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

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

Oct 16, 2022 270 likes •1.94k views

Rankings and Tournaments: A new approach Julio Gonz alez-D az Kellogg School of Management (CMS-EMS) Northwestern University and Research Group in Economic Analysis Universidad de Vigo ........................ (joint with Miguel

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SLIDE 1

Rankings and Tournaments: A new approach

Julio Gonz´ alez-D´ ıaz

Kellogg School of Management (CMS-EMS) Northwestern University and Research Group in Economic Analysis Universidad de Vigo ........................ (joint with Miguel Brozos-V´ azquez, Marco Antonio Campo-Cabana, and Jos´ e Carlos D´ ıaz-Ramos)

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SLIDE 2

Motivation The model Ranking Methods Our contribution

Outline

1

Motivation

2

The model

3

Ranking Methods

4

Our contribution

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 1/24

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SLIDE 3

Motivation The model Ranking Methods Our contribution

Outline

1

Motivation

2

The model

3

Ranking Methods

4

Our contribution

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 2/24

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SLIDE 4

Motivation The model Ranking Methods Our contribution

Main Goal

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 5

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 6

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament Rating vs. Ranking

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 7

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament Rating vs. Ranking

Applications

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 8

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament Rating vs. Ranking

Applications

Rankings of web pages: Google

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 9

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament Rating vs. Ranking

Applications

Rankings of web pages: Google Rankings of scientific journals

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 10

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament Rating vs. Ranking

Applications

Rankings of web pages: Google Rankings of scientific journals Rankings in sports

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 11

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament Rating vs. Ranking

Applications

Rankings of web pages: Google Rankings of scientific journals Rankings in sports Ranking social alternatives

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 12

Motivation The model Ranking Methods Our contribution

Main Goal

Rank the participants in a tournament Rating vs. Ranking

Applications

Rankings of web pages: Google Rankings of scientific journals Rankings in sports Ranking social alternatives Ranking candidates in labor markets

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 3/24

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SLIDE 13

Motivation The model Ranking Methods Our contribution

Example

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 14

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 15

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 16

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 17

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

r1

i :=

  • j

aij

  • k akj

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 18

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

Initially, we can regard all journals as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k akj

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 19

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

Initially, we can regard all journals as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k akj

=

  • j

aij

  • k akj

r0

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 20

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

Initially, we can regard all journals as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k akj

=

  • j

aij

  • k akj

r0

j

The ratings r1 are a better indicator of the real strength of the journals than r0

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 21

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

Initially, we can regard all journals as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k akj

=

  • j

aij

  • k akj

r0

j

The ratings r1 are a better indicator of the real strength of the journals than r0

r2

i :=

  • j

aij

  • k akj

r1

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 22

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

Initially, we can regard all journals as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k akj

=

  • j

aij

  • k akj

r0

j

The ratings r1 are a better indicator of the real strength of the journals than r0

r2

i :=

  • j

aij

  • k akj

r1

j

. . . . . .

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 23

Motivation The model Ranking Methods Our contribution

Example

Ranking Scientific Journals

aij := “number of citations received by i from j”

aij

  • k akj = “percentage of the citations made by j received by i”

Initially, we can regard all journals as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k akj

=

  • j

aij

  • k akj

r0

j

The ratings r1 are a better indicator of the real strength of the journals than r0

r2

i :=

  • j

aij

  • k akj

r1

j

. . . . . . r∞

i

:=

  • j

aij

  • k akj

r∞

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 4/24

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SLIDE 24

Motivation The model Ranking Methods Our contribution

Example: Ranking Scientific Journals (cont)

r∞

i

:=

  • j

aij

  • k akj

r∞

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 5/24

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SLIDE 25

Motivation The model Ranking Methods Our contribution

Example: Ranking Scientific Journals (cont)

r∞

i

:=

  • j

aij

  • k akj

r∞

j

r∞ is just the solution of a linear system of equations

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 5/24

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SLIDE 26

Motivation The model Ranking Methods Our contribution

Example: Ranking Scientific Journals (cont)

r∞

i

:=

  • j

aij

  • k akj

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 5/24

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SLIDE 27

Motivation The model Ranking Methods Our contribution

Example: Ranking Scientific Journals (cont)

r∞

i

:=

  • j

aij

  • k akj

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 Stochastic interpretation

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 5/24

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SLIDE 28

Motivation The model Ranking Methods Our contribution

Example: Ranking Scientific Journals (cont)

r∞

i

:=

  • j

aij

  • k akj

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 Stochastic interpretation

This is the idea of the invariant method (Pinski and Marin, 1976)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 5/24

