Problem Set 10
Economics 703
Spring 2016
Due: Friday April 15
1. Suppose we have two agents, 1 and 2, and each owns one apple. The value to i per
apple is θi where θi ∼ U [0, 1] and θ1 and θ2 are independent. Consider the following
mechanism. The agents simultaneously choose bids. If i’s bid is strictly larger than j’s,
then i pays his bid to j and consumes both apples. If the bids are tied, each agent just
consumes his own apple and no payments are made.
(a) Show that there is a Nash equilibrium where i’s strategy is σi (θi ) = αθi and find α.
(b) Compute the interim payoffs. Does each agent always prefer to participate in the
mechanism (assuming that he can simply consume his own apple if he doesn’t participate)? Is the outcome ex post efficient?
(c) How does this example relate to the Myerson–Satterthwaite Theorem?
2. Suppose we have I bidders in an independent private values auction. The value of
the good to bidder i, θi , is distributed uniformly on [0, 1], iid across bidders. Without
using things we know about first or second price auctions, compute the seller’s expected
revenue for any auction with the property that the good always goes to the bidder with
the highest value and any bidder whose value is 0 has an expected payoff of 0.
3. We have I bidders in an independent private values auction. The value of the object
to bidder i is θi ∼ U [0, 1]. Suppose the seller uses an all–pay auction. That is, each
bidder i puts in a bid bi ∈ [0, ∞). If bi > maxj6=i bj , then i receives the object. If we have
a tie, we randomize uniformly over the bidders who bid the most to determine which gets
the object. But in all cases, all bidders pay their bids.
Show that there is an equilibrium where bidder i’s strategy is σi (θi ) = αθiβ for some
α and β and find α and β. Calculate the seller’s expected revenue in this equilibrium.
Without calculating the revenue for a first price auction, what can you say about how
the revenue in the all–pay auction compares to revenue in the first price auction?
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