The assignment may be submitted late up to 2 days after the deadline with a 15%
penalty per day.
ˆ Students are allowed to work together, but each must submit their own unique final
copy of the assignment.
ˆ Plagiarism will be taken very seriously. See http://academichonesty.unimelb.edu.au.
ˆ Throughout: “NE” refers to Nash Equilibrium/Equilibria; “SPNE” refers to Subgame
Perfect Nash Equilibrium/Equilibria; “PSNE ” refers to Pure-strategy Nash Equilib-
rium/Equilibria.
ˆ Throughout: ”Briefly explain” means explain in no more than 3 sentences.
ECON20005 Assignment 2 1
Question 1 (15 marks) Consider penalty kicks in soccer. Suppose that the kicker has two
pure strategies: kick to the left and kick to the right. The goalie also has two pure strategies:
move to the left and move to the right. The success probabilities of each player are displayed
in the table below. The first (second) number represents the probability that the kicker
(goalie) wins. For instance, if the kicker kicks to the left and the goalie moves to the left,
each has a 50% chance of winning.
Goalie
Left Right
Kicker
Left 50,50 80,20
Right 90,10 20,80
a) Find all the pure strategy and proper mixed strategy Nash equilibria of the game.
b) Draw the best response curves and identify all the NE in part (a) on the graph.
c) What are the expected equilibrium payoffs?
Suppose now that the kicker improves his left free-kick skill, and the probability of scoring
when he kicks to the left and the goalie moves to the left increases to 70% (i.e., the goalie
successfully saves the goal with 30% chance.)
d) What are the NE in the new games? How does it change from part (a)? Provide a brief
intuitive explanation.
e) What are the expected equilibrium payoffs? Compare your answers with part (c) and
explain briefly.
Suppose now that the kicker and goalie have a third action of kicking (or moving) to the
middle. The success probabilities are displayed in the table below.
Goalie
Left Middle Right
Kicker
Left 50,50 90,10 80,20
Middle 40,60 60,40 50,50
Right 90,10 50,50 20,80
f) Find all the pure strategy and proper mixed strategy Nash equilibria of the game.
Question 2 (10 marks) Consider the following game and write it in extensive form.
ECON20005 Assignment 2 2
First, Sofia chooses between actions a and b. She knows that after she makes her decision,
Jack observes which action Sofia has chosen with probability 1/3 and does not observe which
action Sofia has chosen with probability 2/3. In all cases (regardless of whether Jack has
observed that Sofia chose a, or he has observed that Sofia chose b, or he has not observed
any action), Jack chooses between actions α and β. The payoff of each player is 1 after (a, α)
and (b, β) and 0 otherwise.
Question 3 (10 marks) Consider the following game.
a) How many pure strategies does each player have?
b) Write this game in normal form.
c) Find all pure strategy Nash equilibria.
Question 4 (15 marks) Consider the following normal form game:
P1/P2 C D
C 10, 10 5, 14
D 14, 5 α, α
Suppose the game is repeated infinitely.
ECON20005 Assignment 2 3
a) Assuming α = 9, find the minimum value for the discount factor δmin such that (C,C) is
a SPNE of the game when the players use grim strategies.
b) Find the value of α such that the minimum value for the discount factor to sustain the
equilibrium (C,C) is equal to δmin = 1/2. Assume again that the players use grim strategies.
Bonus Question (5 marks) Consider a market in which two firms produce identical products
at marginal costs c1 and c2, respectively. Firms compete by setting prices simultaneously.
The demand function of the market is D(p) = 100 −min{p1, p2}. Hence, the firm with the
lowest price serves the whole market. Assume that firms share the market equally if they
charge the same price. Suppose Firm 1 has a cost advantage. Its marginal cost of production
is c1 = 30 while its rival’s marginal cost of production is c2 = 60. Find the NE of the pricing
game. Show why this is a NE. Assume that prices can be set in increments of 0.1.
ECON20005 Assignment 2 4