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SLIDE 29

Motivation The model Ranking Methods Our contribution

Example: Ranking Scientific Journals (cont)

r∞

i

:=

  • j

aij

  • k akj

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 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)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 5/24

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SLIDE 30

Motivation The model Ranking Methods Our contribution

Example: Ranking Scientific Journals (cont)

r∞

i

:=

  • j

aij

  • k akj

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 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

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SLIDE 31

Motivation The model Ranking Methods Our contribution

Outline

1

Motivation

2

The model

3

Ranking Methods

4

Our contribution

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 6/24

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SLIDE 32

Motivation The model Ranking Methods Our contribution

Primitives

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

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SLIDE 33

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

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SLIDE 34

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

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SLIDE 35

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

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SLIDE 36

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

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SLIDE 37

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 (b1, b2) with b1 ≥ 0, b2 ≥ 0, b1 + b2 = 1

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

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SLIDE 38

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 (b1, b2) with b1 ≥ 0, b2 ≥ 0, b1 + b2 = 1

aij := “number of points achieved by i against j”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

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SLIDE 39

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 (b1, b2) with b1 ≥ 0, b2 ≥ 0, b1 + b2 = 1

aij := “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

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SLIDE 40

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 (b1, b2) with b1 ≥ 0, b2 ≥ 0, b1 + b2 = 1

aij := “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

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SLIDE 41

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 (b1, b2) with b1 ≥ 0, b2 ≥ 0, b1 + b2 = 1

aij := “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

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SLIDE 42

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 (b1, b2) with b1 ≥ 0, b2 ≥ 0, b1 + b2 = 1

aij := “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

slide-43
SLIDE 43

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 (b1, b2) with b1 ≥ 0, b2 ≥ 0, b1 + b2 = 1

aij := “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

aij

  • k akj = “percentage of the points lost by j that were against i”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 7/24

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SLIDE 44

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations Assumptions

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

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SLIDE 45

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations

M := A + At = “matches matrix”

Assumptions

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

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SLIDE 46

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations

M := A + At = “matches matrix” mij = “total number of matches between i and j”

Assumptions

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

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SLIDE 47

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations

M := A + At = “matches matrix” mij = “total number of matches between i and j” si :=

  • j aij
  • j mij = “average score of player i”

Assumptions

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

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SLIDE 48

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations

M := A + At = “matches matrix” mij = “total number of matches between i and j” si :=

  • j aij
  • j mij = “average score of player i”

Assumptions

A is nonnegative

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

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SLIDE 49

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations

M := A + At = “matches matrix” mij = “total number of matches between i and j” si :=

  • j aij
  • j mij = “average score of player i”

Assumptions

A is nonnegative aii = 0

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

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SLIDE 50

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations

M := A + At = “matches matrix” mij = “total number of matches between i and j” si :=

  • j aij
  • j mij = “average score of player i”

Assumptions

A is nonnegative aii = 0 M is irreducible (no incomparable sub-tournaments)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

slide-51
SLIDE 51

Motivation The model Ranking Methods Our contribution

Assumptions

Extra notations

M := A + At = “matches matrix” mij = “total number of matches between i and j” si :=

  • j aij
  • j mij = “average score of player i”

Assumptions

A is nonnegative aii = 0 M is irreducible (no incomparable sub-tournaments) 0 < si < 1

