Assignment 2 Answers
ECON6206, Game Theory and Experiments
February 5, 2013
Directions: Answer all questions as completely as possible. You may work in pairs on the assignment.
If you do so, simply turn in one assignment per pair with both names on the assignment.
1. Consider the following game tree with 4 players:
P1
A
B
P2
W
P2
Y
X
P3
Z
P4
P4
T
9
3
3
5
U
6
2
4
4
P
Q
R
S
2
1
0
3
8
0
4
7
3
1
11
4
6
3
2
8
4
1
3
1
Game 1
Note that the payo¤s are listed in order, so the …rst payo¤ is for Player 1, the second for Player 2, etc.
1
a How many subgames are in this extensive form game (not including the entire game as a subgame,
even though it meets the properties of a subgame)?
Answer:
There are 5 subgames –one starting from each of P2’s information sets, one starting from P3’s information set, and one starting from each of P4’s information sets.
b How many strategies does each player have?
Answer:
P1 has 2 strategies (A, B). P2 has 4 strategies (WY, WZ, XY, XZ). P3 has 2 strategies (T, U). P4
has 4 strategies (PR, PS, QR, QS).
c Find the subgame perfect Nash equilibrium to this extensive form game.
Answer:
P4 would choose P if P2 chose Y. P4 would choose R if P2 chose Z. P3 would choose U if P2 chose
W. P2 would choose W if P1 chose A. P2 would choose Y if P1 chose B. P1 would choose A.
In shorthand:
P1: A
P2: WY
P3: U
P4: PR
2. Consider the following game:
Player 1
Low
Medium
High
Low
7,5
2,0
8,1
Player 2
Medium
2,7
0,2
3,4
High
1,1
5,0
6,2
a Suppose that this game is repeated 37 times. Find a pure strategy Nash equilibrium to this repeated
game.
Answer:
Player 1 has a strictly dominant strategy (in the stage game) of choosing High. Player 2 has a strictly
dominant strategy (in the stage game) of choosing Medium. Thus, Player 1 should choose High at
every stage regardless of what Player 2 chooses and Player 2 should choose Medium at every stage
regardless of what Player 1 chooses.
b Is the Nash equilibrium you found for the …nitely repeated game in part a unique? Explain why or
why not.
Answer:
Yes, it is unique because the players have a strictly dominant strategy in the stage game and we know
that the only SPNE to any …nitely repeated game in which both players have a strictly dominant
strategy is for each player to choose their strictly dominated strategy at every decision node in the
game.
Suppose now that the game is repeated in…nitely.
c Propose a set of strategies such that the outcome repeated in the stage game is the (7; 5) outcome
when both players choose Low.
2
Answer:
Player 1 chooses Low in the …rst stage. If Player 1 ever observes Player 2 choose anything other
than Low, Player 1 would choose High in all subsequent stages, regardless of what Player 2 does after
choosing either Medium or High. If Player 1 has only observed Player 2 choosing Low in the prior
stages, then Player 1 continues to choose Low.
Player 2 chooses Low in the …rst stage. If Player 2 ever observes Player 1 choose anything other than
Low, Player 2 would choose Medium in all subsequent stages, regardless of what Player 1 does after
choosing either Medium or High. If Player 2 has only observed Player 1 choosing Low in the prior
stages, then Player 2 continues to choose Low.
d Determine the minimum discount rate needed by EACH player to ensure that the set of strategies
you have suggested in part c is a subgame perfect Nash equilibrium to the game.
Answer:
For Player 1 we need:
7
1
8+3
1
7
5
8
1
1
5
1
8 +3
For Player 2 we need:
5
1
7+4
1
5
3
7
2
2
3
1
7 +4
So Player 1 would need a discount rate greater than or equal to 15 , while Player 2 would need a discount
rate greater than or equal to 23 . There may be other strategies that support the (Low; Low) outcome
being repeated forever, but this was the easiest one I could …nd.
