Additive conjoint measurement with ordered categories

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Additive conjoint measurement

with ordered categories

Denis Bouyssou Thierry Marchant

CNRS–LAMSADE Paris, France Universiteit Gent Ghent, Belgium ROADEF Nancy February 

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Introduction

Outline

Introduction

Context of today’s talk

preference modelling for MCDA

Central model: additive value functions x % y ⇔ n X i=1 vi(xi) ≥ n X i=1 vi(yi) Remarks

firm theoretical background (Krantz et al., )

many assessment techniques (Keeney & Raiffa, , von Winterfeld & Edwards, )

underliesmanyMCDM techniques

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Introduction

Outline

Introduction

preference modelling for MCDA Central model: additive value functions

x % y ⇔ n X i=1 vi(xi) ≥ n X i=1 vi(yi) Remarks

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Introduction

Outline

Introduction

preference modelling for MCDA Central model: additive value functions

x % y ⇔ n X i=1 vi(xi) ≥ n X i=1 vi(yi) Remarks

underliesmanyMCDM techniques

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Introduction

Outline

Need for extensions

Practical problems

the output of the model is a preference relation, i.e., arelativeevaluation model

there are many cases in which this is not adequate

admission of students to a programme

x is better then y: relative model x is “good”: absolute model

Theoretical problem

can additive value functions be obtained on the basis of a different kind of information?

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Introduction

Outline

Need for extensions

Practical problems

there are many cases in which this is not adequate admission of students to a programme

Theoretical problem

can additive value functions be obtained on the basis of a different kind of information?

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Introduction

Outline

Need for extensions

Practical problems

there are many cases in which this is not adequate admission of students to a programme

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Introduction

Outline

1 Definitions and notation Setting

Model

2 Intuitive sketch Uniqueness

Standard sequences Thomsen and completion

3 Axioms 4 Results Main Results Extensions 5 Discussion Summary Discussion 5 / 28

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Definitions and notation Intuitive sketch Axioms Results Discussion Setting Model

Outline

Model

2 Intuitive sketch Uniqueness

Standard sequences Thomsen and completion 3 Axioms

4 Results

Main Results Extensions

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Framework

Classical conjoint measurement setting N = {1, 2, . . . , n}: set ofattributes

X =Qn

i=1Xi with n ≥ 2: set ofalternatives

notation: (xi, y−i), (xJ, y−J) ∈ X

Primitives: threefold ordered partitions hA , F , U i of X A : set of objects that are “good”

F : set of object that are “neutral” U : set of objects that are “bad” Interpretation

position of objects vis-`a-vis astatus quo objects inA (U ) arenotequivalent

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Framework

X =Qn

F : set of object that are “neutral” U : set of objects that are “bad”

Interpretation

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Framework

X =Qn

F : set of object that are “neutral” U : set of objects that are “bad” Interpretation

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Model

Model: additive value functions with threshold

x ∈    A F U    ⇔ n X i=1 vi(xi)    > = <    0 Interpretation vi is real-valued function on Xi Pn

i=1vi is compared to a threshold

Non-degenerate

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Model

Model: additive value functions with threshold

x ∈    A F U    ⇔ n X i=1 vi(xi)    > = <    0 Interpretation vi is real-valued function on Xi Pn

i=1vi is compared to a threshold

Non-degenerate

hA , F , U i isnon-degenerateifA 6= ∅ and U 6= ∅

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Definitions and notation Intuitive sketch Axioms Results Discussion Uniqueness Standard sequences Thomsen and completion

Outline

Model 2 Intuitive sketch

Uniqueness

3 Axioms 4 Results

Main Results Extensions

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Uniqueness and normalization

a set of functions hvii representing hA , F , U i is not unique

origin of measurement: hvi+ βii give a representation whenPiβi = 0

unit of measurement: hαvii give a representation when α > 0

Suppose that n = 3 so that X = X1× X2× X3

take any (x0

1, x20, x30) ∈F

normalize so that v1(x10) = 0, v2(x20) = 0 and v3(x30) = 0

take any x₁−1∈ X1 such that (x1−1, x 0

2, x30) ∈U

normalize so that v1(x1−1) = −1

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Assess equally-spaced points

