Many items are sold not in markets but in auctions where bidders do not know how much others value the object that is up for bid. We will analyze a straightforward setting like this here — which technically means we are analyzing (for much of this exercise) an incomplete information game of the type covered in Section B of the chapter. The underlying logic of the exercise is, however, sufficiently transparent for you to be able to attempt the exercise even if you have not read Section B of the chapter. Consider the following — known as a second-price sealed bid auction. In this kind of auction, all people who are interested in an item x submit sealed bids (simultaneously). The person whose bid is the highest then gets the item x at a price equal to the second highest bid.
A: Suppose there are n different bidders who have different marginal willingness to pay for the item x. Player is marginal willingness to pay for x is denoted vi. Suppose initially that this is a complete information game —i.e. everyone knows everyone’s marginal willingness to pay for the item that is auctioned.
(a) Is it a Nash equilibrium in this auction for each player i to bid vi?
(b) Suppose individual j has the highest marginal willingness to pay. Is it a Nash Equilibrium? For all players other than j to bid zero and player j to bid vj ?
(c) Can you think of another Nash equilibrium to this auction?
(d) Suppose that players are not actually sure about the marginal willingness to pay of all the other players—only about their own. Can you think of why the Nash equilibrium in which all players bid their marginal willingness to pay is now the most compelling Nash equilibrium?
(e) Now consider a sequential first price auction in which an auctioneer keeps increasing the price of x in small increments and any potential bidder signals the auctioneer whether she is willing to pay that price. (Assume that the signal from bidders to auctioneer is not observable by other bidders.) The auction ends when only a single bidder signals a willingness to pay he price—and the winner then buys the item x for the price equal to his winning bid. Assuming the increments the auctioneer uses to raise the price during the auction are sufficiently small, approximately what will each player’s final bid be?
(f) In equilibrium, approximately what price will the winner of the sequential auction pay?
(g) True or False: The outcome of the sealed bid second price auction is approximately equivalent to the outcome of the sequential (first price) auction.
B: This part provides a real-world example of how an auction of the type analyzed in part A can be used. When I became Department Chair in our economics department at Duke, the chair was annually deciding how to assign graduate students to faculty to provide teaching and research support. Students were paid a stipend by the department but their services were free to the faculty member to whom they were assigned.
(a) Under this system, faculty complained perpetually of a “teaching assistant shortage”. Why do you think this was?
(b) I replaced the system with the following: Aside from some key assignments of graduate students as TAs to large courses, I no longer assigned any students to faculty. Instead, I asked the faculty to submit dollar bids for the right to match with a graduate student. If we had N graduate students available, I then took the top N bids, let those faculty know they had qualified for the right to match with a student and then let the matches take place (with students and faculty seeking each other out to create matches). Every faculty member who had a successful bid was then charged (to his/her research account) a price equal to the lowest winning bid which we called the “market price”. (Students were still paid by the department as before —the charges to faculty accounts simply came into the chair discretionary account and were then redistributed in a lump sum way to all faculty.) Given that we have a large number of faculty, should any individual faculty member think that his/her bid would appreciably impact the “market price”?
(c) In my annual e-mail to the faculty at the beginning of the auction for rights to match with
Students, I included the following line: “For those of you who are not game theorists, please
note that it is a dominant strategy for you to simply bid the actual value you place on the right to match with a student.” Do you agree or disagree with this statement? Why?
(d) Would it surprise you to discover that, for the rest of my term as chair, I never again heard complaints that we had a “TA shortage”? Why or why not?
(e) Why do you think I called the lowest winning bid the “market price”? Can you think of several ways in which the allocation of students to faculty might have become more efficient as a result of the implementation of the new way of allocating students?