Thursday, February 09 2012 @ 04:56 PM EST

Contributed by: Alex Obal

We have two months' distance from the Yu Darvish extravaganza, and pitchers and catchers are still a week away. What better time to ponder the effect a second-price auction would have on the posting system for Japanese players?

We'll call the item up for bids "Yu Darvish's negotiating rights."

The current posting system, as we're all aware, uses a first-price, sealed-bid auction. Teams submit secret bids. The highest bidder wins the right to negotiate with the posted player, and pays the amount of money it bid.
Example: Rangers bid $51 million, Pirates bid $39 million. Rangers win and pay $51 million.

This auction system is not optimal, because each team has an incentive to bid below the maximum amount it's willing to pay for the negotiating rights.

Suppose the Rangers value Darvish's negotiating rights at $55 million. What I mean by this is that if the Nippon Ham Fighters offered Darvish's posting rights in exchange for $55,000,000.01, the Rangers would say no. Call this $55 million figure the Rangers' Willingness To Pay. In this scenario, if the Rangers bid $55 million and win the auction, they get (what they perceive as) a $55-million asset* for $55 million, resulting in a Net Economic Gain of 0. If they lose, they get nothing for $0, resulting in a Net Economic Gain of 0. In other words, if they bid exactly how much they think Darvish is worth, they shouldn't care whether they win the auction.

On the other hand, if they bid $51M and win, they get the (perceived) $55M asset for $51M, resulting in a Net Economic Gain of $4M. This is a pretty good outcome. If they lose, they get nothing for $0. So the Rangers' bidding strategy becomes a kind of guessing game: they have to figure out how far they can go below $55M while still feeling reasonably confident they'll beat all the other bidders.

Every team has an incentive to underbid its actual Willingness To Pay (WTP), no matter how high or low their WTP is. Suppose the Astros think Darvish is wildly overrated. They value his rights at $3M. Even they won't bid the full $3M since, like the Rangers bidding $55M, they can't come out ahead. In theory, they will bid something lower than $3M, to create the remote possibility they do get some Economic Gain when they win the perceived $3M asset. (In practice, they will see rumors of teams offering 17 times their bid and sit the auction out.)

*I know it can be offensive to refer to athletes as 'assets,' but I think it's fair in this starkly utilitarian context. I do generally avoid it, and anyway we're talking about negotiating rights, not a player. In all sincerity, it doesn't bother me when other folks use 'asset.' The word I personally can't stand is 'premium,' as in 'premium talent' as if talented players are carefully harvested status symbols 'coveted' for their pedigree. What are they, beef? Sergio Santos, prized finely aged black angus proven closer. Price: one B+ prospect.

There are alternative forms of sealed-bid auctions which can minimize this guessing element, such as the second-price auction, or Vickrey auction, after Canadian economist and Nobel laureate William Vickrey. In this format, the winning team pays the second-highest bid as its price for Darvish's rights.
Example 1: Rangers bid $51 million, Pirates bid $45 million. Rangers win and pay $45 million.
Example 2: Rangers bid $51 million. Pirates bid $39 million. Rangers win and pay $39 million.

The virtue of this auction is that the best strategy, generally, is to bid exactly the highest price you would be willing to pay. It's a win-win proposition. If your bid is the highest, it doesn't matter how high your bid is; your price is dictated by the second-highest bid. In other words, you can only hurt yourself by bidding lower (or higher) than your WTP. In all of these examples, the Rangers' WTP is $55M:
Example 1A: Rangers bid $51M, Pirates bid $53M, Pirates win + pay $51M (Rangers Net Economic Gain = $0 - value lost by underbidding)
Example 1B: Rangers bid $55M, Pirates bid $53M, Rangers win + pay $53M (Rangers NEG = $2M)
Example 2A: Rangers bid $55M, Pirates bid $49M, Rangers win + pay $49M (Rangers NEG = $6M)
Example 2B: Rangers bid $51M, Pirates bid $49M, Rangers win + pay $49M (Rangers NEG = $6M - nothing gained by underbidding)

Sadly, since the second-place bid does affect the price and it's in certain teams' interest to undercut each other, there may be a gamesmanship issue here as well.  Suppose you're bidding on Daisuke Matsuzaka. You are the Yankees. You think there is a 75% chance the Red Sox will bid in excess of $50M you're not sure. You yourself value Matsuzaka at $30M, and you will feel compelled to sign him if you win the auction, so as to avoid the Wrath of Selig, discussed below. You are 100% certain that the next-highest bid will come from the Pirates and be worth $20M. If you bid $50M, there's a 75% chance you just took $20M out of Boston's pocket. There's a 25% chance you're stuck overpaying by nearly $20M for the rights to the Japanese Steve Trachsel. If you are so cutthroat that you are willing to pay a constant $60 to drain $100 from Boston's pocket, then your expected payoff here is (0.75*$12M) + (0.25*($-20M)) = $+4M and, assuming all your calculations are right, you come out ahead. You could probably make an even more precise bid to maximize your payoff.

