Free Agent $/Win Based on Playoff Probability Added
by stevesommer05 on Jan 13, 2010 9:23 AM EST
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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
- 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.
- 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.
- 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.
- 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.
- 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?
- 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).
- 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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Very well done Steve.
This is the sort of thing I would love to see used more often. Rather than simply saying “wins are worth more here than there in the spectrum,” now we can actually find some decent estimates and use them for transactions.
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This is awesome
And exactly the kind of thing everyone needs…especially certain GMs and owners…
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A good friend of mine used to say, "This is a very simple game. You throw the ball, you catch the ball, you hit the ball. Sometimes you win, sometimes you lose, sometimes it rains." Think about that for a while.
by Stephen Higdon on Jan 13, 2010 11:04 AM EST reply actions
Very nice work!
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?
Looking forward to that study!
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by stevesommer05 on Jan 13, 2010 11:49 AM EST up reply actions
Great work Steve.
You said that the AL curves would be different. I was wondering if you could give a best guess for that? Are we talking a right-ward shift (left-ward?). Would the peaks and troughs be further from the mean? Would the window be shorter (85-87, for example). I get what you’re looking at here, but I’m having trouble extrapolating that to the AL.
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by Andy Hellicksonstine on Jan 13, 2010 1:03 PM EST reply actions
I'll probably do them here the next couple of days (haven't yet being an NL guy)
They would be to the right some I’d guess as the playoff probability inflection point is further right in the AL. Not certain on the width of the windows (which would also affect peaks and valleys).
by stevesommer05 on Jan 13, 2010 1:11 PM EST up reply actions
And there's the issue of the AL East too
which would push teams (especially in that division) even further right (see assumption 4) I’d think
by stevesommer05 on Jan 13, 2010 1:13 PM EST up reply actions
First glance looks like
86-93ish for the window (I have to check my math though).
by stevesommer05 on Jan 13, 2010 1:24 PM EST up reply actions
One thing I've tried to look at but couldn't findt he data for...
Was an “ideal positioning” for each club. Given their revenue, revenue curve based on wins, and an average-intelligence front office, how much should a team spend in order to maximize revenue? Starting at the Marlins end of the spectrum, extra wins are really cheap, but they aren’t worth much in terms of extra revenue. Moving into moderately below-average territory, wins become a bit more expensive, but still don’t return all that much more. At average revenue, wins move up quite a bit in cost, but so does the reward, getting from 81 to high 80s wins. For teams like the Mets/Red Sox, extra wins become really expensive, but their curves are so steep (more wins = a lot more money) they might be worth it.
You’d need a custom revenue/win curve for each team, in other words. And the goal is to figure out where income >> outcome. Guessing there are two possible sweet spots, one with a sub-.500 team for low income teams and one at a low 90s win team for high income teams.
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In this model...
Each team does have its own win-curve.
They are typically s-shaped. Marginal wins represent regular season income as well as potential post-season income (based on playoff probability added), and since all teams that make the playoffs get a World Series share, there’s an added bonus for that.
A sub-.500 team would gain very little by moving 1 win to the right because they would, under most circumstances, still be under .500. There won’t be a noticeable increase in attendance, and they have no prayer of making the playoffs. This is the type of team that would be paying around $4.4M for that win, but likely wouldn’t see that money come back in increased revenue.
High-win teams also gain very little by moving 1 win to the right because they already have a great shot at the playoffs. The difference here is that 1 win might be the difference between actually making the playoffs and not. This is where theory breaks from reality. Though that one win may only add 1 or 2 percent to the team’s playoff probability, in actuality, it was what allowed them to make the playoffs. In this case, marginal revenue is massive.
If the same team didn’t make the playoffs, the marginal revenue might actually be negative.
Because each team rolls with its own curve, it’s kind of impossible to generalize. Strictly looking at $/PPA is one good way to do it, but I question whether it really applies to teams like the Royals and Orioles given their current win-curve positions.
by NoNameOnCard on Jan 13, 2010 7:26 PM EST up reply actions
Right, the revenue/win curve is unique for each team.
