ECOS3012 Strategic Behaviour
Tutorial 3
• You should try your best to solve these questions before coming to the tutorial.
• For the easier questions, you will be asked to share your answers with your classmates.
• For the other questions, you will be invited to share your thoughts on them (even if you
hadn’t figured out the final solution). We will discuss these questions in detail.
Q1. Nash’s Theorem (1950) states that, in any normal form game, there always exists at least one
(possibly mixed) Nash equilibrium if
(i) the number of players is finite, and
(ii) the number of feasible strategies for each player is finite.
In each of the following examples, state which assumption is violated, and why Nash equi-
librium does not exist.
a.easier Two players, each chooses xi ∈ [0,1] and receives payoff
ui =
{
0 if both players choose 1
xi otherwise
b. There are infinitely many players. Each player chooses xi ∈ {0,1} and receives payoff{
xi i f ∑i xi is f inite
−xi i f ∑i xi is in f inite
1
Q2. Consider the following game
Player 2
L R
q 1-q
Player 1
U p -12, 1 8, 8
D 1-p 15, 1 8, -1
a.easier Is there any strictly dominated strategy?
b.easier Is there any weakly dominated strategy?
c. Find all Nash equilibria.
2
Q3. Consider the following game
Player 2
L R
q 1-q
Player 1
T 0 1, 3 1, 0
M p 4, 2 0, 4
B 1-p 0, 5 3, 1
a. Is there any strictly dominated strategy?
b. Find all Nash equilibria. (Easy if you know the answer to (a).)
3