rating, ranking, betting and mathematics r vittal rao
TRANSCRIPT
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.
Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”
Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”
Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”
Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”
Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”
Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”
Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”
Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”
Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”
Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”
Mark: “No, I need your algorithm to rank chess players”
A Scene
Eduardo is entering a dorm room in Harvard University, as Mark isat his laptop.Mark:“Done! Perfect timing. Eduardo is here and he is going tohave the key ingredient”Ed: “Hi Mark”Mark: “Eduardo”Ed: “You and Erica splt up?”Mark: “How did you know that?”Ed: “It is on your blog”Mark: “Yeah”Ed:“Are you alright?”Mark: “I need you”Ed: “I am here for you”Mark: “No, I need your algorithm to rank chess players”
Ed: “Are you OK?”
Mark: “We are ranking girls”Ed: “You mean the other students”Mark: “Yeah”Ed: “Do you think it is such a good idea?”Mark: “I need your algorithm - I need your algorithm”Ed: “With each girl’s base rating 1400...”Ed proceeds to write the formulae (on a window with a crayon)
Ed: “Are you OK?”Mark: “We are ranking girls”
Ed: “You mean the other students”Mark: “Yeah”Ed: “Do you think it is such a good idea?”Mark: “I need your algorithm - I need your algorithm”Ed: “With each girl’s base rating 1400...”Ed proceeds to write the formulae (on a window with a crayon)
Ed: “Are you OK?”Mark: “We are ranking girls”Ed: “You mean the other students”
Mark: “Yeah”Ed: “Do you think it is such a good idea?”Mark: “I need your algorithm - I need your algorithm”Ed: “With each girl’s base rating 1400...”Ed proceeds to write the formulae (on a window with a crayon)
Ed: “Are you OK?”Mark: “We are ranking girls”Ed: “You mean the other students”Mark: “Yeah”
Ed: “Do you think it is such a good idea?”Mark: “I need your algorithm - I need your algorithm”Ed: “With each girl’s base rating 1400...”Ed proceeds to write the formulae (on a window with a crayon)
Ed: “Are you OK?”Mark: “We are ranking girls”Ed: “You mean the other students”Mark: “Yeah”Ed: “Do you think it is such a good idea?”
Mark: “I need your algorithm - I need your algorithm”Ed: “With each girl’s base rating 1400...”Ed proceeds to write the formulae (on a window with a crayon)
Ed: “Are you OK?”Mark: “We are ranking girls”Ed: “You mean the other students”Mark: “Yeah”Ed: “Do you think it is such a good idea?”Mark: “I need your algorithm - I need your algorithm”
Ed: “With each girl’s base rating 1400...”Ed proceeds to write the formulae (on a window with a crayon)
Ed: “Are you OK?”Mark: “We are ranking girls”Ed: “You mean the other students”Mark: “Yeah”Ed: “Do you think it is such a good idea?”Mark: “I need your algorithm - I need your algorithm”Ed: “With each girl’s base rating 1400...”
Ed proceeds to write the formulae (on a window with a crayon)
Ed: “Are you OK?”Mark: “We are ranking girls”Ed: “You mean the other students”Mark: “Yeah”Ed: “Do you think it is such a good idea?”Mark: “I need your algorithm - I need your algorithm”Ed: “With each girl’s base rating 1400...”Ed proceeds to write the formulae (on a window with a crayon)
Eduardo’s Formula!
Ea =1
1 + 10 (Rb−Ra)400
Eb =1
1 + 10 (Ra−Rb)400
Eduardo’s Formula!
Ea =1
1 + 10 (Rb−Ra)400
Eb =1
1 + 10 (Ra−Rb)400
Eduardo’s Formula!
Ea =1
1 + 10 (Rb−Ra)400
Eb =1
1 + 10 (Ra−Rb)400
Mark’s Observation
Mark:“When any two girls are matched up there’s an expectation ofwhich will win based on their current rating, right?And all those expectations are expressed this way”
Mark’s Observation
Mark:“When any two girls are matched up
there’s an expectation ofwhich will win based on their current rating, right?And all those expectations are expressed this way”
Mark’s Observation
Mark:“When any two girls are matched up there’s an expectation ofwhich will win
based on their current rating, right?And all those expectations are expressed this way”
Mark’s Observation
Mark:“When any two girls are matched up there’s an expectation ofwhich will win based on their current rating, right?
