Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion An axiomatic characterization of the prudent order preference function Claude Lamboray SMA - University of Luxembourg CODE - Universit´ e Libre de Bruxelles DIMACS-LAMSADE Workshop on Voting Theory and Preference Modelling October 27, 2006 1 / 27
Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion The context Combination of a profile of linear orders into a set of prudent orders (Arrow and Raynaud, 1986). Minimize the strongest opposition 2 / 27
Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Why characterizing the set of prudent orders? To discover the particularities of the prudence principle. To build a common axiomatic framework for other prudent ranking rules (e.g. Ranked Pairs Rule, Tideman, 1987). To link the results with other ordinal ranking rules (Barbera, 1988, Fortemps and Pirlot, 2004,...). 3 / 27
Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Outline Prudent orders 1 Definitions Example Link with the majority relation Characterization of the Prudent Order Preference Function 2 Axioms Results Extended Prudent Order Preference Function 3 Definition Result Conclusion 4 4 / 27
Prudent orders Definitions Characterization of the Prudent Order Preference Function Example Extended Prudent Order Preference Function Link with the majority relation Conclusion Overview Prudent orders 1 Definitions Example Link with the majority relation Characterization of the Prudent Order Preference Function 2 Axioms Results Extended Prudent Order Preference Function 3 Definition Result Conclusion 4 5 / 27
Prudent orders Definitions Characterization of the Prudent Order Preference Function Example Extended Prudent Order Preference Function Link with the majority relation Conclusion Prudent Orders Basic Notations A profile of q linear orders: u = ( O 1 , O 2 , . . . , O q ) Majority Margins: B ij = { # k : ( a i , a j ) ∈ O k } − { # k : ( a j , a i ) ∈ O k } Cut-Relation: ( a i , a j ) ∈ R >λ ⇐ ⇒ B ij > λ, where λ ∈ {− q , . . . 0 , . . . , q } . 6 / 27
Prudent orders Definitions Characterization of the Prudent Order Preference Function Example Extended Prudent Order Preference Function Link with the majority relation Conclusion Let β be the smallest value such that the corresponding strict cut relation is acyclic: β = min { λ ∈ {− q , . . . , O , . . . , q } : R >λ is acyclic } . Definition of the prudent order preference function PO ( u ) = { O ∈ LO : R >β ⊆ O } = E ( R >β ) . 7 / 27
Prudent orders Definitions Characterization of the Prudent Order Preference Function Example Extended Prudent Order Preference Function Link with the majority relation Conclusion PO ( u ) = E ( R >β ) is a non empty set of linear orders !! Proposition For any acyclic relation R , there exists a profile u of linear orders such that PO ( u ) = E ( R ). Proposition The following statements are equivalent: O P is a prudent order O P ∈ arg min O ∈LO max ( a i , a j ) �∈ O B ij (minimize the strongest opposition) O P ∈ arg max O ∈LO min ( a i , a j ) ∈ O B ij (maximize the weakest link) 8 / 27
Prudent orders Definitions Characterization of the Prudent Order Preference Function Example Extended Prudent Order Preference Function Link with the majority relation Conclusion Example u = ( abcde , adbec , adbec , cdbea , cdbae , bcdae , ecdba ) a b c d e a . -1 -1 -1 3 b 1 . 1 -1 5 1 acbde 7 cadbe c 1 -1 . 3 1 2 abcde 8 bcade d 1 1 -3 . 5 3 9 cabde cdabe e -3 -5 -1 -5 . 4 acdbe 10 cbdae β = 1 5 11 cbade cdbae R >β = R > 1 = 6 bacde 12 bcdae { ( a , e ) , ( b , e ) , ( c , d ) , ( d , e ) } . (but B ( c , d ) ≥ 1 and B ( d , b ) ≥ 1 and B ( b , e ) ≥ 1 !) 9 / 27
