For all the problems on auctions, assume the scenarios described in the lecture notes.
Specifically, assume that players’ values can be strictly ranked, v1 > v2 > v3 . . . vN .
Assume that in case of multiple highest bids, the auctioneer awards the object to the
highest bidder with the lowest index.
Problem 1
a) (Basic) Check whether (b1 = v2 , b2 = v1 , b3 = 0, . . . bN = 0) is a Nash equilibrium in a
second price auction (Hint: Try to reproduce the arguments of the first two paragraphs
on page 4 of the lecture notes). What are the payoffs received by all the players under
this strategy profile. Who wins the object under this strategy profile?
b) (Advanced) Construct a Nash equilibrium under which the N th player (the player
with the lowest value for the object, vN ) wins the object. What are the payoffs received
by all players under this strategy profile.
c) (Advanced) Discuss one way in which the equilibrium outcomes under second price
and first price auctions may differ, qualitatively.
d) (Basic) Name a Nash equilibrium under a second price auction and another one under
a first price auction, such that the auctioneer obtains the same revenue from the sale of
the object under both situations.
Problem 2
a) (Basic) Check that (b1 = v2 , b2 = v2 , b3 = v3 , . . . bN = vN ) is a Nash equilibrium in
a first price auction (Hint: Use similar arguments as on page 4 of the lecture notes but
note that this is a first price auction). What are the payoffs received by all the players
under this strategy profile. Who wins the object under this strategy profile?
b) (Advanced) Is this strategy profile a Nash equilibrium under the second price auction?
Provide arguments for or against.
c) (Basic-advanced) Is the strategy profile (b1 = v1 , b2 = v2 , b3 = v3 , . . . bN = vN ) a
Nash equilibrium in a first price auction? Is it a Nash equilibrium in a second price
auction? Explain why your answer is different for the two auctions.
d) (Advanced) Change the bid of the third player in the profile in part c) to convert the
profile into a Nash equilibrium under first price auction.
1
Problem 3
Find the mixed strategy NE of the Battle of the Sexes (Basic), the Stag-Hunt (Basic),
the Dove-Hawk (Basic) and the two player three actions games (Advanced) below.
Battle of the Sexes
Bach
Stravinsky
Bach
2, 1
0, 0
Stravinsky
0, 0
1, 2
Stag Hunt
Stag
2, 2
1, 0
Stag
Hare
Hare
0, 1
1, 1
Dove Hawk
Hawk
Dove
Hawk
1, 1
0, 4
Dove
4, 0
2, 2
Two Player Three Action
T
M
D
L
2, 3
-1, 6
0, 0
C
0, 0
2, 3
3, 2
2
R
5,1
10, 4
8, 1