Rethinking Pythagorean Winning Percentage

Baseball-reference.com describes Pythagorean winning percentage as "an estimate of a team's winning percentage given their runs scored and runs allowed. Developed by Bill James, it can tell you when teams were a bit lucky or unlucky." Click through for the formula.

An example is the 2009 Washington Nationals. The team finished 59-103. However, the Nationals scored 710 runs while allowing 874 runs, so their Pythagorean W-L is 66-96. The Baseball Reference folks would thus argue that bad luck cost the Nats 7 wins, which is a substantial number in comparison to their actual number of wins.

However, while winning is a function of runs scored and runs allowed, runs in the late innings are also partially a function of runs in earlier innings. Your team's run expectations in a given inning are highly sensitive to what's happened earler in the game.

Consider a game where your team is tied in the top of the seventh. And another where you're losing (or winning) by 10 runs. On average, you'll score more runs in the latter scenario, because both teams will insert inferior relief pitchers, saving their star bullpen guys for another day. Hell, they might even have a position player pitch. Losing by 15 or 5 is just as bad as losing by 10, so it makes sense to conserve baseball resources, even at the expense of the final score in a hopeless game. Any cheap runs you score in such situations won't do much to improve your winning percentage, but Pythagorean winning percentage has no way to account for this.

On the other hand, in a tied game, the other team will do everything possible to prevent you from scoring, because a run or two could mean the ballgame.

Of course, teams also conserve offensive resources in blowouts. Pinch runners and defensive replacements often swap in for the likes of Adam Dunn or Barry Bonds. I suppose it's possible a weaker offense could offset the effect of inferior pitching, or even trump it.

Additionally, batters could be trying less hard in blowouts, just wanting to get the game over with, but I don't think there's any evidence to support this (why not pad your batting average against bad pitchers?).