on Rm strictly increasing with respect to the i-th axis for all i = 1, 2, . . . , m.

  • µ is an m-dimensional distorted probability if it is an at most m

dimensional distorted probability but it is not an at most m − 1 dimensional.

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m-Dimensional Distorted Probabilities > Definition Outline

m-Dimensional Distorted Probabilities: Example

  • Running example: a two dimensional distorted probability

µ(A) = f(P1(A ∩ {L}), P2(A ∩ {M, P}))

  • with partition on X = {M, L, P}
  • 1. Literary subject {L}
  • 2. Science subjects {M, P},
  • probabilities
  • 1. P1({L}) = 1
  • 2. P2({M}) = P2({P}) = 0.5,
  • and distortion function f defined by

1 {L} 0.3 0.9 1.0 ∅ 0.45 0.5 sets ∅ {M}, {P} {M,P} f ∅ 0.5 1

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SLIDE 78

Outline

Distorted Probabilities and Multisets

an approach to define (simple) fuzzy measures on multisets

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DP and Multisets > Multisets Outline

Distorted Probabilities and Multisets

Multisets: elements can appear more than once

  • Defined in terms of countM : X → {0} ∪ N

e.g. when X = {a, b, c, d} and M = {a, a, b, b, c, c, c}, countM(a) = 2, countM(b) = 3, countM(c) = 3, countM(d) = 0.

  • A and B multisets on X, then
  • A ⊆ B if and only if countA(x) ≤ countB(x) for all x in X

(used to define submultiset).

  • A ∪ B:

countA∪B(x) = max(countA(x), countB(x)) for all x in X.

  • A ∩ B:

countA∩B(x) = min(countA(x), countB(x)) for all x in X.

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SLIDE 80

DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Fuzzy measure on multiset: X a reference set, M a multiset on X s.t. M = ∅; then, the function µ from (M, P(M)) to [0, 1] is a fuzzy measure if the following holds:

  • µ(∅) = 0 and µ(M) = 1
  • µ(A) ≤ µ(B) when A ⊆ B and B ⊆ M.

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Fuzzy measure on multiset: X a reference set, M a multiset on X s.t. M = ∅; then, the function µ from (M, P(M)) to [0, 1] is a fuzzy measure if the following holds:

  • µ(∅) = 0 and µ(M) = 1
  • µ(A) ≤ µ(B) when A ⊆ B and B ⊆ M.

How to define fuzzy measures?:

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Fuzzy measure on multiset: X a reference set, M a multiset on X s.t. M = ∅; then, the function µ from (M, P(M)) to [0, 1] is a fuzzy measure if the following holds:

  • µ(∅) = 0 and µ(M) = 1
  • µ(A) ≤ µ(B) when A ⊆ B and B ⊆ M.

How to define fuzzy measures?:

  • Even more parameters

x∈X countM(x) !!

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Fuzzy measure on multiset: X a reference set, M a multiset on X s.t. M = ∅; then, the function µ from (M, P(M)) to [0, 1] is a fuzzy measure if the following holds:

  • µ(∅) = 0 and µ(M) = 1
  • µ(A) ≤ µ(B) when A ⊆ B and B ⊆ M.

How to define fuzzy measures?:

  • Even more parameters

x∈X countM(x) !!

We present two alternative (but related) approaches

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

1st approach: Definition based on a pseudoadditive integral: Nondecreasing function-based fuzzy measures

  • X a reference set, M a multiset on X and µ a ⊕-decomposable fuzzy measure
  • n X. Let f : [0, ∞) → [0, ∞) be a non-decreasing function with f(0) = 0 and

f(m(M)) = 1. Then, we define a fuzzy measure ν on P(M) by νf(A) = f(m(A)) where m is the multiset function m : P(M) → [0, ∞) defined by m(A) = (D)

  • countAdµ.