3. Consider the following game between two investors. Two investors have each deposited 150 with a
bank. The bank has invested these deposits in a long-term project. If the bank is forced to liquidate
its investment before the project matures, a total of 200 can be recovered. If the bank allows the
investment to reach maturity, however, the project will pay out a total of 400.
There are 2 dates at which the investors can make withdrawals from the bank: date 1 is before the
bank’s investment matures; date 2 is after. For simplicity, assume that there is no discounting. If
both investors make withdrawals at date 1 then each receives 100 and the game ends. If only one
investor makes a withdrawal at date 1 then that investor receives 150, the other receives 50, and the
game ends. Finally, if neither investor makes a withdrawal at date 1 then the project matures and
both investors make withdrawal decisions at date 2. If both investors make withdrawals at date 2
then each receives 200 and the game ends. If only one investor makes a withdrawal at date 2 then
that investor receives 250, the other receives 150, and the game ends. If neither investor makes a
withdrawal at date 2 then the bank returns 200 to each investor and the game ends. Note that neither
player observes the withdrawal decision of the other player at either date (in other words, at date 1
the players simultaneously choose to withdraw or not, and the same at date 2 –obviously once date 2
is reached both players know what the other player chose at date 1). The …gure below provides the
game tree:
3
a There are 2 subgames in this game, one of which is the entire game and the other of which is the
game that begins at date 2. Write down the normal form (matrix) version of the subgame that
begins at date 2.
Answer:
The normal form game is a 2x2 (I have already marked the best responses so that we can see the
answer to part b):
Investor 1
W
DW
Investor 2
W
200 ; 200
150; 250
DW
250 ; 150
200; 200
b Find the Nash equilibrium to the date 2 subgame in part a.
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Answer:
Both players have a strictly dominant strategy of choosing W (withdraw), so both players will Withdraw
if they reach the date 2 subgame.
c There are two subgame perfect Nash equilibria to this game. Find them. (Hint: You may want to
use what you know from part b when solving for the SPNE).
Answer:
From part b we know that both players will choose withdraw if they get to play the date 2 subgame,
as withdraw is a strictly dominant strategy for both players. Since the date 1 game ends if either
(or both) players withdraw at date 1, and we now know the outcome to the date 2 subgame, we can
write down the payo¤s to the date 1 game ASSUMING players use the equilibrium strategies at date
2 (which means if both choose DW at date 1 then each will end up with a payo¤ of 200).
Investor 1
W
DW
Investor 2
W
100 ; 100
50; 150
DW
150; 50
200 ; 200
So there are two SPNE to this game:
(1) Both players choose W (withdraw) at date 1 and both choose W (withdraw) at date 2. This leads
to an outcome where both people withdraw at date 1 and receive 100, and the date 2 subgame is not
played.
(2) Both players choose DW (don’t withdraw) at date 1 and both choose W (withdraw) at date 2.
This leads to an outcome where both people don’t withdraw at date 1, the date 2 subgame is played,
and both receive 200 at the end of the game.
d While it is highly unlikely that I would ask you to do this on an exam due to time constraints, you
can write down the entire game in normal form and …nd all PSNE.
Answer:
The normal form version (with the best responses marked) of the extensive form game is:
Investor 1
W, W
W, DW
DW, W
DW, DW
Investor 2
W,W
100 ; 100
100 ; 100
50; 150
50; 150
W, DW
100 ; 100
100 ; 100
50; 150
50; 150
DW, W
150; 50
150; 50
200 ; 200
150; 250
DW, DW
150; 50
150; 50
250 ; 150
200; 200
The strategies for each player are listed as their action in the …rst period …rst, and then action in
the second period last, so that "W, W" for Investor 1 means his strategy is "Withdraw in the 1st
and Withdraw in the 2nd period." The marked best responses show that there are 5 PSNE – every
combination of strategies in which both players withdraw in the …rst period is part of a PSNE, while
the only PSNE that actually involves play in the second period of the game is SPNE #2 in part c
above. However, though there are 4 PSNE that lead to the same payo¤s (100 for each player), only one
of those PSNE is an SPNE (the one where both players W today, and would also choose W tomorrow).
