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

p A , B 1 − p A , B = p A 1 − p A × 1 − p B p B . {\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

p A , B = p A − p A × p B p A + p B − 2 × p A × p B . {\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]

The following notable properties exist:

• If p A = 1 {\displaystyle p_{A}=1} , Log5 will give A a 100% chance of victory.
• If p A = 0 {\displaystyle p_{A}=0} , Log5 will give A a 0% chance of victory.
• If p A = p B {\displaystyle p_{A}=p_{B}} , Log5 will give each team a 50% chance of victory.
• If p A = 1 / 2 {\displaystyle p_{A}=1/2} , Log5 will give A a 1 − p B {\displaystyle 1-p_{B}} probability of victory.
• If p A + p B = 1 {\displaystyle p_{A}+p_{B}=1} , Log5 will give A a ( p A ) 2 / ( ( p A ) 2 + ( p B ) 2 ) {\displaystyle (p_{A})^{2}/((p_{A})^{2}+(p_{B})^{2})} chance of victory.

In addition to head-to-head winning probability, a general formula can be applied to calculate head-to-head probability of outcomes such as batting average in baseball.^[3]

Sticking with the batting average example, let p B {\displaystyle p_{B}} be the batter's batting average (probability of getting a hit), and let p P {\displaystyle p_{P}} be the pitcher's batting average against (probability of allowing a hit). Let p L {\displaystyle p_{L}} be the league-wide batting average (probability of anyone getting a hit) and let p B , P {\displaystyle p_{B,P}} be the probability of batter B getting a hit against pitcher P.

p B , P = p B × p P p L p B × p P p L + ( 1 − p B ) × 1 − p P 1 − p L . {\displaystyle p_{B,P}={\frac {\frac {p_{B}\times p_{P}}{p_{L}}}{{\frac {p_{B}\times p_{P}}{p_{L}}}+(1-p_{B})\times {\frac {1-p_{P}}{1-p_{L}}}}}.}

Or simplified as

p B , P = p B × p P × ( 1 − p L ) ( p B × p P ) − ( p L × p B ) − ( p L × p P ) + p L . {\displaystyle p_{B,P}={\frac {p_{B}\times p_{P}\times (1-p_{L})}{(p_{B}\times p_{P})-(p_{L}\times p_{B})-(p_{L}\times p_{P})+p_{L}}}.}