Conjoint Measurement without Additivity and Transitivity

Transcription

1 Conjoint Measurement without Additivity and Transitivity Denis Bouyssou LAMSADE - Paris - France Marc Pirlot FacultŽ Polytechnique de Mons - Mons - Belgium Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 1

2 Outline n Introduction and Motivation n Models Ð Inter-Attribute Decomposable Models Ð Intra-Attribute Decomposable Models n Extensions/Applications n Conclusion and Open Problems Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 2

3 Introduction Context = Conjoint Measurement n Set of Objects: X Í X 1 x X 2 x Éx X n n Binary relation on this set: Objective = Study/Build/Axiomatise numerical representations of Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 3

4 Introduction Interest of Numerical Representations n Manipulation of n Construction of numerical representations Interest of Axiomatic Analysis n Tests of models n Understanding models Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 4

5 Examples: Cartesian Product Structure x = (x 1, x 2, É, x n ) Î X n MCDM Ð x is an ÒalternativeÓ evaluated on ÒattributesÓ Other examples n DM under uncertainty Ð x is an ÒactÓ evaluated on Òstates of natureó n Economics Ð x is a ÒbundleÓ of ÒcommoditiesÓ n Dynamic DM Ð x is an ÒalternativeÓ evaluated at Òseveral moments in timeó n Social Choice Ð x is a ÒdistributionÓ between several ÒindividualsÓ n x y means Òx is at least as good as yó Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 5

6 Additive Transitive Representation n x y Û å i = 1ui( xi) ³ åi = 1ui( yi) n Basic Model = Additive Utility n Examples: MCDM Weighted sum, Additive utility, Goal programming, Compromise Programming DM under uncertainty: SEU Dynamic DM: Discounting n Properties (among others!) is complete is transitive is independent Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 6

7 Independence Independence: A common consequence on attribute i does not affect preference ( a, x ) ( b, x ) Þ ( a, y ) ( b, y ) -i i -i i -i i -i i Necessity: ( a, x ) ( b, x ) Þ å u ( a ) + u ( x ) ³ å u ( b ) + u ( x ) -i i - i i j ¹ i j j i i j ¹ i j j i i å u ( a ) + u ( y ) ³ å u ( b ) + u ( y ) Þ( a, y ) ( b, y ) j ¹ i j j i i j ¹ i j j i i -i i -i i Þ Weak Independence: Common consequences on attributes other than i does not affect preference ( a, x ) ( a, y ) Þ ( b, x ) ( b, y ) -i i -i i -i i -i i Independence Þ Weak Independence Weak Independence allows to define Òpartial preference relationsó i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 7

8 Triple Cancellation ( xi, a -i) ( yi, b-i) ü and ï ï ( zi, b-i) ( wi, a -i) ý andï ï ( wi, c -i) ( zi, d-i) þï Þ ( x, c ) ( y, d ) i -i i -i Remarks possible generalization to subsets of attributes (replace i by J and Ði by ÐJ) TC and reflexive Þ Independence C m : Cancellation condition of order m Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 8

9 Cancellation Condition of Order m (C m ) 1 2 m 1 2 x, x, ¼, x, y, y, ¼, y ÎX If for all i Î{1,2, ¼,n} 1 i j j m m x y for j = 1, 2, ¼, m - 1 Þ y x m (x, x, ¼, x ) is a permutation of (y, y, ¼, y ) then i 2 i m i 1 i 2 i m Necessity m n j i i i j m n = 1 = 1 j = 1 i = 1 å å u ( x ) = å å u ( y ) i i j Remarks C m+1 Þ C m For no finite m, C m Þ C m+1 C 2 Þ Independence C 3 Þ Transitivity C 4 Þ TC Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot - - 9

10 Axiomatic Analysis: 2 cases n X finite (Scott-Suppes 1958, Scott 1964) + Necessary and sufficient Conditions + Denumerable Set of ÒCancellation ConditionsÓ + No nice uniqueness results n X has a Òrich structureó and behaves consistently in this ÒcontinuumÓ (Debreu 1960, Luce-Tukey 1964) + (Topological assumptions + continuity) or (solvability assumption + Archimedean condition) + A finite (and limited) set of ÒCancellation ConditionsÓ entails the representation (independance, TC) + u i define Òinterval scaleó with common unit Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