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 8/24

slide-52
SLIDE 52

Motivation The model Ranking Methods Our contribution

Outline

1

Motivation

2

The model

3

Ranking Methods

4

Our contribution

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 9/24

slide-53
SLIDE 53

Motivation The model Ranking Methods Our contribution

Tournaments

Ranking methods for tournaments

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 10/24

slide-54
SLIDE 54

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

slide-55
SLIDE 55

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

slide-56
SLIDE 56

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

slide-57
SLIDE 57

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

slide-58
SLIDE 58

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

slide-59
SLIDE 59

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

slide-60
SLIDE 60

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

slide-61
SLIDE 61

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

slide-62
SLIDE 62

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

slide-63
SLIDE 63

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

slide-64
SLIDE 64

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

slide-65
SLIDE 65

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

slide-66
SLIDE 66

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

slide-67
SLIDE 67

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

slide-68
SLIDE 68

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

slide-69
SLIDE 69

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

slide-70
SLIDE 70

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

slide-71
SLIDE 71

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

slide-72
SLIDE 72

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-73
SLIDE 73

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-74
SLIDE 74

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

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

slide-75
SLIDE 75

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

The fair-bets method

aij

  • k aki = “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

slide-76
SLIDE 76

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

The fair-bets method

aij

  • k aki = “points of i against j relative to i’s total number of losses”

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-77
SLIDE 77

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

The fair-bets method

aij

  • k aki = “points of i against j relative to i’s total number of losses”

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k aki

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-78
SLIDE 78

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

The fair-bets method

aij

  • k aki = “points of i against j relative to i’s total number of losses”

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k aki

=

  • j

aij

  • k aki

r0

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-79
SLIDE 79

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

The fair-bets method

aij

  • k aki = “points of i against j relative to i’s total number of losses”

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k aki

=

  • j

aij

  • k aki

r0

j

“ratio victories/losses of i”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-80
SLIDE 80

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

The fair-bets method

aij

  • k aki = “points of i against j relative to i’s total number of losses”

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k aki

=

  • j

aij

  • k aki

r0

j

“ratio victories/losses of i”

r2

i :=

  • j

aij

  • k aki

r1

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-81
SLIDE 81

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

Invariant method: r∞

i

:=

  • j

aij

  • k akj r∞

j

The invariant method rewards victories without punishing for losses

The fair-bets method

aij

  • k aki = “points of i against j relative to i’s total number of losses”

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

aij

  • k aki

=

  • j

aij

  • k aki

r0

j

“ratio victories/losses of i”

r2

i :=

  • j

aij

  • k aki

r1

j

“victories against stronger opponents have more weight” “all losses have the same weight”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 12/24

slide-82
SLIDE 82

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

slide-83
SLIDE 83

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

slide-84
SLIDE 84

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

slide-85
SLIDE 85

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 Bets’ interpretation

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

slide-86
SLIDE 86

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 Bets’ interpretation

Characterization (Sluzki and Volij, 2005)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 13/24

slide-87
SLIDE 87

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 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

slide-88
SLIDE 88

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 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

slide-89
SLIDE 89

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 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

slide-90
SLIDE 90

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 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

slide-91
SLIDE 91

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

slide-92
SLIDE 92

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

j

Problem

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

slide-93
SLIDE 93

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

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

slide-94
SLIDE 94

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

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

slide-95
SLIDE 95

Motivation The model Ranking Methods Our contribution

The Fair-Bets Method

r∞

i

:=

  • j

aij

  • k aki

r∞

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 At

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 14/24

slide-96
SLIDE 96

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

slide-97
SLIDE 97

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

slide-98
SLIDE 98

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 ri in a match against a player with strength rj is given by F(ri − rj).

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

slide-99
SLIDE 99

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 ri in a match against a player with strength rj is given by F(ri − rj). The function F is called rating function

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

slide-100
SLIDE 100

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 ri in a match against a player with strength rj is given by F(ri − rj). The function F is called rating function Bradley and Terry (1952) took the (standard) logistic distribution: F(ri − rj) =

eri eri+erj

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

slide-101
SLIDE 101

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 ri in a match against a player with strength rj is given by F(ri − rj). The function F is called rating function Bradley and Terry (1952) took the (standard) logistic distribution: F(ri − rj) =

eri eri+erj

(F (0) = 1/2, limd→+∞ F (d) = 1, limd→−∞ F (d) = 0) Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

slide-102
SLIDE 102

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 ri in a match against a player with strength rj is given by F(ri − rj). The function F is called rating function Bradley and Terry (1952) took the (standard) logistic distribution: F(ri − rj) =

eri eri+erj

(F (0) = 1/2, limd→+∞ F (d) = 1, limd→−∞ F (d) = 0)