Prudent orders Definitions Characterization of the Prudent Order Preference Function Example Extended Prudent Order Preference Function Link with the majority relation Conclusion Link with the majority relation M is the strict majority relation : ( a i , a j ) ∈ M ⇐ ⇒ B ij > 0 . Let u be a given profile. Let O be any linear order. k O : the smallest number of times that one has to add O to u such that M ( u + k O O ) = O Let k MIN = min O ∈LO k O . Theorem (Debord, 1986) Let u be a profile such that the strict majority relation is not a linear order. O ∈ PO ( u ) if and only if M ( u + k min O ) = O . 10 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Overview Prudent orders 1 Definitions Example Link with the majority relation Characterization of the Prudent Order Preference Function 2 Axioms Results Extended Prudent Order Preference Function 3 Definition Result Conclusion 4 11 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Characterization of the Prudent Order Preference Function A preference function f : O q f : �→ P ( O ) \ ∅ u → f ( u ). 12 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Condorcet Consistency (CC) If M is acyclic, then: f ( u ) ⊆ E ( M ) Strong Condorcet Consistency (SCC) If M is acyclic, then: f ( u ) = E ( M ) SCC implies CC. 13 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion E-invariance (EI) Let u E be a profile such that B ij = 0 ∀ i , j . Then: f ( u + u E ) = f ( u ) Weak homogeneity (WH) If q is odd, then: f ( u ) ⊆ f ( u + u ) Homogeneity (H) If q is odd, then: f ( u ) = f ( u + u ) H implies WH 14 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Definition Let us consider a linear order O and an ordered pair ( a i , a j ). We say that the linear order O ′ is an update of O in favor of pair ( a i , a j ) if O = ( ... a j a i ... ) is such that a j directly precedes a i and O ′ = ( ... a i a j ... ) is obtained by reversing a j and a i in O . Example: Update in favor of ( b , a ): → O ′ = bacde . O = abcde − 15 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Update procedure Let u be a profile. For every pair { a i , a j } do the following: If B ij > 0, then we do an update in favor of pair ( a i , a j ) of a linear order O k of profile u . If B ij = 0, then Do nothing. OR We do an update in favor of pair ( a i , a j ) of a linear order O k of profile u . OR We do an update in favor of pair ( a j , a i ) of a linear order O k of profile u . Let u update be the profile obtained at the end of this procedure 16 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Example u update u update in favor of ( a , b ) O 1 dcba dcab O 2 cabd ( a , c ) acbd O 3 cbda ( b , c ) bcda O 4 dbac ( b , d ) bdac ( c , d ) O 5 dcab cdab O 6 adbc ( d , a ) dabc O 7 abcd n.a. abcd n.a O 8 abcd abcd n.a O 9 abcd abcd O 10 bcda n.a bcda O 11 cbda n.a cbda n.a O 12 dabc dabc O 13 dabc n.a dabc O 14 cbda n.a cbda O 15 bacd n.a. bacd B ( a , b ) = 1; B ( a , c ) = 1; B ( b , c ) = 3; B ( b , d ) = 3; B ( c , d ) = 3; B ( d , a ) = 3 17 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Majority Oriented Profile Convergence (MOPC) Let u be a profile and let u update be the profile obtained using the majority oriented update procedure. Then: f ( u update ) ⊆ f ( u ) . Majority Oriented Profile Invariance (MOPI) Let u be a profile and let u update be the profile obtained using the majority oriented update procedure. If the strict majority relation of u update contains at least one cycle, then: f ( u update ) = f ( u ) . MOPC does not imply MOPI MOPI does not imply MOPC 18 / 27
Prudent orders Characterization of the Prudent Order Preference Function Axioms Extended Prudent Order Preference Function Results Conclusion Theorem The prudent order preference function is the largest preference function (in the sense of the inclusion) which verifies Condorcet Consistency, E-Invariance, Weak Homogeneity and Majority Oriented Profile Convergence. Theorem The prudent order preference function is the only preference function which verifies Strong Condorcet Consistency, E-Invariance, Homogeneity and Majority Oriented Profile Invariance. 19 / 27
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