11 Sample Result on Additive Utility Theorem(Scott 1964): If X is finite then is complete and satisfies C m for m = 2, 3, É iff the additive utility models holds Remarks Ð No nice uniqueness result Ð Proof rests on the Òtheorem of the alternativeó Ð Extension to general sets Jaffray 1974 Ð Fishburn (1997) : bounds on m given X Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

12 Sample Result on Additive Utility Theorem(Luce-Tukey 1964): If n ³ 3 (three essential components) and is an independent weak order satisfies restricted solvability satisfies an archimedean axiom then the additive utility model holds and u i define interval scales with common unit Remarks Ð independence may be replaced by Triple Cancellation Ð With Triple Cancellation result is valid for n = 2 Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

13 Problems n Transitivity and completeness of Ð Experimental violations (May 1954, Luce 1969) Ð Aggregation models in MCDM violating these hypothesis Ð Decision Theory can be conceived without transitivity (Fishburn 1991) Þ Find a more flexible framework n Axiomatic Problems Ð Finite case: Axioms hardly interpretable and testable Ð ÒRich caseó + Respective roles of unnecessary structural conditions and necessary ÒcancellationÓ conditions (Furkhen and Richter 1991) + Asymmetry n = 2 vs. n ³ 3 cases (n = 2 more difficult) Ð Asymmetry Finite vs. ÒRichÓ case n Few Results outside this case (MCDM contribution?) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

14 Possible extensions Additive utility = Additive Transitive Conjoint Measurement 1 2 n Extensions Drop additivity Drop transitivity and/or completeness Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

15 Decomposable Transitive Models Keep transitivity and completeness Ð Drop additivity Krantz et al (1971) x y Û F( u ( x ), u ( x ), ¼, u ( x )) ³ F( u ( y ), u ( y ), ¼, u ( y )) F increasing Advantages n n n n Simple axiomatic analysis Simple proofs Allows to ÒunderstandÓ the Òpure consequencesó of weak independence + transitivity and completeness Drawbacks Transitivity and completeness No nice unicity results Too general? (F is not specified) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

16 Sample Result on Decomposable Transitive Models Theorem(Krantz et al 1971): is a weakly independent weak order (having a numerical representation) iff the decomposable transitive model holds Remarks Ð Necessary and Sufficient conditions for all X Ð Simple proof Ð No asymmetry ÒRichÓ vs. finite, n = 2 vs. n ³ 3 Ð No nice uniqueness result (u i are ÒrelatedÓ ordinal scales) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

17 Additive Non Transitive Models Keep Additivity Ð Drop transitivity and completeness Bouyssou 1986, Fishburn 1990, 1991, Vind 1991 n x y Û å i = 1 pi( xi, yi) ³ 0 ( with additional properties) p i skew symmetric or p i (x i,x i )= 0 Advantages Flexible towards transitivity and completeness Classical results are particular cases Interpretation in terms of Òpreference differencesó Nice unicity results with rich structure: p i define ratio scales with common unit Drawbacks Asymmetry: Finite vs. Rich, n ³ 3 vs. n = 2 (n = 2 simpler!) Complex proofs Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

18 Cancellation Condition of Order m (S m ) 1 2 m 1 2 x, x, ¼, x, y, y, ¼, y ÎX If for all i Î{1,2, ¼,n} {(x, y )} = {(y, x )} then i j i j i j i j j j m m x y for j = 1, 2, ¼, m - 1 Þ y x m m n j = 1 å i i i j i j = 1 0 Necessity : å p ( x, y ) = if p are skew symmetric Remarks S m+1 Þ S m For no finite m, S m Þ S m+1 S 4 Þ TC i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

