Rating, Ranking, Betting and Mathematics R VITTAL RAO

Rating, Ranking, Betting and Mathematics

R VITTAL RAO

[email protected] .ernet.in

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


Ea =1

1 + 10 (Rb−Ra)400

Eb =1

1 + 10 (Ra−Rb)400


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:“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:“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:“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:“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


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


The Social Network



The Social Network


CAST:

Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)

The Social Network


CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)

Andrew Garfield as Eduardo Saverin (cofounder)Justin Timblelake as Sean Parker (cofounder)

The Social Network


CAST:Jesse Eisenberg as Mark Zuckerberg (founder of Facebook)Andrew Garfield as Eduardo Saverin (cofounder)

Justin Timblelake as Sean Parker (cofounder)

The Social Network



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




I Optimization


I Game Theory




I Optimization


I Game Theory




I Optimization


I Game Theory




I Optimization


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)


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)


The four criteria seem to be obvious or self evident - or just PlainCommon Sense

but his result is notArrow’s Impossibility Theorem:






The four criteria seem to be obvious or self evident - or just PlainCommon Sensebut his result is not

Arrow’s Impossibility Theorem:









































BCS (Bowl Championship Series)

In place since 1998Since its inception in 1998, the NCAA’s Bowl Championship Serieshas weathered criticism from nearly all directions.


In place since 1998

Since its inception in 1998, the NCAA’s Bowl Championship Serieshas weathered criticism from nearly all directions.


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 ì games then define

ri =1 + wi

2 + niNote:We have wi + ì = niIf Ti has played 2k games and won exactly half of these, namely kgames, then

ri =1 + k

2 + 2k

=1

2




ri =1 + wi


ri =1 + k

2 + 2k

=1

2



Dynamically adjust the ranking as the turnament progresses

At some stage if Team Ti has played ni games and won wi gamesand lost ì games then define

ri =1 + wi


ri =1 + k

2 + 2k

=1

2



Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games

and won wi gamesand lost ì games then define

ri =1 + wi


ri =1 + k

2 + 2k

=1

2



Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi games

and lost ì games then define

ri =1 + wi


ri =1 + k

2 + 2k

=1

2



Dynamically adjust the ranking as the turnament progressesAt some stage if Team Ti has played ni games and won wi gamesand lost ì games

then define

ri =1 + wi


ri =1 + k

2 + 2k

=1

2




ri =1 + wi

2 + ni

Note:We have wi + ì = niIf Ti has played 2k games and won exactly half of these, namely kgames, then

ri =1 + k

2 + 2k

=1

2




ri =1 + wi

2 + niNote:

We have wi + ì = niIf Ti has played 2k games and won exactly half of these, namely kgames, then

ri =1 + k

2 + 2k

=1

2




ri =1 + wi

2 + niNote:We have wi + ì = ni

If Ti has played 2k games and won exactly half of these, namely kgames, then

ri =1 + k

2 + 2k

=1

2




ri =1 + wi


ri =1 + k

2 + 2k

=1

2




ri =1 + wi


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


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


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


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


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


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


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


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


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.


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.


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.


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.


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?


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?


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?


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?


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?


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,



S =


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,



S =


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,



S =


Elo’s Hypothesis




S =


Elo’s Hypothesis




S =


Elo’s Hypothesis



where K is the constant of proportion.

Elo originally set K = 10For chess games,

S =


Elo’s Hypothesis



where K is the constant of proportion.Elo originally set K = 10

For chess games,

S =


Elo’s Hypothesis




S =


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


Sij =





µij = L

(dij

400

)where

L(x) =1

1 + 10−x

µij =1

1 + 10−((ri )old−(rj )old)

400

Current Version


Sij =





µij = L

(dij

400

)where

L(x) =1

1 + 10−x

µij =1

1 + 10−((ri )old−(rj )old)

400

Current Version


Sij =





µij = L

(dij

400

)where

L(x) =1

1 + 10−x

µij =1

1 + 10−((ri )old−(rj )old)

400

Current Version


Sij =





µij = L

(dij

400

)where

L(x) =1

1 + 10−x

µij =1

1 + 10−((ri )old−(rj )old)

400

Current Version


Sij =





µij = L

(dij

400

)where

L(x) =1

1 + 10−x

µij =1

1 + 10−((ri )old−(rj )old)

400

Current Version


Sij =





µij = L

(dij

400

)where

L(x) =1

1 + 10−x

µij =1

1 + 10−((ri )old−(rj )old)

400

Current Version


Sij =





µij = L

(dij

400

)where

L(x) =1

1 + 10−x

µij =1

1 + 10−((ri )old−(rj )old)

400

Current Version


Sij =





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



Current Version


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


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


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



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


Consider a game in which a weaker player Ti plays against astronger player Tj .

