EconS 503 - Microeconomic Theory II
Homework #6 –
Due date: March 23rd
1. Suppose we have a typical Prisoner’s dilemma where b > a > d > c.
Player 2
Cooperate
Cooperate
Defect
Player 1
a,a
b,c
Defect
c,b
d,d
a) If the game is infinitely repeated, what is the discount rate that is required to ensure that
cooperating can be a Nash equilibrium if a punishment can only last two periods?
b) If the discount rate is .5, and punishments can still only last two periods, what is the range of values
for d to ensure that cooperating can be a Nash equilibrium?
c) Given any discount rate between zero and one, what is the area of possible Nash equilibria in an
infinitely repeated game? (You can either show this graphically or using equations.)
ANSWER:
a) 𝛿 ≥ −.5 + √. 25 +
7
3
𝑏−𝑎
𝑎−𝑑
or 𝛿 + 𝛿 2 ≥
4
3
b) 𝑑 < 𝑎 − 𝑏
c, b
a, a
c)
d, d
b, c
Or, the area given by the intersection of,
𝑦≥𝑑
𝑥≥𝑑
𝑏−𝑎
𝑎−𝑏
𝑐+
𝑥
𝑎−𝑐
𝑎−𝑐
𝑎−𝑐
𝑐−𝑎
𝑦≤𝑎+
𝑎+
𝑥
𝑏−𝑎
𝑏−𝑎
𝑦≤𝑏+
Or if a < .5b + .5c,
𝑏−𝑎
𝑎−𝑑
𝑦≥𝑑
𝑥≥𝑑
𝑦 ≤ 𝑏+𝑐−𝑥
2. Suppose with probability p, we have the following game:
Player 2
L
Player 1
U
D
1,-1
0,2
R
2,0
1,1
And with probability 1 – p we have the following game:
Player 2
l
Player 1
U
D
-1,1
0,2
r
2,0
1,1
So that player 1 does not know the type they are playing against and therefore is not sure of the payoff
structure. However, there is only one type of player 1. Find the Bayesian Nash equilibria of this game.
ANSWER:
If p ≤ .5, {D,Ll} is BNE, if p ≥ .5 {U,Rl} is BNE
3. Suppose that hunter i has a payoff given by, Ui(ei,ej) = ei(1000-ei-ej) – 100ei where ei is player
i’s effort. Hunter j’s cost structure is uncertain to hunter i (but not hunter j), and determined by
nature so that there is a 50% chance hunter j’s payoff is given by, Uj(ej,ei) = ej(1000-ei-ej) – 100ej
and a 50% chance that is it given by Uj(ej,ei) = ej(1000-ei-ej) – 200ej. What is the Bayesian Nash
equilibrium if they simultaneously choose their effort level?
ANSWER:
The BNE is {316.667, 241.667, 291.667}