Is this correct? Thank you Consider the infinitely repeated game in which the fo

Question

Is this correct?

Thank you

Consider the infinitely repeated game in which the following prisoner's dilemma is played in every stage: Prisoner2 CNC Prisoner C: 0,03,-1 NC-1,3 2,2 What is the lowest value of the discount factor such that the prisoners can use trigger strategy with infinite punishment to support (NC, NC) in every period? Please write your answer in digits and round off to 3 decimal places. For example, write 0.333 if your answer is 1/3 0.3330 Question 11 10 pts Continue) What is the lowest value of the discount factor such that the prisoners can use trigger strategy with one-period punishment to support (NC, NC) in every period? (I.e., if there is a deviation from NC to C, then play (C, C) for exactly one period and then resume to (NC, NC) forever after. Please write your answer in digits and round off to at most 3 decimal places. For example, write 0.333 if your answer is 1/3. 0.5000

Explanation / Answer

The nash equilibrium in this situation will be acheived at the stage (NC,NC) giving a payoff (2,2) both prisoners are better off in this case. But that this is not a possible equilibrium for this game, because if Prisoner 1 cooperates, then Prisoner 2 will do better by not cooperating. The only equilibrium is where they both dont cooperate, but then each gets a prison time of 2 years. This is the classic Prisoners’ Dilemma.

The presence of a discount rate and repeated play can be enough to eliminate the inefficiency inherent in the prisoners’ dilemma.

Suppose each prisoner uses a strategy: Cooperate as long as the other prisoner is cooperating. But if the other company ever chooses to not cooperate, then we also wont cooperate.

If D is the Discount rate

and Both prisoners cooperate, they both get

0+D0+D20....=0/(1-D)

If one prisoner defects and chooses to not cooperate, even just once, then it will get

-1 + D3 + D23 + .... = -1+ {3D/(1-D)}

The first payoff is larger if

0 < -1+ {3D/(1-D)}

or

taking 1 to the other side we have 1 < 3D/(1-D)

taking (1-D) to Left side, we have 1-D < 3D or 1<4D

D> 1/4 or 0.25

Hence, the correct answer or lowest value of discount factor such that the prisoner can use the trigger strategy with infinite punishment would be 0.25

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