BIDDING IN SEALED HIGH-BID AUCTIONS

INTRODUCTION

Economists have long studied markets in which large numbers of identical or very similar items are sold. If the markets are competitive, it is the forces of supply and demand that determine the market price. If there is a monopoly seller, the price is raised above the competitive price until marginal revenue and marginal cost are equal.

Setting a fixed price is no longer an efficient way of selling an item if it is significantly different from other items in the marketplace. Take, for example, a Rembrandt that has not been on the market for a long time. Different collectors will place different values on the desirability of adding it to their collections. If the seller knew these values, the optimal strategy would be to negotiate with the collector who valued it the most while threatening to sell it to the collector with the next highest valuation. But the seller does not have this information. So setting a fixed price may mean that there are no bidders. And if he announces a fixed price, is he really committed to that price? If not, bidders may wait for the price to fall. Thus a different way of selling the item is needed.

There are two common ways of selling items when there is great uncertainty about each buyer’s valuation. One is at an open ascending-bid auction. In this, the auctioneer starts the bidding and keeps raising the price when he gets a signal from one of the bidders. The asking price keeps rising until no bidder is willing to accept a higher price. While many art objects are sold by open auction, it is much more common for bids to be tendered privately. Each bidder submits his sealed bid. The bids are unsealed and the buyer who has bid the most is declared the winner. He pays his bid and receives the item.

The strategic considerations in sealed high-bid auctions are subtle. If I value the item at 4.5 (million) I will bid less than that, in order to come out ahead. How much to bid depends on what I believe about my opponents’ valuations of the item. I also realize that they will be bidding less than their valuations. But how much less?

To fix ideas, consider the following DOLLAR BILL auction that might be conducted in the classroom. Two students are invited to get out a dollar bill and look at the second and third digit on the bill’s number. That number will be their prize (in dimes) if they win the auction. Suppose you are bidder 1 and your two digit number is 64. Then the instructor will dig into his wallet and pay you 64*10/100 = $6.40 if you win. If your opponent wins he will receive a payment lying between $0 and $9.90. As bidder 1 you do not know how big his prize is but you do know that it is evenly distributed over the range [$0.00,$9.90].

The two bidders are each invited to write down a sealed bid and submit the bid to the auctioneer (the instructor.) The bids are unsealed and the bidder who has made the highest bid is declared the winner. He must pay his bid to the auctioneer and receives his prize.

Since the mathematic of bidding is complicated (except in a few special cases) the goal here is to invite students into an auction house where they can compete for items and so learn from their own experience what constitutes a good bidding strategy.

The very first game is the DOLLAR BILL auction discussed above. The student and programmed bidder are each assigned a value for the item that is a random number between 0 and 100. The programmed bidder follows a bidding strategy provided him by a consultant. This tells him what to do for every possible valuation. We can write this as a function b(v). Clearly a bidder with a zero valuation will bid zero, that is b(0) = 0. Also no bidder will ever bid more than the item is worth since he would then end up with a loss if he wins the item, that is b(v) < v. Third, if your opponent’s maximum bid is k, you have no incentive to bid much more either since any bid above k guarantees that you will be the winner.

The challenge is to come up with a profitable bid function, knowing that your opponent is following his strategy b(v). For the games in these sessions, bidders’ beliefs about each other are the same. Thus it at least seems plausible that the equilibrium bidding of each buyer will be the same. That is, each player will be choosing the same bidding strategy b(v). This is always true in the theoretical models of equilibrium bidding.

Learning is fast in the ucla auction house because students both compete with the programmed opponent and see their results after the game is played. A student therefore finds out how well he or she did in comparison to others playing the same game. Those who do relatively poorly modify their strategies in an attempt to improve their relative ranking.