Thursday, December 23, 2010

How many wins do baseball teams buy? Some evidence

Pretty much every model of baseball team payroll assumes that wins are worth more the closer you get to playoff contention. If you have 75 wins, buying a 76th win won't help you that much, because the fans don't care a whole lot whether you have 75 wins or 76. But if you have 86 wins, the 87th win is good value, because it puts you closer to the post-season, which is what the fans care about. So that 87th win can net you a lot more money.

One consequence of this hypothesis is that, when they decide how many wins to buy on the free-agent market, teams should *never* settle on any number between, say, 80 and 85. That's because, if the 82nd win is worth buying, or the 85th, then the 86th should *always* be worth buying too. It will generate more revenue than the 82nd win, but cost the same amount of money. (For an imperfect analogy: nobody buys just three tires for their car.)

I think Tango has been somewhat skeptical of this theory ... his hypothesis is that team ownership gives the GM an arbitrary budget, without actually figuring out what the revenue curve would look like or doing a cost/benefit analysis. It sounds implausible to me that that would happen frequently (although I'm sure it could happen occasionally, or even regularly). But neither one of us really has any evidence.

Well, actually, I now have a little bit. A couple of weeks ago, frequent commenter Guy e-mailed me and suggested I look at actual win totals to see if there is a gap between 80 and 85 wins. You wouldn't expect *zero* wins among those teams, of course, since better and worse teams will occasionally fall into the 80-85 range just due to luck, or misjudgment of talent. But you should expect fewer 84s than 88s, for instance.

There is some evidence that that's the case. I looked at raw win totals from 1998 to 2010, and looked at them in various ways. Here are the frequencies of individual wins:

70 wins: 6 teams
71 wins: 9 teams
72 wins: 11 teams
73 wins: 7 teams
74 wins: 12 teams
75 wins: 15 teams
76 wins: 9 teams
77 wins: 8 teams
78 wins: 10 teams
79 wins: 13 teams
80 wins: 11 teams
81 wins: 6 teams
82 wins: 8 teams
83 wins: 17 teams
84 wins: 7 teams
85 wins: 11 teams
86 wins: 15 teams
87 wins: 8 teams
88 wins: 16 teams
89 wins: 12 teams
90 wins: 11 teams
91 wins: 9 teams
92 wins: 9 teams
93 wins: 9 teams
94 wins: 6 teams
95 wins: 15 teams

Nothing obvious looking at it that way ... it is interesting, though, that 81-81 happened less often than any other record within 10 games of .500 -- you'd expect it to be the most. But the highest frequency record was 83-79, just two wins away, so it's probably just coincidence.

We can see more if we group in bins of five wins:

70-74 wins: 45 teams
75-79 wins: 55 teams
80-84 wins: 49 teams
85-89 wins: 62 teams
90-94 wins: 44 teams

Since the 80-84 group is closest to the average of 81, you'd expect that to be the highest group. Instead, the two groups before and after are higher.

The best breakdown to see the pattern is by threes:

71-73 wins: 27 teams
74-76 wins: 36 teams
77-79 wins: 31 teams
80-82 wins: 25 teams
83-85 wins: 35 teams
86-88 wins: 39 teams
89-91 wins: 32 teams
92-94 wins: 24 teams

That shows it neatly. As hypothesized: teams appear more likely to choose either mid-70s, or mid-80s -- less in the middle.

Admittedly, this is weak evidence ... but Bill James did also mention a tendency for teams to peak higher than .500, back in the 1985 Abstract (p. 116).


UPDATE: I reran the study, except that instead of using actual wins, I used expected wins. The "expected" is the actual wins, after subtracting out

-- pythagorean luck
-- runs created luck
-- runs created against luck
-- offense luck (players having career years)
-- pitching luck (pitchers having career years).

The last two categories of luck are very approximate ... the algorithm I used probably isn't perfect. It involves calculating a player's expectation by a weighted average of the two previous and two following years, regressed to the mean a bit. (Details are in Powerpoint slides at my website.)

Anyway, I expected the results for expected wins to be even stronger, but they weren't. Here are the three-year groupings:

71-73 wins: 35 teams
74-76 wins: 44 teams
77-79 wins: 46 teams
80-82 wins: 44 teams
83-85 wins: 55 teams
86-88 wins: 36 teams
89-91 wins: 30 teams
92-94 wins: 13 teams

There's still an effect, but a smaller one. The 80-82 group is smaller than the 77-79 and 83-85 groups, which is something.

