Consider the infinitely repeated game with discount factor E[0,1] of the following variant of the Prisoner's...

Consider the infinitely repeated game with discount factor E[0,1] of the following variant of the Prisoner's Dilemma game:

			Player 2
		L	C	R
	T	(6, 6)	(-1, 7)	(-2, 8)
Player 1	M	(7, -1)	(4, 4)	(-1, 5)
	B	(8, -2)	(5, -1)	(1,1)

A) For which values of the discount factor E[0,1] can the players support the pair of actions (M, C) played in every period?

B) For which values of the discount factor E[0,1] can the players support the pair of actions (T,L) played in every period? Why is your answer different in part A) above?

Solutions

Expert Solution

Assume that both the players follow the grim trigger strategy. This implies punsihment is given forever once any of the player deviates. Hence there are two outcome possible for this subgame: (M, C) in for all periods including the current one or (B, R) in all periods as the punishment is given forever.

a) For the first case, a player’s payoff is 4 + 4? + 4?²+ ... = 4/1?? . If he deviates in first period he will be able to secure 8 in that period by choosing B but will receive only 0 for each period forever when player 2 starts choosing R as a punishment. Hence the payoff is 8 + 0? + 0?² + ... = 8 . The player has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

4/1?? ? 8

? ? 1/2

Hence for ? ? 1/2, players always choose M, C

b) Now we have two outcome possible for this subgame: (T, L) in for all periods including the current one or (B, R) in all periods as the punishment is given forever.

For the first case, a player’s payoff is 6 + 6? + 6?² + ... = 6/1?? . If he deviates in first period he will be able to secure 8 in that period by choosing B but will receive only 0 for each period forever when player 2 starts choosing R as a punishment. Hence the payoff is 8 + 0? + 0?² + ... = 8 . The player has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

6/1?? ? 8

? ? 1/4

Hence for ? ? 1/4, players always choose (T, C)

The discount factor is reduced here because the total payoff from current strategy is more.

Rahul Sunny answered 1 year ago

Next > < Previous

Related Solutions

Two firms are involved in an infinitely repeated game. If they cooperate on pricing, they can...

Two firms are involved in an infinitely repeated game. If they cooperate on pricing, they can split the monopoly profit, each earning $500 per period. But if one firm undercuts slightly, then the firm that undercuts can collect the entire monopoly profit of $1000. The firms interact repeatedly and attempt to cooperate using the following strategy: Let’s start by splitting the monopoly profit. If anyone ever undercuts, we’ll revert to the competitive outcome in the following period, and both of...

Two firms are involved in an infinitely repeated game. If they cooperate on pricing, they can...

Two firms are involved in an infinitely repeated game. If they cooperate on pricing, they can split the monopoly profit, each earning $500 per period. But if one firm undercuts slightly, then the firm that undercuts can collect the entire monopoly profit of $1000.The firms interact repeatedly and attempt to cooperate using the following strategy: Let’s start by splitting the monopoly profit. If anyone ever undercuts, we’ll revert to the competitive outcome in the following period, and both of us...

Consider a repeated prisoner’s dilemma game that will be repeated for one million rounds. 1. If...

Consider a repeated prisoner’s dilemma game that will be repeated for one million rounds. 1. If this game is played by immortal rational utility maximizers, what is the Nash equilibrium for this repeated game? 2. If this game was played as an experiment using human players, would you expect to see this strategy? 3. If this game was played as an experiment with rational utility maximizers who have a maximum lifespan of 500,000 rounds, what is the Nash equilibrium for...

Consider the Stage Game below, and consider the repeated game where players play twice (T =...

Consider the Stage Game below, and consider the repeated game where players play twice (T = 2). Payoﬀs for each agent are simply period one plus period two payoﬀs. L C R T 6,6 0,7 1,2 M 7,0 1,1 2,0 B 2,1 0,1 3,3 (a) Do any strategies dominate any other? (b) What are the two NE of the Stage Game? What is the diﬀerence between the two? (c) Call the TL strategy proﬁle (1 plays T, 2 plays L)...

6. Consider infinitely repeated prisoner’s dilemma C D C 2,2 -1,3 D 3.-1 0,0 (a) For...

6. Consider infinitely repeated prisoner’s dilemma C D C 2,2 -1,3 D 3.-1 0,0 (a) For each δ < 1 find the shortest length of punishment that is needed in order to sustain cooperation, or show that the cooperation cannot be sustained. (b) Using the one-shot deviation principle for each δ argue whether tit-for-tat strategy is an SPE or not. The tit-for-tat strategy is to cooperate in period 1. In period t do whatever your opponent did in period t...

Assume that the stag-hare game is now played repeatedly. What should the discount factor be for...

Assume that the stag-hare game is now played repeatedly. What should the discount factor be for such a game to enforce cooperation? Assume the Grim-Trigger strategy. Show your work to justify your conclusions.

Consider the following two-player game: L C R T 2,2 0,2 0,1 M 2,0 1,1 0,2...

Consider the following two-player game: L C R T 2,2 0,2 0,1 M 2,0 1,1 0,2 B 1,0 2,0 0,0 (a) Find all pure strategy Nash equilibria of this game. (b) Consider the following procedure of iterated elimination of weakly dominated actions : all weakly dominated actions of each player are eliminated at each stage. What are the action profiles that survive this procedure in the above game? I have no problem with solving (a) but (b) is so difficult....

Consider the following variants of the Towers of Hanoi. For each of variant, describe an algorithm...

Consider the following variants of the Towers of Hanoi. For each of variant, describe an algorithm to solve it in as few moves as possible. Prove that your algorithm is correct. Initially, all the n disks are on peg 1, and you need to move the disks to peg 2. In all the following variants, you are not allowed to put a bigger disk on top of a smaller disk. Consider the disappearing Tower of Hanoi puzzle where the largest...

So called “H-Model” in stock valuation is a variant of the following model. 3-stage dividend discount...

So called “H-Model” in stock valuation is a variant of the following model. 3-stage dividend discount model (DDM) 2-stage DDM the Gordon model

Consider the following variant of theBertrand Model of Duopoly. Suppose there are two firms producing the...

Consider the following variant of theBertrand Model of Duopoly. Suppose there are two firms producing the same good and they simultaneously set prices for their product. If firm i sets a price piand firm j sets a price pj, the total quantity demanded for firm i’s product is given by:qi= 10 –pi+ ½ pjEach firm produces exactly the qidemanded by the market. Bothfirms have the same marginal cost of production: c=4. For example, if a firm produces 5 units it...

Subjects