THE BOOK--Playing The Percentages In Baseball

Another way to figure out the marginal $ / marginal win?

Take the standard deviation of the payrolls of all the team, and remove the top (Yanks) and bottom (Marlins).  Normally, you don’t want to do that, but they are aberrations, specifically designed payrolls that are out-of-whack with the rest of the league.  That’s one SD = 21 million$.  If you want to remove the two other extreme teams (Sox, D’Rays), that becomes SD = 19 million$.  If it’s just the Yanks, and you want to bring them down to say 130 million$, and leave all the other teams intact, that’s one SD = 26 million$.

We know from other entries in this blog, that the true talent level of MLB teams is 1 SD = .060 wins per game.  So, over 162 games, that’s 9.72 wins.

26/9.72 = 2.7 million$ per win
21/9.72 = 2.2 million$ per win
19/9.72 = 2.0 million$ per win

There’s half a dozen ways to figure out the marginal $ per marginal win.  And they all always point to the same thing, somewhere close to 2-2.5 million$.

Interesting. We also know the payroll v wins function is not linear. It will be logistic in nature—and this is why the Yankees are projected to win 140 games with a linear function

This was discussed at Fanhome a long while back.  I had proposed P/(P+L), where P = team payroll, and L = league average, as a team’s winning percentage.

Someone else, dackle or Alan, came in, and did a regression, I think, and came up with something similar, like (P+.5L)/(P+2L), or some such.

This has the effect of the ever-increasing marginal $/win.

Here is the result of the non-linear version, alongside the original linear version:

140 108     New York Yankees
102 94     Boston Red Sox
94 89     Los Angeles Angels
94 89     Chicago White Sox
93 88     New York Mets
91 88     Los Angeles Dodgers
89 86     Chicago Cubs
88 86     Houston Astros
87 85     Atlanta Braves
87 85     San Francisco Giants
87 85     St. Louis Cardinals
86 85     Philadelphia Phillies
86 84     Seattle Mariners
84 83     Detroit Tigers
78 79     Baltimore Orioles
78 79     Toronto Blue Jays
77 78     San Diego Padres
76 78     Texas Rangers
74 76     Minnesota Twins
74 76     Washington Nationals
73 75     Oakland Athletics
73 75     Cincinnati Reds
72 74     Arizona Diamondbacks
71 73     Milwaukee Brewers
70 73     Cleveland Indians
66 69     Kansas City Royals
66 69     Pittsburgh Pirates
63 66     Colorado Rockies
60 63     Tampa Bay Devil Rays
50 51     Florida Marlins

Someone else, dackle or Alan, came in, and did a regression, I think, and came up with something similar, like (P+.5L)/(P+2L), or some such.

***

Actually, it was me. The formula was,

xW% = (.6*Lg_Payroll + Team_Payroll)/(2.2*League Payroll + Team_Payroll).

Cool, good stuff David.

The above equation can be expressed in a couple of ways:
Wins/Losses = (P + xL) / ((1 + x)L)

P=payroll, L=league average payroll

Figuring the “x” value is the key.  I originally defaulted to 0, and David has it at .6, which is the much better figure.

win% = (P + xL) / (P + L(2x+1))

So, if someone wants to go through the historical data, you can league adjust by dividing both sides of the equation by L.

So, this:
Wins/Losses = (P + xL) / ((1 + x)L)
becomes:
Wins/Losses = (P/L + x) / (1 + x)

Since P/L is simply the team payroll Indexed to the league, that’s simply “I”.
Wins/Losses = (I + x) / (1 + x)

As you can see, when the payroll index is “1”, you have as many wins as losses.

All that’s left to do is for the grunt work to begin, and solve for “x”.

David: what years did your study cover?

I’m running 1999-2005 payroll data against wins.  My “x” comes in at 2.  This means that is the team’s payroll approaches 0, then their expected win % is .400.  Using David’s .6, that makes it .273.  For all intents and purposes, your minimum payroll index is 0.20, making the minimum win% using the “x=2” as .407, and “x=.6” as .308.

I know we’d like to believe that last one, but that’s simply not the case.

If I look at the bottom 7 teams by payroll index, they averaged .352 for their payroll index.  Their winning percentage was .438.

Using the “x=2”, our expectation given a payroll index of .352 is .439.  Using David’s .6, we get .373.

Clearly, if we use payroll as a proxy for talent level, David is right.  However, this is not what we are doing in our case.  We are taking payroll, which has some over/under spending already built-in.

Therefore, what are we saying?  If a team decides on a payroll budget that is 35% of the league average, then, use the “.6” if they are going to build their team based on a completely free market.  Use the “2”, if they are going to build their team based on the typical pattern.

Very cool, Tom. I think I used 2003-04, but I just eye-balled the .6, IIRC. Two seems very high for “x”, but I’ll take your word for it until I look into it. I’m a little confused by this statement:

“Clearly, if we use payroll as a proxy for talent level, David is right.  However, this is not what we are doing in our case.  We are taking payroll, which has some over/under spending already built-in.”

I don’t necessarily disagree, I just don’t understand what you’re saying (which makes it hard to agree or disagree).

Also, it would be helpful if you could post expected win results with x = 2 as well as x = .6.

Setting the x values to “2.0” and “0.6”, respectively:

97 107 $195,587,711     New York Yankees
88 93 $120,099,824     Boston Red Sox
85 89 $103,472,000     Los Angeles Angels
85 88 $102,750,667     Chicago White Sox
85 88 $101,084,963     New York Mets
84 87 $98,447,187     Los Angeles Dodgers
84 86 $94,424,499     Chicago Cubs
84 86 $92,551,503     Houston Astros
83 85 $90,156,876     Atlanta Braves
83 85 $90,056,419     San Francisco Giants
83 85 $88,891,371     St. Louis Cardinals
83 84 $88,273,333     Philadelphia Phillies
83 84 $87,959,833     Seattle Mariners
82 83 $82,612,866     Detroit Tigers
80 79 $72,585,582     Baltimore Orioles
80 79 $71,915,000     Toronto Blue Jays
80 78 $69,896,141     San Diego Padres
79 78 $68,228,662     Texas Rangers
78 76 $63,396,006     Minnesota Twins
78 76 $63,143,000     Washington Nationals
78 76 $62,243,079     Oakland Athletics
78 75 $60,909,519     Cincinnati Reds
78 75 $59,684,226     Arizona Diamondbacks
77 74 $57,568,333     Milwaukee Brewers
77 73 $56,031,500     Cleveland Indians
75 70 $47,294,000     Kansas City Royals
75 69 $46,717,750     Pittsburgh Pirates
74 67 $41,233,000     Colorado Rockies
73 64 $35,417,967     Tampa Bay Devil Rays
68 54 $14,998,500     Florida Marlins

The standard devation of the first column is 5.3 wins, while the second one is 9.9 wins.  We do know that the “true” standard deviation is 9.7 wins (i.e., .06 x 162).  If payroll would explain 100% of the variation, I’d go with the x=.6 column.  But, that’s not reality, as I’ve previously explained.