19 Sample Results on Additive Non Transitive Models (with skew symmetry) Theorem (Fishburn 1991) If X is finite then is complete and satisfies S m for m = 1, 2, 3, É iff the non transitive additive model holds with p i skew symmetric Remarks Ð No nice uniqueness result Ð Proof rests on the Òtheorem of the alternativeó Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

20 Sample Results on Additive Non Transitive Models (with skew symmetry) Theorem (Fishburn 1991) If n ³ 3 (three essential components) and satisfies restricted solvability complete and satisfies S 4 satisfies an archimedean axiom then the non transitive additive model holds (with skew symmetric p i ) and p i define ratio scales with common unit (if non extremality) Remarks Ð n = 2 is a simpler case Ð S 4 Þ Triple Cancellation on subsets Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

21 Cancellation Condition of Order m (T m ) 1 2 m 1 2 m 1 2 m 1 2 m x, x, ¼, x, y, y, ¼, y, z, z, ¼, z, w, w, ¼, w ÎX If for all i Î{1,2, ¼,n} 1 i 1 i [(x, y ),(x,, y ), ¼,( x, y )] is a permutation of 1 i 1 i i 2 i 2 i m i m [(z, w ),(z,, w ), ¼,( z, w )] i 2 i 2 i m i m Not[ x j y j j j and Not(z w )] for j = 1, 2, ¼, m Necessity m n m n j = 1 åi = 1p i(xi j yi j åj = 1 i = 1p i(zi j wi j å, ) = å, ) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

22 Sample Results on Additive Non Transitive Models (with p i (x i, x i ) = 0 ) Theorem (adapted from Fishburn 1992) If X is finite then is reflexive and independent satisfies T m for m = 1, 2, 3, É iff the non transitive additive model holds with p i (x i, x i ) = 0 Remarks Ð No nice uniqueness result Ð Proof rests on the "theorem of the alternative" Ð Rich case Vind 1991 Ð T 2 Þ RC1 Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

23 Keep additvity Relax Transitivity Additive Transitive n x y Û å i = 1ui( xi) ³ åi = 1ui( yi) n Additive Non Transitive n i 1 x y Û å = p ( x, y ) ³ 0 i i i Keep Transitivity Relax Additivity Decomposable Transitive x y Û F( u ( x )) ³ F( u ( y )) i i i i? Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

24 Keep additvity Relax Transitivity Additive Transitive n x y Û å i = 1ui( xi) ³ åi = 1ui( yi) n Additive Non Transitive n i 1 x y Û å = p ( x, y ) ³ 0 i i i Keep Transitivity Relax Additivity Decomposable Transitive x y Û F( u ( x )) ³ F( u ( y )) i i i i Non Transitive Decomposable Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

25 Non Transitive Decomposable models Trivial model: x y Û F(x, y) ³ 0 ì1 if x y F( x, y) = í î - 1 otherwise. n Inter-attribute Decomposability Decompose F along the various attribute F( x, y) = F( pi( xi, yi)) n Intra-attribute decomposability Decompose p i (x i, y i ) to build ÒcriteriaÓ p ( x, y ) = j ( u ( x ), u ( y )) i i i i i i i i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

26 General Model Inter-Attribute Decomposability x y Û F( p ( x, y ) ) ³ i i i i = 1, 2, ¼, n 0 Problems This model is trivial (under a mild cardinality assumption) is not independent is not reflexive!! Care should be taken in the definition of the models!! Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

27 Inter-Attribute Decomposable Models (M) x y Û F(p 1 (x 1,y 1 ), p 2 (x 2,y 2 ), É, p n (x n,y n )) ³ 0 (M0) (M) with p i (x i, x i ) = 0 and F(0) ³ 0 (M1) (M0) with F non decreasing in all arguments (M1 ) (M0) with F increasing in all arguments (M2) (M1) with p i skew symmetric (M2 ) (M1 ) with p i skew symmetric (M3) (M2) with F odd (M3 ) (M2 ) with F odd Remarks Ð (Mk ) Þ (MkÐ1 ), (Mk) Þ (MkÐ1) Ð All Models are particular cases of (M0) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

