1. Tutorial 1: Game Theory Matthew Robson University of York
Microeconomics 2

2. Question 1
What implication can you draw from the Boeing-Airbus game? Choose the best answer from the following:
Trigger strategy can lead to favourable outcome.
Such a game has infinitely many Nash equilibria.
Nash equilibrium may not necessarily be efficient.
Using a punishment strategy can lead to win-win outcome.
Making a credible commitment can bring advantage for your play of the game.
Airbus
Jumbo No
Boeing
-50, -50
90, -10
-20, 100
0, 0

3. Question 1 Trigger strategy can lead to favourable outcome.
Trigger strategy is used in infinite horizon games (i.e. Tit-for-Tat or Grim), so not particularly useful.
Such a game has infinitely many Nash equilibria.
This game has only two: (Jumbo, No) and (No, Jumbo).
Nash equilibrium may not necessarily be efficient.
Both Nash equilibria are efficient.
Using a punishment strategy can lead to win-win outcome.
The game is simultaneous, but if it wasn’t the punishment wouldn’t be credible.
Making a credible commitment can bring advantage for your play of the game.
Yes. If one firm made a credible commitment, then the other would not make the jumbo jet. This is a coordination game, similar to the Chicken Game.

4. Question 2
Suppose that Alan and Bill will repeat playing the prisoner’s dilemma game given below over an infinite number of periods, where the fixed discount rate common to both players is 𝑟. That is, the payoff 𝑥 received in the second period is worth 𝑥 1+𝑟 in the first period.
Bill
Not Confess
Confess
Alan
-1, -1
-9, 0
0, -9
-6, -6

5. Question 2
If both players adopt a trigger strategy, then in order for the strategy profile (Not Confess, Not Confess) in all periods to be a Nash equilibrium:
r should be greater than -1.
r should be no greater than 3.
r should be no greater than 4.
r should be no greater than 5.
None of the above.

6. Question 2
Grim Trigger strategy: cooperate (do not confess) as long as their opponent has cooperated in the previous period, but if their opponent defects (confess) in the previous period, then defect in all future periods.
As both players are symmetrical we take the point of view of one player, say Bill. We need to know his present value (PV) of defecting in a given period.
When Bill defects he gets a payoff of 0, then in all other periods -6 (due to Alan’s Grim Trigger strategy). So we can model this as:
𝑃𝑉 𝑑𝑒𝑓𝑒𝑐𝑡 =0+ 𝑡=1 ∞ −6 1+𝑟 𝑡 =0− 6 𝑟 =− 6 𝑟
This formula assumes that we defect from the first period… and relies on the fact that:
𝑡=1 ∞ 𝑟 𝑡 = 1 𝑟

7. Question 2 Convergent Series Example: 𝑡=1 ∞ 1 1+𝑟 𝑡 = 1 𝑟
First graph show the value of the function as 𝑡 increases, note is decreases to almost 0
Second graph shows the cumulative sum of the function over time, note it approaches 0.5 which is 1/𝑟

8. Question 2
If instead Bill hadn’t defected, he would get a payoff of -1 in all periods:
𝑃𝑉 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒 =−1+ 𝑡=1 ∞ −1 1+𝑟 𝑡 =− 1+ 1 𝑟 =− 𝑟 𝑟 + 1 𝑟 =− 1+𝑟 𝑟
So for cooperation to be optimal, we thus need:
𝑃𝑉 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒 ≥ 𝑃𝑉 𝑑𝑒𝑓𝑒𝑐𝑡
− 1+𝑟 𝑟 ≥− 6 𝑟
1+𝑟≤6
𝑟≤5
Thus, answer (d) is correct.

