Cournot-Nash Equilibrium

Cournot-Nash Equilibrium

Problem Set I

This problem set is worth 55 points. You may also avail of 2 extra credits if at least 25 students submit by
1:00 pm on the Due Date. Please submit your answers on a separate sheet of paper. Handwritten answer
sheets are allowed. Typed are obviously preferred.

Short Answers

Define the following concepts and explain their importance in your own words. [2.2 points each]

i) Cournot-Nash Equilibrium
ii) External Effects
iii) Best Response
iv) Constitutional Conundrum
v) Pareto efficient and Pareto improvement

Games

A) Consider the following game between Liberty (L) and McKenna (M). Consider the following
statements about the game and determine which are true and which are false. Make sure to explain your
answer. [1.375 points]
i) Liberty has a strictly dominant strategy.
McKenna
Left Right
Liberty
Up (2, 3) (3, 4)
Middle (5, 1) (1, 2)
Down (4, 3) (2, 3)
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ii) Liberty has a weakly dominated strategy
iii) Liberty has no dominant strategies.
iv) McKenna has a strictly dominant strategy.
v) McKenna has a weakly dominant strategy
vi) McKenna has no dominant strategies.
vii) The game has a dominant strategy equilibrium.
viii) The game has a Nash equilibrium.
B) Bill and Ted, two teenagers from San Dimas, California, are playing the game of chicken. Bill drives
south on a one-lane road and Ted drives north along the same road. Each has two strategies: Stay or
Swerve. If one player chooses Swerve he loses face; if both Swerve,they both lose face. If both choose Stay,
they are both killed. Consider the following payoff matrix: [2.2 points each]
Bill
Stay Swerve
Ted
Stay (−3, −3) (2, 0)
Swerve (0, 2) (1, 1)
i) What are the Pareto-efficient outcomes?
ii) What are the best responses? What are the Nash equilibria?
iii) Are there any dominant strategies? If so, which?
iv) Graph the payoff set for this game.
v) What are the conflict and common-interest elements in this game?
C) For each of the following games: (i) identify the Nash equilibrium/equilibria if they exist, (ii) identify
all strictly dominant strategies if there are any, and (iii) identify the Pareto-optimal outcomes and
comment whether they coincide with the Nash Equilibrium(s) you found. Also (iv), would you classify the
game as an invisible hand problem”, a coordination problem”, an assurance game”, a prisoners dilemma”,
or none of these? [2.2 points each]
2
Column Player
(C1) (C2)
Row Player
(R1) (2, 3) (−1, −1)
(R2) (−1, −1) (3, 2)
]
Column Player
(C1) (C2)
Row Player
(R1) (5, 5) (1, 5)
(R2) (5, 1) (4, 4)
]
Column Player
(C1) (C2)
Row Player
(R1) (−1, −1) (−5, 0)
(R2) (0, −5) (−4, −4)
]
Column Player
(C1) (C2)
Row Player
(R1) (1, 0) (1, −1)
(R2) (0, 1) (0, 1)
]
Column Player
(C1) (C2) (C3)
Row Player
(R1) (4, 4) (3, 5) (2, 6)
(R2) (5, 3) (3, 3) (3, 5)
(R3) (6, 2) (5, 3) (4, 4)
D) North (N) and South (S) are selecting environmental policies. The well-being of each is
interdependent, in part due to global environmental effects. Each has a choice of two strategies: Emit or
Restrict emission. Suppose this is just a two-person game between two representative citizens of North and
South. Let the representative citizen in each region have the following utility functions where α, β and γ
are some constants.
North’s utility:
u
N (e
N , eS
) = α(e
N ) + β(e
S
) + γ(e
N ∗ e
S
)
3
(1)
u
S
(e
N , eS
) = α(e
S
) + β(e
N ) + γ(e
N ∗ e
S
)
(2)
Where
(e
N , eS
) = 1 (3)
when the country chooses to emit and where
(e
N , eS
) = 0 (4)
when the country chooses not to emit.
i) Fill in the payoff table representing this game, that is, find the utility for North and South given either
strategy. (Hint:
u
N (e
N , eS
) = α + β + γ (5)
when the country chooses to emit.) [5.5 points]
South
Emit (e
S = 1) Restrict (e
S = 0)
North
Emit (e
N = 1) (u
N (e
N , eS)),u
N (e
N , eS)) (u
N (e
N , eS)),u
N (e
N , eS))
Restrict (e
N = 0) (u
N (e
N , eS)),u
N (e
N , eS)) (u
N (e
N , eS)),u
N (e
N , eS))
ii) Choose values for α, β and γ that make this game a Prisoners’ Dilemma. Show your reasoning. [5.5
points]
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