This is from 1998 to 2007. Going back to 1977, the results are a little more pronounced:

71-73 wins: 78 teams
74-76 wins: 111 teams
77-79 wins: 143 teams
80-82 wins: 127 teams
83-85 wins: 139 teams
86-88 wins: 93 teams
89-91 wins: 72 teams
92-94 wins: 45 teams

There's one element of luck that's not included here: injuries. Expected wins do not include any losses due to "too many" injuries to key players, or gains due to "too few" injuries to key players.

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At Thursday, December 23, 2010 12:50:00 AM, Blogger David Pinto said...


I went back further in time. Shows the same trend back to 1976, the start of the free agent era.

At Thursday, December 23, 2010 12:52:00 AM, Blogger Phil Birnbaum said...

That was quick! Thanks, David!

At Thursday, December 23, 2010 1:06:00 AM, Anonymous mettle said...

Really cool stuff. I followed up using data from '82-. As this graph shows, there is very clearly a dip at 81/82 wins.
There is another fascinating anomaly at 95 and 67 wins. This is wacky because 95+67=162 and while 95=enough to make the playoffs, it isn't clear why you'd get the symmetrical 67-win bump.

From a stats standpoint, you can do a Shapiro-Wilk test to see if wins are normally distributed; p<.01 that the sample is not normal, meaning these dips unlikely random.

BTW, I came here from the insidethebook blog.

At Thursday, December 23, 2010 7:06:00 AM, Blogger Matty D said...

I think this is understating the case- you're not accounting enough for luck. If teams are equally spaced around 81 wins, we'd expect a huge spike there since we're going to get variance from both sides (lucky bad teams and unlucky good teams). For example, if half the teams target 75 wins and half the teams target 87 wins (if they follow a binomial dist) the most likely win total is still 81 wins. That the actual is flat (or possibly dipping) in the middle suggests a much wider spread in true talent.
The predominance of teams in the mid-70s and mid-80s suggests teams target around 70 or 90, just as predicted.

At Thursday, December 23, 2010 8:41:00 AM, Blogger Phil Birnbaum said...

Mettle: Thanks! I suspect that any one-value spike is probably random -- like 81, 95, 67 -- but the overall trend isn't.

I don't know why I didn't do a whole bunch of years ... I think I maybe figured that free agents only got expensive recently, so there would be a more pronounced effect in the later years.

At Thursday, December 23, 2010 8:45:00 AM, Blogger Phil Birnbaum said...


>"if half the teams target 75 wins and half the teams target 87 wins (if they follow a binomial dist) the most likely win total is still 81 wins."

Wow, I didn't realize that was the case. Guess the spread is indeed pretty wide. Thanks.

At Thursday, December 23, 2010 10:44:00 AM, Anonymous Guy said...

Dave Pinto hypothesizes that this could reflect late-season decisions by teams to go for it vs. concede and play for next season. Does anyone have data set with first-half records (or April-July), to see if Phil's pattern is already apparent at the AS break?

At Thursday, December 23, 2010 10:49:00 AM, Blogger Phil Birnbaum said...

Good thought ... but, isn't it more like 2/3 of the way through the season that teams make their last trade?

In any case, I bet that would be pretty small effect. Even if teams trade away 9 WAR (full season WAR) worth of talent, that's only three games over 1/3 the season (or maybe 4, depending on the timing of the trade). That's only one bin of shifting in David's chart.

And that's assuming that EVERY team does this and that it's a BIG effect of 9 WAR (or two stars).

Still worth checking out, of course. And one bin would make SOME difference.

At Thursday, December 23, 2010 12:17:00 PM, Anonymous Bill Petti said...

It is a really interesting question.

My first reaction is it is really difficult to infer intentional action from data that has lots of randomness in it.

I think the way to do this is to strip out those teams that were exceedingly lucky or unlucky based on run differential, expected wins and actual wins.

Then, you have to look at whether the run differential was the result of exceeding randomness--high or low BABIP, etc.

Once you've winnowed the data set down that far you could go even further to look at how each team was constructed. Did payroll expand or contract year-over-year? Did that correlate to an increase or decrease in expected WAR?

There's a lot that needs to get teased out before we can answer the question either way.

Also, would be interested if there is a change before and after, say, early 2000 when front offices seemed to embrace sabermetrics more forcefully. I would bet that if teams were matching budget to expected wins you'd see a tighter correlation after 2000.

At Thursday, December 23, 2010 12:39:00 PM, Anonymous Anonymous said...

Why don't you just look at preseason win total predictions from sportsbooks? I know enough gamblers read this blog that you can probably get the data from someone

At Thursday, December 23, 2010 4:42:00 PM, Blogger Phil Birnbaum said...

Updated the post for "expected wins," inspired by suggestions from the previous two commenters.


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