4. Consider the following normal form game:
Player 1
A
B
C
D
Player
E
22; 22
25; 10
17; 4
6; 1
2
F
6; 4
15; 15
32; 6
9; 4
G
2; 27
4; 3
8; 8
1; 2
H
9; 2
2; 1
5; 3
1; 1
5
a Find all pure and mixed strategy Nash equilibria to the one-shot version of this game. If there are
no MSNE explain why there are none.
Answer:
The only pure strategy Nash equilibrium to the one-shot version of this game is Player 1 chooses C
and Player 2 chooses G. This is shown in the matrix above. There are no MSNE to this game. The
reason why is that we can reach the unique PSNE by using iterated elimination of strictly dominated
strategies. Eliminate D (strictly dominated by C) and H (strictly dominated by G). Now we are left
with a 3x3 (ABC, EFG). Now A is strictly dominated by B, then E is strictly dominated by F, then B
is strictly dominated by C, and then Player 2 is left with choosing F and receiving 6 or G and receiving
8, so F is strictly dominated by G.
b Given the Nash equilibria in part a, …nd a Pareto improving outcome (which means an outcome
where both players are at least as well o¤ as in the equilibrium, and at least one of the two is
strictly better o¤). Suppose the game is repeated in…nitely and that the players use a Nash
reversion punishment strategy (meaning they punish the other player by playing the NE to the
one-shot game forever). Find the minimum discount rates necessary to sustain an outcome path
where your chosen Pareto improving outcome is the result of each stage game.
Answer:
There are 3 Pareto improving outcomes: (A; E), (B; E), and (B; F ).
part of the question depends on which one you chose.
The answer to the remaining
1 (A; E) If you chose this outcome then you need to solve:
Player 1 :
25 +
Player 2 :
27 +
1
P
i
i=1
1
P
i
i=1
3
17
Then Player 1 would need a discount rate
8
8
1
P
i=0
1
P
i
22
(1)
i
22
(2)
i=0
and Player 2 would need a discount rate
5
19 .
2 (B; E) If you chose this outcome then you need to solve:
Player 2 : 15 +
1
P
i
8
i=1
1
P
i
10
(3)
i=0
Note that in this case Player 1 does not have a unilateral defection strategy from this outcome
because for Player 1 choosing B is a best response to Player 2’s choice of E. So for Player 1 any
5
0 would su¢ ce. For Player 2, we need
7.
3 (B; F ) If you chose this outcome then you need to solve:
Player 1 : 32 +
1
P
i=1
i
8
1
P
i
15
(4)
i=0
This is similar to the (B; E) case in that for Player 2 any
0 will su¢ ce to support this outcome
as the SPNE outcome path since F is Player 2’s best response to Player 1’s choice of B. For
17
Player 1 we need
24 .
c Note that Player 1’s highest payo¤ is 32 and occurs when Player 1 plays C and Player 2 plays F.
Again suppose the game is in…nitely repeated. Explain how it is possible for the outcome path
where this outcome (the C, F outcome) is repeated in…nitely to be a SPNE. Be speci…c about
Player 1’s choice of punishment strategy.
Answer:
It seems odd that Player 1 could force Player 2 into accepting a payo¤ below the Nash equilibrium
payo¤, but it is possible. The key is that Player 1 would have to choose a punishment strategy of
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D if Player 2 defected from the C; F outcome. Thus, the set of strategies would be for Player 1 to
choose C at the beginning of the game and as long as Player 1 does not observe a defection by Player
2, and to choose strategy D if Player 1 ever observed a defection by Player 2. Player 2 would choose
F at the beginning of the game and as long as Player 2 does not observe a defection by Player 1, and
to choose strategy F if Player 1 defected. So Player 2 is pretty much going to choose strategy F .
