## Wednesday, July 14, 2010

### Win Shares and Replacement Level

For those of you who have not seen it, the new site The Baseball Gauge (operated by "Railsplitter") has an impressive database of Win Shares, Win Shares Above Bench, and WAR figures for the entirety of major league history. There are a lot of different ways to view the data and I suggest you check it out.

The methodology used to calculate WAR on that site has been discussed at The Book Blog, and I believe that the method used to figure Fielding WAR is based on a faulty conversion of Win Shares to WAR. It's an easy trap to fall into with Win Shares.

To illustrate the Win Shares replacement trap, I'll focus on Offensive Win Shares (OWS is much more straightforward and easily comparable to other metrics like RAR that Fielding WS), but the principle applies to hitting, pitching, and fielding WS.

Suppose you have a league in which there are 25 outs per game and the average team scores 5 runs per game. An 81-81 team that scores and allows 5 runs per game in this league will get credit for 40.5 OWS (assuming the use of 50/150 as the WS margins rather than 52/152 in the original version, and ignoring James' scalar of "3", which is cosmetic and would only serve to complicate comparisons to other metrics in this case).

Suppose this team has no players who contribute "sub-marginal" runs. Then the Win Shares margin is at 5*.5 = 2.5 RG, and this team will have a total of 2.5*162 = 405 marginal runs, meaning 405/40.5 = 10 marginal runs/win. Suppose that this team has four players whom we are interested in, all making 350 outs. One creates 100 runs, another 80 runs, another 70 runs (which happens to be league average), and the last 57.75 (which is 82.5% of the league average, which is defined as the replacement level in Railsplitter's system).

A marginal player will have 2.5 RG, which is 35 runs/350 outs, so we simply subtract 35 to find marginal runs and 57.75 to find RAR. To convert to Win Shares or WAR, we can divide by 10 runs per win:

So we now know exactly what WAR should be for each player. Can we produce the same results by looking only at Win Shares? Railsplitter's approach would be to find the WS produced by an average player (we know this to be 3.5) and multiply by the replacement level (.825): 3.5*.825 = 2.8875. So, we can subtract that 2.8875 from each player's OWS to get our second WAR estimate (WARx):

As you can see, this doesn't work. The differences between players are constant; Alpha is still two wins better than Bravo, who is one win better than Charlie, who is 1.225 wins better than Eko. But the scale is off; everyone is less valuable than we know them to be. Eko, who defines replacement level, is sub-replacement level. Everyone is .6125 WAR below where they should be.

This happens because we've now taken 82.5% of RC above 50% rather than what the replacement level is supposed to represent--82.5% of absolute RC.

This does not demonstrate that Win Shares cannot be converted to WAR (although it will fail for sub-marginal players if they are zeroed out, which is a feature of the original system but can be waived). It just shows that you have to figure a different percentage other than the one you are using.

Here, we know that the ratio of Eko/Charlie is equal to the replacement/average ratio, and it is 65% (2.275/3.5). How can we calculate this without going through the example?

Take the difference between the replacement level (.825) and the margin (.50), and divide by one minus the margin (1 - .5 = .5). This gives (.825 - .5)/(1 - .5) = .65, which we know is the right answer.

Suppose our replacement level was 70% rather than 82.5%. The replacement level in terms of win shares would be (.7 - .5)/(1 - .5) = .4. A different replacement level player, Foxtrot, would create 49 runs, and as you can see, using 40% as the new OWS replacement values matches WAR for this case as well:

One can convert Win Shares to a measure of value above replacement; you just have to keep in mind that the replacement level will be expressed differently as a ratio of win shares than as a ratio of runs created.

ADDENDUM: I forgot to show you what the actual effective replacement level is when one applies 82.5% directly to OWS. Solve this equation for x:

.825 = (x - .5)/(1 - .5)

x = .9125. Applying .825 directly to Win Shares is the equivalent of a "normal" replacement level of .9125, which is roughly a .455 W%--a very high effective replacement level.