The Angry Statistician

In a previous entry I demonstrated one simple way to estimate an exponent for the Pythagorean win expectation . Another nice consequence of a Pythagorean win expectation formula is that it also makes it simple to estimate the run value of a win in baseball, the point value of a win in basketball, the goal value of a win in hockey etc. Let our Pythagorean win expectation formula be \[ w=\frac{P^e}{P^e+1},\] where \(w\) is the win fraction expectation, \(P\) is runs/allowed (or similar) and \(e\) is the Pythagorean exponent. How do we get an estimate for the run value of a win? The expected number of games won in a season with \(g\) games is \[W = g\cdot w = g\cdot \frac{P^e}{P^e+1},\] so for one estimate we only need to compute the value of the partial derivative \(\frac{\partial W}{\partial P}\) at \(P=1\). Note that \[ W = g\left( 1-\frac{1}{P^e+1}\right), \] and so \[ \frac{\partial W}{\partial P} = g\frac{eP^{e-1}}{(P^e+1)^2}\] and it follows \[ \frac{\partial W}{\partial P}(P=1) ...

Given the number of runs scored and runs allowed by a baseball team, what's a good estimate for that team's win fraction? Bill James famously came up with what he called the " Pythagorean expectation " \[w = \frac{R^2}{R^2 + A^2},\] which can also be written as \[w = \frac{{(R/A)}^2}{{(R/A)}^2 + 1}.\] More generally, if team \(i\) scores \(R_i\) and allows \(A_i\) runs, the Pythagorean estimate for the probability of team \(1\) beating team \(2\) is \[w = \frac{{(R_1/A_1)}^2}{{(R_1/A_1)}^2 + (R_2/A_2)^2}.\] We can see that the estimate of the team's win fraction is a consequence of this, as an average team would by definition have \(R_2 = A_2\). Now, there's nothing magical about the exponent being 2; it's a coincidence, and in fact is not even the "best" exponent. But what's a good way to estimate the exponent? Note the structural similarity of this win probability estimator and the Bradley-Terry estimator \[ w = \frac{P_1}{P_1+P_2}.\] Here ...