Consensus measures generated by weighted Kemeny distances on linear - PowerPoint PPT Presentation

Consensus measures generated by weighted Kemeny distances on linear orders e Luis GARC´ David P´ EREZ-ROM´ Jos´ IA-LAPRESTA AN PRESAD Research Group University of Valladolid, Spain COMSOC 2010, (D¨ usseldorf, Germany) September 13, 2010 Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 1 / 22

Introduction Preliminaries Motivation Each member of a committee arranges a set of alternatives by means of a linear order How similar are their opinions? Could we measure consensus? Bosch (2005) introduced the notion of consensus measures in the context of linear orders Garc´ ıa-Lapresta and P´ erez-Rom´ an (2008) extended Bosch’s concept to the context of weak orders Alcalde-Unzu and Vorsatz (2010) have introduced some consensus measures in the context of linear orders (related to some rank correlation indices) Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 2 / 22

Introduction Preliminaries Proposals Since in some decision problems it is not the same to have differences in the top alternatives than in the bottom ones, we introduce weights for distinguishing where these differences occur We consider a class of consensus measures generated by weighted Kemeny distances We analyze some of their properties Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 3 / 22

Introduction Preliminaries Notation V = { v 1 , . . . , v m } set of voters m ≥ 3 X = { x 1 , . . . , x n } set of alternatives n ≥ 3 L ( X ) the set of linear orders on X → R − 1 inverse of R R ∈ L ( X ) �− x i R − 1 x j ⇐ ⇒ x j R x i A profile is a vector R = ( R 1 , . . . , R m ) of linear orders Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 4 / 22

Introduction Preliminaries Codification of linear orders Given R ∈ L ( X ) , o R : X − → { 1 , . . . , n } defines the position of each alternative in R o R = ( o R ( x 1 ) , . . . , o R ( x n )) x 2 x 3 ≡ (3 , 1 , 2 , 4) x 1 x 4 We can identify L ( X ) with S n (the set of permutations on { 1 , . . . , n } ) Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 5 / 22

Distances and Metrics Distance Distance A distance on a set A � = ∅ is a mapping d : A × A − → R satisfying the following conditions for all a, b ∈ A : d ( a, b ) ≥ 0 (non-negativity) 1 d ( a, b ) = d ( b, a ) (symmetry) 2 d ( a, a ) = 0 (reflexivity) 3 If d satisfies the following additional conditions for all a, b ∈ A : d ( a, b ) = 0 ⇔ a = b (identity of indescernibles) 4 d ( a, b ) ≤ d ( a, c ) + d ( c, b ) (triangle inequality) 5 then we say that d is a metric M.M. Deza, E. Deza. Encyclopedia of Distances . Springer-Verlag, 2009 Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 6 / 22

Distances and Metrics Distance Distance Let A ⊆ R n be stable under permutations, i.e., ( a σ 1 , . . . , a σ n ) ∈ A for all ( a 1 , . . . , a n ) ∈ A and σ ∈ S n A distance (metric) d : A × A − → R is neutral if for every σ ∈ S n , it holds d (( a σ 1 , . . . , a σ n ) , ( b σ 1 , . . . , b σ n )) = d (( a 1 , . . . , a n ) , ( b 1 , . . . , b n )) , for all ( a 1 , . . . , a n ) , ( b 1 , . . . , b n ) ∈ A Typical examples of metrics on R n as discrete, Manhattan, Euclidean, Chebyshev and cosine are neutral Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 7 / 22

Distances and Metrics Distance Distance on linear orders Given A ⊆ R n such that S n ⊆ A and a distance (metric) d : A × A − → R , the distance ( metric ) on L ( X ) induced by d is the mapping ¯ d : L ( X ) × L ( X ) − → R defined by ¯ � � d ( R 1 , R 2 ) = d ( o R 1 ( x 1 ) , . . . , o R 1 ( x n )) , ( o R 2 ( x 1 ) , . . . , o R 2 ( x n )) , for all R 1 , R 2 ∈ L ( X ) Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 8 / 22

Distances and Metrics Distance Kemeny metric The Kemeny metric on L ( X ) is the mapping d K : L ( X ) × L ( X ) − → R defined as the cardinality of the symmetric difference between the linear orders. This metric coincides with the metric on L ( X ) induced by the distance d K d K ( R 1 , R 2 ) = ¯ � � d K ( R 1 , R 2 ) = d K ( a 1 , . . . , a n ) , ( b 1 , . . . , b n ) = n � | sgn ( a i − a j ) − sgn ( b i − b j ) | i,j =1 i<j ( a 1 , . . . , a n ) ≡ R 1 ∈ L ( X ) and ( b 1 , . . . , b n ) ≡ R 2 ∈ L ( X ) Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 9 / 22

Distances and Metrics Distance Example Consider four decision makers that rank order the four alternatives of the set X = { x 1 , x 2 , x 3 , x 4 } through the following linear orders and the corresponding codification vectors R 1 R 2 R 3 R 4 R 1 R 2 R 3 R 4 x 1 x 1 x 2 x 2 1 1 2 4 x 2 x 2 x 1 x 4 2 2 1 1 x 3 x 4 x 3 x 3 3 4 3 3 x 4 x 3 x 4 x 1 4 3 4 2 Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 10 / 22

