04 Jun Exercise 87.1 (First-price Sealed-bid Auction) Sh
uestion
Practice Problems
Exercise 87.1 (First-price Sealed-bid Auction) Show that in a Nash equilibrium
of a rst-price sealed-bid auction the two highest bids are the same, one of these bids is
submitted by player 1, and the highest bid is at least v2 and at most v1 . Show also that
any action prole satisfying these conditions is a Nash equilibrium.
Exercise 90.1 (Multiunit Auctions) In some auctions, many units of an object are
available. Such auctions are called multiunit auctions. In a multiunit auction, each
bidder submits a bid for each unit of the object. The bidder who submits the highest
bid for any given unit obtains that unit. Multiunit auctions dier in the prices paid by
the winners. In the discriminatory auction, the price paid for each unit is the winning
bid for that unit. In the uniform-price auction, the price paid for each unit is the same,
equal to the highest rejected bid among all the bids for all units. In the Vickrey auction
a bidder who wins k objects pays the sum of the k highest rejected bids submitted by
the other bidders.
Suppose now that two units of an object are available. There are n bidders. Bidder i
values the rst unit that she obtains at vi and the second unit at wi , where vi > wi > 0.
Each bidder submits two bids; the two highest bids win. Retain the tie-breaking rule
according to which if both bidders submit the same bid for an object, then the bidder
with the lowest name (i.e., bidder 1) obtains it. Show that in discriminatory and
uniform-price auctions, player is action of bidding vi and wi does not dominate all her
other actions, whereas in a Vickrey auction it does. (In the case of a Vickrey auction,
consider separately the cases in which the other players bids are such that player i wins
no units, one unit, and two units when her bids are vi and wi .)
Graded Problems
Exercise BCS 74 (Lowest-Price-Auction) Consider the sealed-bid lowest-priceauction, an auction where the highest bidder wins but he pays the lowest bidders
1
bid. There are n > 2 bidders. The values for the object of the bidders are ordered by
v1 > v2 > … > vn > 0 and known to all bidders. If several bidders bid the highest bid,
then the bidder with the lowest name gets the object. (E.g. if both bidders 3 and 6
have the highest bid, then bidder 3 will get the object because his name, 3, is lower
than 6.) Show that for any bidder i, the bid of vi weakly dominates any lower bid.
Show further that for any bidder i, the bid of vi does not weakly dominate any higher
bid. Next, show that the action prole in which each player bids her valuation is not a
Nash equilibrium. Finally, nd a symmetric Nash equilibrium.
Exercise BCS 75 (Multiobject Auction) There is a seller who has one unit each of
three dierent objects to sell. We name these objects a, b, and c. There are two potential
buyers who have dierent valuations for these objects. These valuations are given in the
following table:
Objects
Buyers
values
a b c
v1
16 10 20
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