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A complete description of comparison meaningful functions

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(1)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

A Complete Description of Comparison

Meaningful Functions

Jean-Luc Marichal

University of Luxembourg

(2)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn. We assume that

Ff1,...,fn(a) = F [f1(a), . . . , fn(a)] (a ∈ S )

(3)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn. We assume that

Ff1,...,fn(a) = F [f1(a), . . . , fn(a)] (a ∈ S )

(4)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives

N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn. We assume that

Ff1,...,fn(a) = F [f1(a), . . . , fn(a)] (a ∈ S )

(5)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn. We assume that

Ff1,...,fn(a) = F [f1(a), . . . , fn(a)] (a ∈ S )

(6)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn. We assume that

Ff1,...,fn(a) = F [f1(a), . . . , fn(a)] (a ∈ S )

(7)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn. We assume that

Ff1,...,fn(a) = F [f1(a), . . . , fn(a)] (a ∈ S )

(8)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn.

We assume that

Ff1,...,fn(a) = F [f1(a), . . . , fn(a)] (a ∈ S )

(9)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Aggregation of measurement scales

We consider

S = {a, b, c, . . .} : a set of alternatives N = {1, . . . , n} : a set of attributes

For any a ∈ S and any i ∈ N, let fi(a) ∈ R be the scoreof a ∈ S according the i th attribute.

fi : S → R is a scale of measurement

We want to obtain anoverall evaluation of a ∈ S by means of an aggregation function Ff1,...,fn : S → R, which depends on f1, . . . , fn. We assume that

(10)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Thus, F is regarded as an aggregation function from Rn to R

: xn+1= F (x1, . . . , xn)

where x1, . . . , xn are the independent variables and xn+1 is the dependent variable.

The general form of F is restricted if we know thescale type of the variables x1, . . . , xn and xn+1 (Luce 1959).

A scale type is defined by the class ofadmissible transformations, transformations which change the scale into an alternative acceptable scale.

xi defines an ordinal scale if the class of admissible transformations consists of the increasing bijections (automorphisms) of R onto R.

(11)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Thus, F is regarded as an aggregation function from Rn to R : xn+1= F (x1, . . . , xn)

where x1, . . . , xn are the independent variables and xn+1 is the dependent variable.

The general form of F is restricted if we know thescale type of the variables x1, . . . , xn and xn+1 (Luce 1959).

A scale type is defined by the class ofadmissible transformations, transformations which change the scale into an alternative acceptable scale.

xi defines an ordinal scale if the class of admissible transformations consists of the increasing bijections (automorphisms) of R onto R.

(12)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Thus, F is regarded as an aggregation function from Rn to R : xn+1= F (x1, . . . , xn)

where x1, . . . , xn are the independent variables and xn+1 is the dependent variable.

The general form of F is restricted if we know thescale type of the variables x1, . . . , xn and xn+1 (Luce 1959).

A scale type is defined by the class ofadmissible transformations, transformations which change the scale into an alternative acceptable scale.

xi defines an ordinal scale if the class of admissible transformations consists of the increasing bijections (automorphisms) of R onto R.

(13)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Thus, F is regarded as an aggregation function from Rn to R : xn+1= F (x1, . . . , xn)

where x1, . . . , xn are the independent variables and xn+1 is the dependent variable.

The general form of F is restricted if we know thescale type of the variables x1, . . . , xn and xn+1 (Luce 1959).

A scale type is defined by the class ofadmissible transformations, transformations which change the scale into an alternative acceptable scale.

xi defines an ordinal scale if the class of admissible transformations consists of the increasing bijections (automorphisms) of R onto R.

(14)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Thus, F is regarded as an aggregation function from Rn to R : xn+1= F (x1, . . . , xn)

where x1, . . . , xn are the independent variables and xn+1 is the dependent variable.

The general form of F is restricted if we know thescale type of the variables x1, . . . , xn and xn+1 (Luce 1959).

A scale type is defined by the class ofadmissible transformations, transformations which change the scale into an alternative acceptable scale.

x defines an ordinal scale if the class of admissible transformations

(15)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Principle of theory construction (Luce 1959)

Admissible transformations of the independent variables should lead to an admissible transformation of the dependent variable.

Suppose that

xn+1= F (x1, . . . , xn)

where xn+1 is an ordinal scale and x1, . . . , xn are independent ordinal scales.