And all those expectations are expressed this way”
Mark’s Observation
Mark:“When any two girls are matched up there’s an expectation ofwhich will win based on their current rating, right?And all those expectations are expressed this way”
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPicturesAdapted from the 2009 book “The accidental Billionnaires” byBen MezrichThe film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPictures
Adapted from the 2009 book “The accidental Billionnaires” byBen MezrichThe film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPicturesAdapted from the 2009 book “The accidental Billionnaires” byBen Mezrich
The film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPicturesAdapted from the 2009 book “The accidental Billionnaires” byBen MezrichThe film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPicturesAdapted from the 2009 book “The accidental Billionnaires” byBen MezrichThe film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:
Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPicturesAdapted from the 2009 book “The accidental Billionnaires” byBen MezrichThe film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)
Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPicturesAdapted from the 2009 book “The accidental Billionnaires” byBen MezrichThe film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)
Justin Timblelake as Sean Parker (cofounder)
The Social Network
Movie relesed in the USA on October 01, 2010 by ColumbiaPicturesAdapted from the 2009 book “The accidental Billionnaires” byBen MezrichThe film portrays the founding of the social networking websiteFACEBOOK and the ensuing legal battles
CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)
Awards
8 Nominations for Academy Awards includingBest Picture, Best Director and Best ActorWon ThreeBest Adapted Screenplay, Best Original Score, and Best FilmEditingIn 68th Golden Globe Awards won the award forBest Moion Picture, Best Diector, Best Screenplay and BestOriginal Score
Awards
8 Nominations for Academy Awards including
Best Picture, Best Director and Best ActorWon ThreeBest Adapted Screenplay, Best Original Score, and Best FilmEditingIn 68th Golden Globe Awards won the award forBest Moion Picture, Best Diector, Best Screenplay and BestOriginal Score
Awards
8 Nominations for Academy Awards includingBest Picture, Best Director and Best Actor
Won ThreeBest Adapted Screenplay, Best Original Score, and Best FilmEditingIn 68th Golden Globe Awards won the award forBest Moion Picture, Best Diector, Best Screenplay and BestOriginal Score
Awards
8 Nominations for Academy Awards includingBest Picture, Best Director and Best ActorWon Three
Best Adapted Screenplay, Best Original Score, and Best FilmEditingIn 68th Golden Globe Awards won the award forBest Moion Picture, Best Diector, Best Screenplay and BestOriginal Score
Awards
8 Nominations for Academy Awards includingBest Picture, Best Director and Best ActorWon ThreeBest Adapted Screenplay, Best Original Score, and Best FilmEditing
In 68th Golden Globe Awards won the award forBest Moion Picture, Best Diector, Best Screenplay and BestOriginal Score
Awards
8 Nominations for Academy Awards includingBest Picture, Best Director and Best ActorWon ThreeBest Adapted Screenplay, Best Original Score, and Best FilmEditingIn 68th Golden Globe Awards won the award for
Best Moion Picture, Best Diector, Best Screenplay and BestOriginal Score
Awards
8 Nominations for Academy Awards includingBest Picture, Best Director and Best ActorWon ThreeBest Adapted Screenplay, Best Original Score, and Best FilmEditingIn 68th Golden Globe Awards won the award forBest Moion Picture, Best Diector, Best Screenplay and BestOriginal Score
RATING and RANKING
Rating
Given n objects a Rating assigns n REAL NUMBERS (positive ornegative), (preferably distinct), one each to these n objects
Rating
Given n objects
a Rating assigns n REAL NUMBERS (positive ornegative), (preferably distinct), one each to these n objects
Rating
Given n objects a Rating assigns n REAL NUMBERS
(positive ornegative), (preferably distinct), one each to these n objects
Rating
Given n objects a Rating assigns n REAL NUMBERS (positive ornegative),
(preferably distinct), one each to these n objects
Rating
Given n objects a Rating assigns n REAL NUMBERS (positive ornegative), (preferably distinct),
one each to these n objects
Rating
Given n objects a Rating assigns n REAL NUMBERS (positive ornegative), (preferably distinct), one each to these n objects
Ranking
Given n objects a Ranking assigns the n positive integers1, 2, 3, · · · , n, one each to these n objectsEvery Rating creates a Ranking by arranging the ratings indescending order
Ranking
Given n objects
a Ranking assigns the n positive integers1, 2, 3, · · · , n, one each to these n objectsEvery Rating creates a Ranking by arranging the ratings indescending order
Ranking
Given n objects a Ranking assigns
the n positive integers1, 2, 3, · · · , n, one each to these n objectsEvery Rating creates a Ranking by arranging the ratings indescending order
Ranking
Given n objects a Ranking assigns the n positive integers1, 2, 3, · · · , n,
one each to these n objectsEvery Rating creates a Ranking by arranging the ratings indescending order
Ranking
Given n objects a Ranking assigns the n positive integers1, 2, 3, · · · , n, one each to these n objects
Every Rating creates a Ranking by arranging the ratings indescending order
Ranking
Given n objects a Ranking assigns the n positive integers1, 2, 3, · · · , n, one each to these n objectsEvery Rating creates a Ranking by arranging the ratings indescending order
Pairwise Comparison
It is well known that human beings have difficulty ranking any setof items of size bigger than fiveHowever we are all particularly adept at pairwise comparisonSuch pairwise comparisons are at the heart of most of the methodsof rating and ranking
Pairwise Comparison
It is well known that human beings have difficulty ranking any setof items of size bigger than five
However we are all particularly adept at pairwise comparisonSuch pairwise comparisons are at the heart of most of the methodsof rating and ranking
Pairwise Comparison
It is well known that human beings have difficulty ranking any setof items of size bigger than fiveHowever we are all particularly adept at pairwise comparison
Such pairwise comparisons are at the heart of most of the methodsof rating and ranking
Pairwise Comparison
It is well known that human beings have difficulty ranking any setof items of size bigger than fiveHowever we are all particularly adept at pairwise comparisonSuch pairwise comparisons are at the heart of most of the methodsof rating and ranking
General Mathematical Techniques
Most of the rating system use one or more of the following:
I Linear Algebra - Linear Systems of Equations, eigenvalues andEigenvectors)
I Optimization
I Statistics (Markov techniques)
I Game Theory
General Mathematical Techniques
Most of the rating system use one or more of the following:
I Linear Algebra - Linear Systems of Equations, eigenvalues andEigenvectors)
I Optimization
I Statistics (Markov techniques)
I Game Theory
General Mathematical Techniques
Most of the rating system use one or more of the following:
I Linear Algebra - Linear Systems of Equations, eigenvalues andEigenvectors)
I Optimization
I Statistics (Markov techniques)
I Game Theory
General Mathematical Techniques
Most of the rating system use one or more of the following:
I Linear Algebra - Linear Systems of Equations, eigenvalues andEigenvectors)
I Optimization
I Statistics (Markov techniques)
I Game Theory
General Mathematical Techniques
Most of the rating system use one or more of the following:
I Linear Algebra - Linear Systems of Equations, eigenvalues andEigenvectors)
I Optimization
I Statistics (Markov techniques)
I Game Theory
ARROW’s IMPOSSIBILITY THEOREM
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes
(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change
(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B
(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection
(Non − dictatorship)
Arrow’s Four Criteria
I Every voter should be free to rank the candidates in any orderhe or she likes(This criterion is known as the Unrestricted Domain criterion)
I If only a subset of the candidates is considered the rankingorder within the subset should not change(Independence of Irrelevant Alternatives)
I If all voters choose candidate A over candidate B, then thevoting system should rank A ahead of B(Pereto Principle)
I No single voter should have disproportionate control over anelection(Non − dictatorship)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sense
but his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is not
Arrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
Arrow’s Theorem (1951)
The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is notArrow’s Impossibility Theorem:
It is IMPOSSIBLE for any
voting system to satisfy all four
common sense criteria simultaneously
(NOBEL PRIZE IN ECONOMICS - 1972)
BCS (Bowl Championship Series)
In place since 1998Since its inception in 1998, the NCAA’s Bowl Championship Serieshas weathered criticism from nearly all directions.
