Assignment 6 -- Four Problems
You may discuss this assignment with classmates, but all your code and
written work must be your own. Reference all sources. Please hand in hard copy.
1.
Suppose there are three (3) bidders competing in an auction.
Bidders' values are independently and identically distributed
uniformly on the unit interval, and there is no reserve.
(a) If this is a first-price sealed bid auction, give a symmetric
Nash equilibrium bidding strategy and find the corresponding expected
revenue to the seller.
(b) Suppose that this is an all-pay auction instead. Give a symmetric
Nash equilibrium and find the corresponding expected revenue to the seller.
(c) Suppose that the seller runs a first-price auction where
there is an entry fee of 0.25 to participate in the auction.
That is, bidders, knowing their values, have to decide
whether or not to pay the seller 0.25, non-refundable, to participate
in the auction. Compute the symmetric equilibrium bidding strategy
and the expected revenue to the seller. Is it a good idea for the
seller to charge an entry fee? Is there a better choice of entry fee than
0.25? Assume for this problem that the value of the item to the seller is zero.
2.
Consider the first-price (FP), second-price (SP), English (E),
and Average-of-Other-Bids (AVO, as in Assignment 3) auctions with
symmetric bidders and affiliated values, as in Milgrom & Weber 82.
Use the Linkage Principle to rank the expected revenue of the AVO
auction with respect to the others, in so far as that's possible.
I'm looking for heuristic arguments, not mathematical proofs.
3.
Two bidders are competing for a single item.
Each of the bidders has a private valuation drawn
independently from an exponential distribution. That
is, the cumulative distribution of bidder i's
value v is F(v) = 1 - e -λ v,
where v can be any nonnegative real number.
Bidder 1's λ parameter is 1; bidder 2's λ parameter is 20.
Assume for this problem that the value of the item to the seller is zero.
(a) Derive an optimal auction, and show in the
v1-v2 plane the regions
where bidder 1 wins, where bidder 2 wins, and where
the seller keeps the object. Also indicate what each
of the two bidders pays when she wins, and where in
the plane the auctioned item is allocated efficiently,
and where it is allocated inefficiently.
[Hint: Use the recipe of Bulow & Roberts 1989.]
(b) Suppose instead that the same two bidders compete in
an English auction with a reserve price (minimum acceptable
bid) of r, which is the same for both bidders.
Show in the v1-v2 plane the regions
where bidder 1 wins, where bidder 2 wins, and where
the seller keeps the object. Also indicate what each
of the two bidders pays when she wins, and where in
the plane the auctioned item is allocated efficiently,
and where it is allocated inefficiently.
(c)
[Extra Credit]
Find the expected revenue in part (a).
(d)
[Extra Credit]
Determine the optimal reserve price r in part (b)
and compute the corresponding expected revenue.
Compare with the expected revenue in the optimal auction
and explain any difference.
4. The following statement appears on the eBay
item pages of seller "ancientauctionhouse.com":
Is this a generous offer? Discuss.
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