Consider an auction for a single good, with two bidders, each with a private value drawn independently from the uniform distribution on [0, 100]. Let values be s1 and s2 , and bids are b1 and b2 .
1. Consider an auction for a single good, with two bidders, each with a private value drawn independently from the uniform distribution on [0, 100]. Let values be s1 and s2 , and bids are b1 and b2 .
(a) Under a first price, sealed bid auction, what are b1 and b2 in equilibrium?
(b) What is the seller’s expected revenue in this setting? (This should be a real number)
(c) Under a second price, sealed bid auction, what are b1 and b2 in equilibrium?
(d) What is the seller’s expected revenue in this setting? (This should be a real number)
Now suppose the seller decides to hold a first price sealed-bid auction, but bids are constrained to be taken from the set {0, 50}. If both bid 0, nobody will be awarded the object; if they both bid 50, the winner will be chosen at random (with equal probabilities).
(e) Does revenue equivalence hold in this setting? Why or why not?
(f) Show that it is a dominant strategy for an agent never to bid 50 unless si ≥ 50. more than their true value.
(g) Therefore, what are equilibrium strategies? 1b, the unconstrained auction?
(h) What is the seller’s expected revenue? Is this more or less than in 1b
(i) What is the expected payoff to each type of bidder? Is there any type that prefers this (constrained) auction to an ordinary first-price auction?
2. Consider an N-bidder auction for K identical objects, with 1 < K < N. Each bidder only wants one object. Private valuations are independent, and drawn from the uniform distribution on [0, 1].
(a) The natural analog to the one-item second-price auction is an auction where the K highest bidders win the items, and they all pay a price equal to the K + 1st-highest bid (the highest losing bid). Show that in this auction, it is a dominant strategy to bid bi = si .
(b) What is the seller’s expected revenue?
(c) The natural analog to the first price sealed bid auction is a sealed bid auction in which the K highest bidders win the items, and they each pay their own bid. Revenue equivalence holds in this auction setting. Calculate the expected revenue in this auction.
3. Some online auctions are run as ascending auctions with a “Buy It Now” option. That is, at any point during the auction, any bidder can immediately claim the object at a predetermined price. Model the ascending portion of the auction as a first price auction: the price rises continuously from 0 until either all but one bidder has dropped out or someone has decided to pay the “Buy It Now” price. Suppose there are two bidders, with independent private values drawn from the uniform distribution on [0, 100]. Suppose the “Buy It Now” price is 50, and that there exists a symmetric equilibrium where both bidders play the following strategy: if si < 50, remain active until the price reaches si and then drop out. If si > 50, remain active until the price reaches g(si ) and then end the auction by bidding the “Buy It Now” price, where g : [50, 100] → [0, 50] is some function.
(a) Show that if g is strictly decreasing, the highest value agent gets the object in equilibrium.
(b) Assuming that such a g exists, show that, conditional on winning, a bidder’s expected payment must be the same in this auction as it would be in a standard ascending auction with no “Buy It Now” price. 1
(c) (YOU DON’T HAVE TO, BUT IF YOU’RE CURIOUS) You can use this to calculate the function g, that is, the price at which each type of bidder with value above 50 jumps to a bid of 50.
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