Imagine an auction that takes place between three bidders. The item in question? An envelope filled with money. All three bidders employ teams of analysts that attempt to ascertain how much money is in the envelope, based on a variety of evidence that isn’t important for this analogy. Each bidder thus arrives at an estimate of the fair value of the envelope. Then they place a single sealed bid. The highest bidder out of the three gets the envelope.
What bidding strategy would you employ? Here’s a bad one: Just bid what your team of analysts calculates as the expected value of what’s in the envelope. The reason this is bad is known as the winner’s curse. If each bidder comes up with an estimate of fair value and bids that number, the winner will be the one with the highest estimate of fair value. In other words, you’ll only win if your estimation of the envelope’s value is higher than everyone else’s, and since you’re always paying exactly what you’re hoping to gain, you’ll tend to lose in the long run.
Allowing for a lot of approximation, this situation describes free agency in major league baseball. Every free agent has an unknowable amount of expected future production. Teams employ armies of analysts who attempt to estimate that production. Then, armed with that knowledge, they make contract offers to that free agent, in competition with other teams.
As I said, there’s a ton of approximation and simplification going on here. Players aren’t envelopes filled with money. Team context matters. Players don’t have to accept the highest bid. Tax regimes aren’t equal, and non-monetary incentives matter, too. Contracts are complex, and there’s no requirement that they be the same number of years, have the same number of options, no trade clauses, or anything of the sort. There’s no agreed-upon universal value system; different players present different value to different teams.
But that doesn’t mean the abstracted case has no use. As we approach the trade deadline, I think there’s one clear one: dispelling the myth that teams refuse to give up much to trade for a player who just signed a big free agent deal — after all, if they valued them enough for a blockbuster, they would have just offered a bigger contract, right? That’s a great soundbite, so you hear it all the time, but it doesn’t jive with established economic theory.
The style of contract negotiation where multiple bidders submit bids and a single seller chooses one of them can be stylized as an auction. “Auction” might sound like a weird way to describe it, but if you stop to think about it, it makes perfect sense. It’s a way for multiple bidders to use their willingness to pay to differentiate themselves to a seller.
The classic auction you think of is an English auction. There’s an auctioneer, and some old people with monocles and paddles. The price keeps going up unit by unit; if you value something more than the current bid price, it’s optimal to bid more for it. In theory, the price will continue to go up until the bidder with the second-highest valuation of the item being auctioned reaches their top valuation and drops out of the bidding. The bidder who has the highest valuation then wins the auction, paying only enough to outbid the valuation held by the second-highest bidder.
A quick example: let’s say that we’re bidding for a Cal Ripken Jr. baseball card. I think it’s worth $250, you think it’s worth $200, and Meg Rowley thinks it’s worth $600. Below $200 dollars, everyone’s bidding. You drop out at $200. I drop out at $250, leaving Meg the winning bidder at either $250 or $251, depending on who bid $250 first. The bidder with the highest valuation won, and the price they paid is the valuation held by the bidder with the second-highest valuation. (A nit-picky academic aside: If you assume that bids can be made in any increment, the winning bidder will pay a fraction of a cent more than the second-highest bidder’s valuation. That’s why it’s expressed as the second-highest valuation; a bid of $250.00000001 is close enough to $250 that there’s no point in distinguishing.)
It doesn’t matter whether Meg thought the card was worth $300, $650, or $10,000. The second-highest bidder’s valuation sets the price. That’s not how free agency works. If Team A offers Player X a $100 million contract, Team B can’t listen in on the phone line and say “$101 million” only for Team A to counter with “$102 million” and so on. Relatively few offers are made. Generally speaking, they’re made without exact knowledge of what the other interested parties are doing. When Team A offers that $100 million contract, they have no way of knowing whether other teams are in the same ballpark as them. Maybe the next-highest offer is $80 million. Maybe there’s already a $130 million offer on the table.
Before I get into the meat of my argument, it’s worth making one thing clear: Money isn’t a proxy for anyone’s value. There’s no way around modeling it that way in these simple abstractions, but they’re just that: abstractions. They aren’t a perfect mirror for the real world. To come up with a model, you have to have some kind of single-unit measure of value, and I’m using dollars for the sake of simplicity. That’s not real life. The optimal amount to offer someone in exchange for their services playing baseball doesn’t say anything about their “worth”; it’s just economic (and free agent contract) shorthand.
Let’s return to free agency. The best way to describe these negotiations, for the purposes of defining a generic game, is a first-price sealed-bid auction. In this style of auction, bidders submit a single sealed bid without knowledge of other bids. The seller then selects the highest price and sells the good to that bidder for that price. It’s not quite a perfect fit – negotiation happens after bids have been submitted, and teams frequently submit multiple offers over time – but it’s a good first-order approximation. And the established strategy is decidedly not “bid what you think the good being auctioned is worth.”
Let’s talk about why. Assume our three-bidder envelope scenario from above. Further assume that the value of the envelope is $100, and that the three teams bidding for the envelope have analysts who independently calculate their own expectation of that value. Those calculations are randomly distributed around $100, with a standard deviation of 15 percentage points.