BCS (Bowl Championship Series)
In place since 1998
Since its inception in 1998, the NCAA’s Bowl Championship Serieshas weathered criticism from nearly all directions.
BCS (Bowl Championship Series)
In place since 1998Since its inception in 1998, the NCAA’s Bowl Championship Serieshas weathered criticism from nearly all directions.
Massey System
Proposed by Kenneth Massey in 1997 (as an undergraduatestudent)Basically based on Least Square Solution of a Linear SystemModified version used in BCS (BOWL CHAMPIONSHIPSERIES)
Massey System
Proposed by Kenneth Massey in 1997 (as an undergraduatestudent)
Basically based on Least Square Solution of a Linear SystemModified version used in BCS (BOWL CHAMPIONSHIPSERIES)
Massey System
Proposed by Kenneth Massey in 1997 (as an undergraduatestudent)Basically based on Least Square Solution of a Linear System
Modified version used in BCS (BOWL CHAMPIONSHIPSERIES)
Massey System
Proposed by Kenneth Massey in 1997 (as an undergraduatestudent)Basically based on Least Square Solution of a Linear SystemModified version used in BCS (BOWL CHAMPIONSHIPSERIES)
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progresses
At some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games
and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi games
and lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games
then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + ni
Note:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:
We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = ni
If Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Colley’s Winning Proportion Ranking
At the beginning each team is given an equal ranking of 12
Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost `i games then define
ri =1 + wi
2 + niNote:We have wi + `i = niIf Ti has played 2k games and won exactly half of these, namely kgames, then
ri =1 + k
2 + 2k
=1
2
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election Campaign
Obama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, and
Obama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants
2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:
“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,”
President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters
afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game.
“If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”
Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
Controversies and Obama
2008 U.S. Presidential Election CampaignObama wa asked to describe one thing about sports he wouldchange, andObama mentioned his disgust with using BCS rating sytem todetermine bowl-game participants2009:“We need a playoff,” President Elect Obama told reporters afterbeing asked about Florida’s 24-14 victory over Oklahoma in BCSchampionship game. “If I’m Utah, or if I’m USC or if I’m Texas, Imight still have some quibbles.”Senator Orrin Hatch of Utah asked the US Justice Department toreview the legality of the BCS for possible violation of antitrustlaws
End of the Road for BCS
2011:The BCS was put on notice that they are being scrutinized by theUS Justice Dpertment2012:On June 26, 2012, it was announced that the Bowl ChampionshipSeries will be replaced by a four-team playoff, effective for the2014-15 season to be known as the College Football Playoff.
End of the Road for BCS
2011:
The BCS was put on notice that they are being scrutinized by theUS Justice Dpertment2012:On June 26, 2012, it was announced that the Bowl ChampionshipSeries will be replaced by a four-team playoff, effective for the2014-15 season to be known as the College Football Playoff.
End of the Road for BCS
2011:The BCS was put on notice that they are being scrutinized by theUS Justice Dpertment
2012:On June 26, 2012, it was announced that the Bowl ChampionshipSeries will be replaced by a four-team playoff, effective for the2014-15 season to be known as the College Football Playoff.
End of the Road for BCS
2011:The BCS was put on notice that they are being scrutinized by theUS Justice Dpertment2012:
On June 26, 2012, it was announced that the Bowl ChampionshipSeries will be replaced by a four-team playoff, effective for the2014-15 season to be known as the College Football Playoff.
End of the Road for BCS
2011:The BCS was put on notice that they are being scrutinized by theUS Justice Dpertment2012:On June 26, 2012, it was announced that the Bowl ChampionshipSeries will be replaced by a four-team playoff, effective for the2014-15 season to be known as the College Football Playoff.