28 Intuition n p i captures Òpreference differencesó between levels of X i n F combines these ÒdifferencesÓ in a consistent way n F increasing and odd brings it ÒcloserÓ to addition n Skew symmetry of p i Þ the ªdifferenceº ( xi, yi) is linked to the ªopposite differenceº ( y, x ) i i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

29 M1 F M0 M1 F non decreasing p i skew symmetric F odd : F( - x) = -F( x) p i skew symmetric : p ( x, y ) = -p ( y, x ) i i i i i i p i skew symmetric F M2 M2 F Odd F Odd F M3 M3 Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

30 Basic Properties (i) If satisfies model (M0) then it is reflexive and independent. (ii) If satisfies model (M1) or (M1 ) then: n [x i i y i for all i Î J Í {1, 2, É, n}] Þ [Not(y J J x J )] (iii) If satisfies model (M2) or (M2 ) then: n i is complete, n [x i i y i for all i Î J] Þ [x J J y J ] (iv) If satisfies model (M3) then it is complete (v) If satisfies model (M3 ) then: n [x i i y i for all i Î J] Þ [x J J y J ] n [x i i y i for all i Î J, x j j y j, for some j Î J] Þ [x J J y J ] Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

31 Axioms: RC1 RC1 i ( xi, a -i) ( yi, b-i) üï andý ( zi, c -i) ( wi, d-i) þï Þ ìï ( zi, a -i) ( wi, b-i) íor îï ( xi, c -i) ( yi, d-i) Interpretation ( x, y ) is either larger or smaller than ( z, w ) i i i i Consequence * i i i i i ( x, y ) ( z, w ) Û [( z, a ) ( w, b ) Þ ( x, a ) ( y, b )] i -i i -i i -i i -i is complete and therefore a weak order Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

32 Axioms: RC2 RC2 i ( xi, a -i) ( yi, b-i) ü ï andý ( yi, c -i) ( xi, d-i) ï þ Þ ì( zi, a -i) ( wi, b-i) ï íor ï î( wi, c -i) ( zi, d-i) Interpretation (x i, y i ) is ÒlinkedÓ to (y i, x i ) Consequence ** i i i i i -i -i ( x, y ) ( z, w ) Û for all a, b, [( z, a ) ( w, b ) Þ ( x, a ) ( y, b )] and i -i i -i i -i i -i [( y, c ) ( x, d ) Þ ( w, c ) ( z, d )] i -i i -i i -i i -i is complete and therefore a weak order Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

33 Results - Denumerable case If X is finite or countably infinite: n (M0) iff reflexivity, independence, n (M1 ) iff reflexivity, independence, RC1, n (M2 ) iff reflexivity, RC1, RC2, n (M3) iff completeness, RC1, RC2, n (M3 ) iff completeness, TC. Non Denumerable case Add a necessary Order Density condition: OD* Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

34 Remarks n (M1)Û (M1 ), (M2)Û (M2 ) n Necessary and Sufficient conditions for all X n Axioms are independent n No nice uniqueness results and Irregular representations n Allow to study the Òpure consequencesó of classical cancellation conditions Ð TC vs. Independence n Adding Òrich structureó + axioms on subsets implies F is additive and uniqueness results (Fishburn 1991, Vind 1991) n Adding transitivity and completeness on M0 implies the Decomposable Transitive Model Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

35 More (technical) remarks n RC1 i Û biorder between X 2 i and X Ði 2 Adding an order density condition implies x y Û p ( x, y ) + P ( x, y ) ³ 0 i i i -i -i -i n n = 2 is a very particular case n TC i + completeness implies x y Û p ( x, y ) + P ( x, y ) ³ 0 with i i i -i -i -i p and P skew symmetric i -i n RC2 Þ Independence n TC, completeness Þ RC1, RC2 Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

36 Remarks n RC1 is NS for: x y Û F( p ( x, y ), p ( x, y ), ¼, pn( xn, yn) ³ with F nondecresing n In all models the function p i can be chosen so as to represent * (or ** ) i i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

