Pythagorean expectations, Pythagenports, and Pythagenpats – a bunch of mumbo jumbo

It seems like everywhere you turn in sports statistics, someone is talking about Pythagorean expectations. The Pythagorean expectation is a formula created by Bill James to estimate team quality in baseball, with runs scored and runs allowed as inputs. Proponents argue that luck plays a big part in close games, making point totals a better measure of team quality than wins. The formula is

Pythagorean = Runs scored^c / (Runs scored^c + Runs allowed^c),

where c is some exponent (usually greater than one) that can be calibrated. This kinda looks like the old Pythagorean formula from grade school (hence the name), though not really. The Pythagorean rises (at a decreasing rate) as you score more and drops (at a decreasing rate) as your opponents score more. In other words, the Pythagorean rewards teams for blowouts and punishes them for getting spanked, since these scoring outcomes may reflect team quality. If you score the same amount as your opponents, your Pythagorean is 0.5. The max is one and the min is zero, like a winning percentage. “Pythagorean wins” are given by the Pythagorean multiplied by the number of games.

Over time, this formula has been exported to basketball and football, and probably other sports. My buddy Adrian the Canadian sent me this week’s DVOA update at Football Outsiders, in which Aaron Schatz informs us that Football Outsiders has upgraded from the Pythagorean expectation to the “Pythagenport,” and Baseball Prospectus (the original Pythagenporters) has moved on to the Pythagenpat! These are identical to the Pythagorean, but allow the exponent c to depend on the number of runs scored and/or the teams involved.

Maybe this sounds reasonable to you, but these formulas make my head hurt. The Pythagenport even gets a logarithm involved. Where do these crazy functional forms come from? The intuition for these stats is simple: