Log5

Log 5 is a formula invented by Bill James^[1] to estimate the probability that team A will win a game, based on the true winning percentage of Team A and Team B. It is equivalent to the Bradley-Terry-Luce model used for paired comparisons, the Elo rating system used in chess and the Rasch model used in the analysis of categorical data.^[2]

Let $p_{i}$ be the fraction of games won by team $i$ and also let $q_{i}=1-p_{i}$ be the fraction of games lost by team $i$ . The Log5 estimate for the probability of A defeating B is $p_{A,B}={\frac {p_{A}-p_{A}\times p_{B}}{p_{A}+p_{B}-2\times p_{A}\times p_{B}}}$ .

A few notable properties

If $p_{A}=1$ , Log5 will always give A a 100% chance of victory
If $p_{A}=0$ , Log5 will always give A a 0% chance of victory
If $p_{A}=p_{B}$ , Log5 will always return a 50% chance of victory for either side
If $p_{A}=1/2$ , Log5 will give A a $1-p_{B}$ probability of victory.

It may also be conveniently rewritten using the odds ratio^[2] as ${\frac {p_{A,B}}{q_{A,B}}}={\frac {p_{A}}{q_{A}}}\times {\frac {q_{B}}{p_{B}}}.$

Here as before $q_{A,B}=1-p_{A,B}$ .

References

↑ "Chancesis: The Origins of Log5". Archived from the original on April 12, 2012. Retrieved 2013-03-07.
1 2 "Baseball, Chess, Psychology and Pychometrics: Everyone Uses the Same Damn Rating System". Retrieved 2013-12-29.

Sports rating systems

Computer models	Advanced Football Analytics Albrecht Matrix Hybrid ARGH Power Ratings Dickinson System Pomeroy College Basketball Ratings Rating Percentage Index (RPI) Sonny Moore Power Ratings TrueSkill

Methods and concepts	Elo rating system Home advantage Log5 Pythagorean expectation Sabermetrics Strength of schedule Win probability

Polls and opinion	AP Poll FWAA-NFF Grantland Rice Super 16 Poll Harris Interactive College Football Poll Legends Poll NAIA Coaches' Poll USA Today Coaches' Poll

People	John Hollinger Bill James Kenneth Massey Ken Pomeroy Jeff Sagarin Nate Silver Peter Wolfe

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