## Free Agent \$/Win Based on Playoff Probability Added

There has been a lot of discussion this off season dealing with spending relative to where a team is on the win curve.  My goal here is simply to do some back of the envelope math to set some ranges on the dollar values that teams should pay for a win based on where they are on the win curve.

ASSUMPTIONS

1. Teams should pay a different cost per win based on how valuable that win will be to them.  For this analysis I'm using playoff probability added (PPA) to define valuable.  If you disagree with either of these then the rest of the article is probably not for you.
2. I used 4.4M per win as the average market value.  Changing this assumption wouldn't change the shape of the curves, just peaks and valleys.
3. The model needs some type of salary floor in order to more closely model reality and take into consideration off the field issues.  I'll be setting the floor as a percentage of the maximum suggested salary (i.e. if I set the floor at 75%, then the teams on the low and high end of the win curve will be modeled at no less than 75% of the \$/win of the teams at the inflection point).  I'll create curves for multiple values as I am uncertain what this number should be.
4. This analysis uses historical playoff probabilities for projected/3rd order wins as the guiding metric.  The next step would be to use division strength to assist in the PPA calculations.
5. Since PPA is the guiding metric, improving a high win team for performance in the playoffs themselves is ignored.  (The Crapshoot Corollary perhaps?)

METHODOLOGY

I went through and calculated the PPA for each one win increase.  I turned each value into a percentage of the max observed PPA, and then took the maximum of that percentage and the assumed percentage floor.  After getting the relative percentages I reverse engineered the final cost per win so that the weighted average* would be our standard \$4.4M/win.

RESULTS

The calculations with a 75% floor yields the following for the National League

The blue line is the \$M/Win for a 1 WAR increase based on the number of projected wins.  Clearly the most important wins to gain from a PPA perspective are those in the 83-88 range.  Expanding that chart to look at other floor values yields

And then one final chart that shows a hybrid of the 80%-90% lines (80% < 75 wins, 90%>=75 wins).

Note:  All of the numbers were calculated using a wins scale of 67-95, everything left of 75 wins was omitted as it was identical to the 75 number.  Clearly here it matters, so I showed it.  For comparison/refernce purposes, here is the 80-90 hybrid with the 90% floor on the original scale

What does this barrage of graphs tell/show us?

1. Identifies windows of projected wins where it would be acceptable to spend above average market value (under whatever set of assumptions you see fit).  Under most of the assumption sets the ranges are from about 83 wins to 88 wins (in the NL, the AL would have its own curves).
2. Analysts can infer ranges of appropriate spending for various teams and their respective locations on the win curve.  For example, the 80-90 hybrid sets the max for most teams at ~market value, but says that teams <75 wins should pay ~\$4M and teams between 84-88 wins can acceptably go above average market value (up to ~\$5M at 85/86 wins).

As an aside the 90% or 80-90 hybrid actual cost is never further than 10-11% from the average of \$4.4M.

Additional aside.  The chart / numbers change based on the assumed production level of the contract.  Basically a 4 win player would shift the bump in the curve to the left ~4 wins [update, I should look at my own chart before making statements, the curve peak is actually shifted left only 2 wins to 84]...  See here for related charts.

FINAL THOUGHTS / NEXT STEPS

I think these curves are another tool to look at some of the contracts given out this off-season.  I think their primary use would be in examining the contracts that are likely marginal (i.e. likely to be around break even) or were marginal (if doing retrospective analysis).

I think the next step would be to see if there is any historical evidence to support any of this.  Have teams in the middle of the projected win curve spent more than teams on the edges?

*Weighted average was a fairly unscientific normal-ish curve centered at 81 projected wins.

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