(x₁0, x₂0, x₃0) ∈F v1(x10) = v2(x20) = v3(x30) = 0 (x₁−1, x20, x30) ∈U v1(x1−1) = −1 (x₁−1, x₂1, x₃0) ∈F ⇒ v2(x21) = 1 (x10, x21, x3−1) ∈F ⇒ v3(x3−1) = −1 (x₁−1, x₂2, x₃−1) ∈F ⇒ v2(x22) = 2 (x10, x22, x3−2) ∈F ⇒ v3(x3−2) = −2 (x₁−1, x20, x 1 3) ∈F ⇒ v3(x31) = 1 (x₁0, x₂−1, x₃1) ∈F ⇒ v2(x2−1) = −1 (x₁−1, x₂−1, x32) ∈F ⇒ v3(x31) = 2 (x₁0, x₂−2, x₃2) ∈F ⇒ v2(x2−1) = −2 11 / 28

(x₁0, x₂0, x₃0) ∈F v1(x10) = v2(x20) = v3(x30) = 0 (x₁−1, x20, x30) ∈U v1(x1−1) = −1 (x₁−1, x₂1, x₃0) ∈F ⇒ v2(x21) = 1 (x10, x21, x3−1) ∈F ⇒ v3(x3−1) = −1 (x₁−1, x₂2, x₃−1) ∈F ⇒ v2(x22) = 2 (x10, x22, x3−2) ∈F ⇒ v3(x3−2) = −2 (x₁−1, x20, x31) ∈F ⇒ v3(x31) = 1 (x₁0, x₂−1, x₃1) ∈F ⇒ v2(x2−1) = −1 (x₁−1, x₂−1, x32) ∈F ⇒ v3(x31) = 2 (x₁0, x₂−2, x₃2) ∈F ⇒ v2(x2−1) = −2 11 / 28

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Diagonal standard sequences

(x₁0, x₂0, x₃0) ∈F v1(x10) = v2(x20) = v3(x30) = 0 (x₁−1, x20, x30) ∈U v1(x1−1) = −1 (x₁−1, x₂1, x₃0) ∈F ⇒ v2(x21) = 1 (x10, x21, x3−1) ∈F ⇒ v3(x3−1) = −1 (x₁−1, x₂2, x₃−1) ∈F ⇒ v2(x22) = 2 (x10, x22, x3−2) ∈F ⇒ v3(x3−2) = −2 (x₁−1, x20, x31) ∈F ⇒ v3(x31) = 1 (x0, x−1, x1) ∈F ⇒ v (x−1) = −1 (x₁0, x₂−2, x₃2) ∈F ⇒ v2(x2−1) = −2

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Diagonal standard sequences

(x₁0, x₂0, x₃0) ∈F v1(x10) = v2(x20) = v3(x30) = 0 (x₁−1, x20, x30) ∈U v1(x1−1) = −1 (x₁−1, x₂1, x₃0) ∈F ⇒ v2(x21) = 1 (x10, x21, x3−1) ∈F ⇒ v3(x3−1) = −1 (x₁−1, x₂2, x₃−1) ∈F ⇒ v2(x22) = 2 (x10, x22, x3−2) ∈F ⇒ v3(x3−2) = −2 (x₁−1, x20, x31) ∈F ⇒ v3(x31) = 1 (x₁0, x₂−1, x₃1) ∈F ⇒ v2(x2−1) = −1 (x₁−1, x₂−1, x32) ∈F ⇒ v3(x31) = 2 (x₁0, x₂−2, x₃2) ∈F ⇒ v2(x2−1) = −2 11 / 28

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X2 X3 x1 2 x22 x23 x24 x₂−1 x₂−2 x₂−3 x₂−4 x1 3 x2 3 x3 3 x4 3 x₃−1 x₃−2 x₁−1 x₁0

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X2 X3 x1 2 x22 x23 x24 x₂−1 x₂−2 x₂−3 x₂−4 x1 3 x2 3 x3 3 x4 3 x₃−1 x₃−2 x₃−3 x₃−4 x₁−1 x₁0

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Conditions

Ordering the elements of Xi

the function vi orders the element of Xi

this weak order is compatible with hA , F , U i

Thinness

categoryF is thin

strict compatibility with hA , F , U i Archimedean condition

a diagonal standard sequence is able to “reach” all diagonal points Structural conditions

solvability: we can find (x0

1, x20, x30) ∈F

influence of attribute 1: we can find x₁−1 below x0 1

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Conditions

this weak order is compatible with hA , F , U i Thinness

categoryF is thin

strict compatibility with hA , F , U i

Archimedean condition

a diagonal standard sequence is able to “reach” all diagonal points Structural conditions

1, x20, x30) ∈F

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Conditions

categoryF is thin

a diagonal standard sequence is able to “reach” all diagonal points

Structural conditions

1, x20, x30) ∈F

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Conditions

categoryF is thin

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Thomsen condition

Problems

how do we extend the construction to X1?

is the construction sound?

