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Nary a year has gone by in recent memory in which I can’t remember people talking about the growth in player salaries. Every offseason, it seems, we expect the free agent market to explode. In years with weak free agent classes we anticipate that the low supply will drive up wages; when there are several premium players available we say that the high talent level will lubricate teams’ wallets, or something. Then, after we see a couple second- or third-tier free agents get significantly more than anyone had expected, there’s a fear that that year’s inflation will be even higher than we’d thought. It’s one of the many cyclical talking points in the community of baseball analysis (I say this with love).

The typical explanation for why salaries have grown is the ever-increasing pool of money in baseball. I certainly wouldn’t deny that it’s a factor, but there’s more to it than that. Any economist worth his or her salt will tell you that having more money doesn’t necessarily mean that it’s rational to spend more money. And besides, MLB team owners aren’t your typical corporate executives whose overarching goal is to maximize profits — much (if not most) of the value of a win to an organization comes not from increasing revenues but from the sheer joy of winning. We on the outside can’t tell teams how they should value a win (as distinct from how much they should pay for them) when its worth is largely subjective.

But if there’s more behind salary inflation than the influx of money in the game, what else is going on? I’ve developed a theory that I think can explain both what’s driving the increases in the cost of a win and why the inflationary pressure manifests itself in inconsistent ways. At the heart of my explanation is the economic phenomenon known as the “winner’s curse.”

The Winner’s Curse
The concept of the winner’s curse has permeated the realm of baseball analysis this year to the point where a full explanation may be gratuitous. As a quick refresher, I’ll borrow the scenario outlined by Gary Huckabay (read his full article for a more thorough definition):

The Winner's Curse is a term borrowed from the oil industry. It stems from the system of auctions of oil rights to parcels of land. (It may have earlier origins than that, but if so, I'm not aware of them.) Oil companies had to submit bids for the rights to drill for oil in particular parcels of land. The value of those rights is directly driven by the amount of oil on a particular parcel of land. More oil meant more money, which means companies could afford to bid more for the rights to drilling a particular parcel.

Huckabay describes four companies that each take samples from the land to estimate how much oil they would be able to extract. The company that gets the most oil from its sample will end up placing the highest bid. But the average of the four sample outputs would be a better predictor of how much oil the land would produce than the highest individual sample. The winning bidder is basing its offer on an overly optimistic estimate of how much oil it will extract from the land, meaning the sale price is probably too high. Winning is thus said to be a “curse” because there is a good reason why no one else was willing to match the top bid.

The same concept can apply to any bidding war over a good with an uncertain value. When a player is available as a free agent, 30 different MLB teams, each with its own group of talent evaluators, come up with their own estimates of how much he is worth (or at least that he isn’t worth signing at all). Signing a player usually means making the highest offer, and making the highest offer always means being willing to pay more than 29 other teams. Unless the signing team’s evaluation of the player is better than the collective wisdom of the rest of the league, the player is particularly well-suited to his new team, or the signee has a well-over-50th-percentile breakout, any free agent acquisition is therefore destined to provide a lower return on investment than the club anticipates.

Optimism and Inflation
Imagine a fictional free agent market in which there are 100 wins worth of production available. (Quantify this with WARP, pitchers’ win-loss records, RBI divided by caught stealings, whatever — every player contributes to his team’s record in some way, and winning games is the goal of every organization.) In past years, the combined utility functions of each team have established the cost of a win on the free agent market to be about $1 million. Assuming no major changes in the economic value of a win to league owners, in a market with perfect information we would expect that the cost of a win would hold steady at $1 million and that that year’s free agent class would combine to earn $100 million.

But not all teams project players equally — and that’s where the winner’s curse comes in. Say that each team overestimates the value of the player(s) it signs by 30 percent (I think this figure is conservative). There are still only 100 wins for sale, but the sum of the wins each team thinks it’s buying is 130. If teams think they’re getting 130 wins then they will pay for the equivalent of 130 wins. At an assumed market value of $1 million per win, the league will shell out $130 million for only 100 wins. Thus, the cost of a win will end up rising by 30 percent to $1.3 million.

There’s one other factor that we should consider here: the asymmetry of information and differences of opinion across baseball. While most wouldn’t put it quite so brazenly, I think it’s fair to say that every team thinks its process and evaluation are better than those of its average competitor. Every time a team takes an action, it does so because it thinks it is a smart move. So perhaps it’s unfair to say every team plans to pay market value on free agent deals.

