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Microeconom´ıa II
Maestr´ıa en Econom´ıa
Problem Set I
This problem set is due on Thursday 5th, March (at the end of the midterm!)
1. Player 1 owns an indivisible object which she is going to sell to one of her two friends, players
2 and 3, by means of an auction. For each i ∈ {2, 3}, let vi be player i’s valuation of the object.
Player i knows vi , but the other players only know that either vi = vH or vi = vL , with equal
probability. Assume that v2 , v3 are independent and that vH > vL > 0. The auction takes place
as follows. First, player 1 chooses a “reservation’ price r ∈ {0, vL }. After observing r, each player
i ∈ {2, 3} submits a bid bi ∈ {0, vL , vH } in a sealed envelope. Finally, the envelopes are opened
and one of the following outcomes takes place. If max {b2 , b3 } < r, then player 1 keeps the object
and no payments are made. If max {b2 , b3 } ≥ r and b2 6= b3 , then player 1 sells the object to the
highest bidder at a price equal to the highest bid. If max {b2 , b3 } ≥ r and b2 = b3 , then with equal
probability player 1 sells the object to player 2 or player 3 at a price equal to their bids.
Represent this situation as a game in extensive form and identify its formal elements. [20pts]
2. Consider the extensive-form game represented as
L
r A
1b
PPP
P
PP
P
M
PP R
PP
PP
PP
PP
P
rp p p p p p p p p p p p p p p p p p2p p p p p p p p p p p p p p P
p pP
p r
@
@
@
@
l
l
@r
@r
@
@
@
@
p r
p r
rp p p p p p p p p p 1p p p p p p p p p @
rp p p p p p p p p p 1p p p p p p p p p @
B
B
B
B
x By
x By
x By
x By
B
B
B
B
BBr
BBr
BBr
BBr
r
r
r
r
B
D
C
E
F
G
H
I
(a) What is the set of player 1’s strategies? What is the set of player 2’s?
(b) Show that for any behavior strategy that player 1 might play, there is a mixed strategy that
generates the same probability distribution over the terminal nodes {A,B, . . . ,I} for any mixed
strategy choice of player 2.
(c) Show that the converse is also true: for any mixed strategy that player 1 might play, there is a
behavior strategy that generates the same probability distribution over the terminal nodes for any
1
mixed strategy choice of player 2.
(d) Suppose that we merge the information sets of player 1’s second round of moves (so that all
four nodes are now in a single information set). Which of the two results in (b) and (c) still holds?
[25pts]
3. Consider the following two-player game. First, player 1 selects a number x, which must be greater
than or equal to zero. Player 2 observes x. Then, simultaneously and independently, player 1 selects
a number y1 and player 2 selects a number y2 . Player 1’s payoff is u1 = y1 y2 + xy1 − y12 − (x3 /3)
and player 2’s payoff is u2 = −(y1 − y2 )2 . Represent this game in extensive form.
[15pts]
4. Consider the following situation. Cristiano Ronaldo (CR7) wishes to leave R. Madrid and then
sign up with either F.C. Barcelona (F CB), Bayern Munich (BM ) or Chelsea (C). Both BM and
C have already told CR7 (and the media as well!) that they want to hire him. CR7 has yet to
hear anything from F CB and he has only a week to determine which club he will be playing for
the next season. F CB has announced that their Board of Directors will meet tonight and state
tomorrow the names of the players the club has interest in. CR7 has to decide whether to approach
F CB with an expression of interest or wait until F CB’s official announcement tomorrow. In the
meeting, F CB’s Board of Directors will be called upon to consider two alternatives: either hiring
an outside expert to assess CR7’s potential contribution to the squad or dropping all aspirations
of hiring CR7 (due to his large salary) without even consulting any expert. If F CB hires the
expert, then they will make an offer to CR7 if the expert’s assessment is positive, and decline if
the expert does not give the okay. The expert, if hired, will not be informed of whether or not CR7
has approached F CB with an expression of interest. Thus, if CR7 does not receive an offer from
F CB, he does not know whether this is so because the expert did not give the okay or because the
club did not consult an expert. After a week, regardless of whether he receives an offer from F CB,
CR7 must decide which club he will be playing for the next season, F CB (if he has received an
offer from them), BM , or C.
Represent this game in extensive form and identify its formal elements.
[20pts]
5. Player 1 owns an indivisible object which she is going to sell to one out of four possible buyers.
For each i ∈ {2, 3, 4, 5}, let vi be player i’s valuation of the object. All buyers submit their bids
simultaneously and the object is sold to the buyer who makes the highest offer. However, the
winner pays for the object the amount of the second highest offer.
Identify all formal elements, including strategies and payoffs, of this situation as an strategic
(or normal form) game.
[20pts]
2