# 1) Put together an example of a 2×2 game (two players with each having two options) that has NO pure-strategy Nash equilibria. (Hint:

1) Put together an example of a 2×2 game (two players with each having two options) that has NO pure-strategy

Nash equilibria. (Hint: just try some values and then change them as you need to; as an example think about

the Rock, Scissors paper which has no pure strategy NE).

2) Using your own words, explain how the concept of elimination of dominated strategies differs from the

concept of Nash equilibrium.

3) Indicate whether the following statement is TRUE or FALSE and explain your answer: If one player is the

leader in a game it means that s/he has the first move in the game. The key thing is that this move can be

observed by the follower before making his/her own decision, otherwise it would be a simultaneous move

game that cannot be solved by backwards induction.

4) MULTIPLE CHOICE (identify the one best answer below and explain your reasoning for each option): Recall

the Game of chicken from the previous problem set. If James is the leader in the game then: (hint: you can

draw the game tree to see it better)

a. There are two equilibrium outcomes that coincide with the Nash equilibria.

b. The unique equilibrium outcome is James playing Rooster and Buzz playing Chicken.

c. The equilibrium outcome is the same as if Buzz were the leader.

d. All of the above.

e. None of the above.

PART B (INDEPENDENT THINKING AND APPLICATION)

5) (KEY QUESTION) Consider the following strategic interaction between two Australia telecommuncation

companies deciding to set the prices of their ‘unlimited calls’ mobile package.

Optus

Low Medium High

Telstra

Low 2, 3 13, 1 17, -8

Medium 1, 10 10, 8 15, 3

High -10, 19 3, 16 11, 9

a. Put yourself in the shoes of a CEO of these companies. Try to explain the business reasons behind the

relationships between the various payoffs (obviously, this is just an example, they may differ in the

real world and change over time). For example, why is Low in the payoff matrix the best response to

the opponent playing High? Why is the payoff from (High, High) higher for both players than from

(Medium, Medium)?

b. State all the dominated strategies in the game, by which strategy they are dominated, and whether

weakly or strictly. What is the equilibrium outcome by dominance, if any?

c. What are the pure-strategy Nash equilibria of this game?

d. In the simultaneous game, is there anything interesting or surprising in this game? Does it remind

you a different game we have examined? Explain.

e. Now assume Telstra is the leader (first mover), what is the equilibrium outcome? Is Telstra’s

leadership an advantage or disadvantage for Telstra compared to the simultaneous game? Is Telstra’s

leadership an advantage or disadvantage for Optus compared to the simultaneous game? Explain.

6) (KEY QUESTION) Consider the following one-shot simultaneous game (three actions for each player is the

maximum we will consider in this subject, but once you get on top of this, you will see that even large games

are equally easy to solve):

Cindy

D E F

Phil

A 6, 4 1, 2 5, 5

B 5, 8 4, 8 6, 2

C 6, -2 2, 0 5, -1

a. Before solving the game, put yourself in the position of Phil and write down your action. Then

independent of that, put yourself in the position of Cindy and write down your action.

b. State all the dominated strategies in the full game, by which strategy they are dominated, and

whether weakly or strictly.

c. What is the equilibrium outcome by dominance (by elimination of dominated strategies), if any?

d. What are the pure strategy Nash equilibria of this game? Pick one and explain precisely (prove) why

it is the Nash equilibrium.

e. Argue which NE is more likely and why. You can then relate this argument to your play in part a.

f. Prove that the (C, D) outcome is not a NE.

g. Assume Phil is the leader and Cindy the follower. Solve the game by backwards induction. What is

the equilibrium outcome? Explain your steps.

7) (REAL-WORLD APPLICATION) We practice this because it is a very valuable skill to be able to summarize

real world situations as games). Choose a strategic situation from your life and summarize it as a two player

game.

a. Make sure you specify the players, their strategies, the timing, and the payoffs.

b. Solve the game using dominance and Nash equilibria.

c. Discuss whether there is something interesting/surprising in the solution.

Consider whether alternative timing (leadership of either player) may produce different outcomes, and whether

it is an advantage or disadvantage for the leader.

8) (CRITICAL THINKING & REFLECTION) You can choose ONE from the following two tasks:

a. Write one good question about any issues covered in this problem set. You will ask and discuss

it in your tutorial so try to think of the answer too.

b. Find on the internet some academic study or media article (newspaper or blog etc.) that relates to

any of the covered topics. explain how.