Let’s throw in a 10 percent discount on the assumed market price of a win to account for what each team considers to be its superior analysis; now each team is willing to pay only $900,000 per win. But the winner’s curse effect still applies, and teams think they’re buying 130 wins at $900,000 each when there are really only 100 wins for sale. So even if teams sign only deals that they think are bargains, the cost of a win would rise by 17 percent from $1 million to $1.17 million.

This is just one stage in the cycle. What happens in a future offseason once the league realizes that the cost of a win has gone up? They’ll adjust their behavior accordingly. With a 10 percent expected discount from allegedly superior knowledge, teams will plan to pay $1.053 million per win. But if the 30 percent average overestimation holds, then the empirical cost of a win will end up inflating by 17 percent again, from $1.17 million to $1.369 million. And the next time it’ll rise to $1.602 million. And so on.

The basic mechanism at work is that teams overestimate their respective information and strategic advantages relative to the rest of the league. Every organization may think it is ahead of the curve, but it’s impossible for 100 percent of the league to be above average. Even if teams collectively realized they were overrating the players they signed and shaded down what they were willing to pay per expected to win to account for their own biases, in the aggregate they would be too conservative in their adjustments because they would underestimate the magnitudes of their respective optimisms.

This leads me to a conclusion that, unless/until it is disproven, I’d like to offer as the humbly named Pollis’ Theorem (historians, please note that there are two L’s in “Pollis”):

So long as MLB teams overestimate their own respective information and strategic advantages, player salaries will continue to inflate.

Modeling Inflation
No matter how much you buy into my logic, I should note that the numbers I came up with for the sizes of the effects are completely made up. Generally speaking, the size of inflation I in any given year can be expressed by:

where B is the amount by which teams overrate the players they sign (i.e., the effect of the winner’s curse) and D is the discount by which teams shade down their bids to ensure that they project each deal they sign to have a better-than 100-percent return on investment. So long as the difference between B and D is greater than their product (if you play with the numbers it’s hard to come up with plausible values for which that wouldn’t hold), the price of a win will inflate.

From this, one could project the price of a win in an offseason in which we anticipate some inflation. The new price of a win W’ can be projected by:

where W is the previously established market price of a win.

If this process repeats itself every year and B and D remain constant, then the cost of a win Wy in y years from now can be projected by:

However, I don’t think this last model is very realistic. Most years, the inflationary effects on player salaries are probably lower than they seem — we remember the major overpays as anecdotal evidence that the market has gone insane, but we forget that the inevitable bargains factor into the aggregate too. Not to mention that (I assume) not every team has a specific, ever-shifting number in mind for the precise cost of a win, and even if they did these dynamic estimates wouldn’t be identical across the league.

Reactionary Self-Awareness
Instead of imagining a highly dynamic market marked by continuous, constant inflation, I propose another idea. Once a clear consensus of the cost of a win takes hold across the league, we would see a great inflationary event as teams use the empirical market price in combination with their overly optimistic projections of the players they sign. But the compounding effects wouldn’t begin a year later, because it takes a few seasons for teams to realize that the market has shifted. We may see some minor fluctuations in the market with a slight inflationary trend in the seasons after the watershed price leap, but it would take time for teams to appreciate the inflation that has occurred and to adjust their strategies accordingly.

This theory presents something of a catch-22: The Happening is triggered not by teams deciding to approach the market differently but by the collective realization that they already were. As soon as teams start to plan and act in accordance with the market conditions they have already been dealing with, the price of a win jumps again.

There is some empirical evidence to support this idea. About a year ago, when I calculated my own numbers for the cost of a win dating back to 1996, the results described a market characterized by lots of year-to-year noise and slight inflation punctuated by rare paradigmatic leaps in the equilibrium price. Specifically I observed major market shifts in 2000, when the cost of a win jumped from $2.2 million to $3.6 million, and 2007, when it soared from $4.5 million to $6.3 million. From 2001-2006 and 2008-2013 the price of a win moved up and down but stayed within $1 million of the anchoring points set in 2000 and 2007, respectively.

I’ve been at a loss as to what happened in those two years that dramatically changed the way teams evaluate players. This theory provides an answer: nothing. Those just happened to be the points at which the market reacted to its own new self-awareness.