Elo’s Rating
Chess Rating
Arpad Elo (1903-1992)An avid Chess PlayerCreated an efficient method to rate and rank chess playersHis system was approved byUSCF (United States Chess Federation)FIDE (Federaton Internationale des Eches) - The World ChessFederationBecame popular outside chess world and adapted to rate othersports and competitive situations
Chess Rating
Arpad Elo (1903-1992)
An avid Chess PlayerCreated an efficient method to rate and rank chess playersHis system was approved byUSCF (United States Chess Federation)FIDE (Federaton Internationale des Eches) - The World ChessFederationBecame popular outside chess world and adapted to rate othersports and competitive situations
Chess Rating
Arpad Elo (1903-1992)An avid Chess Player
Created an efficient method to rate and rank chess playersHis system was approved byUSCF (United States Chess Federation)FIDE (Federaton Internationale des Eches) - The World ChessFederationBecame popular outside chess world and adapted to rate othersports and competitive situations
Chess Rating
Arpad Elo (1903-1992)An avid Chess PlayerCreated an efficient method to rate and rank chess players
His system was approved byUSCF (United States Chess Federation)FIDE (Federaton Internationale des Eches) - The World ChessFederationBecame popular outside chess world and adapted to rate othersports and competitive situations
Chess Rating
Arpad Elo (1903-1992)An avid Chess PlayerCreated an efficient method to rate and rank chess playersHis system was approved by
USCF (United States Chess Federation)FIDE (Federaton Internationale des Eches) - The World ChessFederationBecame popular outside chess world and adapted to rate othersports and competitive situations
Chess Rating
Arpad Elo (1903-1992)An avid Chess PlayerCreated an efficient method to rate and rank chess playersHis system was approved byUSCF (United States Chess Federation)
FIDE (Federaton Internationale des Eches) - The World ChessFederationBecame popular outside chess world and adapted to rate othersports and competitive situations
Chess Rating
Arpad Elo (1903-1992)An avid Chess PlayerCreated an efficient method to rate and rank chess playersHis system was approved byUSCF (United States Chess Federation)FIDE (Federaton Internationale des Eches) - The World ChessFederation
Became popular outside chess world and adapted to rate othersports and competitive situations
Chess Rating
Arpad Elo (1903-1992)An avid Chess PlayerCreated an efficient method to rate and rank chess playersHis system was approved byUSCF (United States Chess Federation)FIDE (Federaton Internationale des Eches) - The World ChessFederationBecame popular outside chess world and adapted to rate othersports and competitive situations
The Basic Formulation
Based on the premise that“the performance of a player is a normally distributed randomvariable”Keep Updating the rating after each game(ri )old is the rating of Player Pi before he plays a game(ri )new is the rating of Player Pi after he plays the gameHow are (ri )old and (ri )new related?
The Basic Formulation
Based on the premise that“the performance of a player is a normally distributed randomvariable”
Keep Updating the rating after each game(ri )old is the rating of Player Pi before he plays a game(ri )new is the rating of Player Pi after he plays the gameHow are (ri )old and (ri )new related?
The Basic Formulation
Based on the premise that“the performance of a player is a normally distributed randomvariable”Keep Updating the rating after each game
(ri )old is the rating of Player Pi before he plays a game(ri )new is the rating of Player Pi after he plays the gameHow are (ri )old and (ri )new related?
The Basic Formulation
Based on the premise that“the performance of a player is a normally distributed randomvariable”Keep Updating the rating after each game(ri )old is the rating of Player Pi before he plays a game
(ri )new is the rating of Player Pi after he plays the gameHow are (ri )old and (ri )new related?
The Basic Formulation
Based on the premise that“the performance of a player is a normally distributed randomvariable”Keep Updating the rating after each game(ri )old is the rating of Player Pi before he plays a game(ri )new is the rating of Player Pi after he plays the game
How are (ri )old and (ri )new related?
The Basic Formulation
Based on the premise that“the performance of a player is a normally distributed randomvariable”Keep Updating the rating after each game(ri )old is the rating of Player Pi before he plays a game(ri )new is the rating of Player Pi after he plays the gameHow are (ri )old and (ri )new related?
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new gameS − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that time
S is the score in the new gameS − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new game
S − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new gameS − µ is the variation (could be positive or negative or zero)
The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new gameS − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new gameS − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new gameS − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.
Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new gameS − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10
For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Elo’s Hypothesis
µ is the mean score of Pi till that timeS is the score in the new gameS − µ is the variation (could be positive or negative or zero)The correction factor to the old ratng of Pi is proportional to thevariation S − µ, that is,
(ri )new = (ri )old + K (S − µ)
where K is the constant of proportion.Elo originally set K = 10For chess games,
S =
1 if Pi wins12 if Pi draws0 if Pi loses
Modifications
As more data became available it was found thatnormal distribution assumption is not a good assumptionA more suitable assuption is the Logistic DistributionThis changes the formula aboveIn 1917 Bob Runyan adapted for International FootballJeff Segarin (USA Today) adapted for American Football
Modifications