The specific distribution can be chosen depending on the discipline

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

slide-103
SLIDE 103

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 ri in a match against a player with strength rj is given by F(ri − rj). The function F is called rating function Bradley and Terry (1952) took the (standard) logistic distribution: F(ri − rj) =

eri eri+erj

(F (0) = 1/2, limd→+∞ F (d) = 1, limd→−∞ F (d) = 0)

The specific distribution can be chosen depending on the discipline In chess tournaments also a logistic distribution has proved to fit the observed data quite well

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 15/24

slide-104
SLIDE 104

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-105
SLIDE 105

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-106
SLIDE 106

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-107
SLIDE 107

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood Properties:

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-108
SLIDE 108

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood Properties: Excellent asymptotic properties

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-109
SLIDE 109

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood Properties: Excellent asymptotic properties

The ranking proposed for A is the inverse of the one proposed for At

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-110
SLIDE 110

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood Properties: Excellent asymptotic properties

The ranking proposed for A is the inverse of the one proposed for At

Problems:

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-111
SLIDE 111

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood Properties: Excellent asymptotic properties

The ranking proposed for A is the inverse of the one proposed for At

Problems: Typically reduces to solve a nonlinear system of equations (high computational cost)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-112
SLIDE 112

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood Properties: Excellent asymptotic properties

The ranking proposed for A is the inverse of the one proposed for At

Problems: Typically reduces to solve a nonlinear system of equations (high computational cost) Even mild misspecifications on the function F may lead to severe asymptotic bias

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-113
SLIDE 113

Motivation The model Ranking Methods Our contribution

The Maximum Likelihood Approach

Given a tournament A and a rating function F, choose the vector

  • f ratings r under which the probability of A being realized, when

the matches in M are played, is maximized Select the ratings under which A has maximum likelihood Properties: Excellent asymptotic properties

The ranking proposed for A is the inverse of the one proposed for At

Problems: Typically reduces to solve a nonlinear system of equations (high computational cost) Even mild misspecifications on the function F may lead to severe asymptotic bias What about non-asymptotic behavior?

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 16/24

slide-114
SLIDE 114

Motivation The model Ranking Methods Our contribution

Outline

1

Motivation

2

The model

3

Ranking Methods

4

Our contribution

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 17/24

slide-115
SLIDE 115

Motivation The model Ranking Methods Our contribution

Idea

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-116
SLIDE 116

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-117
SLIDE 117

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Fair-Bets: The iterative method (linear system) to use all the information of the tournament

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-118
SLIDE 118

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Fair-Bets: The iterative method (linear system) to use all the information of the tournament Maximum-Likelihood: The idea of using the rating function to reward results according to their statistic relevance

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-119
SLIDE 119

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Fair-Bets: The iterative method (linear system) to use all the information of the tournament Maximum-Likelihood: The idea of using the rating function to reward results according to their statistic relevance (F is non-linear)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-120
SLIDE 120

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Fair-Bets: The iterative method (linear system) to use all the information of the tournament Maximum-Likelihood: The idea of using the rating function to reward results according to their statistic relevance (F is non-linear)

—WE DO NOT LIKE

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-121
SLIDE 121

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Fair-Bets: The iterative method (linear system) to use all the information of the tournament Maximum-Likelihood: The idea of using the rating function to reward results according to their statistic relevance (F is non-linear)

—WE DO NOT LIKE

Fair-Bets: Asymmetric treatment of victories with respect to losses

(formalized through the axiom concerning A and At)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-122
SLIDE 122

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Fair-Bets: The iterative method (linear system) to use all the information of the tournament Maximum-Likelihood: The idea of using the rating function to reward results according to their statistic relevance (F is non-linear)

—WE DO NOT LIKE

Fair-Bets: Asymmetric treatment of victories with respect to losses

(formalized through the axiom concerning A and At)

