Rankings and Tournaments: A new approach Julio Gonz alez-D az - PowerPoint PPT Presentation

Motivation The model Ranking Methods Our contribution Example: Ranking Scientific Journals (cont) a ij � r ∞ := r ∞ � i j k a kj j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Stochastic interpretation This is the idea of the invariant method (Pinski and Marin, 1976) The invariant method is the core of Google’s PageRank method (Page et al., 1998) Characterized axiomatically by Palacios-Huerta and Volij (2004) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 5/24

Motivation The model Ranking Methods Our contribution Outline Motivation 1 The model 2 Ranking Methods 3 Our contribution 4 Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 6/24

Motivation The model Ranking Methods Our contribution Primitives Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) The result of each individual match is a pair ( b 1 , b 2 ) with b 1 ≥ 0 , b 2 ≥ 0 , b 1 + b 2 = 1 Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) The result of each individual match is a pair ( b 1 , b 2 ) with b 1 ≥ 0 , b 2 ≥ 0 , b 1 + b 2 = 1 a ij := “number of points achieved by i against j ” Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) The result of each individual match is a pair ( b 1 , b 2 ) with b 1 ≥ 0 , b 2 ≥ 0 , b 1 + b 2 = 1 a ij := “number of points achieved by i against j ” Should we use the invariant method? Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) The result of each individual match is a pair ( b 1 , b 2 ) with b 1 ≥ 0 , b 2 ≥ 0 , b 1 + b 2 = 1 a ij := “number of points achieved by i against j ” Should we use the invariant method? NO Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) The result of each individual match is a pair ( b 1 , b 2 ) with b 1 ≥ 0 , b 2 ≥ 0 , b 1 + b 2 = 1 a ij := “number of points achieved by i against j ” Should we use the invariant method? NO: match, victory, loss Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) The result of each individual match is a pair ( b 1 , b 2 ) with b 1 ≥ 0 , b 2 ≥ 0 , b 1 + b 2 = 1 a ij := “number of points achieved by i against j ” Should we use the invariant method? NO: match, victory, loss Before, it was not bad to cite another journal. Now, this represents a loss Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Primitives A tournament is given by: A set of n players (denoted by N ) The pairwise results of a number of matches among them (contained in an n × n matrix A ) The result of each individual match is a pair ( b 1 , b 2 ) with b 1 ≥ 0 , b 2 ≥ 0 , b 1 + b 2 = 1 a ij := “number of points achieved by i against j ” Should we use the invariant method? NO: match, victory, loss Before, it was not bad to cite another journal. Now, this represents a loss a ij k a kj = “percentage of the points lost by j that were against i ” � Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations Assumptions Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations M := A + A t = “matches matrix” Assumptions Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations M := A + A t = “matches matrix” m ij = “total number of matches between i and j ” Assumptions Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations M := A + A t = “matches matrix” m ij = “total number of matches between i and j ” � j a ij s i := j m ij = “average score of player i ” � Assumptions Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations M := A + A t = “matches matrix” m ij = “total number of matches between i and j ” � j a ij s i := j m ij = “average score of player i ” � Assumptions A is nonnegative Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations M := A + A t = “matches matrix” m ij = “total number of matches between i and j ” � j a ij s i := j m ij = “average score of player i ” � Assumptions A is nonnegative a ii = 0 Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations M := A + A t = “matches matrix” m ij = “total number of matches between i and j ” � j a ij s i := j m ij = “average score of player i ” � Assumptions A is nonnegative a ii = 0 M is irreducible (no incomparable sub-tournaments) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Assumptions Extra notations M := A + A t = “matches matrix” m ij = “total number of matches between i and j ” � j a ij s i := j m ij = “average score of player i ” � Assumptions A is nonnegative a ii = 0 M is irreducible (no incomparable sub-tournaments) 0 < s i < 1 Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

Motivation The model Ranking Methods Our contribution Outline Motivation 1 The model 2 Ranking Methods 3 Our contribution 4 Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 9/24

Motivation The model Ranking Methods Our contribution Tournaments Ranking methods for tournaments Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

Motivation The model Ranking Methods Our contribution Tournaments Ranking methods for tournaments Scores ranking (axiomatized by Rubinstein, 1980) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

Motivation The model Ranking Methods Our contribution Tournaments Ranking methods for tournaments Scores ranking (axiomatized by Rubinstein, 1980) Fair bets (Pinsky and Narin, 1976; axiomatized in Sluztky and Volij, 2005 and 2006) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

Motivation The model Ranking Methods Our contribution Tournaments Ranking methods for tournaments Scores ranking (axiomatized by Rubinstein, 1980) Fair bets (Pinsky and Narin, 1976; axiomatized in Sluztky and Volij, 2005 and 2006) Maximum likelihood approach (Bradley and Terry, 1952) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

Motivation The model Ranking Methods Our contribution Tournaments Ranking methods for tournaments Scores ranking (axiomatized by Rubinstein, 1980) Fair bets (Pinsky and Narin, 1976; axiomatized in Sluztky and Volij, 2005 and 2006) Maximum likelihood approach (Bradley and Terry, 1952) Recursive performance (this paper) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

Motivation The model Ranking Methods Our contribution Tournaments Ranking methods for tournaments Scores ranking (axiomatized by Rubinstein, 1980) Fair bets (Pinsky and Narin, 1976; axiomatized in Sluztky and Volij, 2005 and 2006) Maximum likelihood approach (Bradley and Terry, 1952) Recursive performance (this paper) Examples Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

