In this problem we examine a second-price, sealed-bid auction. Assume that there are two bidders who have independent, private values vi , which are either 1 or 7. For each bidder, the probabilities of vi = 1 and vi = 7 are each 1 2 . So there are four possible pairs of the bidders’ values (v1, v2): (1, 1), (1, 7), (7, 1), and (7, 7). Each pair of values has probability .

Assume that if there is a tie at a bid of x for the highest bid the winner is selected at random from among the highest bidders and the price is x.

(a) For each pair of values, what bid will each bidder submit, what price will the winning bidder pay, and how much profit (the difference between the winning bidder’s value and price he pays) will the winning bidder earn?

(b) Now let’s examine how much revenue the seller can expect to earn and how much profit the bidders can expect to make in the second-price auction. Both revenue and profit depend on the values, so let’s calculate the average of each of these numbers across all four of the possible pairs of values. (Note that in doing this we are computing each bidder’s expected profit before the bidder knows his value for the object.) What is the seller’s expected revenue in the second-price auction? What is the expected profit for each bidder?

(c) The seller now decides to charge an entry fee of 1. Any bidder who wants to participate in the auction must pay this fee to the seller before bidding begins and, in fact, this fee is imposed before each bidder knows his or her own value for the object. The bidders know only the distribution of values and that anyone who pays the fee will be allowed to participate in a second-price auction for the object. This adds a new first stage to the game in which bidders decide simultaneously whether to pay the fee and enter the auction, or to not pay the fee and stay out of the auction. This first stage is then followed by a second stage in which anyone who pays the fee participates in the auction. We assume that after the first stage is over, both potential bidders learn their own value for the object (but not the other potential bidder’s value for the object) and that they both learn whether or not the other potential bidder decided to enter the auction.

Let’s assume that any potential bidder who does not participate in the auction has a profit of 0; if no one chooses to participate, then the seller keeps the object and does not run an auction; if only one bidder chooses to participate in the auction then the seller runs a second-price auction with only this one bidder (and treats the second highest bid as 0); and finally, if both bidders participate, the second-price auction is the one you solved in part (a).

Is there an equilibrium in which each bidder pays the fee and participates in the auction? Give an explanation for your answer.

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