As more data became available it was found that
normal distribution assumption is not a good assumptionA more suitable assuption is the Logistic DistributionThis changes the formula aboveIn 1917 Bob Runyan adapted for International FootballJeff Segarin (USA Today) adapted for American Football
Modifications
As more data became available it was found thatnormal distribution assumption is not a good assumption
A more suitable assuption is the Logistic DistributionThis changes the formula aboveIn 1917 Bob Runyan adapted for International FootballJeff Segarin (USA Today) adapted for American Football
Modifications
As more data became available it was found thatnormal distribution assumption is not a good assumptionA more suitable assuption is the Logistic Distribution
This changes the formula aboveIn 1917 Bob Runyan adapted for International FootballJeff Segarin (USA Today) adapted for American Football
Modifications
As more data became available it was found thatnormal distribution assumption is not a good assumptionA more suitable assuption is the Logistic DistributionThis changes the formula above
In 1917 Bob Runyan adapted for International FootballJeff Segarin (USA Today) adapted for American Football
Modifications
As more data became available it was found thatnormal distribution assumption is not a good assumptionA more suitable assuption is the Logistic DistributionThis changes the formula aboveIn 1917 Bob Runyan adapted for International Football
Jeff Segarin (USA Today) adapted for American Football
Modifications
As more data became available it was found thatnormal distribution assumption is not a good assumptionA more suitable assuption is the Logistic DistributionThis changes the formula aboveIn 1917 Bob Runyan adapted for International FootballJeff Segarin (USA Today) adapted for American Football
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
Current Version
Suppose Pi plays against Pj
Sij =
1 if Pi beats Pj12 if game is a draw0 if Pi loses
µij = The Score Pi is expected to score against Pj
dij = (ri )old − (rj)old
( the difference in rating at the start of the game)
µij = L
(dij
400
)where
L(x) =1
1 + 10−x
µij =1
1 + 10−((ri )old−(rj )old)
400
The Logistic Curve L(x)
Current Version
(ri )new = (ri )old + K (Sij − µij)
The K Factor Used For Chess:K = 25 for new players (upto 30 recognized games)K = 15 for those with more than 30 gamesK = 10 for those who have recahed 2400 points at some point oftime
Current Version
(ri )new = (ri )old + K (Sij − µij)
The K Factor Used For Chess:K = 25 for new players (upto 30 recognized games)K = 15 for those with more than 30 gamesK = 10 for those who have recahed 2400 points at some point oftime
Current Version
(ri )new = (ri )old + K (Sij − µij)
The K Factor Used For Chess:
K = 25 for new players (upto 30 recognized games)K = 15 for those with more than 30 gamesK = 10 for those who have recahed 2400 points at some point oftime
Current Version
(ri )new = (ri )old + K (Sij − µij)
The K Factor Used For Chess:K = 25 for new players (upto 30 recognized games)
K = 15 for those with more than 30 gamesK = 10 for those who have recahed 2400 points at some point oftime
Current Version
(ri )new = (ri )old + K (Sij − µij)
The K Factor Used For Chess:K = 25 for new players (upto 30 recognized games)K = 15 for those with more than 30 games
K = 10 for those who have recahed 2400 points at some point oftime
Current Version
(ri )new = (ri )old + K (Sij − µij)
The K Factor Used For Chess:K = 25 for new players (upto 30 recognized games)K = 15 for those with more than 30 gamesK = 10 for those who have recahed 2400 points at some point oftime
Better Reward For beating stronger Team
Consider a game in which a weaker player Ti plays against astronger player Tj . This means at the time the game starts
ri < rj
In the Elo system of rating,the reward that the weaker player Ti gets if he beats Tj is,greater than the reward that stronger player Tj gets if he beats Ti
Better Reward For beating stronger Team
Consider a game in which a weaker player Ti plays against astronger player Tj .
This means at the time the game starts
ri < rj
In the Elo system of rating,the reward that the weaker player Ti gets if he beats Tj is,greater than the reward that stronger player Tj gets if he beats Ti
Better Reward For beating stronger Team
Consider a game in which a weaker player Ti plays against astronger player Tj . This means at the time the game starts
ri < rj
In the Elo system of rating,
the reward that the weaker player Ti gets if he beats Tj is,greater than the reward that stronger player Tj gets if he beats Ti
Better Reward For beating stronger Team
Consider a game in which a weaker player Ti plays against astronger player Tj . This means at the time the game starts
ri < rj
In the Elo system of rating,the reward that the weaker player Ti gets if he beats Tj is,
greater than the reward that stronger player Tj gets if he beats Ti
Better Reward For beating stronger Team
Consider a game in which a weaker player Ti plays against astronger player Tj . This means at the time the game starts
ri < rj
In the Elo system of rating,the reward that the weaker player Ti gets if he beats Tj is,greater than the reward that stronger player Tj gets if he beats Ti
Better Reward For beating stronger Team
Consider a game in which a weaker player Ti plays against astronger player Tj . This means at the time the game starts
ri < rj
In the Elo system of rating,the reward that the weaker player Ti gets if he beats Tj is,greater than the reward that stronger player Tj gets if he beats Ti
The Rewards
Reward for Ti if he beats Tj = K (Sij − µij)
= K
{1− 1
1 + 10−αij
}where
αij =(ri )old − (rj)old
400
Similarly
Reward for Tj if he beats Ti = K (Sji − µji )
= K
{1− 1
1 + 10−αji
}where
αji =(rj)old − (ri )old
400
The Rewards
Reward for Ti if he beats Tj = K (Sij − µij)
= K
{1− 1
1 + 10−αij
}where
αij =(ri )old − (rj)old
400
Similarly
Reward for Tj if he beats Ti = K (Sji − µji )
= K
{1− 1
1 + 10−αji
}where
αji =(rj)old − (ri )old
400
The Rewards
Reward for Ti if he beats Tj = K (Sij − µij)
= K
{1− 1
1 + 10−αij
}where