But the cost/per win curve is the same for each team. If you overlay the two, you can (theoretically) see where each team should ideally sit to maximize profits.
Now, there’s probably a third factor, which is that the revenue/win curve will change over time depending on success/marketing/star power/stadium/etc. My compliments to anyone who makes headway on that…
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Vince Gennaro did.
He wrote a book about it. I have sample size concerns about a lot of the work, though. He ultimately determined that baseline revenues (the revenue estimate floor) decrease after 3 consecutive poor seasons. Of course, this is franchise relative. He also determined that accretion occurs after roughly 3 consecutive good seasons.
Here’s a link to his book: Diamond Dollars: The Economics of Winning in Baseball
I’m not sure I agree that cost/win curve is the same for each team. I think they have essentially the same shape, but the Yankees and Red Sox are teams that are far more likely to overpay for a guy who takes them from 95 wins to 96 wins than the Marlins would.
If the cost per win were normalized to estimated revenues, then the curves might be the same for each team, but I’ve got a feeling a lot of guys in charge of this stuff aren’t the best at finance math.
by NoNameOnCard on Jan 14, 2010 11:59 AM EST up reply actions
Awesome, I'll have to read that book.
Regarding:
I’m not sure I agree that cost/win curve is the same for each team. I think they have essentially the same shape, but the Yankees and Red Sox are teams that are far more likely to overpay for a guy who takes them from 95 wins to 96 wins than the Marlins would.
Whether each team would pay for those wins is a different question than how much those wins actually cost, I think. The Yankees are more likely to spend the money on those wins (well, not necessarily not those wins, but wins in general) because the return in revenue is a lot higher. The decisions to spend the money is based off comparing the two curves. Is the cost worth the reward?
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I meant to say...
In the example of a bidding war, even though the wins may be worth more to a team like the Rays, the Yankees could outbid the Rays, increasing their own cost per win.
Similarly, a team like the Royals may bail on a 1-win player if his price gets anywhere near $3.5M/win. A team’s cost per win is how much it spends for its wins. If a team refuses to spend, its cost per win is definitely going to be below market value.
Position on the win-curve, I think, necessarily affects a team’s cost per win.
by NoNameOnCard on Jan 14, 2010 4:29 PM EST up reply actions
We might be talking past each other, but I still don't agree.
Similarly, a team like the Royals may bail on a 1-win player if his price gets anywhere near $3.5M/win. A team’s cost per win is how much it spends for its wins. If a team refuses to spend, its cost per win is definitely going to be below market value.
Are you talking free agents here? If so, the Royals can spend less on them because they can wait for cheaper options (not that they do). Usually these are 1-3 WAR guys. The Yankees will also grab some of these, but more often they don’t have the holes to plug 1-3 WAR guys into. Why? Because they’re already higher on the win curve.
When I talk cost/win, I’m not just talking free agents, but all salaries. At sub-Marlins level, you’re basically only paying guys in years 1-3, so it’s close to $0/win (in marginal dollars). The real Marlins are a step above that, adding in some smaller arbitration awards. Then teams like the A’s who pay some players all the way through arbitration and sign a couple free agents. Then teams who pay a handful of free agents. Then the big boys who have to spend a lot on free agents in order to improve even more. As you get better, you have to pay more, because you’ve exhausted the cheaper alternatives. (Development and drafting should be worked in there somewhere.)
In the example of a bidding war, even though the wins may be worth more to a team like the Rays, the Yankees could outbid the Rays, increasing their own cost per win.
Why would the Yankees outbid the Rays if the wins are more valuable to the Rays? They wouldn’t. The reason the Yankees have so much money to spend is because each additional win is worth so much more to them. So while spending money to go from 81 wins to 95+ might be really costly, it’s worth it to the Yankees because each win returns more that it costs. It’s not worth it to the Rays, because they won’t see much money back at all for each additional win.
