CHAPTER 13 • Game Theory and Competitive Strategy 519 formats generate nearly identical outcomes (The outcomes should differ in theory only by a dollar or two.) To illustrate, suppose that there are three bidders whose valuations are $50, $40, and $30, respectively, and furthermore the auctioneer and the bidders have complete information about these valuations In an English auction, if your valuation was $50 you would offer a winning bid of $40.01 in order to win the bidding from the individual whose reservation price was $40.00 You would make the identical bid in a sealed-bid auction Even in a world of incomplete information, we would expect similar results Indeed, you know that as a seller, you should be indifferent between an oral English auction and a second-price sealed-bid auction, because bidders in each case have private values Suppose that you plan to sell an item using a sealed-bid auction Which should you choose, a first-price or a secondprice auction? You might think that the first-price auction is better because the payment is given by the highest rather than the second-highest bid Bidders, however, are aware of this reasoning and will alter their bidding strategies accordingly: They will bid less in anticipation of paying the winning bid if they are successful The second-price sealed-bid auction generates revenue equal to the secondhighest reservation price However, the revenue implications of a first-price sealed-bid auction for the seller are more complicated because the optimal strategy of bidders is more complex The best strategy is to choose a bid that you believe will be equal to or slightly above the reservation price of the individual with the second-highest reservation price.21 Why? Because the winner must pay his or her bid, and it is never worth paying more than the second-highest reservation price Thus, we see that the first-price and second-price sealed-bid auctions generate the same expected revenue Common-Value Auctions Suppose that you and four other people participate in an oral auction to purchase a large jar of pennies, which will go to the winning bidder at a price equal to the highest bid Each bidder can examine the jar but cannot open it and count the pennies Once you have estimated the number of pennies in the jar, what is your optimal bidding strategy? This is a classic common-value auction, because the jar of pennies has the same value for all bidders The problem for you and other bidders is the fact that the value is unknown You might be tempted to what many novices would in this situation— bid up to your own estimate of the number of pennies in the jar, and no higher This, however, is not the best way to bid Remember that neither you nor the other bidders knows the number of pennies for certain All of you have independently made estimates of the number, and those estimates are subject to error—some will be too high and some too low Who, then, will be the winning bidder? If each bidder bids up to his or her estimate, the winning bidder is likely to be the person with the largest positive error—i.e., the person with the largest overestimate of the number of pennies THE WINNER’S CURSE To appreciate this possibility, suppose that there are actually 620 pennies in the jar Let’s say the bidders’ estimates are 540, 590, 615, 650, and 690 Finally, suppose that you are the bidder whose estimate is 21 To be more exact, the best strategy is to choose a bid that you believe will be equal to or slightly above the second-highest expected reservation price conditional on your value being the highest