Log5

Log 5 is a formula invented by Bill James 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.

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 as $\frac{p_{A,B}}{q_{A,B}} = \frac{p_A}{q_A}\times \frac{q_B}{p_B}.$

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