The Knapsack Problem and the Winner’s Curse
As I wait to see if Valve allows Halflife 2 to drop at midnight Eastern across the country (do it! do it!) , I wanted to write a little about two of the problems that face teams in the off-season and drive free agent prices that don’t enter into a lot of discussions: player scarcity as it affects roster construction, and the errors of player valuation. Many of you already know all this, so please– don’t mind me.
Players aren’t paid according to their talents. Given two left-fielders, one who contributes 20 runs/year while the other contributes 10, generally speaking, the first will get more than twice what the second one does. A third player, who adds 30 runs/year, will get more than 3x the worst, and probably more than 2x the first.
This isn’t only because great players are more scarce, though that is true. It’s because the problem of roster construction forces teams to pursue that scarce quality, and so bid it up higher than you would expect given a strict relationship of salary to contribution (say, $1m/win added).
Here’s why. Teams have only 25 spots on their roster. Now, take the pretty universally accepted concept of replacement talent and build a team of six-year minor league free agents, floatsam and jetsam… on the cheap, you’ve got a team that’ll lose over a hundred games, and you don’t have to pay them anything.
How do you get better? Every upgrade on that team requires you to find a player better than the one you already have — and they get harder and harder for you to find, the greater the improvement at one position you want.
Many people advocate the Gillick-Mariners philosophy: put together enough good players and pay them modestly, and succeed as a team. Essentially, find 25 players who are each a win over replacement, and you’re a .500 team. It sounds so easy when you put it like that.
But in real life, if that was cheap and efficent, every team would do that, driving up the bidding on those guys. And then some teams would pick up some +2 win players on the cheap, punting every third position… you see where this is headed. Because teams don’t exist in a vacuum, because free agents are almost always in the decline of their careers, and because of the draft and how players move from team to team, any ideal team construction theory doesn’t translate to the real world.
Each year, every team heads into the off-season knowing two things: what they have, and what they can spend to fill their needs. That budget may include farm prospects, players on the team, money… but look at the Mariners.
What they have: you know this part
What they can spend: a bunch of cash, potentially players
Now, say the M’s need a third baseman and only a third baseman, and they have $20m to spend, which can’t be carried over year-to-year. Their options are:
Justin Leone, 1 win over Bloomquist, $300,000
Koskie (now Steve Kelley-endorsed!), 3 wins, $1,000,000
Beltre, 7 wins, $12m
Leone is clearly the best value choice. But if the team has only one slot to fill, the best way to improve the team is to sign Beltre, even as you recognize that you’re wildly overpaying for performance.
This is part of why the best free agents get such huge deals. There are very few teams that have the flexibility to add a good value player wherever that player may play — and those that do don’t have the money to spend, because they suck (which is why they can add anywhere).
It’s also why pitchers, despite being risky, can get such huge deals: each year there are several teams with money who need at least one more starter, and they’ll bid against each other for the guy who can upgrade their team the most.
Which brings us to the Winner’s Curse. In essence, if four teams with equal footing bid for a player, the team that wins is the one that figured that player would make the greatest contribution — and so is the one most likely to have overestimated that player’s future value.
Or, to put this another way: say something goes up for auction. All four bid on it, and one wins. They immediately decide to try and sell it to the losing bidders, who dropped out at different prices, for the winning price. None will take it — they all think the item has a lower actual value than the winner paid.
Similarly, teams that win free agents and then wish to move them frequently have to pick up the tab to move them.
Put these together and you get Chan Ho Park deals: teams competing in a pitching-scarce free agent market for the commodity that can best improve them, with the one with the most unrealistic and optimisitc view of the player winning the bidding.
Yet to assemble a team that wins more than 81 games, and particularly once you want to reach and win in the playoffs, a team must make these decisions. And so GMs are faced with a horrible decision: which of these terrible deals is the least terrible? If I really want to sign Beltre, I’m not going to have to just overpay for his future contribution, I’m going to have to overpay from what any other team will overpay.
I don’t envy the general manager who has to try and explain to an ownership group why one player can be worth $10, $12, $25 million dollars.