37 Example: Additive Utility n Additive utility n x y Û å i = 1ui( xi) ³ åi = 1ui( yi) n Interpretation x y Û F( p ( x, y ) ) ³ with F = å and p ( x, y ) = u ( x ) - u ( y ) n i i i i = 1, 2, ¼, n 0 i i i i i i i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

38 Example: ELECTRE I (Roy 1968) x y Û ì( åi: x y ki i ki s i )/( å i i = 1 ) ³ ï íand ïnot( xivy î i i) n x y Û u ( x ) - u ( y ) ³ -q i i i i i i i x Vy Û u ( x ) - u ( y ) < v i i i i i i i s ³ 1 2 n Interpretation x y Û F( pi( xi, yi) i = 1, 2, ¼, n) ³ 0 with F = å and ìk i if ui( xi) - ui( yi) ³ -q ï s pi( xi, yi) = í- s k i if - v u i( x i) - u i( y i) < - q ï 1 - î-m otherwise Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

39 TACTIC Vansnick 1986 (Adaptation) ìr åi: x k i y i ³ å i i : ï x y Û íand ï înot( xivy i i) x y Û u ( x ) - u ( y ) > q i i i i i i i x Vy Û u ( x ) - u ( y ) < v i i i i i i i r ³ 1 i y x i i i k i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

40 Application: Compensation vs. Noncompensation RC1 i implies * i i i i i [( x, y ) ( z, w ), for all i] Þ [ x y Û z w] Fishburn's 1976 Definition of Noncompensation [( x y ) Û( z w ), ( y x ) Û( w z )] Þ [ x y Û z w] i i i i i i i i i i i i Formally very similar definitions Fishburn's Noncompensation only at most three distinct equivalence classes of * i the comparison of preference differences is only based on i All Methods are ÒNoncompensatoryÓ in our more general sense Clue = number of equivalence classes of i * Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

41 ELECTRE I PI P Ñ V u ( x ) - u ( y ) ³ -q i i i i - v u ( x ) - u ( y ) < -q i i i i u ( x ) - u ( y ) < -v i i i i is reflexive, independent and satisfies RC1 and RC2 Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

42 TACTIC P u ( x ) - u ( y ) > q i i i i I - q u ( x ) - u ( y ) q i i i i P Ñ V - v u ( x ) - u ( y ) < -q i i i i u ( x ) - u ( y ) < -v i i i i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

43 Additive Utility ui( xi) - ui( yi) ÎI 1 u ( x ) - u ( y ) ÎI 2 i i i i ui( xi) - ui( yi) ÎIk -1 u ( x ) - u ( y ) ÎI i i i i k... Many Equivalence Classes Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

44 Difficulty n In all models the ÒweightÓ of the preference difference is computed with respect to an underlying ÒcriterionÓ n i has nice properties (semi order) n Study ÒIntra-Attribute DecomposabilityÓ pi( xi, yi) =ji( ui( xi), ui( yi)) u i (x i ) u i (y i ) Criterion Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

45 Additive Difference Model (Tversky 1969) n i 1 x y Û å = F ( u ( x ) - u ( y )) ³ 0 i i i i i F i increasing and odd n Introduction of Intra-Attribute Decomposability may be intransitive (but is complete) i are weak orders n Axioms (Fishburn 1992) Rich Structure Complex proofs Nice uniqueness results (u i define interval scale) n Extensions (Bouyssou 1986) Non decreasing F i (allows for semi orders on each attribute) F i not odd but F i (0) = 0 (allows for incomplete ) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

46 Additive Utility Additive Difference Additive Non Transitive Decomposable Transitive? Non Transitive Decomposable Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

47 Additive Utility Inter Additive Intra Decomposable Additive Difference Inter Additive Not Intra Decomposable Additive Non Transitive Decomposable Transitive? Inter Decomposable Intra Decomposable Non Transitive Decomposable Inter Decomposable Not Intra Decomposable Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

48 Additive Transitive Model F i (u i (x i ) - u i (y i )) u i (x i ) - u i (y i ) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

49 ELECTRE I without Discordance F i (u i (x i ) - u i (y i )) k i -q 0 u i (x i ) - u i (y i ) -sk i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