Maximum-Likelihood: The problems derived from the non-linearity of

the system to be solved

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-123
SLIDE 123

Motivation The model Ranking Methods Our contribution

Idea

—WE LIKE

Fair-Bets: The iterative method (linear system) to use all the information of the tournament Maximum-Likelihood: The idea of using the rating function to reward results according to their statistic relevance (F is non-linear)

—WE DO NOT LIKE

Fair-Bets: Asymmetric treatment of victories with respect to losses

(formalized through the axiom concerning A and At)

Maximum-Likelihood: The problems derived from the non-linearity of

the system to be solved (lack of robustness on F, computation costs)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 18/24

slide-124
SLIDE 124

Motivation The model Ranking Methods Our contribution

Recursive Performance

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-125
SLIDE 125

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-126
SLIDE 126

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-127
SLIDE 127

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-128
SLIDE 128

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

The recursive performance method

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-129
SLIDE 129

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

The recursive performance method

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-130
SLIDE 130

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

The recursive performance method

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

mij

  • k mik

+ F −1(si)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-131
SLIDE 131

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

The recursive performance method

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • + F −1(si)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-132
SLIDE 132

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

The recursive performance method

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-133
SLIDE 133

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

The recursive performance method

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

exhibited strength Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-134
SLIDE 134

Motivation The model Ranking Methods Our contribution

Recursive Performance

Suppose that player i achieves a score of sij = aij

mij against j

F −1(sij) represents the difference of strength between i and j given by these results

mij

  • k mik =“percentage of the games played by i that were against j”

The recursive performance method

Initially, we can regard all players as equally strong: r0 := (1, . . . , 1)

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

exhibited strength

r1

i = “performance of i”

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 19/24

slide-135
SLIDE 135

Motivation The model Ranking Methods Our contribution

Recursive Performance

The recursive performance method

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

exhibited strength Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 20/24

slide-136
SLIDE 136

Motivation The model Ranking Methods Our contribution

Recursive Performance

The recursive performance method

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

exhibited strength

The ratings r1 are a better indicator of the real strength of the players than r0

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 20/24

slide-137
SLIDE 137

Motivation The model Ranking Methods Our contribution

Recursive Performance

The recursive performance method

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

exhibited strength

The ratings r1 are a better indicator of the real strength of the players than r0

r2

i :=

  • j

mij

  • k mik

r1

j+F −1(si)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 20/24

slide-138
SLIDE 138

Motivation The model Ranking Methods Our contribution

Recursive Performance

The recursive performance method

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

exhibited strength

The ratings r1 are a better indicator of the real strength of the players than r0

r2

i :=

  • j

mij

  • k mik

r1

j+F −1(si) . . .

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 20/24

slide-139
SLIDE 139

Motivation The model Ranking Methods Our contribution

Recursive Performance

The recursive performance method

r1

i :=

  • j

mij

  • k mik

+ F −1(si) =

  • j

mij

  • k mik

r0

j

  • average opponent

+ F −1(si)

exhibited strength

The ratings r1 are a better indicator of the real strength of the players than r0

r2

i :=

  • j

mij

  • k mik

r1

j+F −1(si) . . . r∞ i

:=

  • j

mij

  • k mik

r∞

j +F −1(si)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 20/24

slide-140
SLIDE 140

Motivation The model Ranking Methods Our contribution

Recursive Performance

r∞

i

:=

  • j

mij

  • k mik

r∞

j + F −1(si)

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 21/24

slide-141
SLIDE 141

Motivation The model Ranking Methods Our contribution

Recursive Performance

r∞

i

:=

  • j

mij

  • k mik

r∞

j + F −1(si)

The recursive performance is well defined

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 21/24

slide-142
SLIDE 142

Motivation The model Ranking Methods Our contribution

Recursive Performance

r∞

i

:=

  • j

mij

  • k mik

r∞

j + F −1(si)

The recursive performance is well defined r∞ is just the solution of a linear system of equations

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 21/24

slide-143
SLIDE 143

Motivation The model Ranking Methods Our contribution

Recursive Performance

r∞

i

:=

  • j

mij

  • k mik

r∞

j + F −1(si)

