MLB : How much does a win "cost"?
From the article,
"We feel that the team has underachieved," [General Manager Mike] Rizzo said. "We feel we have a better ballclub than we've shown on the field."Easy to say in the heat of the moment, when you're looking for a scapegoat or a reason for having dumped Acta. But, is it true? Has the team underachieved?
Washington has many problems: the worst ERA in the National Leage (5.21), the most errors committed in baseball (82), a rotation full of rookies, they rank at the bottom of most offensive categories, etc. But, in my opinion, most of these problems are simply the result of having one of the lowest team salaries in the majors: around $59 million, 27th out of the 30 teams. And this begs the question: what is the relationship (if any) between team salary and wins?
We can find team salaries these days readily available on the internet and plot that against the current number of wins for each team. The result looks like this:
Several assumptions need to be listed and justified. First, the Yankees and Marlins are notorious outliers but for competing reasons: the Yankees are willing to spend nearly twice as much as its nearest competitor (always have to stay ahead of the Red Sox) and the Marlins have the ability to do more with less. After removing these two, we're left with 15 NL teams and 13 AL teams. If we're going to compare leagues, it would help to remove two NL teams: the most obvious choices are the Cubs and Mets, both grouped together and around $20 million higher than the Phillies. I'll show at the end that the removal of these four teams leads to little change in the correlation for the data. Last, I chose to run the trend line through the origin: if you didn't spend a dime, you'd have no team and win zero games. The result is a trend line that fits the data very well for 26 of the 30 teams.
Any measure of performance should, by definition, include roughly the same number of teams "over-performing" and "under-performing". By "over-performing", we mean that the team is getting more wins than the team salary should suggest; by "under-performing", we mean that, for this salary, the team might be expected to have more wins. Thus, investigating the distribution of AL and NL teams above and below the trend line, we see:

Seven AL teams above the line (over-performing) and seven AL teams below the line (under-performing); a note about the Yankees later. Nine NL teams are over-performing, one is on the line and six are under-performing. So, the teams are spread fairly evenly on each side of the line; the NL is slightly over-performing, but we'll see in the next graph that this is mostly because of one division (the NL West).
Finally, we look at the six divisions of MLB. Do all teams in a division follow a similar plan? Or are any of the divisions being won by a team willing to pay a higher salary? Grouping the teams by divisions looks like this:

If we count the number of over-performing teams in the AL, we find three (out of five) in the East, two (out of five) in the Central and 2 (out of 4) in the West. Meaning, the AL is not just evenly divided as a league, it's also fairly even within each of the divisions; in other words, in the AL, you get what you pay for. The exception, of course (and when is it not), is the Yankees whose salary is so outlandishly high they would have to have a record of 100 wins and negative 20 losses to sit on the trend line; clearly, the Yankees are proof of the Law of Diminishing Returns (LoDR), you can't buy more wins than the number of games you play.
In the NL, however, the situation is different. Counting the number of over-performers, we find one (out of five) in the East (the Marlins*, of course), four (out of 5) in the Central and five (out of five) in the West. The West makes for certainly the most interesting division right now in baseball, with no team spending more than $100 million but three of the five on pace for 90+ wins and all five with a mathematical chance at a winning record.
Returning to the LoDR, we see all six division leaders grouped together near the $100-$110 million range (boxed section), with around 50-55 wins at the All-Star break. Thus, while one would expect there to be a minimum amount of payroll needed to compete (unless you have the best scouts in the majors like the Marlins), this data appears to support the idea that there's also a maximum payroll; beyond $120 million, you don't really get any extra value and may be filling your team with stars that can't find playing time and egos that clash. Smart money would say, emulate the Cardinals, whose $89 million salary (13th highest and nearly a third of the Yankees) has them ~50 wins and the lead in the Central. Also don't overlook: the Rays, the Rangers, the Giants, the Rockies (who also fired their manager) and the King of More With Less, the Marlins.
But, the question remains: were the Nationals under-performing? Not even close. In fact, they fall nearly perfectly on the trend line: 26 wins is about what you should expect with a salary of $59 million.
And how strong is the correlation between salary and wins (where 1 means very large positive correlation and 0 means no correlation)? If we exclude the four outliers, the correlation is around 0.65; with the outliers, around 0.45 -- both of these values suggest a strong correlation between salary and wins. So my advice to the Nationals owners and GM would be: first, lay off Acta, he wasn't the problem (or, at worst, one in a sea of many); second, open your wallets. $59 million these days is merely the ante, you better chip in another $20 or $40 if want to still be sitting at the table come October.
UPDATE: Originally, this said "the Braves" by mistake.