9. Question 3
Consider the following two player (Alice and Bob) game, where each player has two strategies.
Bob
Left
Right
Alice Up
2, 2
8, 4
Down
3, 9
5, 9

10. Question 3 Which of the following statements is correct?
There exist infinitely many mixed strategy Nash equilibria where Alice chooses Down with probability 1 while Bob can choose Left with any probability 𝑞∈ , 1 and choose Right with probability 1 − 𝑞.
The game has only pure strategy Nash equilibria.
The game has two Nash equilibria: One is 𝑈𝑝, 𝑅𝑖𝑔ℎ𝑡 , and the other is a mixed strategy equilibrium 1, which means Alice will choose Up with probability 1 while Bob play Left with probability and Right with probability
𝐷𝑜𝑤𝑛, 𝐿𝑒𝑓𝑡 cannot be a Nash equilibrium.
None of the above.

11. Question 3
Let p be the probability that Alice plays up, and q be the probability that Bob plays left.
Alice’s expected values are:
𝐸 𝐴𝑙𝑖𝑐𝑒 𝑢𝑝 =2𝑞+8 1−𝑞 =8−6𝑞
𝐸 𝐴𝑙𝑖𝑐𝑒 𝑑𝑜𝑤𝑛 =3𝑞+5 1−𝑞 =5−2𝑞
Alice plays down with probability 1 if 5−2𝑞>8−6𝑞, that is 𝑞> 3 4 .
In other words, Alice’s best response if Bob chooses any 𝑞∈ 3 4 ,1 is to play down with probability 1 (i.e., she chooses p=0).

12. Question 3
What is Bob’s best response if Alice plays down with probability 1 (p=0)?
If Alice plays down, Bob’s payoff is 9, whether he plays left or right, so he is indifferent between left and right. Any randomised strategy of his, including any 𝑞∈ 3 4 ,1 , is therefore a best response to Alice’s playing of down.
Because there are infinitely many possible values of 𝑞 satisfying 𝑞∈ 3 4 ,1 , there are infinitely many mixed strategies where Alice plays Down with probability 1 and Bob plays Left with any probability 𝑞∈ 3 4 ,1 , so answer (a) is correct.

13. Question 4
Consider the following two-player (Alice and Bob) game, where each player has two strategies.
Bob
Left
Right
Alice Up
3, 5
9, 5
Down
2, 8
5, 9

14. Question 4 Which of the following statements is false?
(Up, Left) is a Nash equilibrium.
(Up, Right) is a Nash equilibrium.
7 19 , is a mixed strategy Nash equilibrium which means Alice chooses Up with probability and Down with probability while Bob plays Left with probability and Right with probability
There exist infinitely many mixed strategy Nash equilibria where Alice chooses Up with probability 1 while Bob can choose Left with any probability 𝑞∈ 0, 1 and choose Right with probability 1 − 𝑞.
(Down, Right) is not a Nash equilibrium.

15. Question 4 Let q be the probability that Bob plays left.
Alice’s expected values are:
𝐸 𝐴𝑙𝑖𝑐𝑒 𝑢𝑝 =3𝑞+9 1−𝑞 =9−6𝑞
𝐸 𝐴𝑙𝑖𝑐𝑒 𝑑𝑜𝑤𝑛 =2𝑞+5 1−𝑞 =5−3𝑞
Alice plays up with probability 1 if 9−6𝑞>5−3𝑞, that is 𝑞< Note that is greater than 1. That is, for any 𝑞∈ 0,1 , Alice’s best response is to play up with probability 1. Alice will always play up with probability 1, no matter Bob’s choice of q.
Therefore, (c) is wrong. All the other statements are correct.

16. Question 5
The Stag Hunt game is based on a story told by Jean Jacques Rousseau in his book Discourses on the Origin and Foundation of Inequality Among Men (1754). The story goes something like this: “Two hunters set out to kill a stag. One has agreed to drive the stag through the forest, and the other to post at a place where the stag must pass. If both faithfully perform their assigned stag-hunting tasks, they will surely kill the stag and each will get an equal share of this large animal. During the course of the hunt, each hunter has an opportunity to abandon the stag hunt and to pursue a hare. If a hunter pursues the hare instead of the stag he is certain to catch the hare and the stag is certain to escape. Each hunter would rather share half of a stag than have a hare to himself.” The matrix below shows payoffs in a stag hunt game. If both hunters hunt stag, each gets a payoff of 4. If both hunt hare, each gets 3. If one hunts stag and the other hunts hare, the stag hunter gets 0 and the hare hunter gets 3.