In the case where each team bids 100% of their calculated value, they each win a third of the time (obviously). On average, the sale price is 112.7% of $100 – oof! Imagine being one of those teams of analysts and suggesting this plan to your boss. “We’re going to bid in an auction. We’ll win a third of the time. On average, we’ll be overpaying by 12.7%. Oh, and we’ll only pay less than the envelope is worth 12.5% of the time that we win.” This is an obviously abysmal plan.
The clear problem here is that you shouldn’t bid an amount such that you’ll never be excited about winning. If you always pay 100% of what you think a thing is worth, the only way you end up winning is if a) you undervalue the item in question and b) both of your rivals in this game do as well, and by more than you did. That doesn’t happen very often. A better strategy is to bid an amount lower than you think the item is worth, but still close to the value, so that you can still win some percentage of the time without paying vastly more than its value.
To do a bit better than broad generalizations, I wrote a Python script that simulates this auction. That’s where I got the 112.7% number, as well as the 12.5%. That’s with each of the three teams bidding 100% of their calculated value in the auction. To figure out alternative strategies, I can just change the bid.
For example, if Team A bids 88.8% of its estimate while the other two teams bid 100% of theirs, things change meaningfully. Now the results look like this:
Implied Values, Team A 88.8%
| Team | Bidding Strategy | Hit Rate | Average Price Paid | Bargain% |
|---|---|---|---|---|
| A | 88.8% | 14.0% | 104.6% | 32.7% |
| B | 100% | 43.0% | 110.6% | 17.9% |
| C | 100% | 43.0% | 110.6% | 17.9% |
A quick explainer on the columns: bidding strategy refers to what percentage of their calculated fair value a given team bids in the auction. Hit rate is how frequently a given team wins. Average price paid is what percentage of true value (100%) each team pays, on average, across all its winning bids. Bargain percentage is the percentage of winning bids that provide positive value, i.e. where the winning bid is less than 100%.
Now, Team A’s strategy looks meaningfully better to me than their two rivals. They’re winning auctions less frequently, sure, but winning wasn’t so great when it was almost never a good deal. If this is a repeated game (many auctions over time), like free agency, you’d expect Team B and Team C to rein in their strategies. What if they, too, started bidding 88.8% of their estimate in an attempt to rein in costs?
Implied Values, All Teams 88.8%
| Team | Bidding Strategy | Hit Rate | Average Price Paid | Bargain% |
|---|---|---|---|---|
| A | 88.8% | 33.3% | 100.1% | 51.2% |
| B | 88.8% | 33.3% | 100.1% | 51.2% |
| C | 88.8% | 33.3% | 100.1% | 51.2% |
That 88.8% figure wasn’t chosen at random; it’s the ratio that, in this example, produces an expected cost of roughly 100% for each bidder if they all follow the same rule. Roughly 50% of the time, the price paid ends up being a bargain, which follows logically. If you want to counter the winner’s curse, you have to bid less than your expected value, and that holds for everyone involved in the bidding.
This isn’t what economists call a stable equilibrium. Now that Team A’s rivals are bidding less aggressively, Team A can bid even less aggressively than the rivals and capture some expected profits, at the cost of winning the auction less frequently:
Implied Values, Team A Conservative
| Team | Bidding Strategy | Hit Rate | Average Price Paid | Bargain% |
|---|---|---|---|---|
| A | 85.0% | 25.2% | 97.3% | 62.3% |
| B | 88.8% | 37.4% | 99.3% | 54.5% |
| C | 88.8% | 37.4% | 99.3% | 54.5% |
Now, on average, is this deal worth it for Team A? If all they care about is maximizing excess value, sure. If they’re targeting some minimum amount of value added – imagine this past year’s Giants, who had money to spend and wanted to add some talented players with it – being more passive than breakeven might be a bad strategy, because it has a chance of leaving you with nothing.
Interestingly, Team A bidding less aggressively makes Team B and Team C’s outcomes look better, even with a static bidding strategy of 88.8%. As Team A gets even less aggressive, things continue to look rosier:
Implied Values, Team A Opportunistic
| Team | Bidding Strategy | Hit Rate | Average Price Paid | Bargain% |
|---|---|---|---|---|
| A | 75.0% | 8.7% | 90.0% | 88.4% |
| B | 88.8% | 45.6% | 97.6% | 60.6% |
| C | 88.8% | 45.6% | 97.6% | 60.6% |
Maybe that’s a Tampa Bay style of strategy. Come in low, knowing you’ll usually miss. When you do hit, you’re probably clearing a good deal. On the other hand, if one of the bidders gets extremely conservative, maybe it makes sense for another bidder to get aggressive to take advantage:
Implied Values, Team B Aggressive
| Team | Bidding Strategy | Hit Rate | Average Price Paid | Bargain% |
|---|---|---|---|---|
| A | 75.0% | 7.2% | 90.5% | 87.3% |
| B | 92.0% | 52.9% | 100.0% | 51.7% |
| C | 88.8% | 40.0% | 56.7% | 57.7% |
Team A’s timid bidding means that the winner’s curse is lessened. Plenty of times, Team B will win not because it has the highest valuation, but because Team A just isn’t competing enough. That opens room to get more and more aggressive in bidding relative to modeled value. Now Team B is winning the auction a full half the time without losing money on average.