Thomsen condition (a1, x2, x3) ∈F and (x2, x3) ∼23(y2, y3) (b1, y2, z3) ∈F and (y2, z3) ∼23(z2, x3) ) ⇒ (x2, z3) ∼23(z2, y3) Interpretation (x2, x3) ∼23(y2, y3) ⇔    (a1, x2, x3) ∈A ⇔ (a1, y2, y3) ∈A (a1, x2, x3) ∈F ⇔ (a1, y2, y3) ∈F (a1, x2, x3) ∈U ⇔ (a1, y2, y3) ∈U 14 / 28

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Thomsen condition

Problems

is the construction sound? Thomsen condition (a1, x2, x3) ∈F and (x2, x3) ∼23(y2, y3) (b1, y2, z3) ∈F and (y2, z3) ∼23(z2, x3) ) ⇒ (x2, z3) ∼23(z2, y3) Interpretation (x2, x3) ∼23(y2, y3) ⇔    (a1, x2, x3) ∈A ⇔ (a1, y2, y3) ∈A (a1, x2, x3) ∈F ⇔ (a1, y2, y3) ∈F (a1, x2, x3) ∈U ⇔ (a1, y2, y3) ∈U

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Thomsen condition

Problems

is the construction sound? Thomsen condition (a1, x2, x3) ∈F and (x2, x3) ∼23(y2, y3) (b1, y2, z3) ∈F and (y2, z3) ∼23(z2, x3) ) ⇒ (x2, z3) ∼23(z2, y3) Interpretation (x2, x3) ∼23(y2, y3) ⇔    (a1, x2, x3) ∈A ⇔ (a1, y2, y3) ∈A (a1, x2, x3) ∈F ⇔ (a1, y2, y3) ∈F (a1, x2, x3) ∈U ⇔ (a1, y2, y3) ∈U 14 / 28

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X2 X3 x2 2 x23 x24 x₂−1 x₂−2 x₂−3 x₂−4 x1 3 x2 3 x3 3 x4 3 x₃−1 x₃−2

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X2 X3 x2 2 x23 x24 x₂−1 x₂−2 x₂−3 x₂−4 x1 3 x2 3 x3 3 x4 3 x₃−1 x₃−2 x₃−3 x₃−4

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X2 X3 x2 2 x23 x24 x₂−1 x₂−2 x₂−3 x₂−4 x1 3 x2 3 x3 3 x4 3 x₃−1 x₃−2

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Definitions and notation Intuitive sketch Axioms Results Discussion

Outline

Uniqueness

3 Axioms 4 Results Main Results Extensions 5 Discussion Summary Discussion 16 / 28

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A -linear

hA , F , U i is A -linear on I ⊂ N if (xI, a−I) ∈A and (yI, b−I) ∈A    ⇒    (yI, a−I) ∈A or (xI, b−I) ∈A Strong linearity

A -linear, F -linear and AF -linear, for all I ⊂ N

xI %I yI ⇔ for all a−I ∈ X−I,

(

(yI, a−I) ∈A ⇒ (xI, a−I) ∈A

(yI, a−I) ∈F ⇒ (xI, a−I) ∈AF

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F -linear

hA , F , U i is F -linear on I ⊂ N if (xI, a−I) ∈F and (yI, b−I) ∈F    ⇒    (yI, a−I) ∈AF or (xI, b−I) ∈AF Strong linearity

(

(yI, a−I) ∈A ⇒ (xI, a−I) ∈A

(yI, a−I) ∈F ⇒ (xI, a−I) ∈AF

%I is complete ⇔A -linear, F -linear, AF -linear on I ⊂ N

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AF -linear

hA , F , U i is AF -linear on I ⊂ N if (xI, a−I) ∈A and (yI, b−I) ∈F    ⇒    (yI, a−I) ∈A or (xI, b−I) ∈AF Strong linearity

(

(yI, a−I) ∈A ⇒ (xI, a−I) ∈A

(yI, a−I) ∈F ⇒ (xI, a−I) ∈AF

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Linearity

(

(yI, a−I) ∈A ⇒ (xI, a−I) ∈A

(yI, a−I) ∈F ⇒ (xI, a−I) ∈AF

%I is complete ⇔A -linear, F -linear, AF -linear on I ⊂ N

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Linearity

A -linear, F -linear and AF -linear, for all I ⊂ N (

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Thinness

hA , F , U i is thinI if (xI, a−I) ∈F and (yI, a−I) ∈F    ⇒ ( (xI, b−I) ∈A ⇔ (yI, b−I) ∈A (xI, b−I) ∈U ⇔ (yI, b−I) ∈U Strong thinness

thinI, for all I ⊂ N

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Thinness

hA , F , U i is thinI if (xI, a−I) ∈F and (yI, a−I) ∈F    ⇒ ( (xI, b−I) ∈A ⇔ (yI, b−I) ∈A (xI, b−I) ∈U ⇔ (yI, b−I) ∈U Strong thinness

