The Knapsack Problem and the Winner’s Curse

DMZ · November 15, 2004 at 10:08 pm · Filed Under Mariners 

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.


24 Responses to “The Knapsack Problem and the Winner’s Curse”

  1. U.S.S. Mariner » Pedro on December 15th, 2004 3:50 am
    […] n over our other options. Then, based on the team’s individual situation (see “Knapsack Problem“) and whether that player will push them into the playoffs (where the team makes a […]

  2. The Ancient Mariner on November 15th, 2004 10:50 pm

    That’s it in a nutshell: “Which of these terrible deals is the least terrible?” Superbly put.

  3. tyler on November 15th, 2004 11:05 pm

    absolutely fascinating… the link and its subsequent links were excellent info from both an educational and athletic perspective.

  4. Conor Glassey on November 15th, 2004 11:33 pm

    Hey Derek – Nice read, which reminds me…How come we haven’t seen a “Breaking Balls” column in over a month? Does it have something to do with your top-secret other projects? When do we get to find out about those?

  5. Cap on November 15th, 2004 11:36 pm

    Midnight Pacific!

  6. DMZ on November 16th, 2004 1:36 am

    How come we haven’t seen a “Breaking Balls” column in over a month? Does it have something to do with your top-secret other projects? When do we get to find out about those?

    Good questions, Conor. I have no answers to give you right now. Sorry.

  7. Gary on November 16th, 2004 1:56 am

    Players like Beltre, Beltran or at another level Barry Bonds, Albert Pujols: they’re overpaid only if you assume that the relationship between wins and salary is linear.

    In fact, it’s geometric because there’s only 25 guys allowed on a roster. In this restricted universe, we have seen that the number of players who can contribute 7XB (where B is Bloomquist) is far fewer than than those who can contribute 1XB, 2XB or 3XB.

    Say the Yankees acquire 25 players who are all 7XB, giving them the potential to win an amazing total of 175 games in a 162 game season. If another team could get 175 Bloomquists on the roster (assuming they could also get all the extra at bats needed) to balance out the 25 X 7B on the Yankees, then they could tie the Yankees, theoretically.

    But in fact the universe is 25 players and a game has only 27 outs (you can’t give the Willie B’s sufficient at bats). So a player who can contribute 3X or 4X more than Bloomquist has a geometric, not an linear impact. These players “deserve” more than 3X or 4X times Bloomquist, hence they are not, strictly speaking, overpaid.

    Which is to say if Beltre could actually contribute 7XB, the Mariners should snap him up.

  8. tede on November 16th, 2004 2:27 am

    So why was Dan Rohn passed over for the coaching staff?

    Is Hargrove afraid of having a potential successor on board (although Baylor sorta qualifies)?

    I think the guy did pretty good in Tacoma under the circumstances Bavasi put him under.

  9. Paul Weaver on November 16th, 2004 3:56 am

    #6, Gary makes a great point about the non-linear relationship of players’ value. But Derek’s point is still valid, because the highest bidder is going to get the player, hence they will generally be overvalued (even if it is over their geometric value as opposed to linear value). Also, these are players that have shown what they can do, i.e. past their peak.

    #7, at least we’ve got Rohn still in the system. I think he is a great coach, but I will temper that by saying we shouldn’t over value his ’04 season. If you look at AA for ’03, and consider the pick up of Bucky Jacobsen, we had a pretty good class ’04 AAA team coming in.

  10. Chris Caldwell on November 16th, 2004 6:07 am

    Wow. I try to take a study break from cramming for my exam this morning and I still learn something mathematical. Thanks a lot, Derek.

  11. Mark on November 16th, 2004 7:18 am

    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.

    I’m not so sure. Baseball isn’t a rational market. It’s far more ego-driven than other markets we’re familiar with. The owners, for the most part, aren’t guys who depend on baseball revenues for their livelihood; they’re millionaires or billionaires who struck it rich somewhere else and bought a team as more of a hobby than a business. They then sink funds into the team for the same reason that a guy soups up the car in his garage: not because it’s the most rational, reasonable path to success, but because it’s fun and he likes doing it.

    For that reason you can’t expect every team, or even most teams, to follow rational principles in organizing their payroll. Even if there’s one approach that clearly seems to be the best, there will always be a certain number of owners who ignore it and do things their way. So the question for a GM is actually: what’s the best approach given the likely behavior of other teams in the league? Not unlike a poker game; if you know some of the players at the table are almost certain to bluff, even when it doesn’t make much sense for them to do so, you should revise your own strategy to adjust to/take advantage of that.

