Thursday, June 24, 2010

Vickrey and the "True" Value of FAAB

Rotoman gently chided me for my post Tuesday about Vickrey auctions.
The goal of the Vickrey auction is to induce bidders to make true bids. Vickrey's insight was that in a closed auction bidders often underbid the true value of the asset for fear of overbidding and looking foolish. By assuring the bidders that they'll pay just one unit more than someone else bid, they are freer to bid the actual value they give the object for sale.
After reading the linked article again, I'm not sure that the Tout Wars model truly fits the Vickrey model.
Vickrey's original paper considered only auctions where a single, indivisible good is being sold. In this case, the terms Vickrey auction and second-price sealed-bid auction are equivalent, and are used interchangeably. When multiple identical units (or a divisible good) are being sold in a single auction, the most obvious generalization is to have all winning bidders pay the amount of the highest non-winning bid. This is known as a uniform price auction. The uniform-price auction does not, however, result in bidders bidding their true valuations as they do in a second-price auction unless each bidder has demand for only a single unit.
The likelihood that there are leagues out there that use some kind of bi-weekly or monthly FAAB where players are auctioned off one-by-one that use a true Vickrey model is high. Tout Wars, however, does not use a true Vickrey method. Every player we want to buy is put up for bid simultaneously. In this case, the "true" value is distorted (per Vickrey's theory, at least) by the fact that we are not bidding on a "single, indivisible" good but rather on multiple "goods".

Putting aside whether or not Tout Wars (or any Roto league) truly fits the Vickrey method, a larger challenge I have is that it is far more difficult to translate FAAB bids into some kind of in-season value.

Let's go back to Billy Almon Brown Graduate (BABG) for a moment, and look at the stats they purchased at their 2009 auction, the stats the American League put up on the whole, and the stats BABG wound up with at the end of the season.


BABG Hitters Auction2,2739,0021,304.270$2,184

A.L. Hitters Total2,558 10,4231,541.267$2,512

BABG Hitters Total2,4499,8521,476.268$2,393


BABG Pitchers Auction6484624.181.358$848

A.L. Pitchers Total1,1475574.461.403$1,313

BABG Pitchers Total843 5214.291.364$1,046

BABG used its FAAB to buy $407 of additional stats (out of an available $793 of available stats) from the free agent pool. BABG spent $929 of its $1200 in FAAB (including adjustments for players lost to the American League). Each FAAB dollar spent brought a return of $0.438.

The best free agent hitter in the A.L. last year was Scott Podsednik, who earned $24. His "market" bid, then, would have been $55. The best free agent pitcher was Andrew Bailey, who earned $38. His "market" bid was $87.

Bailey's season, though, is atypical for a free agent pitcher. The next best A.L. pitcher - Jason Frasor - had a more "typical" year, earning $21. His "market" bid would have been $48.

Coming all the way back to Rotoman's comment and Conor Jackson, then, in theory we overbid on any hitter and almost any pitcher when we cross the $50 barrier with our FAAB. Based on ROI, we should be bidding consistently and frequently throughout the season.

The paradox here is that if everything goes right, you buy a balanced team, and don't have any injuries, you are almost always better off than if you buy a team with holes and/or injuries and have to churn through your FAAB all year to try and find this year's Bailey or Podsednik (you might not). This brings me back to what I said earlier. I'm not sure Vickrey reflects the true value of a player any more than a closed bidding system does simply because we're bidding far more on the perceived value of a player than we actually are on what the player will be worth at the end of the season.

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