SABANCI UNIVERSITY
ECON 201 – A
GAMES AND STRATEGY
SECOND MIDTERM
EXAMINATION ANSWERS
April 3, 2008
(X)
Instructor: Mehmet Baç
Time allowed: 45 minutes, 100 points
READ CAREFULLY.
..
1
(
20 points
)
Olga and Irina, two neighbors, will simultaneously plant flowers in their
gardens. Let
x
denote the number of flowers that Olga plants and
y
, the number of
flowers than Irina plants. The payoffs of Olga and Irina are as follows:
U
Olga
= 10xy
– x
2
and
U
Irina
= 10y /x – y
2
/2.
The terms x
2
and 0.5y
2
represent the players’ cost of planting flowers.
(i) [5p] What kind of externality is there between Olga and Irina? Do they like each other’s
flowers? Explain by using the payoff functions above.
(ii) [10p] Draw the best response functions of the players on the same figure.
(iii) [5p] Find the Nash equilibrium numbers of flowers (x*, y*). Show your calculations.
Answer
:
(i) Olga’s payoff is increasing in y, thus, Olga enjoys Irina’s flowers, whereas Irina’s payoff is
decreasing in x, thus, Irina dislikes Olga’s flowers. There is a positive externality from Irina to
Olga, a negative externality from Olga to Irina.
(ii) Olga’s best response is found by taking the firstorder derivative of her payoff with respect to
x:
10y – 2x = 0, thus, x = 5y. Similarly, Irina’s best response is: 10/x – y = 0, or y = 10/x. These
best response functions look like:
y
y = 10/x
Nash Eq
x = 5y
0
x
(iii) The Nash equilibrium is found by solving the best response functions: x = 5(10/x) or, x =
(50)
0.5
or, x = 5.(2)
0.5
. Given this, y = 10 / 5.(2)
0.5
= 2
0.5
.
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(
40 points
) Consider the following game.
Player 2
Player 1
L
M
R
U
0 , 3
4 , 1
1 ,
2
D
2 ,
0
1 , 4
0 ,
1
(i)
[10p] Is any of player 1’s or player 2’s strategies
not
rationalizable? Explain.
(ii)
[10p] Denoting by
p
the probability that player 1 plays U, write player 2’s expected
payoffs from
each
of his three pure strategies.
(iii)
[10p] Obtain player 2’s best response function (denote by
q
L
, q
M
, and
q
R
the
probability with which player 2 plays L, M and R, respectively). Show that player
2’s best response involves
q
R
= 0.
(iv)
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 Fall '12
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 Game Theory, best response, Olga, Irina

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