Birnbaum: When log5 works and when it doesn’t

From SABR member Phil Birnbaum at Sabermetric Research on January 7, 2016:

Team A, with an overall winning percentage talent of .600, plays against a weaker Team B with an overall winning percentage of .450. What’s the probability that team A wins?

In the 1980s, Bill James created the “log5” method to answer that question. The formula is

P = (A – AB)/(A+B-2AB)

… where A is the talent level of team A winning (in this case, .600), and B is the talent level of team B (.450).

Plug in the numbers, and you get that team A has a .647 chance of winning against team B.

That makes sense: A is .600 against average teams. Since opponent B is worse than average, A should be better than .600.

Team B is .050 worse than average, so you’d kind of expect A to “inherit” those 50 points, to bring it to .650. And it does, almost. The final number is .647 instead of .650. The difference is because of diminishing returns — those “.050 lost wins” are what B loses to *average* teams because it’s bad. Because A is better than average, it would have got some of those .050 wins anyway because it’s good, so B can’t “lose them again” no matter how bad it is.

In baseball, the log5 formula has been proven to work very well.