Long before I went to grad school, I worked for a construction company that specialized in building water and wastewater treatment plants.  They got work by submitting bids.  That’s how I got interested in auctions and I eventually wrote my dissertation on auctions.  Since Swiss has been beating the drum for content, I thought I’d write a little about auctions. I’m not going to the write out the proofs this time because writing equations in Word makes me cranky.

Economists separate auctions into private value and common value auctions.  The difference is driven by how the object for sale is valued.  In a private value auction, there is no true, objective value.  The value of the item for sale depends on the individual.  In a common value auction, there is a true value that is the same to everyone and it doesn’t depend on the individual.

For a private value auction, consider a painting by an unknown artist.  I think it’s beautiful and you think it’s ugly.  The value depends on our private opinions – my value does not affect your value.  For a common value auction, consider an oil lease.  The amount of oil under the ground doesn’t change depending on who owns it.

First, I’ll talk about private value auctions.  What is neat about private value auctions, is that as long as you meet certain criteria, the format doesn’t matter, they all raise the same revenue for a given number of bidders.  This is known as the revenue equivalence theorem. (And thus, the bidders get the same surplus (profit) defined as the difference between the winner’s value and the price paid.)

What I mean by format, is the auction rules.  A first price sealed bid is like a silent auction at a charity event.  Bids are written down and submitted.  They are opened and the high bidder wins and pays their own bid.  A first price descending auction uses a clock.  Bidders watch the clock count down and the first bidder to stop the clock gets the item and pays their own bid.  This is also called a Dutch auction because the Dutch flower auctions use this mechanism.

A sealed bid second price auction (or Vickrey auction) is like a first price sealed bid auction in that the high bidder wins, but the winning bidder pays the second highest bid.  I’m not aware of second price sealed bid auctions used in the wild.  Vickrey developed it as it allows economists to investigate various aspects of behavior. A second price ascending auction (English) auction can be an open outcry auction like eBay or the type that Sloopy runs or even an ascending clock auction (Japanese auction).  In these auctions, the high bidder wins and pays the winning bid, but it is equivalent to paying the second highest bid. It is easiest to see the equivalence if you think about an ascending clock auction.  The price rises and the bidders drop as it reaches their values.  The winner is the last one standing and pays the price at which the last bidder dropped out – equivalent to the second highest bid.

Auctions can be used to model other phenomena as well. In an all pay auction the high bidder wins, but all the bidders pay their own bids. This model is often used to analyze things like lobbying.  And yes, this format will yield the same revenue as a first price sealed bid auction.

The criteria for revenue equivalence are:

  • The bidders are risk neutral
  • The values are independently and identically distributed
  • the bidder with the highest value (and thus bid) wins (bids are monotonically increasing in value)
  • the lowest value has an expected surplus of zero.

Now, revenue equivalence doesn’t mean strategically equivalent.  People bid differently under the different formats.  The first price sealed bid and the Dutch auction are strategically equivalent (bidders follow the same strategy), and the second price sealed bid and the English auction are strategically equivalent in the private value setting.

In the first price sealed bid auction and the Dutch auction, since winners pay their own bid, they don’t want to bid their actual value.  If they did, they wouldn’t have any surplus.  Instead, they bid slightly less than their value.  How much less is determined by the number of bidders.  As the number of bidders increases, they bid closer and closer to their own value.  Their actual values are not revealed in the auction process.

The second price auction (whether sealed bid or ascending) is what economists and game theorists call a truth telling mechanism.  In many economic transactions, it is in the interest of the parties to conceal information.  If I let the car salesman know my top number, he will do all he can to get me to pay that amount.  I’m sure you can all think of other examples.  Second price private value auctions are truth telling mechanisms because it is your interest to bid your actual value. If you were to bid more than your value, you run the risk than the second highest bid is also above your actual value and instead of a surplus, you have a loss.  If you bid less than your value, you run the risk that the high bid is above your bid, but less than your value, and thus you have lost the item and the surplus.  So, the best strategy is to just bid your value, revealing the value in the process.

Next, let’s consider common value auctions. Common value auctions have some interesting effects.  Most famously, common value auctions are subject to the winner’s curse.  By winner’s curse, I mean winning is bad news. The phenomenon was first reported by oil companies.   Here’s what’s happening:

Imagine we have three oil companies bidding on an oil lease in a first price sealed bid auction. The bidders have examined the lease and taken readings (Hi Agent Sloper!).  Based on the readings they have an estimate of the true value and place their bids based on those estimates.  Assuming bids are monotonically increasing in the estimates, Bidder C wins and learns that the other bidders think the true value is lower than Bidder C’s estimate. Oops.

In a common value setting, there is no revenue equivalence across auction formats. A first price sealed bid auction does not yield the same revenue as a second price sealed bid auction.  The strategic equivalences I mentioned above do not necessarily hold either.  While a first price sealed bid auction is still the same as a Dutch auction, a second price sealed bid auction and an ascending auction are not strategically equivalent.  This is because in an ascending auction I can learn more about your estimate, which lets me refine my own bid. However, the second price sealed bid auction is still a truth telling mechanism.

Let’s try an example called the envelope game that I often used in class.  I have two envelopes and I put some money into each envelope.  Next, I show the contents of one envelope to Player A (call it $X) and the contents of the second envelope to Player B (call it $Y).  The two players will bid for the two envelopes in a second price sealed bid auction.  The winner gets both envelopes ($X+$Y) and pays me the second bid.

What should be Player A’s bid?  What should be Player B’s bid? Post your answers in the comments and I’ll put the answer in the comments after about an hour.