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

1st approach: Definition based on a pseudoadditive integral: Nondecreasing function-based fuzzy measures

  • X a reference set, M a multiset on X and µ a ⊕-decomposable fuzzy measure
  • n X. Let f : [0, ∞) → [0, ∞) be a non-decreasing function with f(0) = 0 and

f(m(M)) = 1. Then, we define a fuzzy measure ν on P(M) by νf(A) = f(m(A)) where m is the multiset function m : P(M) → [0, ∞) defined by m(A) = (D)

  • countAdµ.
  • Rationale of the definition: (C)
  • χAdµ = µ(A)

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

1st approach: Definition based on a pseudoadditive integral: Nondecreasing function-based fuzzy measures

  • X a reference set, M a multiset on X and µ a ⊕-decomposable fuzzy measure
  • n X. Let f : [0, ∞) → [0, ∞) be a non-decreasing function with f(0) = 0 and

f(m(M)) = 1. Then, we define a fuzzy measure ν on P(M) by νf(A) = f(m(A)) where m is the multiset function m : P(M) → [0, ∞) defined by m(A) = (D)

  • countAdµ.
  • Rationale of the definition: (C)
  • χAdµ = µ(A)
  • Properties:

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DP and Multisets > Fuzzy Measure Outline

if A ⊆ B by the monotonicity of the integral m(A) ≤ m(B) → monotonicity condition of the fuzzy measure fulfilled

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

2nd approach: Definition based on prime numbers1:

  • Define

n(A) :=

  • x∈X

φ(x)countA(x), where φ is an injective function from X to the prime numbers, and let h be a non-decreasing function from N to [0, 1] satisfying h(1) = 0 and h(n(M)) = 1. We define the prime number-based fuzzy measure νφ,h(A) = h(n(A)).

1and using the unique factorization of integers into prime numbers København, Danmark, 2013 47 / 58

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

2nd approach: Definition based on prime numbers1:

  • Define

n(A) :=

  • x∈X

φ(x)countA(x), where φ is an injective function from X to the prime numbers, and let h be a non-decreasing function from N to [0, 1] satisfying h(1) = 0 and h(n(M)) = 1. We define the prime number-based fuzzy measure νφ,h(A) = h(n(A)).

Properties: if A = B by the unique factorization n(A) = n(B) if A ⊆ B by the factorization n(A) < n(B) → monotonicity condition of the fuzzy measure fulfilled

1and using the unique factorization of integers into prime numbers København, Danmark, 2013 47 / 58

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Properties:

  • Fuzzy measures based on prime-number are a particular case of the

1st approach

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Properties:

  • Fuzzy measures based on prime-number are a particular case of the

1st approach

  • Neither the 1st nor the 2nd approach represent all possible fuzzy

measures

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Properties:

  • Fuzzy measures based on prime-number are a particular case of the

1st approach

  • Neither the 1st nor the 2nd approach represent all possible fuzzy

measures

  • It seems that there is some parallelism between prime-number based

fuzzy measures and distorted probabilities

  • f and the distortion
  • φ and the probability distribution

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DP and Multisets > Fuzzy Measure Outline

Distorted Probabilities and Multisets

Properties:

  • Fuzzy measures based on prime-number are a particular case of the

1st approach

  • Neither the 1st nor the 2nd approach represent all possible fuzzy

measures

  • It seems that there is some parallelism between prime-number based

fuzzy measures and distorted probabilities

  • f and the distortion
  • φ and the probability distribution
  • Can we establish a relationship??

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • ν a fuzzy measure according to Approach 1 on a proper finite set

(M, P(M)) = (X, 2X). Then ν is a distorted probability on (X, 2X).

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • ν a fuzzy measure according to Approach 1 on a proper finite set

(M, P(M)) = (X, 2X). Then ν is a distorted probability on (X, 2X).

  • Same for Approach 2 (primer-number definition)

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • ν a fuzzy measure according to Approach 1 on a proper finite set

(M, P(M)) = (X, 2X). Then ν is a distorted probability on (X, 2X).

  • Same for Approach 2 (primer-number definition)
  • This is easy to prove (consists on defining the probability distribution)

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • ν a fuzzy measure according to Approach 1 on a proper finite set

(M, P(M)) = (X, 2X). Then ν is a distorted probability on (X, 2X).

  • Same for Approach 2 (primer-number definition)
  • This is easy to prove (consists on defining the probability distribution)
  • So, Approach 1 and Approach 2 equal to or more general than

distorted probabilities

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • Can we prove something else? much more general? almost the same?

exactly the same?