Would anything like this ever happen in practice? Probably not. It's doubtful GMs would ever disagree to this extent about a player's value. Even the lopsided Matsuzaka auction had a respectable second bid of $39 million from the Mets.

Which auction system favors the NPB teams more? It depends. The Nippon Ham Fighters, obviously, want to get as close to the highest bidder's WTP as possible. Ideally, the system would let them take all the GMs out drinking individually, figure out that the top bidder is the Rangers who will pay $55M and not a cent more, and then give Texas the rights for $55M. Of course, since that system is so unfavorable to the MLB owners, Bud Selig would never allow it.

But what's closer to the "isolated drunk GM" auction  the first-price auction or the second-price auction? I figure it depends on how much disagreement on player value there is among MLB teams (bigger disagreement = first-price probably better; the Red Sox were totally smitten with Daisuke Matsuzaka) and how good the MLB teams are at figuring out their competitors' bids (very good = second-price probably better; not good = first-price probably better). I suspect Matsuzaka would have fetched less if he'd been subject to the second-price auction, and Darvish would have fetched more. However, in the absence of data on the actual GMs' WTPs and strategies, any answer to this question will amount to hack theorizing, which is unfortunate, because I think this is by far the most interesting question here.

Anyway, this is a nice simple picture, but it's not complete.  

Complicating factor #1: The winning team doesn't actually have to pay anything either way. This opens the door to defensive bidding. The Angels may decide they value not having to face Darvish in their division  say they're willing to pay $3 million to keep him away, but don't actually want to pay the exorbitant costs to get Darvish for themselves. In theory, they can bid $60 million to win the auction, then tearfully announce that they couldn't settle on a contract. The Angels get the $3M 'benefit' (avoiding Darvish to Texas) and pay nothing for it, resulting in a cool Net Economic Gain of $3M. This strategy works regardless of the auction rules, obviously, as long as the Angels aren't required to pay anything once they win the auction.  

This comes with its own set of complications. First, section 13 of the Japanese-US player agreement as posted on the JPBPA website** gives Bud Selig a range of remedies if it looks like somebody has "undermined" the posting process. Selig gets, among other powers, "... the authority to revoke a U.S. Major League Club's exclusive negotiation rights with respect to a Japanese Player (and, subject to the Japanese Club's approval ... to award such rights to the next highest bidder, if any)." The Angels have to be stealthy. They can't just bid $792 trillion their bid has to look credible. They might also lose some fan goodwill by getting their fans' hopes up in winning the auction. Still, if they're willing to live with the fan backlash, there's really no reason not to place the dishonest high bid. If it works, it forestalls the Yu invasion by one year. If it doesn't, no harm, no foul, right?

Well, maybe not. There may also be an unspoken threat of sanctions from MLB if a team appears to be overbidding with no intention of actually signing Darvish. It's hard to say precisely how valuable it is to avoid the Wrath of Selig, but there must be good reasons to do so. In theory, there could also be objective sanctions: the posting system could be designed so that the winning team must pay X% of its bid to charity, or to the revenue-sharing pool, if it fails to sign Darvish. It's doubtful MLB would publicly announce such a rule, though, since it would give extra leverage to Darvish's agent in negotiating the contract.

Complicating factor #2: The second-price auction lets the public know two teams' bids instead of one. More importantly, it compels every team to bid its actual WTP, so those secret valuations become public knowledge. Thus, the Vickrey auction nudges the murky market for Japanese baseball players toward perfect information.

Who does this favor? It depends. I do think there's a tendency toward groupthink in baseball front offices, which could swing the effect either way. If the Darvish auction had ended with the Rangers bidding their true WTP of $75 million (say) and the second-place team, heavily rumored to be the Blue Jays, bidding their true WTP of $65 million, that would suggest to the other 28 teams that some sharp front offices think Japanese players are undervalued. You'd expect the Yankees and Red Sox to take note, and for the next posting team to cash in. On the other hand, if the Darvish auction ended with the Rangers bidding their true WTP of $51.7M and the SPTHRTBTBJ bidding $35M, this would be bad news for future posting teams, at least in the short term. This guy puts up sub-2 ERAs year in and year out and that's all he fetched? Anyway, this whole paragraph is pretty trivial. The answer to this question turns on what teams' actual WTPs were. No one knows that's the point.

Formally announcing two teams' WTPs also gives a clear signal of one or two teams' intentions. As it stands, we don't know what the Jays bid. Richard Griffin asserts the bid was above $50 million, "sources in baseball insist." Needless to say, I am skeptical. At any rate, one reason this matters is that the Jays' front office seems to prize secrecy. Presumably they would be very annoyed if their competitors could tell whether they were seriously in on Darvish. Did they lose by 25 cents, or were they so constrained that they couldn't even go above $30M? In general, if a specific team thinks Japanese players are a market inefficiency, and they're right, the improved information will likely disable their competitive advantage in a hurry by making enormous bids look perfectly normal.

The moral of the story is that the season needs to start already.