Short Problems About Nash Equilibrium

1. Miller Lite and Bud Light dominate the U.S. market for light beer. Each of them can choose whether to advertise or not advertise. If one firm advertises and the other does not, the firm doing the advertising gets a larger share of
the market and higher profits. If both firms advertise, their market shares remain the same as when neither
advertises (the only difference is that both must pay the advertising cost). Their profits in millions of dollars are
shown below.
a. Does Bud Light have a dominant strategy? What about Miller Lite? Explain.
b. What is the Nash equilibrium of this game?
2. [Mankiw, p. 387, #6] You and a classmate are assigned a project on which you will receive one combined grade.
You each want to receive a good grade, but you also want to avoid hard work. In particular, here is the situation:
i. If both of you work hard, you both get an A, which gives each of you 40 units of happiness.
ii. If only one of you works hard, you both get a B, which gives each of you 30 units of happiness.
iii. If neither of you works hard you both get a D, which gives each of you 10 units of happiness.
iv. Working hard costs 25 units of happiness.
a. Fill in the payoffs in the decision box on the next page.
b. What is the likely outcome? Explain.
c. If you get this classmate as your partner on a series of projects throughout the year, rather than only once,
how might that change the outcome you predicted in part (b)?
d. Another classmate cares more about good grades: He gets 50 units of happiness for a B, and 80 units of
happiness from an A (he still only gets 10 units from a D). Working hard still costs 25 units of happiness. If
this classmate were your partner (but your preferences were unchanged), how would your answers to parts (a)
and (b) change? Which of the two classmates would you prefer as a partner? Would he also want you as a
partner?
Bud Light
Miller Lite
Advertise Do not advertise
Advertise
Do not advertise
B: $3
M: $3
B: $2
M: $5
B: $5
M: $2
B: $4
M: $4
2
3. [Based on Mankiw, p. 388, #10] Use the Jack & Jill duopoly example we did in class to answer the following
question. As a reminder, the demand schedule for water is given below.
Quantity
(gallons) Price
Total
Revenue
(and
profit)
0 120 0
10 110 1,100
20 100 2,000
30 90 2,700
40 80 3,200
50 70 3,500
60 60 3,600
70 50 3,500
80 40 3,200
90 30 2,700
100 20 2,000
110 10 1,100
120 0 0
Suppose that Jack and Jill are at the duopoly's Nash Equilibrium (40 gallons each; 80 gallons total produced)
when a third person, John, discovers a water source and joins the market as a third producer (he also has a zero
marginal cost of production).
a. Jack and Jill propose that the three of them continue to produce a total of 80 gallons, splitting the market
three ways. If John agrees to this, how much profit will he make?
b. After agreeing to the proposed deal, John is considering increasing his production by 10 gallons. If he
does, and Jack and Jill stick to the agreement, how much profit will John make? What does this tell you
about the proposed agreement?
c. What is the Nash equilibrium for this market with three producers? How does it compare to the Nash
equilibrium with two producers?
Your decision
Classmate's
Decision
Work Shirk
Work
Shirk
You:
Classmate:
You:
Classmate:
You:
Classmate:
You:
Classmate: