# Log5 Method and Juggling Starters

BASEBALL

Someone at the motownsports message board wondered what the difference would be if you avoided an ace to ace match up on opening day, or in a play off series.  This led me to rediscover the log5 method (the link goes to an article explaining how to use this to estimate chances a team wins a playoff series) introduced by Bill James his 1981 Abstract.

Let’s suppose the aces on two teams are equally good and each team normally wins 70% of the games they start.  Let’s assume the second best pitchers are equally good and they each win 60% of the games.  I’m going to ignore the 4% advantage a home team typically has in an MLB game.

If you go ace versus ace and second best pitcher versus second best pitcher then each game is equally likely to be won by either team.  There is a 25% for either team of winning two games, and a 50% chance of a split.

But how do you compare two good pitchers?  If one pitcher wins 70% of the time and the other wins 60% of the time against a typical team then how do they do against one another?

You can estimate this using the log5 method.  (The link goes to an article that makes the rule intuitive to me.)  Let p be the chance the first pitcher typically wins (p = 0.7 here).  Let q be the chance the second pitcher typically wins (q = 0.6 here.)  The chance the team with the better pitcher wins is given by (p(1-q))/(p(1-q)+q(1-p)).

For our scenario (0.7)(0.4)/((0.7)(0.4)+(0.6)(0.3)) = 28/46 = 61%.   The Ace will beat the second best pitcher 61% of the time.  (Well his team will win- he may not get the W.)

So if you juggle the starters the chance of a sweep becomes (0.69)(0.31) = 21.4% for either team.  The chance of a split has increased to 57.2%.

So with a five game series if all pitchers on both teams are rested maybe the road team should juggle its starters to increase the chance of winning at least one.  Then the team is more likely to get two home playoff games and the associated revenue.

The log5 method assumes that the overall win percentage in a league is 0.500.  With adjustments in the formula you can account for different means and thus use it for batter-pitcher match ups as well.