This means at the time the game starts

ri < rj




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



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



ri < rj




ri < rj


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


= K

{1− 1

1 + 10−αij

}where


400

Similarly


= K

{1− 1

1 + 10−αji

}where


400

The Rewards


= K

{1− 1

1 + 10−αij

}where


400

Similarly


= K

{1− 1

1 + 10−αji

}where


400

The Rewards


= K

{1− 1

1 + 10−αij

}where


400

Similarly


= K

{1− 1

1 + 10−αji

}where


400

The Rewards


= K

{1− 1

1 + 10−αij

}where


400

Similarly


= K

{1− 1

1 + 10−αji

}where


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


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


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


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


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


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


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


µij =1

1 + 10−αij

(where, recall, αij = ((ri )old − (rj)old)

µji =1


=⇒µij + µji =

1

1 + 10−αij+

1

1 + 10αij

= 1


µij =1


µji =1

1 + 10αij

(since αji = −αij)

=⇒µij + µji =

1

1 + 10−αij+

1

1 + 10αij

= 1


µij =1


µji =1


=⇒

µij + µji =1

1 + 10−αij+

1

1 + 10αij

= 1


µij =1


µji =1


=⇒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


=n∑

k=1


=n∑

k=1

(rk)old + K (1− 1)

=n∑

k=1

(rk)old

n∑k=1


(rk)new + (ri )new +

(rj)new

=∑k 6=i ,j

(rk)old


=n∑

k=1


=n∑

k=1

(rk)old + K (1− 1)

=n∑

k=1

(rk)old

n∑k=1



=∑k 6=i ,j

(rk)old


=n∑

k=1


=n∑

k=1

(rk)old + K (1− 1)

=n∑

k=1

(rk)old

n∑k=1



=∑k 6=i ,j

(rk)old

+ {(ri )old + K (Sij − µij)}

+ {(rj)old + K (Sji − µji )}

=n∑

k=1


=n∑

k=1

(rk)old + K (1− 1)

=n∑

k=1

(rk)old

n∑k=1



=∑k 6=i ,j

(rk)old


=n∑

k=1


=n∑

k=1

(rk)old + K (1− 1)

=n∑

k=1

(rk)old

n∑k=1



=∑k 6=i ,j

(rk)old


=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



=∑k 6=i ,j

(rk)old


=n∑

k=1


=n∑

k=1

(rk)old + K (1− 1)

=n∑

k=1

(rk)old

n∑k=1



=∑k 6=i ,j

(rk)old


=n∑

k=1


=n∑

k=1

(rk)old + K (1− 1)

=n∑

k=1

(rk)old

n∑k=1



=∑k 6=i ,j

(rk)old


=n∑

k=1


=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


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


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


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


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


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


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


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


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


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


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


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 3: A loses to B


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:





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:





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:





























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


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


The Spread matrix



The Spread matrix


(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


(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


(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


(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


(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



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


Suppose

x =

x1x2...xn

be any rating of the teams.

The Rating Spread Matrix:X = (xij)n×n where xij = xi − xj


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


S = X


Ideal Situation


S = X


Ideal Situation


S = X


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



‖A‖F =

√√√√ n∑i=1

n∑j=1

(aij)2





‖A‖F =

√√√√ n∑i=1

n∑j=1

(aij)2





‖A‖F =

√√√√ n∑i=1

n∑j=1

(aij)2





‖A‖F =

√√√√ n∑i=1

n∑j=1

(aij)2



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


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}


n



r =1

n{S1 + S2 + · · ·+ Sn}


n



r =1

n{S1 + S2 + · · ·+ Sn}


n



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



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



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



r =1

n{S1 + S2 + · · ·+ Sn}


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!!


It is not clear if in real life Mark Zuckerberg used Elo’s formula torate girls at Harvard!

Only he knows!!


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”



Similar Views

Benjamin Franklin, in Poor Richard’s Almanack, 1741, wrote:“Beauty, like supreme dominion

Is but supported by opinion”



Similar Views




Similar Views


David Hume’s Essays, Moral and Political, 1742, include:

“Beauty in things exists merely in the mind which contemplatesthem.”


Similar Views




Similar Views



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




Beauty and the Beast

Poets: “Beauty of a girl cannot be ranked”Arrow: “Beastly politicians cannot be ranked”


Poets: “Beauty of a girl cannot be ranked”

Arrow: “Beastly politicians cannot be ranked”


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”


A Commercial



A Commercial


“There are some things science can’t rate, rank or bet on,

for everything else there is Mathematics”

A Commercial



Documents

Rating, Ranking, Betting and Mathematics R VITTAL RAO