αij =(ri )old − (rj)old
400
Similarly
Reward for Tj if he beats Ti = K (Sji − µji )
= K
{1− 1
1 + 10−αji
}where
αji =(rj)old − (ri )old
400
The Rewards
Reward for Ti if he beats Tj = K (Sij − µij)
= K
{1− 1
1 + 10−αij
}where
αij =(ri )old − (rj)old
400
Similarly
Reward for Tj if he beats Ti = K (Sji − µji )
= K
{1− 1
1 + 10−αji
}where
αji =(rj)old − (ri )old
400
The Rewards
Reward for Ti if he beats Tj = K (Sij − µij)
= K
{1− 1
1 + 10−αij
}where
αij =(ri )old − (rj)old
400
Similarly
Reward for Tj if he beats Ti = K (Sji − µji )
= K
{1− 1
1 + 10−αji
}where
αji =(rj)old − (ri )old
400
An Example
Ti has rating 1700, Tj has rating 2100Suppose Ti beats Tj :
Reward for Ti = K
(1− 1
1 + 10−1700−2100
400
)= K
(1− 1
1 + 101
)≈ K (1− 0.09)
= (0.91)K
An Example
Ti has rating 1700, Tj has rating 2100
Suppose Ti beats Tj :
Reward for Ti = K
(1− 1
1 + 10−1700−2100
400
)= K
(1− 1
1 + 101
)≈ K (1− 0.09)
= (0.91)K
An Example
Ti has rating 1700, Tj has rating 2100Suppose Ti beats Tj :
Reward for Ti = K
(1− 1
1 + 10−1700−2100
400
)= K
(1− 1
1 + 101
)≈ K (1− 0.09)
= (0.91)K
An Example
Ti has rating 1700, Tj has rating 2100Suppose Ti beats Tj :
Reward for Ti =
K
(1− 1
1 + 10−1700−2100
400
)= K
(1− 1
1 + 101
)≈ K (1− 0.09)
= (0.91)K
An Example
Ti has rating 1700, Tj has rating 2100Suppose Ti beats Tj :
Reward for Ti = K
(1− 1
1 + 10−1700−2100
400
)
= K
(1− 1
1 + 101
)≈ K (1− 0.09)
= (0.91)K
An Example
Ti has rating 1700, Tj has rating 2100Suppose Ti beats Tj :
Reward for Ti = K
(1− 1
1 + 10−1700−2100
400
)= K
(1− 1
1 + 101
)
≈ K (1− 0.09)
= (0.91)K
An Example
Ti has rating 1700, Tj has rating 2100Suppose Ti beats Tj :
Reward for Ti = K
(1− 1
1 + 10−1700−2100
400
)= K
(1− 1
1 + 101
)≈ K (1− 0.09)
= (0.91)K
An Example
Ti has rating 1700, Tj has rating 2100Suppose Ti beats Tj :
Reward for Ti = K
(1− 1
1 + 10−1700−2100
400
)= K
(1− 1
1 + 101
)≈ K (1− 0.09)
= (0.91)K
An Example (Continued)
Similarly, interchanging the roles of i and j , we get if Tj beats Ti
then
Reward for Tj = K
(1− 1
1 + 10−2100−1700
400
)= K
(1− 1
1 + 10−1
)≈ K (1− 0.91)
= (0.09)K
An Example (Continued)
Similarly, interchanging the roles of i and j , we get if Tj beats Ti
then
Reward for Tj = K
(1− 1
1 + 10−2100−1700
400
)= K
(1− 1
1 + 10−1
)≈ K (1− 0.91)
= (0.09)K
Total expected Score is One
µij =1
1 + 10−αij(where, recall, αij = ((ri )old − (rj)old)
µji =1
1 + 10αij(since αji = −αij)
=⇒µij + µji =
1
1 + 10−αij+
1
1 + 10αij
= 1
Total expected Score is One
µij =1
1 + 10−αij
(where, recall, αij = ((ri )old − (rj)old)
µji =1
1 + 10αij(since αji = −αij)
=⇒µij + µji =
1
1 + 10−αij+
1
1 + 10αij
= 1
Total expected Score is One
µij =1
1 + 10−αij(where, recall, αij = ((ri )old − (rj)old)
µji =1
1 + 10αij
(since αji = −αij)
=⇒µij + µji =
1
1 + 10−αij+
1
1 + 10αij
= 1
Total expected Score is One
µij =1
1 + 10−αij(where, recall, αij = ((ri )old − (rj)old)
µji =1
1 + 10αij(since αji = −αij)
=⇒
µij + µji =1
1 + 10−αij+
1
1 + 10αij
= 1
Total expected Score is One
µij =1
1 + 10−αij(where, recall, αij = ((ri )old − (rj)old)
µji =1
1 + 10αij(since αji = −αij)
=⇒µij + µji =
1
1 + 10−αij+
1
1 + 10αij
= 1
The Total score is One
Clearly we have
Sij + Sji = 1
The Total score is One
Clearly we have
Sij + Sji = 1
The Total Rating is Invariant
If Ti and Tj play then only the ratings of Ti and Tj change afterthe game depending on the reward they get. Hence
The Total Rating is Invariant
If Ti and Tj play then only the ratings of Ti and Tj change afterthe game depending on the reward they get. Hence
n∑k=1
(rk)new =
∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new +
(ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new +
(rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}
+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old
+ K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
n∑k=1
(rk)new =∑k 6=i ,j
(rk)new + (ri )new + (rj)new
=∑k 6=i ,j
(rk)old
+ {(ri )old + K (Sij − µij)}+ {(rj)old + K (Sji − µji )}
=n∑
k=1
(rk)old + K {(Sij + Sji )− (µij + µji )}
=n∑
k=1
(rk)old + K (1− 1)
=n∑
k=1
(rk)old
SPREAD BETTING
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gambling
McNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.
He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.
His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.
He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in Chicago
While gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,
betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,
but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.
He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940s
Charles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.
He started the new method of trading and changed the way peoplebet
Charles K. McNeil (16 August 1903 - April 1981)
Inventor of the point spread in sports gamblingMcNeil earned a Master’s Degree from the University of Chicago.He then taught Mathematics at the Riverdale Country School inNew York and in Connecticut.His students included John F. Kennedy.He was also a securities analyst in ChicagoWhile gambling on the side, he developed the point spread,betting not on the probability of the final outcome,but on the expected difference in score.He eventually opened his own bookmaking operation in the 1940sCharles McNeil method is used today in different areas: anythingfrom basketball to poker.He started the new method of trading and changed the way peoplebet
Beat The Spread
Since the time of McNeil,“BEATING THE SPREADis the HOLY GRAIL of the betting world
Beat The Spread
Since the time of McNeil,
“BEATING THE SPREADis the HOLY GRAIL of the betting world
Beat The Spread
Since the time of McNeil,“BEATING THE SPREAD
is the HOLY GRAIL of the betting world
Beat The Spread
Since the time of McNeil,“BEATING THE SPREADis the HOLY GRAIL of the betting world
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game.
Thebookmaker announces the betting as follows:A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:
A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:A will beat B by 10 Points
Suppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
What does “Beating the spread mean?