Let A(R) be the automorphism group of R.

For any φ1, . . . , φn∈ A(R), there is Φφ1,...,φn ∈ A(R) such that F [φ1(x1), . . . , φn(xn)] = Φφ1,...,φn[F (x1, . . . , xn)]

(16)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Principle of theory construction (Luce 1959)

Admissible transformations of the independent variables should lead to an admissible transformation of the dependent variable.

Suppose that

xn+1= F (x1, . . . , xn)

where xn+1 is an ordinal scale and x1, . . . , xn are independent ordinal scales.

Let A(R) be the automorphism group of R.

For any φ1, . . . , φn∈ A(R), there is Φφ1,...,φn ∈ A(R) such that F [φ1(x1), . . . , φn(xn)] = Φφ1,...,φn[F (x1, . . . , xn)]

(17)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Principle of theory construction (Luce 1959)

Admissible transformations of the independent variables should lead to an admissible transformation of the dependent variable.

Suppose that

xn+1= F (x1, . . . , xn)

where xn+1 is an ordinal scale and x1, . . . , xn are independent ordinal scales.

Let A(R) be the automorphism group of R.

For any φ1, . . . , φn∈ A(R), there is Φφ1,...,φn ∈ A(R) such that F [φ1(x1), . . . , φn(xn)] = Φφ1,...,φn[F (x1, . . . , xn)]

(18)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Principle of theory construction (Luce 1959)

Admissible transformations of the independent variables should lead to an admissible transformation of the dependent variable.

Suppose that

xn+1= F (x1, . . . , xn)

where xn+1 is an ordinal scale and x1, . . . , xn are independent ordinal scales.

Let A(R) be the automorphism group of R.

For any φ1, . . . , φn∈ A(R), there is Φφ1,...,φn ∈ A(R) such that F [φ1(x1), . . . , φn(xn)] = Φφ,...,φ [F (x1, . . . , xn)]

(19)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn define the same ordinal scale.

Then the functional equation simplifies into

F [φ(x1), . . . , φ(xn)] = Φφ[F (x1, . . . , xn)]

Equivalently, F fulfills the condition (Orlov 1981) F (x1, . . . , xn) 6 F (x10, . . . , xn0)

m

F [φ(x1), . . . , φ(xn)] 6 F [φ(x10), . . . , φ(xn0)] F is said to becomparison meaningful(Ovchinnikov 1996)

(20)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn define the same ordinal scale.

Then the functional equation simplifies into

F [φ(x1), . . . , φ(xn)] = Φφ[F (x1, . . . , xn)]

Equivalently, F fulfills the condition (Orlov 1981) F (x1, . . . , xn) 6 F (x10, . . . , xn0)

m

F [φ(x1), . . . , φ(xn)] 6 F [φ(x10), . . . , φ(xn0)] F is said to becomparison meaningful(Ovchinnikov 1996)

(21)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn define the same ordinal scale.

Then the functional equation simplifies into

F [φ(x1), . . . , φ(xn)] = Φφ[F (x1, . . . , xn)]

Equivalently, F fulfills the condition (Orlov 1981) F (x1, . . . , xn) 6 F (x10, . . . , xn0)

m

F [φ(x1), . . . , φ(xn)] 6 F [φ(x10), . . . , φ(xn0)]

F is said to becomparison meaningful(Ovchinnikov 1996)

(22)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn define the same ordinal scale.

Then the functional equation simplifies into

F [φ(x1), . . . , φ(xn)] = Φφ[F (x1, . . . , xn)]

Equivalently, F fulfills the condition (Orlov 1981) F (x1, . . . , xn) 6 F (x10, . . . , xn0)

m

F [φ(x1), . . . , φ(xn)] 6 F [φ(x10), . . . , φ(xn0)]

F is said to becomparison meaningful(Ovchinnikov 1996)

(23)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn are independent ordinal scales.

Recall that the functional equation is

F [φ1(x1), . . . , φn(xn)] = Φφ1,...,φn[F (x1, . . . , xn)]

Equivalently, F fulfills the condition

F (x1, . . . , xn) 6 F (x10, . . . , xn0) m

F [φ1(x1), . . . , φn(xn)] 6 F [φ1(x10), . . . , φn(xn0)] We say that F isstrongly comparison meaningful

(24)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn are independent ordinal scales.

