Name:........................................................................
ECO 5341
Spring 2016
Homework 1
Solutions
Total Points: 100
This homework assignment will only be accepted if the answers are provided on the
space following each question. Treat this assignment like an exam. Write your
answers only on the space provided following each question. Do not use separate
sheets. Do not write your answers on other sheets of paper. For full credit, please be
coincise and tidy. If your answer is illegible and not well organized, you will lose
points!
Question 1 (30 points): Consider the situation represented by the following bimatrix:
L
M R
U 1,0 2,5 -2,-1
D 2,1 2,1 -1,0
(a) (10 points) Write down the strategic form of this game.
Answer:
Players: row player (player 1), column player (player 2)
Strategies: S1 = {U, D }, S2 = { L, M, R}
Payoffs:
u1 (U, L) = 1
u1 (U, M) = 2
u1 (U, R) = −2,
u1 ( D, L) = 2
u1 ( D, M) = 2
u1 ( D, R) = −1
u2 (U, L) = 0
u2 (U, M) = 5
u2 (U, R) = −1
u2 ( D, L) = 1
u2 ( D, M ) = 1
u2 ( D, R) = 0.
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(b) (10 points) Is there a strictly dominant strategy equilibrium of this game? Explain.
Answer: No. Neither player has a strictly dominant strategy. To see this, let us write
down the best responses for each player. For player 1, we have
BR1 ( L) = D
BR1 ( M ) = D, U
BR1 ( R) = D
When player 2 plays M, strategy U is also a best response for player 1. Therefore, player
1 does not have a strictly dominant strategy.
For player 2, we have
BR2 (U ) = M
BR2 ( D ) = L, M
Note that strategy R is never a best response for player 2. Indeed, strategy R is strictly
dominated for player 2 by strategy M. This follows because
u2 (U, M ) = 5 > u2 (U, R) = −1
u2 ( D, M ) = 1 > u2 ( D, R) = 0
However, strategy M does not strictly dominate strategy L for player 2 since
u2 (U, M) = 5 > u2 (U, L) = 0 but
u2 ( D, M) = 1 = u2 ( D, L).
Hence, player 2 does not have a strictly dominant strategy.
Therefore, the game does not have a strictly dominant strategy equilibrium.
2
(c) (10 points) Is there a weakly dominant strategy equilibrium of this game? Explain.
Answer: Yes, (D,M) is the unique weakly dominant strategy equilibrium of the game.
D is the only weakly dominant strategy of player 1, and M is the only weakly dominant
strategy of player 2.
To see this, recall from part (b) that strategy R is strictly dominated for player 2 by
strategy M. This follows because
u2 (U, M ) = 5 > u2 (U, R) = −1
u2 ( D, M ) = 1 > u2 ( D, R) = 0
Furthermore, strategy M weakly dominates strategy L for player 2 since
u2 (U, M) = 5 > u2 (U, L) = 0 but
u2 ( D, M) = 1 = u2 ( D, L).
Hence strategy M is a weakly dominant strategy for Player 2.
For player 1, we have
u1 ( D, L) = 2 > u1 (U, L) = 1
u1 ( D, R) = −1 > u1 (U, R) = −2
u1 ( D, M) = 2 = u1 (U, M)
Hence strategy D is a weakly dominant strategy for Player 1.
Since both players have a unique weakly dominant strategy, (D,M) is the unique
weakly dominant strategy equilibrium of the game
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Question 2 (30 points) Consider the following game of “divide the dollar.” There
is a dollar to be split between two players. Player 1 makes an offer (an offer by Player
1 specifies how much he would like Player 2 to have). Without observing Player 1’s
offer, Player 2 specifies what would be an acceptable offer. Players’ choices have to be
in increments of 25 cents, i.e., 0 cents, 25 cents, 50 cents, 75 cents, and $1. If Player 1’s
offer is at least as large as what is acceptable to Player 2, then there is an agreement.
Otherwise we say that there is no agreement. If there is an agreement Player 2 gets
the amount offered by Player 1, while Player 1 gets the rest of the dollar. If there is no
agreement neither player gets anything.
a) (15 points) Illustrate this game in a bi-matrix.
Answer:
0
25
50
75
100
0
(100, 0)
(75, 25)
(50, 50)
(25, 75)
(0, 100)
25
(0, 0)
(75, 25)
(50, 50)
(25, 75)
(0, 100)
50
(0, 0)
(0, 0)
(50, 50)
(25, 75)
(0, 100)
75
(0, 0)
(0, 0)
(0, 0)
(25, 75)
(0, 100)
100
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 100)
The row player is player 1, and the column player is player 2. s1 = x means player 1
offers x cents to player 2, and s2 = x means player 2 only accepts an offer greater than
or equal to x cents.
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(b) (15 points) Find all the pure strategy Nash Equilibria of this game.
Answer: There are six pure strategy Nash Equilibria: (0, 0), (25, 25), (50, 50), (75, 75),
(100, 100) and (0, 100).