50 F i (u i (x i ) - u i (y i )) k i -v -q 0 u i (x i ) - u i (y i ) -sk i -M Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

51 F i (u i (x i ) - u i (y i )) k i -v -q 0 u i (x i ) - u i (y i ) -sk i "Sophisticated Discordance" Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

52 TACTIC without Discordance F i (u i (x i ) - u i (y i )) rk i -q 0 +q u i (x i ) - u i (y i ) -k i Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

53 TACTIC with Discordance F i (u i (x i ) - u i (y i )) rk i -v -q 0 +q u i (x i ) - u i (y i ) -k i -M Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

54 Intra-Attribute Decomposability n Idea: Use previous theorems and find conditions such that p ( x, y ) = j ( u ( x ), u ( y )) i i i i i i i i where j is non decreasing in 1st and non increasing in 2nd i n Coming close to the additive difference model without implying subtractivity n Use of general theorem on numerical representation of valued relations (Doignon et al 1986) (generated by p i ) n Difficulty = Irregularity of the representations in the previous models Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

55 n (D) n (D1) n (D2) n (D12) Models p ( x, y ) = j ( u ( x ), v ( y )) i i i i i i i i (D) with j i non decreasing in 1st argument (D) with j i non increasing in 2nd argument (D1) and (D2) n (D12=) (D12) with u º v Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

56 Preliminary Results n Model (D) is trivial (under mild cardinality assumptions) n With (M2 ), (M3) and (M3 ) (D1)Û (D2)Û (D12) Û (D12=) n Seven models of interest (M1 ) with (D1), (D2), (D12) and (D12=) (M2 ) with (D12=) (M3) with (D12=) (M3 ) with (D12=) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

57 Axioms AC1 i AC2 i ( xi, a -i) ( yi, b-i) ü ï if andý Þ ( zi, c -i) ( wi, d-i) ï þ ( xi, a -i) ( yi, b-i) ü ï if andý Þ ( zi, c -i) ( wi, d-i) ï þ ì( zi, a -i) ( yi, b-i) ï íor ï î( xi, c -i) ( wi, d-i) ì( xi, a -i) ( wi, b-i) ï íor ï î( zi, c -i) ( yi, d-i) ( xi, a -i) ( yi, b-i) ü ì( wi, a -i) ( yi, b-i) ï ï AC3 i if andý Þ í or ( zi, c -i) ( xi, d-i) ï ï þ î( zi, c -i) ( wi, d-i) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

58 Axioms AC1 i Upward Dominance AC2 Downward Dominance Not incompatible i ( xi, a -i) ( yi, b-i) ü ï if andý Þ ( zi, c -i) ( wi, d-i) ï þ ì( zi, a -i) ( yi, b-i) ï íor ï î( xi, c -i) ( wi, d-i) ( xi, a -i) ( yi, b-i) ü ì( xi, a -i) ( wi, b-i) ï ï if andý Þ íor ( zi, c -i) ( wi, d i) ï ï - þ î( zi, c -i) ( yi, d-i) ( xi, a -i) ( yi, b-i) ü ì( wi, a -i) ( yi, b-i) ï ï AC3 i if andý Þ í or ( zi, c -i) ( xi, d-i) ï ï þ î( zi, c -i) ( wi, d-i) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

59 Consequences * * * i i i i i i i i i i i i AC1 Û is right linear Û [ Not( y, z ) ( x, z ) Þ ( x, w ) ( y, w )] * * * i i i i i i i i i i i i AC2 Û is left linear Û [ Not( z, x ) ( z, y ) Þ ( w, y ) ( w, x )] * ** i i i Û i Û i AC1, AC2, AC3 is strongly linear is strongly linear Remark In all Inter-Attribute Models these axioms are independent Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

60 Results - Denumerable case If X is finite or countably infinite n (M1 -D1) iff reflexivity, independence, RC1, AC1, n (M1 -D2) iff reflexivity, independence, RC1, AC2, n (M1 -D12) iff reflexivity, independence, RC1, AC12, n (M1 -D12=) iff reflexivity, independence, RC1, AC123 Non Denumerable case: Add necessary Order Density conditions: OD* and (ODTR* or ODTL* or ODT*) Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

