Economics (Game Theory)

For all the problems on auctions, assume the scenarios described in the lecture notes.
Specically, assume that players’ values can be strictly ranked, v
1
> 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 (b
1 =v
2
; b
2 =v
1
; b
3
= 0; :: : bN= 0) is a Nash equilibrium in a
second price auction (Hint: Try to reproduce the arguments of the rst two paragraphs
on page 4 of the lecture notes). What are the payos received by all the players under
this strategy prole. Who wins the object under this strategy prole?
b) (Advanced) Construct a Nash equilibrium under which theNth player (the player
with the lowest value for the object,vN) wins the object. What are the payos received
by all players under this strategy prole.
c) (Advanced) Discuss one way in which the equilibriumoutcomes under second price
and rst price auctions may dier, qualitatively.
d) (Basic) Name a Nash equilibrium under a second price auction and another one under
a rst 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 (b
1 =v
2
; b
2 =v
2
; b
3 =v
3
; :: : bN=vN) is a Nash equilibrium in
a rst price auction (Hint: Use similar arguments as on page 4 of the lecture notes but
note that this is a rst price auction). What are the payos received by all the players
under this strategy prole. Who wins the object under this strategy prole?
b) (Advanced) Is this strategy prole a Nash equilibrium under the second price auction?
Provide arguments for or against.
c) (Basic-advanced) Is the strategy prole (b
1 =v
1
; b
2 =v
2
; b
3 =v
3
; :: : bN=vN) a
Nash equilibrium in a rst price auction? Is it a Nash equilibrium in a second price
auction? Explain why your answer is dierent for the two auctions.
d) (Advanced) Change the bid of the third player in the prole in part c) to convert the
prole into a Nash equilibrium under rst 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 Hare
Stag 2, 2 0, 1
Hare 1, 0 1, 1
Dove Hawk
Hawk Dove
Hawk 1, 1 4, 0
Dove 0, 4 2, 2
Two Player Three Action
L C R
T 2, 3 0, 0 5,1
M -1, 6 2, 3 10, 4
D 0, 0 3, 2 8, 1
2