The recursive performance is well defined r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 21/24

slide-144
SLIDE 144

Motivation The model Ranking Methods Our contribution

Recursive Performance

r∞

i

:=

  • j

mij

  • k mik

r∞

j + F −1(si)

The recursive performance is well defined r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 Symmetric treatment of victories and losses. The ranking proposed for A is the inverse of the ranking proposed by At

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 21/24

slide-145
SLIDE 145

Motivation The model Ranking Methods Our contribution

Recursive Performance

r∞

i

:=

  • j

mij

  • k mik

r∞

j + F −1(si)

The recursive performance is well defined r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 Symmetric treatment of victories and losses. The ranking proposed for A is the inverse of the ranking proposed by At The proposed rating is robust in F

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 21/24

slide-146
SLIDE 146

Motivation The model Ranking Methods Our contribution

Recursive Performance

r∞

i

:=

  • j

mij

  • k mik

r∞

j + F −1(si)

The recursive performance is well defined r∞ is just the solution of a linear system of equations The ranking induced by r∞ is independent of r0 Symmetric treatment of victories and losses. The ranking proposed for A is the inverse of the ranking proposed by At The proposed rating is robust in F According to the proposed rating, the performance of each player coincides with his own rating

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 21/24

slide-147
SLIDE 147

Motivation The model Ranking Methods Our contribution

Round Robin

When restricting attention to round robin tournaments we get:

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 22/24

slide-148
SLIDE 148

Motivation The model Ranking Methods Our contribution

Round Robin

When restricting attention to round robin tournaments we get:

Scores = Maximum Likelihood = Recursive Performance = Fair Bets

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 22/24

slide-149
SLIDE 149

Motivation The model Ranking Methods Our contribution

Some numeric examples

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 23/24

slide-150
SLIDE 150

Motivation The model Ranking Methods Our contribution

Some numeric examples

A1

(av.scores)

s RP ML FB     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 1.010 1.099 0.703 0.633 1.037 1.099 0.703 0.1 −1.017 −1.099 0.078 0.1 −1.031 −1.099 0.078

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 23/24

slide-151
SLIDE 151

Motivation The model Ranking Methods Our contribution

Some numeric examples

A1

(av.scores)

s RP ML FB     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 1.010 1.099 0.703 0.633 1.037 1.099 0.703 0.1 −1.017 −1.099 0.078 0.1 −1.031 −1.099 0.078 A2

(av.scores)

s RP ML FB     1 90 0.9 1 0.9 10 0.1 0.1     0.892 1.127 1.099 0.703 0.633 0.971 1.099 0.703 0.1 −1.049 −1.099 0.078 0.1 −1.048 −1.099 0.078

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 23/24

slide-152
SLIDE 152

Motivation The model Ranking Methods Our contribution

Some numeric examples

A1

(av.scores)

s RP ML FB     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 1.010 1.099 0.703 0.633 1.037 1.099 0.703 0.1 −1.017 −1.099 0.078 0.1 −1.031 −1.099 0.078 A2

(av.scores)

s RP ML FB     1 90 0.9 1 0.9 10 0.1 0.1     0.892 1.127 1.099 0.703 0.633 0.971 1.099 0.703 0.1 −1.049 −1.099 0.078 0.1 −1.048 −1.099 0.078 A3

(av.scores)

s RP ML FB     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 1.087 1.055 0.637 0.667 1.082 1.227 0.764 0.1 −1.085 −1.140 0.071 0.1 −1.084 −1.142 0.070

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 23/24

slide-153
SLIDE 153

Motivation The model Ranking Methods Our contribution

Directions for future research

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 24/24

slide-154
SLIDE 154

Motivation The model Ranking Methods Our contribution

Directions for future research

Axiomatic analysis of the recursive performance ranking method

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 24/24

slide-155
SLIDE 155

Motivation The model Ranking Methods Our contribution

Directions for future research

Axiomatic analysis of the recursive performance ranking method Develop comparative studies of the different ranking methods in applied settings

