Put together an example of a 2x2 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).
1) Put together an example of a 2x2 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.
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.
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