Explaining Bill James' Win Shares

I have to admit, the first time I saw his formula, I cringed. I thought he completely missed the boat. Then after brainstorming with my gang at Fanhome, I came to realize that his concept is actually sound. He still makes one glaring error, and a few minor ones, but we'll get to those later.

His formula
James uses the following formula to come up with "win shares" win% for a team.
RS - LG/2 = marginal off runs = Off (where LG is the league average runs scored)
1.5 * LG - RA = marginal def runs = Def
(Off + Def) / (2 * LG) = win%
Expanding all this and we get
(RS - LG/2) + (1.5 * LG - RA) / (2 * LG) = win%
which filters down to
(RS - RA) / (2 * LG) + .5 = win%

That's all I want to say about that for this article.

Working from the ground-up

Let's say you have a team that scores 5 runs and allows 5 runs in a 5-run league. Obviously, this team will win 50.0% of their games (or 81 wins), and each of the offense and defense will get equal credit for this, or 40.5 wins each. Now, we also know that a team that scores about 2.5 runs per game and allows 5.0 runs per game will end up with about a 25.0% winning record (or 40.5 wins). (see http://www.geocities.com/tmasc/winsrpg.txt)

Now, the question is, how do you split this up into offense and defense? Well, we know that the "controlled variable", the defense in this case, was worth 40.5 wins, when they were allowing 5 runs per game. Since we didn't change this part of the team, then the defense should still be worth 40.5 wins. Since the team overall is worth 40.5 wins, then it stands to reason that the missing part, the part that we changed, the offense of 2.5 runs per game, is worth ZERO wins. This is why he is using "marginal runs over" 2.5 (or 50% of the league average). That is, 2.5 runs will cause a change to .25 wins, or 10 runs / wins.

So, let's come up with our chart
Runs Scored per gameRuns Allowed per gameLeague Runs per gameRS - LGLG - RAWin%Win% - .500

As you can see, the first loss of 0.2 runs scored (going from 5 runs to 4.8 runs) caused the team to win .017 less games. The runs/win converter in this case is 11.7 (.2/.017). Going from 5.0 runs to 4.6 runs gives us a runs/win converter of 11.1. And from 5.0 to 2.5 gives is 9.9. The more you stray from that .500 level, the less win impact each run will have. (Scoring the 7th, 8th, and 9th runs of the game will not give you the same payoff as scoring the 4th, 5th, and 6th.)

The key point

We can say then that, GENERALLY, within the context of a .500 team, the run/win converter is 10 to 11. That is, there is a LINEAR relationship between runs and wins, as long as you don't deviate too much from the .500 team.

An example

Let's take a ficitious team that scores 5.4 runs per game (or 874.8 runs in a 162 game season). Let's say that all those runs were created equally by 9 players, or 97.2 runs each. Let's also say that this team allowed as many runs as they scored, and therefore would have won 81 games. Since baseball is half offense (hitting and running) and half defense (pitching and fielding), we can reasonably say that the offense was responsible for 40.5 wins. Since each of our 9 hitters are equals, then each hitter is responsible for 4.5 wins.

Let's now say that this team scores instead 826.2 runs (5.1 runs or 48.6 less than the .500 level), becauses we traded Garry Templeton for Rey Ordonez. This player is worth 48.6 runs less than average. This team will be expected to have a win% of about .472 (or 76.5 wins, or 4.5 wins less than .500). Now, the only thing we changed about this team was this one player. Since the team scored 48.6 less runs, then we can say that the runs created of this player went from 97.2 to 48.6 runs. This one player also caused the team to win 4.5 less games.

Since we didn't change the defense, then the defense is still worth 40.5 wins, while the offense is now worth 36.0 wins. The 8 hitters didn't change, and so they keep their 4.5 wins each, or 36 wins in all. The defense plus these 8 hitters account for 76.5 wins. The team won 76.5 games. Therefore, there are no unaccounted for wins. This last player that we changed, Rey Ordonez, is therefore worth 0 hitting wins. Therefore, we can say that any run scored above 48.6 runs will have a positive win impact, and any run scored below 48.6 runs will have a negative win impact. Mathematically speaking, we get
ScenarioOffense TotalOffense of 8Offense of 9th hitterDefenseTeam Wins
Solving for x, and we get zero. The change in runs from 97.2 to 48.6 (or 48.6 runs) caused that 4.5 less wins (or 10.8 runs/win). While the baseline LOOKS like it's 48.6 and work upwards, it is actually 97.2 and work downwards.

Replacement Level

As it works out, a TEAM that scores at the rate of 48.6 runs per player (or 437.4 runs for a season, or 2.7 runs per game will win almost 25.0% of their games. And so, it looks like we are comparing our hitters to a "baseline" win% level of 25.0%. However, this is not the case, because Bill James' methodology is based on the MARGINAL effect on the .500 team. This last example, the 2.7 example, does not apply in this case. To be more clear, it is an ACCIDENT of the formula that a player's Runs Created is compared against 50% of the league average.

50% of league average

As you can see, the comparison point is 48.6 runs / 97.2 league runs, or 50% of league average. Again, this number should be based on the run context of the .500 team. If the team plays in a 4 run per game (rpg) context, then you have a different formula. In actual fact, the replacement level here is not 50%, but 39%! For Bill James' purposes, and for ease, I suppose that using a 50% of league average baseline is sufficient.

The drawbacks

Where are the loss shares? Assume that the league average is 80 RC / 400 outs. You have Larry Walker getting 90 RC in 300 outs, and you have Albert Belle getting 100 RC in 400 outs. According to James' win shares, Walker would get 90 - 30 = 60 marginal runs, which is 6 wins created, and 18 win shares. (He has a 3 shares to win converter). Belle would get 100 - 40 = 60 ..... 18 win shares, the same.

Now, if we introduce a 150% of league average baseline, and the difference from that is the loss created, we can get loss shares. If we add in loss shares, Walker would get 90 - 90 = 0, and Belle would get 120 - 100 = 20 .... 6 loss shares.

So, you have Walker with 18-0 and Belle at 18-6. Who's better?

If we did this with Linear Weights, Walker would be 90 - 60 = +30 runs = +3 wins = +9 win shares. Belle would be 100 - 80 = +20 runs = +2 wins = +6 win shares.

18-0, is 9 above .500. 18-6 is 6 above .500. As you can see Win Shares & Loss Shares is EXACTLY like Linear Weights. However, you need those damn Loss Shares to tell the whole picture.

And since we can derive Win Shares from Linear Weights, then why go through this whole exercise? Linear Weights is far easier to calculate.