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Problem Set 5 The following is a modification to the Prisoner’s Dilemma game, where each player has an additional action

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Problem Set 5

1. The following is a modification to the Prisoner’s Dilemma game, where each player has an additional action. In addition to “Cooperate” (C) and Defect (D), they can also choose “Nuclear” (P). If any player chooses the “Nuclear Option”, all payoff is eliminated for every player.

The stage game payoffs for this Modified Prisoner’s Dilemma are given below:

Player 2 Player 1	Cooperate	Defect	Nuclear
Cooperate	(4,4)	(1,5)	(0,0)
Defect	(5,1)	(2,2)	(0,0)
Nuclear	(0,0)	(0,0)	(0,0)

 

1. Find the Nash Equilibria of the Stage Game.

 

Suppose the Modified Prisoner’s Dilemma stage game is repeated twice over Periods 1 and 2. Consider the following Cooperation Strategy Profile for each player:

Player i

 plays “Cooperate” (C

) in Period 1. In Period 2, Player i

 chooses “Defect” (D

) if the (C,C)

 outcome was realized in Period 1. If any outcome other than (C,C)

 was realized in Period 1, then, Player i

 chooses “Nuclear” (N

) in Period 2.

 

 

2. How many subgames are there in Period 2? How many subgames does this twice-repeated game have in total?

 

3. Show that the considered strategy profile produces a Nash Equilibrium in all Period 2 subgames.

 

4. What is the payoff to each player if both players followed the cooperation strategy? Assume there is no discounting of future payoffs, i.e., δ=1

   

   .

 

5. Use the best single deviation method to show that the suggested cooperation strategy profile is a Period 1 best response.

 

6. Suppose players discounted Period 2 payoff by a factor δ<1

   

   , what is the lowest value of δ

   

   , that will support the given cooperation strategy as a Subgame Perfect Nash Equilibrium?

 

 

 

 

2. Consider the Bertrand Pricing game in Problem Set 3:

 

Firm 2   Firm 1	High Price	Low Price
High Price	($60,$60)	($0,$80)
Low Price	($80,$0)	($40,$40)

 

1. If this game is repeated over 10 periods, what is the Subgame Perfect Nash Equilibrium?

 

Suppose the two firms compete with each other in Period 1. After that in every successive period, there is 90%

 chance, or probability p=910

 that the firms will compete again in the following period. With the remaining 10%

probability the market is terminated and both firms get $0

 payoff.

 

 

2. Can the Grim Trigger Strategy be sustained as Subgame Perfect Nash Equilibrium? Show why or why not. Assume that there is no discounting of future payoffs, i.e. δ=1

   

   .

The Grim Trigger Strategy is as follows: Firm i

 chooses “High Price” in Period 1. After that, in every subsequent period, if the market exists, the firm chooses “High Price” if the “High Price, High Price” outcome was realized in the previous period and the “Low Price” if any other outcome was realized in Period 1.

 

 

 

3. Consider the Bertrand Pricing game given in Question 2. Suppose the two firms interact which each other for infinite periods and future payoffs are discounted by discount factor δ

   

   .

1. Describe a Tit-for-Tat strategy that firms could use to support collusion to maintain monopoly profits.

 

2. How high does δ

   

    have to be in order for collusion to be successful with the Tit-for-Tat strategy?

 

4. Consider a Cournot model of competition between two firms who simultaneously choose quantity. The market demand is given by

p=150-Q.

 

Cost for each firm from q

 is Cq=30q

.

 

The reaction function for Firm i

, given Firm j, j≠i

 quantity choice is qj

 is

 

qi=60-12qj.

 

 

1. Find the Cournot Nash Equilibrium quantities for each firm and their profits.

 

2. What is the monopoly outcome in quantity and profits?

 

3. If Firm 1 and Firm 2 were to collude with each other by equally sharing profits, how much would each firm produce and what would be their profits?

 

Suppose the two firms competed with each for infinite periods; consider the following Forgiving Trigger strategy that firms follow to enforce collusion with each other.

Each firm chooses q1M=q2M=30

 in Period 1. For subsequent periods, if both firms chose q1M=q2M=30

 in the previous period then they continue to produce q1M=q2M=30.

 (Cooperation Profile)

 

If any of the firms produced a different quantity in the previous period, then both firms choose the Cournot Nash quantity, q1*=q2*=40

 for T

 periods. After T

 periods, they revert back to collusion and choose q1M=q2M=30

. (Punishment Profile)

 

Assume that δ=910.

 

4. What is the discounted payoff in the cooperation profile?

 

5. If a firm deviating from this strategy, in the current period, what is the best possible deviation quantity it can choose and what would its payoff in the current period be?

 

Suppose the punishment period is T=1

.

 

 

6. Given this period of punishment, what is the discounted payoff during the punishment phase. (Hint: There is no need to solve the expression.)

 

7. What is the discounted payoff during the forgiveness period after the period of punishment? Hint: Use infinite geometric series formula.

 

8. Determine whether the forgiving trigger strategy for the given T  supports collusion as a SPNE.

 

1. Redo parts f), g) and h) for a punishment of two periods followed by forgiveness, i.e., with T=2 .