17. Question 5
If you are sure that the other hunter will hunt stag, what is the best thing for you to do?
If you are sure that the other hunter will hunt hare, what is the best thing for you to do?
Does either hunter have a dominant strategy in this game? No. If so, what is it? If not explain why not.
This game has two pure strategy Nash equilibria. What are they?
Hunter B
Hunt Stag
Hunt Hare
Hunter A
4, 4
0, 3
3, 0
3, 3

18. Question 5
If you are sure that the other hunter will hunt stag, what is the best thing for you to do? Hunt Stag
If you are sure that the other hunter will hunt hare, what is the best thing for you to do? Hunt Hare
Does either hunter have a dominant strategy in this game? No. If so, what is it? If not explain why not. No, the best response depends on other player
This game has two pure strategy Nash equilibria. What are they? Both Hare or Both Stag
Hunter B
Hunt Stag
Hunt Hare
Hunter A
4, 4
0, 3
3, 0
3, 3

19. Question 5
Is one Nash equilibrium better for both hunters than the other? Yes If so, which is the better equilibrium?
If a hunter believes that with probability ½ the other hunter will hunt stag and with probability ½ he will hunt hare, what should this hunter do to maximize his expected payoff?
Can you think of any real life economic or political situations that can be modelled as a Stag Hunt Game? Give an example.
Hunter B
Hunt Stag
Hunt Hare
Hunter A
4, 4
0, 3
3, 0
3, 3

20. Question 5
Is one Nash equilibrium better for both hunters than the other? Yes If so, which is the better equilibrium? Yes, both hunt stag
If a hunter believes that with probability ½ the other hunter will hunt stag and with probability ½ he will hunt hare, what should this hunter do to maximize his expected payoff? Then 𝑬 𝑺𝒕𝒂𝒈 =𝟒 𝟏 𝟐 +𝟎 𝟏 𝟐 =𝟐 and 𝑬 𝑯𝒂𝒓𝒆 =𝟑 𝟏 𝟐 +𝟑 𝟏 𝟐 =𝟑  Hare.
Can you think of any real life economic or political situations that can be modelled as a Stag Hunt Game? Give an example. Pirates Mutiny, MPs not wanting to be in a minority vote, Hockey Players
Hunter B
Hunt Stag
Hunt Hare
Hunter A
4, 4
0, 3
3, 0
3, 3

21. Question 6
State two main assumptions of game theory, and explain why these assumptions are crucial for the application of game theory to microeconomic analysis.
Rationality: The player has well-defined preferences over the set of possible outcomes and implements the best available strategy to optimize her outcome according to her preferences.
Complete information: The structure of the game and the payoff functions of the players are commonly known by all players.
These assumptions make it possible to analyse games in the way we analyse them. If players weren’t rational it would be much harder to analyse and predict players’ choices in a game. Without complete information, players would not be able to respond optimally to the choice that they expect their opponent to take. They would not be able to predict their opponents’ choices because they would not know their opponents’ payoffs.
Rationality is also consistent with our usual assumption on how agents behave in microeconomic theory, and thus makes Game Theory more easily applicable to the analysis of economic problems.

22. Question 7 Matching pennies
Construct a game (a concrete numerical example) such that it has a unique Nash equilibrium, and moreover this Nash equilibrium is a mixed strategy equilibrium.
Matching pennies
Unique Nash Equilibrium: Mixed Strategy (0.5, 0.5) for both Bob and Alice
Bob H T
Alice
1, -1
-1, 1