    All of this is a lot of words to make a pretty simple point: the Gillick strategy is a good one if the situation warrants it. You have to be smart in deploying it, though; any plan that’s followed reflexively is likely to go sour once the environmental circumstances shift.

  12. ajp on November 16th, 2004 8:02 am

    Gary, there just has to be something fundamentally wrong with an arguiment that includes the statement, “…you can’t give the Willie B’s sufficient at bats.”

  13. Zzyzx on November 16th, 2004 8:14 am

    You’re missing another point here. Different teams have different amounts of money to spend. If the Yankees have 10x as much money as the Devil Rays to spend, spending twice as much on a player as the Rays would have been willing to do could be a good deal for them.

  14. Mark on November 16th, 2004 8:24 am

    Different teams have different amounts of money to spend.

    They also have different amounts of money to earn. A team that own its own cable network could see a much, much better return on its investment if it spends heavily on high market-profile players — who can be expected to generate buzz, viewership, and advertising revenue — than a team that depends largely on in-stadium revenues. There’s only so many tickets you can sell, but a television audience is almost limitless.

    No doubt about it, this is a complicated equation, with lots of variables that are hard to pin down.

  15. DMZ on November 16th, 2004 9:07 am

    Come on now, I did go so far as to speckle the post with caveats and notes on assumptions like if teams are “on equal footing” and the like. I’m well aware of differences in the marginal value of wins between different teams. I wanted to illustrate the basic principles at work that drive star player valuation (and over-valuation) is all. Let’s not get too overly wonky just yet.

  16. everett on November 16th, 2004 9:16 am

    Thats the problem with having a readership with above-average intelligence… They’re always looking to take the next step (or three). Great read.

  17. Zzyzx on November 16th, 2004 9:21 am

    Don’t worry, I knew that you knew that.

  18. Thor on November 16th, 2004 9:25 am

    I think the $10,000 question that this brings up is, why don’t teams do more trading to balance themselves? I mean, if salaries really work this way (i.e. decreasing returns to dollars spent on a single player), then why don’t teams try really hard to become balanced, 2.5 win/player teams? DMZ makes the point that they’re only going after one player at a time, so they spend all their money on that one player. But instead of buying one 7 win share player for $13, why not trade away one of your 1 win share guys (who gets $1) and sign 2 4 win share players for $5 a piece? (i.e. now you have the same 8 wins for $3 million less…)

  19. chris w on November 16th, 2004 9:26 am

    The point is that, because of the large numbers of variables in this equation, including team-to-team variations, it’s almost impossible to accurately generalize about players’ “values” in relation to their salaries. You need the context. The bottom line is that the MOST important thing for a front office to do is understand its own context and how it interacts with all those variables and to consistently take steps in light of that understanding. That requires organizational direction, and that’s where the Ms fail.

  20. Evan on November 16th, 2004 10:22 am

    Gary – even with the geometric relationship, the Winner’s Curse still guarantees the player will be overpaid.

    The only way to avoid overpaying players is to pursue players other teams don’t want. Billy Beane used to get away with that, but now even he has competition.

  21. ajp on November 16th, 2004 11:25 am

    The best way to avoid overpaying is to develop your own star players, buying out their first years of free agency with below market, long-term contracts tendered in their 1st six years.

  22. Patrick R on November 16th, 2004 1:34 pm

    20. Hank Blalock comes to mind.

  23. The Ancient Mariner on November 16th, 2004 4:45 pm

    Re #17: Thor, take two teams. Team A starts four 7+ players and four 1+ players; Team B starts eight 4+ players. Which team is going to have an easier time improving itself? Obviously, Team A; Team B is going to win more than it loses, but they’re going to have a difficult time trying to contend, because any improvement is going to be very expensive. Team A, on the other hand, can go out and sign two or three 4+ players without too much difficulty, which will bring significant improvement.

  24. Paul Weaver on November 17th, 2004 5:22 am

    #22, AM: But Team A is paying so much more for the four 7+ that it can’t afford the improvement….unless they’re the Yankees….#17 was making a point about how to get more bang for the buck.

    You’re right that team A is in the better position (for improvement), but team B will have more financial flexibility and is more applicable to mid-market teams.

    Team C has 25 1+ players, most of whom are below drinking age, and hopes they emerge into a cheap world series team in 3 years.