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Thomsen

hA , F , U i on X satisfies the Thomsen condition if (xi, xj, a−ij) ∈F & (xi, xj) ∼ij (yi, yj)

(yi, zj, b−ij) ∈F & (yi, zj) ∼ij (zi, xj)

⇒ (xi, zj) ∼ij (zi, yj)

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Definitions and notation Intuitive sketch

Axioms

Results Discussion

Archimedean and Unrestricted solvability

hA , F , U i satisfies the Archimedean condition if a diagonal standard sequence that is strictly bounded must be finite

Unrestricted solvability

hA , F , U i satisfies unrestricted solvability if, for all i ∈ N and all x−i ∈ X−i,

(xi, x−i) ∈F , for some xi ∈ Xi

unrestricted solvability is a strong condition

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Axioms

Results Discussion

Archimedean and Unrestricted solvability

unrestricted solvability is a strong condition

can be weakened to restricted solvability with some connectedness spice

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Axioms

Results Discussion

Archimedean and Unrestricted solvability

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Definitions and notation Intuitive sketch Axioms Results Discussion Main Results Extensions

Outline

Uniqueness

Standard sequences Thomsen and completion 3 Axioms 4 Results Main Results Extensions 5 Discussion Summary Discussion 21 / 28

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Main result

Theorem, B & Marchant ()

hA , F , U i is an ordered partition on X = X1× X2× · · · × Xn,n ≥ 3.

Suppose that hA , F , U i is

non-degenerate

satisfies unrestricted solvability satisfies strong linearity satisfies strong thinness

satisfies Archimedean condition (if n = 3) satisfies Thomsen

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Main result

Suppose that hA , F , U i is non-degenerate

satisfies unrestricted solvability

satisfies strong linearity satisfies strong thinness

Then there is an additive representation hviii∈N of hA , F , U i

The functions hviii∈N are unique up to the choice of origins and unit

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Main result

satisfies unrestricted solvability

satisfies strong linearity

satisfies strong thinness

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Main result

satisfies unrestricted solvability satisfies strong linearity

satisfies strong thinness

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Main result

satisfies Archimedean condition

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Main result

satisfies Archimedean condition

(if n = 3) satisfies Thomsen

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Main result

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Extensions

Questions

do you have something to say for the case oftwoattributes?

yes: but uses very different techniques (ordinal vs conjoint measurement)

do you have to supposestronglinearity andstrongthinness

no: working on singletons and pairs suffices

do you have to supposeunrestricted solvability (which forces vi to be

unbounded)

yes and no: yes in an algebraic setting, no withconnectedness spice

do you have to use conditions onthe frontierF?

Uniqueness

Standard sequences Thomsen and completion 3 Axioms 4 Results Main Results Extensions 5 Discussion Summary Discussion 24 / 28

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Summary

Additive conjoint measurement

additive value functions with tight uniqueness properties an be obtained on the basis of rather poor information: hA , F , U i

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Usefulness to MCDM

All this is theory. . . but

axioms lead to an assessment technique

axiomatic analysis as a tool to compare models axiomatic analysis as a tool to create new techniques

Growing literature on the foundations of sorting methods decomposable models and decision rules, GMS () ELECTRE TRI

surprising relation to a Sugeno integral, B & Marchant (a, b)

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Usefulness to MCDM

All this is theory. . . but

axioms lead to an assessment technique

axiomatic analysis as a tool to compare models axiomatic analysis as a tool to create new techniques Growing literature on the foundations of sorting methods decomposable models and decision rules, GMS () ELECTRE TRI

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References

Bouyssou, D., Marchant, Th. (2009a)

Ordered categories and additive conjoint measurement on connected sets

Journal of Mathematical Psychology, forthcoming.

Bouyssou, D., Marchant, Th. (2009b)

Additive conjoint measurement with ordered categories

Working paper, submitted.

Bouyssou, D., Marchant, Th. (2009c) Biorders and bi-semiorders with frontiers

Working paper, submitted.

Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A. (1971) Foundations of measurement, vol. 1: Additive and polynomial representations

Academic Press, New York.

Vind, K. (1991)

Independent preferences

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