Allen complains about the quality of the HDMI adapters in the lecture room. The university hears Allen’s concern and invites firms to bid for a place to supply HDMI adapters on campus. Specifically, the university designs an auction to invite bids from different firms. There are n > 1 bidders, indexed by i = 1, . . . , n. Each bidder i’s value of supplying HDMI adapters on campus is vi 2 [0, 1], independently distributed according to
a cumulative distribution function F, with F0 = f.
(a) Suppose that the university uses a first-price, sealed-bid auction. In such
an auction, each firm submits a sealed bid of value to the university. The
highest bidder wins and pays the university its bid. Define each firm’s
payoff to the university. Define each firm’s strategy. Define a symmetric
monotone Bayesian Nash equilibrium. Solve for all symmetric monotone
Bayesian Nash equilibria.
(b) Suppose that the university uses a second-price, sealed-bid auction. In
such an auction, each firm submits a sealed bid of value to the university.
The highest bidder wins and pays the second-highest bid to the university.
Define each firm’s payoff. Define each firm’s strategy.
i. Compared to the first-price sealed bid auction in which the university
receives the highest bid, in the second-price auction the university
receives only the second highest bid. Is it obvious that a first-price,
sealed-bid auction must yield a higher revenue for the university than
a second-price, sealed-bid auction? Why or why not?
ii. Solve for all Bayesian Nash equilibria.
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(c) Suppose that the university uses a Dutch auction. In such an auction,
the university begins by calling price 1 and begins to reduce it gradually
and continuously towards 0. The first bidder to raise his hand wins the
auction at the current price. Solve for all symmetric monotone Bayesian
Nash equilibria.

  

9. (12%) LECTURE 2122-As is well known, Allen complains about the quality of the HDMI adapters in the lecture room. The university hears Allen's concern and invites firms to bid for a place to supply HDMI adapters on campus. Specifically, the university designs an auction to invite bids from different firms. There are n > 1 bidders, indexed by i = 1,..., n. Each bidder i's value of supplying HDMI adapters on campus is v, (0, 1], independently distributed according to a cumulative distribution function F, with F = f. (a) Suppose that the university uses a first-price, sealed-bid auction. In such an auction, each firm submits a sealed bid of value to the university. The highest bidder wins and pays the university its bid. Define each firm's payoff to the university. Define each firm's strategy. Define a symmetric monotone Bayesian Nash equilibrium. Solve for all symmetric monotone Bayesian Nash equilibria. (b) Suppose that the university uses a second-price, sealed-bid auction. In such an auction, each firm submits a sealed bid of value to the university. The highest bidder wins and pays the second-highest bid to the university. Define each firm's payoff. Define each firm's strategy. i. Compared to the first-price sealed bid auction in which the university receives the highest bid, in the second-price auction the university receives only the second highest bid. Is it obvious that a first-price, sealed-bid auction must yield a higher revenue for the university than a second-price, sealed-bid auction? Why or why not? ii. Solve for all Bayesian Nash equilibria.