Dollars and Jens
Friday, September 23, 2011
A classic result from auction theory
This isn't the least bit timely, except that it was covered in one of my classes recently.

The King of Elbonia wants to build a new moat. The Elbonian Construction Group can do the job for $1 million, but the Construction Group of Elbonia has a new technique that brings their costs down. Unfortunately, nobody outside of CGE is quite sure how much, and the insiders can't quite be trusted to be honest about this information; the King's Counselors, though, are well known to believe that CGE could build it for somewhere between $800,000 to $1 million, with any dollar figure in between equally likely; the quality would be the same. Since Elbonia is located in Auction Theorist's World, all agents are known to keep their agreements, to be risk-neutral, to have no costs associated with putting together a bid, and not to anticipate any gain or loss after this deal as a consequence of it; for example, CGE does not worry that giving up information on their costs now will reduce their chance to exploit its privacy in the future. (None of these features of ATW are necessary for the qualitative effect being illustrated, though they will make it bigger, and much easier to calculate.)

The usual thing to do in a situation like this is to put it out for bid, in which case CGE would simply bid $999,999; everyone already knows that CGE is the low-cost producer. As an expected price optimizer, however, the King instead proposes the following: CGE will be hired to build the moat for $900,000, if it wishes to do it for that price; otherwise, ECG will be paid $1,000,000. If CGE can do it for $950,000, of course, the King is spending an extra $50,000 to have ECG do it; why would he do that? Well, with his proposal, there's a 50% chance that he'll save $99,999; the other 50% of the time he only loses $1 based on what CGE would actually bid, so that, overall, he saves an expected value of $50,000 (in round numbers) doing it his way. And, by doing it his way, he keeps CGE honest; there's no reliable way to get CGE to announce its costs while offering to pay no more than that. If he offered CGE two different rates, depending on its costs, CGE would have no reason not to claim to need the higher price. If he raises the price above $900,000, he's losing more, in expectation, than he's gaining.

Suppose neither company's costs are known exactly; then it gets more complicated. If ECG's price is somewhere between $1 million and $1,200,000, again uniformly distributed, the optimal plan turns out to be to ask both firms their price and hire ECG if its price is less than $100,000 more than CGE's price; CGE will be paid $925,000 if it's willing to do the job for less than $900,000, but would have no reason to admit to having lower costs than that; the $25,000 is necessary to keep the company from padding its costs. If ECG comes in right at $1,000,000 (and is hired), ECG will be paid $1,025,000—the $25,000 is, again, to keep it from inflating its costs. The essential point is that, even though ECG is known to be the higher-cost bidder, by committing to hire ECG if its price is close enough to CGE's, CGE can be made to compete with it—to the King's profit, at least in expectation.

As one final note, this is related to placing a reserve price on an auction to sell something you really don't want, anyway. In some sense, the King (in the original story) is selling the right to be paid $1,000,000 to build him a moat, and is asking CGE to bid for it; as far as he knows, it may be worth up to $200,000 to CGE, and he's insisting that CGE pay at least $100,000. If there were more than one potential bidder in the same position as CGE, each with its own cost, they would have to bid against each other, as well as the "reservation price"; with only one bidder, there's no reason to divulge anything but a willingness (or not) to pay the $100,000 for the contract.

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