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If you had to pick one number over the history of baseball to convert runs into wins, it would be 10. Across 140 seasons, every 10 runs a player adds or subtracts adds about one win. So if we were doing this by hand, we would use 10 runs per win and be done with it. We are using computers, though, so we can compute more exact values for every player and season.

With update 2.1, we are now on our third method for converting runs to wins. Version 1.0 used a R/W estimation based on the league's runs per game. With Version 2.0, we introduced a variable runs-to-wins estimation which varied with the leagues runs per game and the player's runs added or subtracted from average per game. This was an appropriate step, but the approximation we were using broke down for extreme players like 1985 Gooden, 2009 Greinke or 1999-2004 Bonds. We noticed this issue, but didn't quite follow through on realizing it was a problem until after posting the numbers. Version 2.1, fixed this issue, but it complicates the calculation considerably, so we've split the explanation onto a separate page.

The league context obviously plays a major role in how many runs translate into a win. A typical Babe Ruth season is going to win you more games in 1912 than it would in 1933. Teams scored many fewer runs in 1912, so the value of each run is heightened.



The runs-to-wins estimators are related to the pythagorean formula for baseball.


This is using an exponent of two, but extensive research by a sabermetrician who goes by the nom de plume of Patriot has shown that using a variable exponent based on the run-scoring environment of the league does better. So we get a better estimate called:


Our Previous Runs to Wins calculations.






For Pitchers, we do something similar. One twist is that we compute outs recorded per game for the season (capped at 26.8) and then pad the remainder of the game with league average run prevention. This way we more accurately model how the player affects the runs-to-win conversion. We still start with Runs/Win = 2 * (RunsPerGameForBothTeams)^.715.




Version 2.1, the underlying system is the that of runs added and subtracted changing the number of wins for a team. This is modeled very well by the PythagenPat formula cited earlier. To apply this here, we find the number of runs per game the player is better or worse than average. In this case, for Halladay, we have 59.7 runs in 32 games or 1.866 runs/game. So our exponent X = (53.6*.15447 - 1.866).285=1.698. And then we set RS=4.14 and RA=4.14-1.866 = 2.27 and plug into the PythagenPat formula, W-L% = .735. So an otherwise average team facing an average team should win 73% of the time with 2011 Halladay starting for them. Then to get wins above average: WAA = (.735-.500)*32 games = 7.52 WAA. We then do a similar calculation for a replacement player's runs allowed to get the difference between average and replacement level and add that here to get WAR. For Halladay's # of innings pitched and run-scoring environment this is 1.7 wins, so WAR = 1.7 + 7.52 = 9.22 WAR.

Halladay WAR: version 1.0, 8.57 WAR; version 2.0, 10.25 WAR; version 2.1, 9.22 WAR. As you can see this estimate makes a big difference, and with 2.1, our estimate is now very tight and technically correct.

A couple nice things about this new method. The W-L% is a display of WAR as a rate stat. Note this is just for games the player plays in, so a top starter will always have a higher W-L% than a top position player, but position players are in a lot more games. If you want the team's W-L% per 162-game season, add in enough .500 games to get the season total. One other nice thing is that it handles correctly the effect of the player's runs on run environment, and the differences between high and low run scoring environments. We've ditched the approximation for the real thing.