You can play around with this style of analysis endlessly. Team C might actually have room to get less aggressive themselves at this point, since they’re generally going to beat Team A anyway. If they back off, they can win a ton of auctions while still getting meaningful positive value on the ones they win:
Implied Values, Team C Counter
| Team | Bidding Strategy | Hit Rate | Average Price Paid | Bargain% |
|---|---|---|---|---|
| A | 75.0% | 9.7% | 89.5% | 89.5% |
| B | 92.0% | 63.0% | 98.6% | 57.0% |
| C | 83.0% | 27.4% | 94.0% | 74.2% |
If teams have to act without knowing their rivals’ strategy, there’s no strong-form equilibrium to be found. Game theorists have calculated what’s called a Bayesian-Nash equilibrium for one form of this auction when auction valuations are drawn from a continuous uniform distribution, but that’s not what we’re dealing with here. In any case, the right behavior for a given team depends on the behavior of others, but in every case, the optimal bid is less than 100% of calculated value.
This makes sense intuitively. Imagine a GM winning the auction to sign an impact player. If the “every team bids up to its indifference point” crowd are correct, that GM’s reaction should be just that: indifference. “I like my team the same as I liked it before signing Bryce Harper because I made a bid of exactly what I am willing to pay to the point where his deal has no surplus value.” That seems dumb on its face. Teams don’t bid for free agents because, if their bid is accepted, they’ll be indifferent. They do it because they want to add that player at that price. They’d prefer to win as opposed to lose the bidding. Otherwise they wouldn’t bid that much!
If teams are acting as economically rational actors, they should rue missing out on free agents fairly often. To leave yourself room to come out ahead, you have to sometimes miss on bargains. Teams are no fools. They understand this concept. I’m willing to wager that, some significant fraction of the time, teams see the terms for a free agent who just signed and think “Ooh, we missed on that one.” When you’re bidding in the dark, that has to be the case if you want to pick a winning strategy in the long run.
For a variety of reasons, this abstracted example isn’t a perfect reflection of free agency. I picked three teams rather than four or five arbitrarily. I don’t have any particular reasoning behind my 15% standard deviation selection; the real variation in projections is likely smaller than that, though I don’t have access to team valuation models to say that with any certainty. Cut the variance term from 15% to 7.5%, and the bidding strategy that produces no excess value moves up from 88.8% for each team to 94.5%. There’s nothing special about those numbers; I’m just using them to show how the math works rather than saying they exactly represent reality.
The very concept that every team has a consistent valuation framework is probably wrong; they all no doubt have some version of it, but players output hits and runs and strikeouts and walks, not dollars. It’s all very indirect, and different teams probably handle that process in extremely different fashions. Should you account for marketing value? Blocking a prospect? A team’s place on the win curve?
A marquee player changes the equation even more. Sure, in theory you’re playing a repeated game, and making good decisions in the long run adds up. But each free agent is unique. You don’t get to bid on Harper 15 times and look at how you did in aggregate; there’s only one of him and he’s not a free agent every year. That might cause teams to diverge from “optimal” long-run behavior; players aren’t fungible, and there really might be no replacing the guy you miss. What are you going to do, trade for him?
I also don’t think that the calculations are done on the terms I’m describing here. Teams almost certainly don’t calculate up some grid of expected production value and discount from there. I assume it happens more organically: A GM goes to their team of contract specialists and says something along the lines of “come up with a contract offer for Player X that will make us happy if we sign him.” More or less wiggle room might get added based on how badly the team needs that particular player, whether the owner is a fan, or whatever other factors you can think of. Game theory never needs to explicitly come into the discussion.
I’m not claiming that I’ve solved the equation. I don’t think I ever will, in fact. Probably, no one can solve this problem perfectly. But I think the general conclusion is inescapable. Teams absolutely expect to get a positive benefit when a free agent accepts their contract offer. A meaningful fraction of free agents sign deals that pay them less per contribution than some arbitrary fair value, normalized across all free agents, would suggest. Mathematically, it just has to be that way.
What should you take away from this article? It’s basically this: stop thinking that a free agent contract is a perfect reflection of exactly what the league, as a whole, thinks a given player’s contributions are worth. Nothing about the way free agency works suggests that conclusion – it’s a logical fallacy. It feels like any auction should find the fair value of the thing being auctioned, but that’s not how it works. Auctions find the auction clearing price, which generally includes some expected profit for the buyer.
Enough competition can erode that expected profit to roughly zero, but even then, an expectation of zero implies that about half of the time, the buyer will be getting a bargain. Other teams know that, and while “what did this guy get in free agency” is a useful data point for working out a player’s value in trade, it’s definitely not the end of the argument. If you want to figure out what teams would give up to get a player, don’t just lean on precedent. Start from first principles and figure it out. The shortcut of “oh they were a free agent so I can assume they are being paid perfectly efficiently” just doesn’t work.
