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.
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?
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.
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.
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.
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.
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.
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…