Distances and Metrics Distance Example Consider four decision makers that rank order the four alternatives of the set X = { x 1 , x 2 , x 3 , x 4 } through the following linear orders and the corresponding codification vectors R 1 R 2 R 3 R 4 R 1 R 2 R 3 R 4 x 1 x 1 x 2 x 2 1 1 2 4 x 2 x 2 x 1 x 4 2 2 1 1 x 3 x 4 x 3 x 3 3 4 3 3 x 4 x 3 x 4 x 1 4 3 4 2 d K ( R 1 , R 2 ) = ¯ d K ( R 1 , R 2 ) = d K ((1 , 2 , 3 , 4) , (1 , 2 , 4 , 3)) = 2 | − 1 − ( − 1) | + | − 1 − ( − 1) | + | − 1 − ( − 1) | + + | − 1 − ( − 1) | + | − 1 − ( − 1) | + + | − 1 − 1 | Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 10 / 22

Distances and Metrics Distance Example Consider four decision makers that rank order the four alternatives of the set X = { x 1 , x 2 , x 3 , x 4 } through the following linear orders and the corresponding codification vectors R 1 R 2 R 3 R 4 R 1 R 2 R 3 R 4 x 1 x 1 x 2 x 2 1 1 2 4 x 2 x 2 x 1 x 4 2 2 1 1 x 3 x 4 x 3 x 3 3 4 3 3 x 4 x 3 x 4 x 1 4 3 4 2 d K ( R 1 , R 3 ) = ¯ d K ( R 1 , R 3 ) = d K ((1 , 2 , 3 , 4) , (2 , 1 , 3 , 4)) = 2 | − 1 − 1 | + | − 1 − ( − 1) | + | − 1 − ( − 1) | + + | − 1 − ( − 1) | + | − 1 − ( − 1) | + + | − 1 − ( − 1) | Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 10 / 22

Distances and Metrics Distance Weighted Kemeny distances Let w = ( w 1 , . . . , w n − 1 ) ∈ [0 , 1] n − 1 be a weighting vector such that w 1 ≥ · · · ≥ w n − 1 and � n − 1 i =1 w i = 1 . The weighted Kemeny distance on L ( X ) associated with w is the mapping ¯ d K, w : L ( X ) × L ( X ) − → R defined by  n 1 ¯ � � sgn � � a σ 1 − a σ 1 � � b σ 1 − b σ 1 �� d K, w ( R 1 , R 2 ) = − sgn w i  � i j i j 2  i,j =1 i<j  n � � sgn b σ 2 − b σ 2 a σ 2 − a σ 2 � � � � �� + w i − sgn  ,  � i j i j i,j =1 i<j where ( a 1 , . . . , a n ) ≡ R 1 ∈ L ( X ) , ( b 1 , . . . , b n ) ≡ R 2 ∈ L ( X ) and σ 1 , σ 2 ∈ S n are such that R σ 1 = R σ 2 ≡ (1 , 2 , . . . , n ) 1 2 Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 11 / 22

Distances and Metrics Distance Example Consider two linear orders, their corresponding codification vectors, the permutations σ 1 = (3 , 1 , 2 , 4) and σ 2 = (3 , 2 , 4 , 1) and a weighting vector w = ( w 1 , w 2 , w 3 ) R σ 1 R σ 1 R σ 2 R σ 2 R 1 R 2 R 1 R 2 1 2 2 1 x 3 x 3 2 4 1 1 1 1 x 1 x 2 3 2 2 4 2 3 x 2 x 4 1 1 3 2 3 4 x 4 x 1 4 3 4 3 4 2 1 ¯ � � � � � � �� d K, w ( R 1 , R 2 ) = w 2 | − 1 − 1 | + | − 1 − 1 | + w 2 | − 1 − 1 | + w 3 | − 1 − 1 | 2 = 3 w 2 + w 3 Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 12 / 22

Distances and Metrics Distance Example Consider two linear orders, their corresponding codification vectors, the permutations σ 1 = (3 , 1 , 2 , 4) and σ 2 = (3 , 2 , 4 , 1) and a weighting vector w = ( w 1 , w 2 , w 3 ) R σ 1 R σ 1 R σ 2 R σ 2 R 1 R 2 R 1 R 2 1 2 2 1 x 3 x 3 2 4 1 1 1 1 x 1 x 2 3 2 2 4 2 3 x 2 x 4 1 1 3 2 3 4 x 4 x 1 4 3 4 3 4 2 1 ¯ � � � � � � �� d K, w ( R 1 , R 2 ) = w 2 | − 1 − 1 | + | − 1 − 1 | + w 2 | − 1 − 1 | + w 3 | − 1 − 1 | 2 = 3 w 2 + w 3 Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 12 / 22

Distances and Metrics Distance Example Consider two linear orders, their corresponding codification vectors, the permutations σ 1 = (3 , 1 , 2 , 4) and σ 2 = (3 , 2 , 4 , 1) and a weighting vector w = ( w 1 , w 2 , w 3 ) R σ 1 R σ 1 R σ 2 R σ 2 R 1 R 2 R 1 R 2 1 2 2 1 x 3 x 3 2 4 1 1 1 1 x 1 x 2 3 2 2 4 2 3 x 2 x 4 1 1 3 2 3 4 x 4 x 1 4 3 4 3 4 2 1 ¯ � � � � � � �� d K, w ( R 1 , R 2 ) = w 2 | − 1 − 1 | + | − 1 − 1 | + w 2 | − 1 − 1 | + w 3 | − 1 − 1 | 2 = 3 w 2 + w 3 Garc´ ıa-Lapresta, P´ erez-Rom´ an Weighted Kemeny distances on linear orders 12 / 22