61 Results - Denumerable case If X is finite or countably infinite n (M2 -D12=) iff reflexivity, RC12, AC123 n (M3-D12=) iff completeness, RC12, AC123, n (M3 -D12=) iff completeness, TC, AC123. Non Denumerable case: Add necessary Order Density conditions: OD*, ODT* Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

62 Remarks n Necessary and sufficient conditions n Independent axioms n AC3 i + (AC1 i or AC2 i ) Þ i is a semi order n With all Inter-Attribute Models AC1 i,ac2 i,ac3 i Þ i * defines an homogeneous family of semi orders Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

63 Summary of Results (Denumerable case) (D1) (D2) (D12) (D12=) AC1 AC2 AC12 AC123 (M1 ) ref+indep+rc1 x x x x (M2 ) ref+rc12 x (M3 ) RC12+comp. x (M3 ) TC+comp. x Non denumerable case: add necessary order density conditions Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

64 Extensions: Rough Sets n Greco-Matarazzo-Slowinski: Nontransitive Conjoint Measurement is the theoretical framework for rough approximations of outranking relations Ð formal definition of Òpreference differencesó Ð relation is build through rules combining preference differences Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

65 Extensions: Particular Cases n Nontransitive Decomposable Conjoint Measurement models contain as particular cases : Ð MCDM methods aiming at building a crisp preference relation (MAUT, ELECTRE I, TACTIC, etc.) Ð rules of thumb put forward by experimental psychologists È lexicographic semiorder È conjunctive È disjunctive È etc. Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

66 Extensions: Valued relations n Non Transitive Decomposable models can easily be extended to the valued case f( x, y) = F( p ( x, y ), p ( x, y ), ¼, p ( x, y ) a x y Û f( x, y) ³ a n Example: PROMETHEE n f( x, y) = F ( u ( x ) - u ( y )) å i = 1 i i i i i n n n Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

67 Sample Model for (ordinal) Valued Relations ( ) a aîa x ay Û F( p1( x1, y1), p2( x2, y2), ¼, pn( xn, yn) ³ with F nondecreasing, p ( x, y ) = 0 i i i a Similar models corresponding to (M2), (M3) (M3 ) + D12= Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

68 Sample Result on valued relations n The preceding models obtains when X and A are finite iff ( a ) aîa is a nonincresaing family of independent relations (A is a well - ordered set of indices) satisfying RC aa ( xi, a -i) a( yi, b-i) üï zi a i a wi b i RCaa andý Þ ì ï (, - ) (, - ) íor ( zi, c - i) a ( wi, d-i) þï îï ( xi, c - i) a ( yi, d-i) for all i Î{1,2, ¼,n} and all a, a ÎA n Add Order Density when X is not denumerable + Lower semi-continuity when A is not finite. Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

69 Extensions n (Dis)similarity indices s( x, y) = s( y, x) = F( p ( x, y ), p ( x, y ), ¼, p ( x, y ) n Perny (1998) : Filtering by indifference n n n Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

70 Summary Non Transitive Decomposable models n imply substantive requirements on n may be axiomatised in a simple way avoiding the use of a denumerable number of conditions in the finite case and of unnecessary structural assumptions in the infinite case n allow to study the Òpure consequencesó of cancellation conditions in the absence of transitivity, completeness and structural requirements on X n are sufficiently general to include as particular cases most aggregation rules that have been proposed in the literature n provide insights on the links and differences between methods Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot

71 Future Work n Many technical Open Problems Ð Additive non transitive Ð Rich Structure Ð Intra-Attribute Decomposability and increasingness Ð Other interesting models? Ð Specific form of F (Min, Max, OWA, etc.) n Aggregation Theory of Homogeneous families of semi orders Ð valued relations Ð difference measurement Ð conjoint measurement n ÒModel-freeÓ tests of MCDM: RC12, AC123 Conjoint Measurement without Additivity and Transitivity - Bouyssou/Pirlot