ECON 402: Solutions to Practice Questions
Ron Siegel
Fall 2020
1. Consider the following game.
P1
P2
(1, 3)
A
(3,−2)
B
L
P2
(1,−1)
a
P1
(2,−1)
l
(4,−2)
r
b
R
(a) What is the set of strategies for each player?
The set of strategies for player 1 is:
S1 = {Ll, Lr,Rl, Rr}.
For player 2:
S2 = {Aa,Ab,Ba,Bb}.
(b) Write the game in normal form (as a matrix).
Aa Ab Ba Bb
Ll 1, 3 1, 3 3,−2 3,−2
Lr 1, 3 1, 3 3,−2 3,−2
Rl 1,−1 2,−1 1,−1 2,−1
Rr 1,−1 4,−2 1,−1 4,−2
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(c) Find the rationalizable strategies. For each rationalizable strategy find a belief
over the other player’s rationalizable strategies to which this strategy is a best
response.
There are no dominated strategies. Thus, Ll and Lr, for example, allow to
attain a maximal payoff for player 1 given strategy Ba of player 2, Rl - given
strategy Aa, Rr - given strategy Bb. For player 2, Aa and Ab give the maximal
payoffs given, for example, strategy Ll of player 1, Ba and Bb- given Rl. There
are many different beliefs that rationalize these strategies. One example for
beliefs which rationalize every strategy are as follows: for Ll - θ2 = (1, 0, 0, 0),
for Lr - θ2 = (0, 0, 1, 0), for Rl - θ2 = (0, 0, 1, 0), for Rr - θ2 = (0, 0, 0, 1). For
Aa - θ1 = (1, 0, 0, 0), for Ab - θ1 = (1, 0, 0, 0), for Ba - θ1 = (0, 0, 0, 1), for
Bb - θ1 = (0, 0, 1, 0). Every strategy can be supported as a best response to a
rationalizable strategy.
Also, observe that θ2 = (1, 0, 0, 0) rationalizes player 1’s all strategies and, sim-
ilarly, θ1 = (0, 0, 1, 0) rationalizes player 2’s all strategies.
(d) Find the Nash equilibria.
Player 1’s best response to s2 = Aa, s2 = Ab, s2 = Ba, and s2 = Bb are
BR1(Aa) = {Ll, Lr,Rl, Rr}, BR1(Ab) = {Rr}, BR1(Ba) = {Ll, Lr}, and
BR1(Bb) = {Rr}, respectively.
Player 2’s best response to s1 = Ll, s1 = Lr, s1 = Rl, and s1 = Rr are
BR2(Ll) = {Aa,Ab}, BR2(Lr) = {Aa,Ab}, BR2(Rl) = {Aa,Ab,Ba,Bb}, and
BR2(Rr) = {Aa,Ba}, respectively.
The payoff matrix below summarizes all best responses.
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Aa Ab Ba Bb
Ll 1, 3 1, 3 3,−2 3,−2
Lr 1, 3 1, 3 3,−2 3,−2
Rl 1,−1 2,−1 1,−1 2,−1
Rr 1,−1 4,−2 1,−1 4,−2
In order to find the Nash equilibria, we must find the strategy profiles (s1, s2),
where s1 is a best response to s2 for player 1 and s2 is a best response to s1 for
player 2. Therefore, the set of Nash equilibira is:
NE = {(Ll, Aa), (Lr,Aa), (Rl,Aa), (Rr,Aa), (Ll, Aa), }
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2. Consider the following game: With Player 2 out of the room, Player 1 chooses a ball
of one of two colors—red or blue—and puts that ball inside a box. Player 2 then
comes in the room and tries to guess the color of the ball, by declaring R (for red)
or B (for blue) (Player 2 cannot see the color of the ball, since it is inside the box).
Both players get a payoff of 10 if Player 2 correctly guesses the color; both get a
payoff of 5 if Player 2’s guess is incorrect.
(a) Draw the extensive form of this game, labeling all parts. (Be careful about
information sets.)
P1
(10, 10)
R
(5, 5)
B
red
(5, 5)
R
(10, 10)
B
blue
P2
(b) Write the game in normal form.
R B
red 10, 10 5, 5
blue 5, 5 10, 10
(c) For each strategy, explain whether or not it is a dominated strategy.
R is not dominated because it gives higher payoff given strategy red, B is not
dominated because it gives higher payoff given strategy blue. Similarly, red is
not dominated as it gives higher payoff given strategy R, blue is not dominated
as it gives higher payoff given strategy B. So, there are no dominated strategies.
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Now consider the following modification of the game above: Before guessing the
color, Player 2 can choose to open the box or leave it shut. If Player 2 chooses
to leave the box shut, the payoffs of the game are as before; if Player 2 chooses
to open the box (thereby observing the color of the ball) before guessing the
color, then Player 1’s payoff is still the same as before (10 if the guess is correct
and 5 if it is incorrect), but Player 2’s payoff is the payoff of Player 1 minus 5.