Note that Player 1 is threatening to punish with a strictly dominated strategy, although this choice of
punishment for Player 1 does guarantee him a higher payo¤ than Nash reversion.
5. Consider a developer who wishes to purchase k parcels of land. If the developer purchases all k parcels,
the developer receives a payment of D. If the developer does not purchase all k parcels, the developer
receives a payment of 0. The developer must purchase each parcel of land from the landowner who
owns the land.
Consider k landowners who each own a parcel of land. That parcel has value of vi to the landowner,
where vi U [0; D
k ]. The individual landowners know their own value for the land but the developer
does not. Also, the landowners do NOT know the values of other landowners. Note that given these
restrictions we have D kvi for any possible set of vi .
The game can be modeled as a sequential game. The developer makes an o¤er wi to each landowner.
Each landowner only observes his own wi and must make a decision to accept or reject that wi . If all
k landowners accept their own o¤er wi , then the landowners each receive wi as a payment from the
k
P
developer; the developer pays an amount
wi , and the developer receives a payment of D. If ANY
i=1
landowner chooses to reject wi , then the developer makes no payment to any landowner and acquires
no parcels of land – the developer receives 0 but pays 0. The landowners, who still own their land,
receive vi .
For simplicity, assume the seller sets wi = wj for all i; j. The developer maximizes expected utility,
and receives D kw if aggregation is successful (which occurs only if all k landowners accept the o¤er)
v
e
and 0 if not. Note that Pr (e
v > v) for the uniform distribution U 0; D
. Assume the developer
k is D
k
is risk neutral. Find a subgame perfect Nash equilibrium to this game with k landowners. Be sure to
set up the developer’s expected utility function correctly.
Answer:
Each of the landowners will accept any w vi and reject any o¤ers w < vi . The developer, knowing
this, then maximizes expected utility (Ud in the notation below) by choosing w:
max Ud
w
@Ud
@w
=
=
0
=
0
=
(D
w
D
k
w
D
k
(D
w
kw)
!k
!k
D
k
!k
( k) + (D
kw) k
( k) + (D
kw) k
1
kw) k D
w
kD
k
k
0 = (D kw)
D = kw + w
D
w =
k+1
w
D
k
w
D
k
!k
!k
1
1
D
k
1
1
D
k
w
The key to solving the developer’s problem is to correctly specify the probability that the o¤er is
accepted. For one individual that probability is w
D , while for k individuals the probability that k
o¤ers are accepted is
w
D
k
k
k
. Because the developer only receives a payo¤ when aggregation occurs,
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and because payments are only made when aggregation occurs, we do not need to worry about what
happens when less than k landowners accept an o¤er (those terms are multiplied by zero).
So the SPNE to this game is:
All k landowners accept any o¤er w
vi .
The developer o¤ers each landowner w =
D
k+1 .
What this means (in terms of the outcome) is that the developer does not always acquire the land.
Substituting in for w in
k
k+1
k
w
D
k
k
, we …nd that the developer only acquires the land with probability
.
6. Here are 2 extensive form games. Answer the questions below.
Game tree 1
Game tree 2
8
Find the SPNE for each game.
Answer:
In the …rst game the SPNE is that Player 1 chooses Fight, and Player 2 chooses Enter if 1 chooses
Accommodate and Out if 1 chooses Fight.
The SPNE to the second game is that Player 1 chooses A, Player 2 chooses C if 1 chooses A, and
Player 3 chooses F if 1 chooses A and 2 chooses D and H if 1 chooses B. In shorthand, the SPNE is
A,C,F,H, which leads to payo¤s of 1 for Player 1, 10 for Player 2, and 2 for Player 3. Those payo¤s
come from following the equilibrium path from A to C (and then the game ends).
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