Nothing original here, just an interesting way to visualize it.
The “win value” of a run refers to how adding one affects the predicted number of wins at seasons end. The “rule of thumb” is that every 10 runs—10 more, 10 less, or any combination that adds to 10—is worth 1 win.
The Sabermetrics Nobel Prize-winning formula for predicting wins based on runs is James’ Pythagorean Win Pct. Formula. It holds:
The formula has an R2 just a tad below 0.90 when applied to all the AL/NL teams since 1900. Nice!
There are lots of interesting things to say about why it works (hint: it has to do with likelihood ratios), and how it can be made to work better by futzing with the value of the exponent. As run scoring increases, so does the exponent that makes the formula work best (it’s 14 in basketball). As the run environment in baseball has variously contracted and expanded, exponents from around 1.75 to 2.0 have been empirically best, and a value of 1.8-something is more accurate than 2 across the span of AL/NL seasons. But by such a small amount (< 1% in variance explained overall or in any era), that it’s just not worth the bother.
The main reason JPT works better than anything proposed before it is that it treats the win value of runs as nonlinear—hence the exponent. The more games over .500 a team is, the more runs it has to score to add a win; the more below, the fewer. Not if they are playing a game against each other, obviously, but if they’ve already played a bunch (against themselves or anyone else) and you are trying to figure out how many more they would have won by increasing their runs scored/allowed differential.
This is a bit of a simplification. The deciding value isn’t win percentage but the runs-scored-runs-allowed differential. Using calculus, you can figure out that the benefit of 10 runs goes up as the differential declines until it peaks at roughly -205–at which point, the value of 10 runs starts to drop again. But a -205 differential between runs scored and allowed is worse than the average last place team experiences, and even if a team got outscored by substantially more than that, a 10-run improvement would still buy it more “win value” than it would the league’s best (which would derive less and less value from 10 runs as its differential increased–all the way to infinity, more or less). Pretty cool.
Visualizing the impact of this non-linearity along a practically sensible range reveals some interesting things. Like how poor your predictions will be if you rely on the “10 runs = 1 win” heuristic to try to predict team records.
How much 10 runs is worth varies about 30% across teams by seasons end. That’s what the plot on the left shows.
And what the histogram on the right says is that if you apply it to all the teams over the course of AL/NL history, the “10 = 1” heuristic will overestimate wins 2/3 of the time and by 10% on average.
Of course, no one said you should use it this way, and it’s actually a perfectly serviceable rule of thumb if you don’t have a spreadsheet on hand (as it were) to do JPT calculations.
I was pondering all of this in connection with my goal to figure out where the R2 body is buried in the translation of runs to WAR. The prevailing WAR systems don’t use the “10 = 1” rule, but they do use a uniform “win value” rate every year when they translate their estimates of players’ run production or suppression into team wins. (Sean Smith doesn’t; but I’m pretty sure pretty much everything he does differently from FanGraphs, Baseball Reference, et al., works better.)
Anyway, here’s a cool implication of the Jamesian Pythagorean Win Pct formula: you can increase MLB “utility” by transferring runs from strong teams to weak ones!
If, say, the uber rich 1927 Yankees (runs/runs allowed Δ +376; JPT predicted record 112-42, actual 110-44-1) decided to donate, oh, 100 runs to the penurious 1962 Mets (Δ -331; 48-154 predicted, 40-120-1 actual), the Mets would be predicted to win 12 more games and the Yankees to win only 7 fewer.
So there’d be a net gain in “win utility” of 5—and the Yankees would still win the pennant by double digits.
Something’s off here, obviously. No matter how we redistributed runs, in the real world wins and losses would have to balance out (at least if the run transfers were made within seasons). They don’t under JPT—but that has no noticeable impact on its value as a predictor.
A place where things might actually work out this way, though, is in the domain of human income. The science of economics tells us that the enjoyment of money is nonlinear: the “diminishing marginal utility of wealth” means that a rich person gets less happiness out of every additional $1 than a poor person does.
So if we take $1 from the rich person and give it to the poor one, the utility gain of the latter will more than offset the utility loss of the former. Wealth redistribution increases net happiness (if we think dollars are a good proxy for utility; mainstream economists do).
Now this definitely links back to JPT. If the Yankees were forced to give, say, $100 million to the strapped Pittsburgh Pirates, the former’s performance, and their fans’ enjoyment, wouldn’t fall nearly as much as the latter’s performance, and resulting fan enjoyment, would rise!
Anyone else think Bernie Sanders should be MLB Commissioner?