Log5

Log5 is a method of estimating the probability that team A will win a game against team B, based on the odds ratio between the estimated winning probability of Team A and Team B against a larger set of teams.

Let ${\displaystyle p_{A}}$ and ${\displaystyle p_{B}}$ be the average winning probabilities of team A and B and let ${\displaystyle p_{A,B}}$ be the probability of team A winning over team B, then we have the following odds ratio equation

${\displaystyle {\frac {p_{A,B}}{1-p_{A,B}}}={\frac {p_{A}}{1-p_{A}}}\times {\frac {1-p_{B}}{p_{B}}}.}$

One can then solve

${\displaystyle p_{A,B}={\frac {p_{A}-p_{A}\times p_{B}}{p_{A}+p_{B}-2\times p_{A}\times p_{B}}}.}$

The name Log5 is due to Bill James[1] but the method of using odds ratios in this way dates back much further. This is in effect a logistic rating model and is therefore equivalent to the Bradley–Terry_model used for paired comparisons, the Elo rating system used in chess and the Rasch model used in the analysis of categorical data.[2]

A few notable properties exist:

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