Prince and Albert in a Can: The Winner’s Curse

In early December, Albert Pujols signed a ten-year deal with the Los Angeles Angels of Anaheim, worth around $254 million – one of the largest contracts in Major League Baseball (MLB) history. The 31-year-old first baseman is arguably the best hitter in baseball; he has laid claim to multiple MVPs, Gold Gloves and Silver Sluggers, not to mention the Rookie of the Year award. Since 2001 (his rookie season), he has had a batting average over .300 and had more than 100 RBIs in every single season except the most recent (he batted a mere .299 with only 99 RBIs), something that no other player has ever done.

Prince Fielder has thus far had an equally impressive career. He swatted 230 home runs in a little more than six seasons with the Milwaukee Brewers, including 50 long balls in the 2007 season. He has a career on-base percentage of .390 and a slugging percentage of .540, a rare ability to both reach base and hit the ball hard. Though Fielder remains unsigned (as of December 29, 2011) and his trophy case is not quite as full as that of Pujols (Fielder has a pair of Silver Slugger awards), he figures to receive a contract nearly as lucrative.

It hardly seems that either of these ballplayers could be over-valued, but they each have opportunities to deliver a winner’s curse.

The winner’s curse is a tendency for an auction’s winning bid to be greater than the intrinsic value of the item being auctioned. It’s a term from economic theory pertaining to auctions where the value of the item is unknown and the amount of each bid is known only to the one who places it. This condition clearly applies to the market for free agent baseball players; at best, teams can only estimate the value of a player based on his historic performance, and free agent contract negotiations are frequently done in private, such that one team does not know what another is bidding for a player. It is for this reason that teams can suffer from the winner’s curse when signing free agents.

When a player is a free agent, any of the 30 MLB teams can bid for his services. Each team independently evaluates the player’s value and thereby assigns a dollar amount they would be willing to pay for his services. Assuming the player’s production is worth roughly the same to all teams, each team will arrive at an estimate of the player’s value that is close to the player’s intrinsic value. Since teams do not know the player’s intrinsic value (who could predict how many home runs Fielder will hit in the next eight years?) each team’s estimate will be slightly different. On average, the estimates of the player’s value will likely accurately represent his actual value. But the winner of the auction is not the one who places an average bid, it’s the one who places the highest bid – and the one who places the highest bid will have thereby overpaid for the player’s services.

In fact, you sometimes hear players talk about this phenomenon when discussing free agent negotiations or vying for a high draft selection in a new player draft: You don’t need to convince every team that you are an outstanding player who deserves a huge contract; you only need to convince one team. Even if nearly every team accurately assesses the value of a player’s production at $18 million per year, the one team that inaccurately thinks the player is worth $20 million per year will be the one who wins the contract, and will pay too much. Though they likely aren’t aware of it, the players are describing the winner’s curse.

While one could find a long list of players who have been given fat contracts only to underperform, I’ll leave you with one of my favorites. In 2007, the Chicago Cubs signed outfielder Alfonso Soriano to an eight-year contract worth about $136 million. Soriano was coming off a career year in 2006, and had a great first year with the Cubs (batted .299 with 33 home runs). However, his production regressed significantly thereafter; in 2009 he batted just .241 with only 20 home runs and hasn’t improved much since. The Cubs badly overestimated the value of having Soriano on their team, got a bad case of winner’s curse and are consequently paying him much more than his numbers would dictate. The real kicker is that Soriano has a no-trade clause in his contract, which means the Cubs are effectively stuck with him through the 2014 season.

Will Pujols’ and Fielder’s future performance not stack up with their enormous contracts? I suppose that remains to be seen.

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If you made it this far and are still interested, this website has an excellent applet you can tinker with demonstrating how the winner’s curse works. The example is that you are a CFO deciding what to bid on a takeover of a small privately-held company whose value is unknown. It lets you plug in potential values of the company and then submit a bid; it then generates 20 random scenarios and tells you if your bid won and how much your profit/loss is.

Comments

[Extreme Fajita]

From the applet:
"Consider the expected value of the firm. Since any value between 0 and 1000 is equally likely, on average, the firm is worth 500. However, if you bid 500, notice that you will only win if V < 500. But, if the firm is worth at most 500, it is worth on average only 250, so you overpaid! Even with the synergies, 250x(1.5) = 375 is much less than you bid. This is the winner's curse! To be successful, you have to recognize that the expected value of the firm is irrelevant. All that should concern you is the expected value of the firm if you win, which depends on your bid. "

I don't get it. If the firm is worth between 0 and $1000, and it's worth is a random outcome with all dollar values equally probable, then the expected value is $500. Not that it is worth at most $500, but that it is expected that it IS $500 (and that it is worth at most $1000), so computing another expected value calculation ($250) makes no sense. In this example, all rational competing bidders would use expected value and all come up with a valuation of $500.

Its value IS known in a sense, since the probability is the only information you have on it, and repeated trials would eventually produce the expected value due to the law of large numbers, erasing any overpayment if all bidders bid the expected value.

What I'm saying is that your example is much better than theirs.

[Spudboy]

I'd also challenge the assumption that any value between 0 and 1000 is equally likely. Wouldn't you expect a bell curve, with values near 500 being more likely than those at the extremes?