(d) Repeat parts a and b.
P1
P2
(10, 5)
R0
(5, 0)
B0
open
(10, 10)
R1
(5, 5)
B1
not open
red
(5, 5)
R1
(10, 10)
B1
not open
P2
(5, 0)
R2
(10, 5)
B2
open
blue
P2
P2
For this part, the payoff matrix is flipped. Now player 2 is the rows player,
player 1 is the columns player.
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red blue
Open,R0, R1, R2 5, 10 0, 5
Open,R0, R1, B2 5, 10 5, 10
Open,R0, B1, R2 5, 10 0, 5
Open,R0, B1, B2 5, 10 5, 10
Open,B0, R1, R2 0, 5 0, 5
Open,B0, R1, B2 0, 5 5, 10
Open,B0, B1, R2 0, 5 0, 5
Open,B0, B1, B2 0, 5 5, 10
Not Open,R0, R1, R2 10, 10 5, 5
Not Open,R0, R1, B2 10, 10 5, 5
Not Open,R0, B1, R2 5, 5 10, 10
Not Open,R0, B1, B2 5, 5 10, 10
Not Open,B0, R1, R2 10, 10 5, 5
Not Open,B0, R1, B2 10, 10 5, 5
Not Open,B0, B1, R2 5, 5 10, 10
Not Open,B0, B1, B2 5, 5 10, 10
(e) Find the rationalizable strategies. For each rationalizable strategy find a belief
over the other player’s rationalizable strategies to which this strategy is a best
response.
It is not the case that for the columns player in the matrix above either red
always gives strictly larger payoffs, or blue does so. So, the columns player
does not have dominated strategies. For the rows player, (Open,R0, R1, R2),
(Open,R0, B1, R2), (Open,B0, R1, R2), (Open,B0, B1, R2) are dominated by
(Not Open,R0, R1, R2); (Open,B0, R1, B2), (Open,R0, B1, B2) are dominated
by (Not Open,R0, B1, R2); (Open,R0, R1, B2) , (Open,R0, B1, B2) are domi-
nated by an equal probability mix of (Not Open,R0, R1, R2) and (Not Open,R0, B1, R2);
Hence, all the strategies involving Open are dominated. The reduced game is
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red blue
Not Open,R0, R1, R2 10, 10 5, 5
Not Open,R0, R1, B2 10, 10 5, 5
Not Open,R0, B1, R2 5, 5 10, 10
Not Open,R0, B1, B2 5, 5 10, 10
Not Open,B0, R1, R2 10, 10 5, 5
Not Open,B0, R1, B2 10, 10 5, 5
Not Open,B0, B1, R2 5, 5 10, 10
Not Open,B0, B1, B2 5, 5 10, 10
In this game (Not Open,R0, R1, R2), (Not Open,R0, R1, B2),(Not Open,B0, R1, R2),
(Not Open,B0, R1, B2) are supported as best response by the the belief θcolumns player =
(1, 0), all the other rows player’s strategies are supported as best response by
the belief θcolumns player = (0, 1). red is rationolizable for example by a belief
θrows player = (1, 0, ..., 0), blue - for example by a belief θrows player = (0, 0, 1, ..., 0).
(f) Find the Nash equilibria.
Because the Nash equilibria will survive under the iterated elimination of dom-
inated strategies procedure, instead of the original game, we can focus our at-
tention to the rationalizable strategies in order to find the Nash equilibria.
Following payoff matrix shows us the best responses of both players. For in-
stance, when we look at the first cell in the matrix, because the first 10 is under-
lined, it means that Not Open,R0, R1, R2 is a best response to scolumns player =
red for the rows player. Similarly, because the second 10 in the same cell is
underlined, we understand that red is a best response to srows player = Not
Open,R0, R1, R2 for the columns player.
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red blue
Not Open,R0, R1, R2 10, 10 5, 5
Not Open,R0, R1, B2 10, 10 5, 5
Not Open,R0, B1, R2 5, 5 10, 10
Not Open,R0, B1, B2 5, 5 10, 10
Not Open,B0, R1, R2 10, 10 5, 5
Not Open,B0, R1, B2 10, 10 5, 5
Not Open,B0, B1, R2 5, 5 10, 10
Not Open,B0, B1, B2 5, 5 10, 10
As a result, the set of Nash equilibria of the game is:
NE = {(NotOpen,R0, R1, R2 − red) , (NotOpen,R0, R1, B2 − red) ,(
NotOpen,R0, B1, R2 − blue) , (NotOpen,R0, B1, B2 − blue) ,(
NotOpen,B0, R1, R2 − red) , (NotOpen,B0, R1, B2 − red) ,(
NotOpen,B0, B1, R2 − blue) , (NotOpen,B0, B1, B2 − blue)}
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