What This Means
Aside from establishing a more satisfying answer for why player salaries are increasing than just the increasing amount of money in the game, pinning inflation on the winner’s curse should change our understanding of when and how market shifts manifest themselves. If I’m right that inflation in the price of a win is more discrete than continuous (admittedly I’m less sure of that than I am of the more general point that overvaluing players leads to inflation), then using constant inflationary factors to assess the long-term value of a player or evaluate a multi-year contract does not do justice to the complexities of the market. At a basic level, I don’t know how one could accurately predict where the market will go in the future if such knowledge is in itself enough to cause a fundamental market shift.

Another wrinkle to consider: Just as the collective realization that the price of a win has gone up leads to another market shift, widespread understanding of this mechanism would change the discontinuous nature of salary inflation. If my theory is correct, the existence of the phenomenon hinges on teams not realizing that their too-optimistic projections of player performances are driving salaries up leaguewide. I don’t think we’ll ever get to a point where there are no major differences in how teams project players, but if teams know to anticipate the effects of the winner’s curse, they might become quicker to react to the market shifts it creates. Take this to its logical extreme and someday soon we might see this inflationary effect take hold on a year-to-year basis.

I wish I could end this by offering some concrete advice to teams about how they should optimize their approaches to the player market, but I don’t have anything profound to say. I suppose this reinforces the notion that there is immense value in having superior information about the available players and being skilled at reading the market at large (and thus that teams should invest more heavily in front office talent), though of course that’s nothing new. But one of the fundamental goals of sabermetrics is to achieve a better understanding of how the game works, and in that vein I think it’s important to appreciate the spell that the winner’s curse casts on the baseball economy.

Thank you for reading

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Richie
12/24
Good stuff, thank you. Arcane such that I can't foresee many comments, but I'm sure there's enough of this type of us in the BP audience to appreciate this. I enjoyed the writing, too.
Bob1475
12/24
Yes, auctions can contribute to price inflation but if there is no available cash prices fall. The primary factor in dalary inflation remains more available cash. You may not find this "satisfactory" but it remains a mathematical reality.
Marcgiz
12/24
I'm going to go look for the Nobel Prize winner who found that in accepting bids the best bid to accept was the second lowest. It seems to me that some of the same sort of informational pressures, among other factors, are at work if a government or business take the lowest bid.
Marcgiz
12/24
Hmm - Found this which is close but not what I was looking for: http://whoswholegal.com/news/features/article/29137/the-art-identifying-the-economically-advantageous-tender-use-relative-evaluation-models-public-utilities-procurement/ Still, many of the issues are the same.
Plucky
12/29
The main problem from a GM's prespective, the mentality required to always avoiding winner's curse leads to paralysis. If a GM refuses to acquire any player for which another team has superior information, then he will make zero trades and sign zero free agents. Let's play that conversation: Owner: So I see payroll is looking like it will be under the budget I gave you. Congrats on putting together the team so efficiently. I've been looking at a yacht upgrade for a few years now and I think this will finally let me pull the trigger. GM: Thanks, the free agent market this offseason was absurdly overpriced, so I figured the only way to win was to not play. Owner: I applaud your discipline. I do have a couple questions though- Last year LF and 2B were kind of disasters, and yet we are going into this season with the same guys. We ought to be in the wildcard mix so that seems to me like a bit of a problem. I don't want to tell you how to do your job, but there were several available upgrades on the market, you had said in October you were going to pursue them. What gives? GM: Well, in the case of John Doe, we had extensive talks with his agent, and he showed some definite interest. We were a day away from making him a hard offer, but that all changed though when the Mets came in with an offer we thought undervalued him by $2-3m a year. Owner: I don't follow... GM: Sorry, it's a complicated mix of game theory, signalling theory, and auction theory. Basically, the Mets offer clearly demonstrated to us that we had erred in our evaluation, probably due to overestimating his defensive ability Owner: Woah, let me get this straight- you decided not to sign an obvious upgrade at a position of need that I gave you the budget for because the Mets' lowball offer convinced you that your own player evaluation was wrong? GM: That's...um...I don't...um... that's not really a fair way to characterize the process Owner: Why are you still my GM? GM:... I don't think there is a single person involved in baseball who actually thinks assembling a team through free agency is superior or more efficient than developing home-grown talent. GM's already know that free agency is a less-good way to fix your holes. And yet every offseason teams who are plausibly in the playoff hunt have holes that need fixing and can either spend money or spend prospects to fix them. Teams in the hunt want to upgrade and players who are FA's want to get paid. Someone has to win the auction, and everyone playing has more and more money to play with.