Motivation The model Ranking Methods Our contribution Tournaments Ranking methods for tournaments Scores ranking (axiomatized by Rubinstein, 1980) Fair bets (Pinsky and Narin, 1976; axiomatized in Sluztky and Volij, 2005 and 2006) Maximum likelihood approach (Bradley and Terry, 1952) Recursive performance (this paper) Examples Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Responsiveness with respect to the beating relation Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Responsiveness with respect to the beating relation Independence of irrelevant matches Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Responsiveness with respect to the beating relation Independence of irrelevant matches !?!? Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Responsiveness with respect to the beating relation Independence of irrelevant matches !?!? Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Responsiveness with respect to the beating relation Independence of irrelevant matches !?!? Problems Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Responsiveness with respect to the beating relation Independence of irrelevant matches !?!? Problems Many ties Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Scores Ranking The scores ranking Rank the players according to the vector s Characterization for Round Robin (Rubinstein, 1980) Anonymity Responsiveness with respect to the beating relation Independence of irrelevant matches !?!? Problems Many ties Only makes sense for Round-Robin tournaments (because of IIA). Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 11/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method a ij k a ki = “points of i against j relative to i ’s total number of losses” � Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method a ij k a ki = “points of i against j relative to i ’s total number of losses” � Initially, we can regard all players as equally strong: r 0 := (1 , . . . , 1) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method a ij k a ki = “points of i against j relative to i ’s total number of losses” � Initially, we can regard all players as equally strong: r 0 := (1 , . . . , 1) a ij � r 1 i := � k a ki j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method a ij k a ki = “points of i against j relative to i ’s total number of losses” � Initially, we can regard all players as equally strong: r 0 := (1 , . . . , 1) a ij a ij � � r 1 r 0 i := = � � j k a ki k a ki j j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method a ij k a ki = “points of i against j relative to i ’s total number of losses” � Initially, we can regard all players as equally strong: r 0 := (1 , . . . , 1) a ij a ij � � r 1 r 0 i := = “ratio victories/losses of i ” � � j k a ki k a ki j j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method a ij k a ki = “points of i against j relative to i ’s total number of losses” � Initially, we can regard all players as equally strong: r 0 := (1 , . . . , 1) a ij a ij � � r 1 r 0 i := = “ratio victories/losses of i ” � � j k a ki k a ki j j a ij � r 2 r 1 i := � j k a ki j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method � a ij Invariant method: r ∞ k a kj r ∞ := � i j j The invariant method rewards victories without punishing for losses The fair-bets method a ij k a ki = “points of i against j relative to i ’s total number of losses” � Initially, we can regard all players as equally strong: r 0 := (1 , . . . , 1) a ij a ij � � r 1 r 0 i := = “ratio victories/losses of i ” � � j k a ki k a ki j j a ij � “victories against stronger opponents have more weight” r 2 r 1 i := � j “all losses have the same weight” k a ki j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Bets’ interpretation Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Bets’ interpretation Characterization (Sluzki and Volij, 2005) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Bets’ interpretation Characterization (Sluzki and Volij, 2005) Responsiveness with respect to the beating relation Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Bets’ interpretation Characterization (Sluzki and Volij, 2005) Responsiveness with respect to the beating relation Anonymity Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Bets’ interpretation Characterization (Sluzki and Volij, 2005) Responsiveness with respect to the beating relation Anonymity Quasi-flatness preservation Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j r ∞ is just the solution of a linear system of equations The ranking induced by r ∞ is independent of r 0 Bets’ interpretation Characterization (Sluzki and Volij, 2005) Responsiveness with respect to the beating relation Anonymity Quasi-flatness preservation Negative responsiveness to losses Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j Problem Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j Problem Asymmetric treatment of victories with respect to losses (because of negative responsiveness to losses) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j Problem Asymmetric treatment of victories with respect to losses (because of negative responsiveness to losses) Violates the axiom: Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

Motivation The model Ranking Methods Our contribution The Fair-Bets Method a ij � r ∞ := r ∞ � i j k a ki j Problem Asymmetric treatment of victories with respect to losses (because of negative responsiveness to losses) Violates the axiom: The ranking proposed for A is the inverse of the one proposed for A t Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

Motivation The model Ranking Methods Our contribution The Maximum Likelihood Approach Paired comparison analysis (Statistics) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

Motivation The model Ranking Methods Our contribution The Maximum Likelihood Approach Paired comparison analysis (Statistics) Completely different approach Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

Motivation The model Ranking Methods Our contribution The Maximum Likelihood Approach Paired comparison analysis (Statistics) Completely different approach Assume that there is a distribution function F such that the expected score of a player with strength r i in a match against a player with strength r j is given by F ( r i − r j ) . Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

Motivation The model Ranking Methods Our contribution The Maximum Likelihood Approach Paired comparison analysis (Statistics) Completely different approach Assume that there is a distribution function F such that the expected score of a player with strength r i in a match against a player with strength r j is given by F ( r i − r j ) . The function F is called rating function Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

Motivation The model Ranking Methods Our contribution The Maximum Likelihood Approach Paired comparison analysis (Statistics) Completely different approach Assume that there is a distribution function F such that the expected score of a player with strength r i in a match against a player with strength r j is given by F ( r i − r j ) . The function F is called rating function Bradley and Terry (1952) took the (standard) logistic distribution: e ri F ( r i − r j ) = e ri + e rj Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24