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • Can we prove something else? much more general? almost the same?

exactly the same? Not so surprising theorem: Fuzzy measures based

  • n

prime numbers on proper sets are equivalent to distorted probabilities

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • Can we prove something else? much more general? almost the same?

exactly the same? Not so surprising theorem: Fuzzy measures based

  • n

prime numbers on proper sets are equivalent to distorted probabilities → probabilities and prime numbers play the same role →

x∈A px and x∈A φ(x) play the same role

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • Can we prove something else? much more general? almost the same?

exactly the same? Not so surprising theorem: Fuzzy measures based

  • n

prime numbers on proper sets are equivalent to distorted probabilities → probabilities and prime numbers play the same role →

x∈A px and x∈A φ(x) play the same role

Much more surprising theorem: Fuzzy measures based

  • n

Approach 1 on proper sets are equivalent to distorted probabilities

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • Can we prove something else? much more general? almost the same?

exactly the same? Not so surprising theorem: Fuzzy measures based

  • n

prime numbers on proper sets are equivalent to distorted probabilities → probabilities and prime numbers play the same role →

x∈A px and x∈A φ(x) play the same role

Much more surprising theorem: Fuzzy measures based

  • n

Approach 1 on proper sets are equivalent to distorted probabilities Surprising corollary: Approach 1 and approach 2 are equivalent.

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DP and Multisets > Fuzzy Measure Outline

Results

Properties:

  • Can we prove something else? much more general? almost the same?

exactly the same? Not so surprising theorem: Fuzzy measures based

  • n

prime numbers on proper sets are equivalent to distorted probabilities → probabilities and prime numbers play the same role →

x∈A px and x∈A φ(x) play the same role

Much more surprising theorem: Fuzzy measures based

  • n

Approach 1 on proper sets are equivalent to distorted probabilities Surprising corollary: Approach 1 and approach 2 are equivalent. Proof based on some results on number theory about the existence

  • f k prime numbers in certain intervals (Bertrand’s postulate).

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DP and Multisets > Fuzzy Measure Outline

Results

An example to satisfy curiosity:

  • µ distorted probability p = (0.05, 0.1, 0.2, 0.3, 0.35), g(x) = x2.

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DP and Multisets > Fuzzy Measure Outline

Results

An example to satisfy curiosity:

  • µ distorted probability p = (0.05, 0.1, 0.2, 0.3, 0.35), g(x) = x2.
  • Representation with prime numbers and appropriate function

φ(x1) = 17 ∈ [16.0, 32.0001] φ(x2) = 367 ∈ [362.041, 724.081] φ(x3) = 185369 ∈ [185366.0, 370732.0] φ(x4) = 94907801 ∈ [9.49078 × 107, 1.89816 × 108] φ(x5) = 2147524151 ∈ [2.14752 × 109, 4.29505 × 109]

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DP and Multisets > m-Dimensional Outline

m-Dimensional DP for multisets

How to solve the problem that not all fuzzy measures for multisets are distorted probabilities ?

  • Same approach as before: m-dimensional prime number-based fuzzy

measure

DP Unconstrained fuzzy measures

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DP and Multisets > m-Dimensional Outline

m-Dimensional DP for multisets

m-dimensional prime number-based fuzzy measure

  • µ is an at most m-dimensional prime number-based fuzzy measure if

µ(A) = f(n1(A ∩ X1), . . . , nm(A ∩ Xm)) where, {X1, X2, · · · , Xm} is a partition of X, ni(A) =

x∈Xi φ(x)countA(x) with φi injective functions from Xi

to the prime numbers f is a strictly increasing function with respect to the i-th axis for all i = 1, 2, . . . , m. µ is an m-dimensional prime number-based fuzzy measure if it is an at most m dimensional distorted probability but it is not an at most m−1 dimensional.

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DP and Multisets > m-Dimensional Outline

m-Dimensional DP for multisets

Properties:

  • All fuzzy measures are at most |X|-dimensional prime number-based

fuzzy measures.

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SLIDE 109

Outline

Integral

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Integral > Definitions Outline

Integral

Definition

  • Boundary measures:
  • µ+(A) = A · M for all A ⊆ X
  • µ−(A) = A ∩ M for all A ⊆ X
  • They satisfy:

µ−(A) ≤ µ+(A) and, therefore, (C)

  • fdµ− < (C)
  • fdµ+

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Outline

Summary

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Summary Outline

Summary

Summary:

  • Brief justification of the use of non-additive (fuzzy) measures
  • Introduction to distorted probabilities
  • Extensions
  • m-dimensional distorted probabilities
  • Fuzzy measures for multisets

Vicen¸ c Torra; Aggregation functions København, Danmark, 2013 58 / 58