0
25
50
75
100
0
(100, 0)
(75, 25)
(50, 50)
(25, 75)
(0, 100)
25
(0, 0)
(75, 25)
(50, 50)
(25, 75)
(0, 100)
50
(0, 0)
(0, 0)
(50, 50)
(25, 75)
(0, 100)
75
(0, 0)
(0, 0)
(0, 0)
(25, 75)
(0, 100)
100
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 100)
BR2 (0) = {0, 25, 50, 75, 100}
BR2 (25) = {0, 25}
BR2 (50) = {0, 25, 50}
BR2 (75) = {0, 25, 50, 75}
BR2 (100) = {0, 25, 50, 75, 100} .
BR1 (0) = 0
BR1 (25) = 25
BR1 (50) = 50
BR1 (75) = 75
BR1 (100) = {0, 25, 50, 75, 100}
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Question 3 (30 points) (Altruistic players in the Prisoners’ Dilemma) Each of two
players has two possible actions, cooperate (C) and defect (D). The monetary payoff to
each action profile are given in the following table
D
C
D C
1,1 3,0
0,3 2,2
However, the above table does not represent the true preferences of the players, as they
care about what the other player earns as well as their own earnings. In particular, if
mi ( a) is the amount of money earned by player i and m j ( a) is the amount of money
earned by player j; then the payoff to player i is mi ( a) + αm j ( a) ; where α ≥ 0. Of course,
when α = 0; the payoffs coincide with the ones in the above table. For example, player
1’s payoff to action profile (C,C) is
u1 (C; C ) = 2 + 2α
and player 1’s payoff to action profile (C,D) is
u1 (C; D ) = 0 + 3α
(a) (15 points) Write down the strategic form of this game for α = 1: Is this game a
Prisoners’ Dilemma game?
Answer: For α = 1 the game table is as follows:
D
C
D C
2,2 3,3
3,3 4,4
Players: Player 1, Player 2. Strategy Sets: S1 = S2 = { D, C }. Payoffs are below. This
is not a Prisoners’ Dilemma game. For the game to be a Prisoners’ Dilemma, we need
D to be a strictly dominant strategy for each player. However, (C) is a strictly dominant
strategy for both players.
u1 ( D; D ) = 2 < u1 (C; D ) = 3
u1 ( D; C ) = 3 < u1 (C; C ) = 4
u2 ( D; D ) = 2 < u2 ( D; C ) = 3
u2 (C; D ) = 3 < u2 (C; C ) = 4
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(b) (15 points) Find the range of values of α for which the resulting game is the
Prisoners’ Dilemma. For values of α for which the game is not the Prisoners’ Dilemma,
find its Nash equilibria.
Answer: For any α ≥ 0, the game table is as follows:
D
C
D
1 + α, 1 + α
3α, 3
C
3, 3α
2 + 2α, 2 + 2α
For any α ≥ 0, the payoff from full cooperation (C,C) is always higher than the payoff
from (D,D) for each player. Thus for this game to be a Prisoners’ Dilemma game, we
only need to make sure that D is a strictly dominant strategy for each player.
D is a strictly dominant strategy for player 1 if and only if:
1
and
2
1
3 > 2 + 2α ⇒ α <
2
1 + α > 3α ⇒ α <
D is a strictly dominant strategy for player 2 if and only if:
1
and
2
1
3 > 2 + 2α ⇒ α <
2
1 + α > 3α ⇒ α <
These conditions are the same for the two players since the game is symmetric.
Thus for α ∈ [0, 21 ) the game is a Prisoners’ Dilemma game.
When the game is not a Prisoners’ Dilemma game, i.e., α ≥ 12 , there are two cases.
(i) If α = 12 , the game table is as follows:
D
D 32 , 23
C 32 , 3
C
3, 32
3, 3
In this case all the pure strategy profiles are Nash Equilibria, i.e., there are four pure
strategy Nash Equilibria: (D,D), (D,C), (C,D), (C,C).
(ii) If α > 21 , C is now a strictly dominant strategy for both players, so there is only
one Nash Equilibrium: (C,C).
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Question 4 (10 points)
Find all the pure strategy Nash Equilibria of the following game.
T
M
B
L
2,3
2,3
1,2
C
1,3
2,4
0,5
R
5,1
6,0
1,5
Answer: The two pure strategy Nash Equilibria are (T,L) and (M,C).
First, B is strictly dominated by both T and M for Player 1. To see this note that
BR1 ( L) = T, M
BR1 (C ) = M
BR1 ( R) = M
BR2 ( T ) = L, C
BR2 ( M ) = C
BR2 ( B) = C, R
and
u1 ( T, L) = 2 > u1 ( B, L) = 1
u1 ( T, C ) = 1 > u1 ( B, C ) = 0
u1 ( T, R) = 5 > u1 ( B, R) = 1
u1 ( M, L) = 2 > u1 ( B, L) = 1
u1 ( M, C ) = 2 > u1 ( B, C ) = 0
u1 ( M, R) = 6 > u1 ( B, R) = 1
Now note that after eliminating B for player 1, R becomes strictly dominated for
player 2 in the following game by both L and C
T
M
L
C
2,3 1,3
2,3 2,4
R
5,1
6,0
After eliminating R, we have
T
M
L
2,3
2,3
C
1,3
2,4
In the reduced 2x2 game it can be easily seen that there are two pure strategy Nash
Equilibria: (T,L) and (M,C).
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