Session D

Asymmetric Auctions

In Sessions A-C each bidder has the same beliefs about the opposing bidders. Such auctions are called "symmetric auctions." In the Session D you are asked to bid knowing that your opponent (or two opponents) is drawing his value from a different distribution. To illustrate, in the auction of a painting, if bidder 1 already has a large Monet collection his possible values of the next Monet to come up for auction are likely to be lower than the Getty Museum trying to get its second major Monet. Each bidder's valuation is a draw from some distribution just as in the first two sessions.

As the first example, suppose that bidder 1’s valuation is equally likely to be anywhere from zero to 300 (thousand) dollars while bidder 2’s valuation is equally likely to be anywhere between zero and 100 (thousand) dollars. How should they bid? For a clue, suppose first that both bidders valuations are uniformly distributed over the interval [0,100]. The maximum bid in this case is bS(100). Suppose that bidders continue to bid in this way when buyer 1’s valuation are on the interval [0,300]. When his valuation is 100 he bids bS(100) and wins with probability 1. Thus he has no incentive to ever bid higher. Thus his best response to bidder 2 is to bid bS(100) whenever his valuation is more than 100. The probability of this is 2/3.

Now consider bidder 2. If his value is 100 and he bids bS(100) he beats his opponent 1/3 of the time and ties him 2/3 of the time. In the latter case the winner is determined by lottery. But if he bids just slightly more than bS(100) he wins with probability 1. Thus his expected gain jumps by raising his bid. The same is true for bidder 2 if his valuation is somewhat less than 100. Thus the asymmetry results in higher bids by bidder 2. Bidder 1 responds by bidding higher as well.

Summing up, if you are the “weak” bidder with valuations on [0,100], the stronger your opponent the higher you should bid in order to maximize your expected payoff.