Suppose two Teams A and B are playing a basketball game. Thebookmaker announces the betting as follows:A will beat B by 10 PointsSuppose the match score is as follows:
I CASE 1: A beats B with score line 58− 47
I CASE 2: A beats B with score line 61− 54
I CASE 3: A loses to B
My Bet on A
Suppose I bet on AIn Case 1, I win the bet because A beats B by more than 10 PointsIn Case 2, I lose the bet because even though A beats B it is notby more than 10 PointsIn Case 3, I lose the bet because A loses to B
My Bet on A
Suppose I bet on A
In Case 1, I win the bet because A beats B by more than 10 PointsIn Case 2, I lose the bet because even though A beats B it is notby more than 10 PointsIn Case 3, I lose the bet because A loses to B
My Bet on A
Suppose I bet on AIn Case 1, I win the bet because A beats B by more than 10 Points
In Case 2, I lose the bet because even though A beats B it is notby more than 10 PointsIn Case 3, I lose the bet because A loses to B
My Bet on A
Suppose I bet on AIn Case 1, I win the bet because A beats B by more than 10 PointsIn Case 2, I lose the bet because even though A beats B it is notby more than 10 Points
In Case 3, I lose the bet because A loses to B
My Bet on A
Suppose I bet on AIn Case 1, I win the bet because A beats B by more than 10 PointsIn Case 2, I lose the bet because even though A beats B it is notby more than 10 PointsIn Case 3, I lose the bet because A loses to B
My Bet on B
Suppose I bet on BIn Case 1 I lose because A has beaten B by more than 10 pointsIn Case 2 I win because even though B lost he has lost it by lessthan 10 pointsIn Case 3 I win because anyway B has won
My Bet on B
Suppose I bet on B
In Case 1 I lose because A has beaten B by more than 10 pointsIn Case 2 I win because even though B lost he has lost it by lessthan 10 pointsIn Case 3 I win because anyway B has won
My Bet on B
Suppose I bet on BIn Case 1 I lose because A has beaten B by more than 10 points
In Case 2 I win because even though B lost he has lost it by lessthan 10 pointsIn Case 3 I win because anyway B has won
My Bet on B
Suppose I bet on BIn Case 1 I lose because A has beaten B by more than 10 pointsIn Case 2 I win because even though B lost he has lost it by lessthan 10 points
In Case 3 I win because anyway B has won
My Bet on B
Suppose I bet on BIn Case 1 I lose because A has beaten B by more than 10 pointsIn Case 2 I win because even though B lost he has lost it by lessthan 10 pointsIn Case 3 I win because anyway B has won
Handling Ties
To handle ties the book makers usually give the spread as 7.5 or8.5 etc.
Handling Ties
To handle ties the book makers usually give the spread as 7.5 or8.5 etc.
Using the Spread For Rating
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,
For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other once
Let sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )
Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sji
Note that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −Sji
The Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:
S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n
(diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Spread matrix
Consider a tournament with n Teams T1, T2,· · · ,Tn,For simplicity assume each plays the other onceLet sij be the score of Ti against Tj
(Obvioulsiy sji denotes the score of Tj against Ti )Sij = sij − sjiNote that Sij = −SjiThe Spread Matrix S:S = (Sij)n×n (diagonal entries are defined to be zero)S is a SKEW SYMMETRIC MATRIX
The Rating Spread matrix
Suppose
x =
x1x2...xn
be any rating of the teams.The Rating Spread Matrix:X = (xij)n×n where xij = xi − xj
The Rating Spread matrix
Suppose
x =
x1x2...xn
be any rating of the teams.
The Rating Spread Matrix:X = (xij)n×n where xij = xi − xj
The Rating Spread matrix
Suppose
x =
x1x2...xn
be any rating of the teams.The Rating Spread Matrix:X = (xij)n×n where xij = xi − xj
Ideal Situation
The ideal situation is to get a rating system X such that
S = X
The best way is to find the error S − X and find “the” x thatminimzes this error
Ideal Situation
The ideal situation is to get a rating system X such that
S = X
The best way is to find the error S − X and find “the” x thatminimzes this error
Ideal Situation
The ideal situation is to get a rating system X such that
S = X
The best way is to find the error S − X and find “the” x thatminimzes this error
Ideal Situation
The ideal situation is to get a rating system X such that
S = X
The best way is to find the error S − X and find “the” x thatminimzes this error
How do we Quantify the Error?
For A an n× n real matrix we define the “FROBENIUS NORM”,
‖A‖F =
√√√√ n∑i=1
n∑j=1
(aij)2
So get a rating r such that
‖S −R‖ < ‖S − X‖for any other rating x 6= r
How do we Quantify the Error?
For A an n× n real matrix we define the “FROBENIUS NORM”,
‖A‖F =
√√√√ n∑i=1
n∑j=1
(aij)2
So get a rating r such that
‖S −R‖ < ‖S − X‖for any other rating x 6= r
How do we Quantify the Error?
For A an n× n real matrix we define the “FROBENIUS NORM”,
‖A‖F =
√√√√ n∑i=1
n∑j=1
(aij)2
So get a rating r such that
‖S −R‖ < ‖S − X‖for any other rating x 6= r
How do we Quantify the Error?
For A an n× n real matrix we define the “FROBENIUS NORM”,
‖A‖F =
√√√√ n∑i=1
n∑j=1
(aij)2
So get a rating r such that
‖S −R‖ < ‖S − X‖for any other rating x 6= r
How do we Quantify the Error?
For A an n× n real matrix we define the “FROBENIUS NORM”,
‖A‖F =
√√√√ n∑i=1
n∑j=1
(aij)2
So get a rating r such that
‖S −R‖ < ‖S − X‖for any other rating x 6= r
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given bythe “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix SIn simple language,(The Rating of Team Ti namely ri )=(Total Scores of Ti in all games) -(Total opponents’ score against Ti )
n
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given by
the “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix SIn simple language,(The Rating of Team Ti namely ri )=(Total Scores of Ti in all games) -(Total opponents’ score against Ti )
n
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given bythe “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix SIn simple language,(The Rating of Team Ti namely ri )=(Total Scores of Ti in all games) -(Total opponents’ score against Ti )
n
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given bythe “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix SIn simple language,(The Rating of Team Ti namely ri )=(Total Scores of Ti in all games) -(Total opponents’ score against Ti )
n
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given bythe “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix S
In simple language,(The Rating of Team Ti namely ri )=(Total Scores of Ti in all games) -(Total opponents’ score against Ti )
n
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given bythe “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix SIn simple language,
(The Rating of Team Ti namely ri )=(Total Scores of Ti in all games) -(Total opponents’ score against Ti )
n
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given bythe “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix SIn simple language,(The Rating of Team Ti namely ri )=
(Total Scores of Ti in all games) -(Total opponents’ score against Ti )n
Can we find such an r?