Ranking Participants in Tournaments Gonz´ alez-D´ ıaz et al. 24/24

slide-156
SLIDE 156

Rankings and Tournaments: A new approach

Julio Gonz´ alez-D´ ıaz

Kellogg School of Management (CMS-EMS) Northwestern University and Research Group in Economic Analysis Universidad de Vigo ........................ (joint with Miguel Brozos-V´ azquez, Marco Antonio Campo-Cabana, and Jos´ e Carlos D´ ıaz-Ramos)

slide-157
SLIDE 157

Some tournaments

Return

slide-158
SLIDE 158

Some tournaments

A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −−

Return

slide-159
SLIDE 159

Some tournaments

A1

(av.scores)

s ++ ++ ++     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −−

Return

slide-160
SLIDE 160

Some tournaments

A1

(av.scores)

s ++ ++ ++     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A3

(av.scores)

s ++ ++ ++     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 − −− − −− − −− 0.667 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −−

Return

slide-161
SLIDE 161

Some tournaments

A1

(av.scores)

s ++ ++ ++     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A3

(av.scores)

s ++ ++ ++     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 − −− − −− − −− 0.667 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −−

Return

slide-162
SLIDE 162

Some tournaments

A1

(av.scores)

s ++ ++ ++     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A3

(av.scores)

s ++ ++ ++     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 − −− − −− − −− 0.667 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −−

Return

slide-163
SLIDE 163

Some tournaments

A1

(av.scores)

s ++ ++ ++     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 − −− − −− − −− 0.633 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −− A3

(av.scores)

s ++ ++ ++     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 − −− − −− − −− 0.667 − −− − −− − −− 0.1 − −− − −− − −− 0.1 − −− − −− − −−

Return

slide-164
SLIDE 164

Some tournaments

A1

(av.scores)

s ++ ++ ++     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 1.010 1.099 0.703 0.633 1.037 1.099 0.703 0.1 − 1.017 − 1.099 0.078 0.1 − 1.031 − 1.099 0.078 A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 1.127 1.099 0.703 0.633 0.971 1.099 0.703 0.1 − 1.049 − 1.099 0.078 0.1 − 1.048 − 1.099 0.078 A3

(av.scores)

s ++ ++ ++     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 1.087 1.055 0.637 0.667 1.082 1.227 0.764 0.1 − 1.085 − 1.140 0.071 0.1 − 1.084 − 1.142 0.070

Return

slide-165
SLIDE 165

Some tournaments

A1

(av.scores)

s ++ ++ ++     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 1.010 1.099 0.703 0.633 1.037 1.099 0.703 0.1 − 1.017 − 1.099 0.078 0.1 − 1.031 − 1.099 0.078 A2

(av.scores)

s ++ ++ ++     1 90 0.9 1 0.9 10 0.1 0.1     0.892 1.127 1.099 0.703 0.633 0.971 1.099 0.703 0.1 − 1.049 − 1.099 0.078 0.1 − 1.048 − 1.099 0.078 A3

(av.scores)

s ++ ++ ++     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 1.087 1.055 0.637 0.667 1.082 1.227 0.764 0.1 − 1.085 − 1.140 0.071 0.1 − 1.084 − 1.142 0.070

Return

slide-166
SLIDE 166

Some tournaments

A1

(av.scores)

s RP ML FB     1 0.9 0.9 1 0.9 0.1 0.1 0.1     0.7 1.010 1.099 0.703 0.633 1.037 1.099 0.703 0.1 − 1.017 − 1.099 0.078 0.1 − 1.031 − 1.099 0.078 A2

(av.scores)

s RP ML FB     1 90 0.9 1 0.9 10 0.1 0.1     0.892 1.127 1.099 0.703 0.633 0.971 1.099 0.703 0.1 − 1.049 − 1.099 0.078 0.1 − 1.048 − 1.099 0.078 A3

(av.scores)

s RP ML FB     0.9 90 0.9 1.1 0.9 10 0.1 0.1     0.891 1.087 1.055 0.637 0.667 1.082 1.227 0.764 0.1 − 1.085 − 1.140 0.071 0.1 − 1.084 − 1.142 0.070

Return