Economics 100C April 6, 2010

Have you ever bid for anything on eBay? Yes, frequently Yes, but not frequently No

Equilibrium bids in first and second price sealed bid auctions with private values. Second price; bid true value. First price; bid less than true value. With uniform distributions if values and n bidders, equilibrium in first price auction has bidders bid fraction (n-1)/n of values.

Expected revenues with 2 bidders. Second-price auction—Expected value of second highest bid. First price auction---Expected value of half of highest bid. If values are uniformly distributed on an interval [0,A], these both turn out to equal A/3. See posted notes.

Expected revenues with n bidders. Second-price auction—Expected value of second highest bid. First price auction---Expected value of half of highest bid. If values are uniformly distributed on an interval [0,A], these both turn out to equal A(n-1)/(n+1). See notes.

An Oil Auction This illustrates a “common value auction” in which different bidders have partial information about the value of object being auctioned. Two bidders. Each has explored half of the oil field. Whole oil field is up for bids.

Instructions: Classroom Experiment Form groups of 3. Choose one person to be Auctioneer, one to be Player A and one to be Player B. Players A and B not allowed to talk to each other. Value of the oil field the sum of last two digits of A’s perm number and last two digits of B’s Players A and B write their perm numbers and bids on slip of paper and pass them to auctioneer. High bidder gets oilfield. Profits are value of oil field minus bid. Low bidder’s profits are zero. Auctioneer records total value, bids and profits.

Is this oilfield auction a common value auction or a private values auction?

Answer This is a common values auction. The oilfield is worth the same amount to whoever gets it. The only difference between the bidders is that they have different bits of information. This would be a private values auction if e.g. one firm could use the oilfield more effectively than the other.

In this auction, if you are Player A and your side of the oil field is worth 60, what is the expected value of the whole oil field? 60 90 110 120 150

Expected Value or whole field if my side is worth 60 Value of other side is a random variable uniformly distributed on [0,100] Expected value of other side is 50. Expected total value is 60+50=110.

What happens if bidders bid expected values? Each bidder would bid xi+50 where xi is the value of his own side. Winner would be person with higher xi Suppose that A wins and has value xA Given that he won, it must be that xB<xA What is the expected value of xB conditional on xB<xA? xA/2 So expected value of oil field is xA/2+ xA= 3xA/2< xA+50. On average, winning bidder would lose money.

The winner’s curse In this auction, you would on average lose money if you bid as high as your expected value. The expected value conditional on winning the auction is lower than the expected value. This effect is called the winner’s curse.

Buying Montana I will sell a contract in which I promise to pay $1 for every 100,000 people who live in Montana. The sale will be by an English auction. Top bidder pays me his bid. I pay top bidder $1 for every 100,000 people who live in West Montana.

Was there aWinners Curse? If so, why?

See you Thursday Don’t forget your clicker.