A little bit of simple Linear Algebra and calculus shows that suchan r exists and is given bythe “centroid” of the columns of S
r =1
n{S1 + S2 + · · ·+ Sn}
where Sj denotes the j-th column of the Spread matrix SIn simple language,(The Rating of Team Ti namely ri )=(Total Scores of Ti in all games) -(Total opponents’ score against Ti )
n
Conclusion
Tell Us The Truth Mark
It is not clear if in real life Mark Zuckerberg used Elo’s formula torate girls at Harvard!Only he knows!!
Tell Us The Truth Mark
It is not clear if in real life Mark Zuckerberg used Elo’s formula torate girls at Harvard!
Only he knows!!
Tell Us The Truth Mark
It is not clear if in real life Mark Zuckerberg used Elo’s formula torate girls at Harvard!Only he knows!!
Shakespeare
“Good Lord Boyet, my beauty, though but mean,Needs not the painted flourish of your praise:Beauty is bought by judgement of the eye,Not utter’d by base sale of chapmen’s tongues”(Love ′s Labours Lost, 1588)(Nor rated by heartless mathematicians’ formulae!!)
Shakespeare
“Good Lord Boyet, my beauty, though but mean,
Needs not the painted flourish of your praise:Beauty is bought by judgement of the eye,Not utter’d by base sale of chapmen’s tongues”(Love ′s Labours Lost, 1588)(Nor rated by heartless mathematicians’ formulae!!)
Shakespeare
“Good Lord Boyet, my beauty, though but mean,Needs not the painted flourish of your praise:
Beauty is bought by judgement of the eye,Not utter’d by base sale of chapmen’s tongues”(Love ′s Labours Lost, 1588)(Nor rated by heartless mathematicians’ formulae!!)
Shakespeare
“Good Lord Boyet, my beauty, though but mean,Needs not the painted flourish of your praise:Beauty is bought by judgement of the eye,
Not utter’d by base sale of chapmen’s tongues”(Love ′s Labours Lost, 1588)(Nor rated by heartless mathematicians’ formulae!!)
Shakespeare
“Good Lord Boyet, my beauty, though but mean,Needs not the painted flourish of your praise:Beauty is bought by judgement of the eye,Not utter’d by base sale of chapmen’s tongues”
(Love ′s Labours Lost, 1588)(Nor rated by heartless mathematicians’ formulae!!)
Shakespeare
“Good Lord Boyet, my beauty, though but mean,Needs not the painted flourish of your praise:Beauty is bought by judgement of the eye,Not utter’d by base sale of chapmen’s tongues”(Love ′s Labours Lost, 1588)
(Nor rated by heartless mathematicians’ formulae!!)
Shakespeare
“Good Lord Boyet, my beauty, though but mean,Needs not the painted flourish of your praise:Beauty is bought by judgement of the eye,Not utter’d by base sale of chapmen’s tongues”(Love ′s Labours Lost, 1588)(Nor rated by heartless mathematicians’ formulae!!)
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominionIs but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line“Beauty is in the eye of the beholder”
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:
“Beauty, like supreme dominionIs but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line“Beauty is in the eye of the beholder”
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominion
Is but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line“Beauty is in the eye of the beholder”
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominionIs but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line“Beauty is in the eye of the beholder”
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominionIs but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:
“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line“Beauty is in the eye of the beholder”
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominionIs but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line“Beauty is in the eye of the beholder”
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominionIs but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line
“Beauty is in the eye of the beholder”
Similar Views
Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominionIs but supported by opinion”
David Hume’s Essays, Moral and Political, 1742, include:“Beauty in things exists merely in the mind which contemplatesthem.”
The person who is widely credited with coining the saying in itscurrent form is Margaret Wolfe Hungerford (ne Hamilton), whowrote many books, often under the pseudonym of ’The Duchess’.In Molly Bawn, 1878, there’s the line“Beauty is in the eye of the beholder”
Beauty and the Beast
Poets: “Beauty of a girl cannot be ranked”Arrow: “Beastly politicians cannot be ranked”
Beauty and the Beast
Poets: “Beauty of a girl cannot be ranked”
Arrow: “Beastly politicians cannot be ranked”
Beauty and the Beast
Poets: “Beauty of a girl cannot be ranked”Arrow: “Beastly politicians cannot be ranked”
A Commercial
“There are some things money can’t buy,for everything else there is Mastercard”
“There are some things science can’t rate, rank or bet on,for everything else there is Mathematics”
A Commercial
“There are some things money can’t buy,
for everything else there is Mastercard”
“There are some things science can’t rate, rank or bet on,for everything else there is Mathematics”
A Commercial
“There are some things money can’t buy,for everything else there is Mastercard”
“There are some things science can’t rate, rank or bet on,for everything else there is Mathematics”
A Commercial
“There are some things money can’t buy,for everything else there is Mastercard”
“There are some things science can’t rate, rank or bet on,
for everything else there is Mathematics”
A Commercial
“There are some things money can’t buy,for everything else there is Mastercard”
“There are some things science can’t rate, rank or bet on,for everything else there is Mathematics”