The only way you’ll see the Rays approach the 85-95 win level is when they beat the system and don’t have to spend the money that an average-intelligence front office would (think Pirates/Reds over the past 20 years). For those teams, sitting at 75 wins might maximize their profits.
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by Sky Kalkman on Jan 14, 2010 10:39 PM EST up reply actions
The model is built around free agent market concepts.
“Adding a win” refers to signing a free agent worth 1 win. It’s strictly a concept of marginal values. From that standpoint, every time you add a free agent, you can easily measure his marginal cost and [estimated] marginal revenue. Because it is a model based on marginal values, you can apply the model to trades as well.
When a team is able to add wins from the farm, the cost is practically zero, like you said, which lowers that team’s cost/win. I’m just not sure what that does for your point. It seems to support my assertions about the Royals and Yankees and who spends how much to get those wins.
Why would the Yankees outbid the Rays if the wins are more valuable to the Rays? They wouldn’t. The reason the Yankees have so much money to spend is because each additional win is worth so much more to them. So while spending money to go from 81 wins to 95+ might be really costly, it’s worth it to the Yankees because each win returns more that it costs. It’s not worth it to the Rays, because they won’t see much money back at all for each additional win.
This concept is inherently flawed because risk aversion plays a large role. The Yankees can afford to miss. Teams like the Rays and Marlins can not. This is particularly true of the Yankees whose game-to-game revenues are virtually maxed out. They receive no win-level benefit from adding that extra win UNLESS it gets them to the playoffs; unfortunately, there’s no guarantee of that.
Without playoff appearances, marginal revenue starts to decline around 92 (I don’t recall exactly) wins and reaches roughly ZERO around 105 wins. The reason the Yankees would outspend the Rays is because of the near certainty of making the playoffs. Their win-curve is so far to the right that they can take that risk.
You can argue that risk is part of the value (cost and/or revenue) equation, and to a point it is, but that’s just the theoretical model. In real life, a team like the Rays – that must compete in the same division as the Yankees and Red Sox – is far less likely to count on revenue generated by a playoff appearance. Because the model builds in playoff probability added (as measured by the team’s expected win total), even though the model may suggest that the 91st win is actually worth about $6M (random number, probably not accurate), to the Rays it may only be worth about $4M.
More abstractly: because of they play in the AL East, they might require a theoretical 2-win player to achieve 1 real win.
by NoNameOnCard on Jan 15, 2010 1:03 AM EST up reply actions
If I remember the book
(I can’t find in whatever moving box it sits in now), you only get to see a select number of teams actual curves. While bits and pieces of all of the teams are revealed throughout, I remember not having one for every team being my true disappointment with the book.
Of course, I could be making all of this up and should really just dig out the book to check.
The Crawfishboxes
A good friend of mine used to say, "This is a very simple game. You throw the ball, you catch the ball, you hit the ball. Sometimes you win, sometimes you lose, sometimes it rains." Think about that for a while.
by Stephen Higdon on Jan 16, 2010 12:31 AM EST up reply actions
You're right.
The book was more about the model’s construction and design as opposed to the results and data from his work.
by NoNameOnCard on Jan 16, 2010 3:13 PM EST up reply actions
Nice work, Steve. I love these types of studies.
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Looks like a good start
is there any thought toward extending the “bump” to include playoff-win probability added? maybe this only applies to perennial contenders, but adding an extra playoff win would also be more valuable than adding an ordinary win.
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by SagehenMacGyver47 on Jan 14, 2010 12:45 PM EST reply actions
It was/is bumping around
in the back of my mind. If I look at it, it would be after a couple of other projects I’ve started on.
by stevesommer05 on Jan 14, 2010 12:57 PM EST up reply actions
Here is a study done to project attendance
Games behind will still bring attendance — being close matters some, be maybe not enough. ~6000 tickets less for each game behind. Say $50 revenue per ticket, then $300,000 lost for each game behind. with the rough number it is not enough.
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