Recall that the functional equation is

F [φ1(x1), . . . , φn(xn)] = Φφ1,...,φn[F (x1, . . . , xn)]

Equivalently, F fulfills the condition

F (x1, . . . , xn) 6 F (x10, . . . , xn0) m

F [φ1(x1), . . . , φn(xn)] 6 F [φ1(x10), . . . , φn(xn0)] We say that F isstrongly comparison meaningful

(25)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn are independent ordinal scales.

Recall that the functional equation is

F [φ1(x1), . . . , φn(xn)] = Φφ1,...,φn[F (x1, . . . , xn)]

Equivalently, F fulfills the condition

F (x1, . . . , xn) 6 F (x10, . . . , xn0) m

F [φ1(x1), . . . , φn(xn)] 6 F [φ1(x10), . . . , φn(xn0)]

We say that F isstrongly comparison meaningful

(26)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Assume x1, . . . , xn are independent ordinal scales.

Recall that the functional equation is

F [φ1(x1), . . . , φn(xn)] = Φφ1,...,φn[F (x1, . . . , xn)]

Equivalently, F fulfills the condition

F (x1, . . . , xn) 6 F (x10, . . . , xn0) m

F [φ1(x1), . . . , φn(xn)] 6 F [φ1(x10), . . . , φn(xn0)]

We say that F isstrongly comparison meaningful

(27)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Purpose of the presentation

To provide a complete description of comparison meaningful functions

To provide a complete description of strongly comparison meaningful functions

(28)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Purpose of the presentation

To provide a complete description of comparison meaningful functions

To provide a complete description of strongly comparison meaningful functions

(29)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

Aggregation of measurement scales Comparison meaningful functions Strongly comparison meaningful functions

Purpose of the presentation

To provide a complete description of comparison meaningful functions

To provide a complete description of strongly comparison meaningful functions

(30)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(31)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(32)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(33)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(34)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(35)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R

- continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(36)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(37)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant

such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(38)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(39)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(40)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(41)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔



∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(42)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n}

such that F (x1, . . . , xn) = xk

(43)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that

F (x1, . . . , xn) = xk

(44)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The continuous case

First result (Osborne 1970, Kim 1990)

F : Rn → R iscontinuousand strongly comparison meaningful

⇔









∃ k ∈ {1, . . . , n}

∃ g : R → R - continuous

- strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent (agreeing), i.e., F (x , . . . , x ) = x

⇔

 ∃ k ∈ {1, . . . , n} such that F (x , . . . , x ) = x

(45)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(46)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004)

F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(47)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(48)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(49)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(50)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant

such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(51)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(52)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(53)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(54)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔



∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(55)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n}

such that F (x1, . . . , xn) = xk

(56)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that

F (x1, . . . , xn) = xk

(57)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The nondecreasing case

Second result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isnondecreasingand strongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly increasing or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(58)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(59)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004)

F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(60)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(61)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(62)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(63)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant

such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(64)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(65)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(66)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(67)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔



∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(68)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n}

such that F (x1, . . . , xn) = xk

(69)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that

F (x1, . . . , xn) = xk

(70)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The continuous case The nondecreasing case The general case

The general case

Third result (Marichal & Mesiar & R¨uckschlossov´a 2004) F : Rn → R isstrongly comparison meaningful

⇔





∃ k ∈ {1, . . . , n}

∃ g : R → R strictly monotonic or constant such that

F (x1, . . . , xn) = g (xk)

+idempotent

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = xk

(71)

Introduction Strongly comparison meaningful functions Comparison meaningful functions Invariant functions

The symmetric case The nonsymmetric case The nonidempotent case The noncontinuous case

Comparison meaningful functions

First result (Orlov 1981) F : Rn → R is -symmetric

- continuous

- internal, i.e., minixi 6 F (x1, . . . , xn) 6 maxixi - comparison meaningful

⇔

 ∃ k ∈ {1, . . . , n} such that F (x1, . . . , xn) = x(k)

where x(1), . . . , x(n) denote theorder statisticsresulting from reordering x1, . . . , xn in the nondecreasing order.

Next step : suppress symmetry and relax internality into idempotency

Références

Télécharger maintenant ( PDF - 215 Page - 1.78 MB )

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