Consider the 2-player strategic game represented by the following payoff matrix. Player 1’s actions are denoted by T and B and player 2’s actions are denoted by L and R, respectively. Each cell shows the payoff outcomes for the respective pair of actions chosen by the two players. The first numbers in the cells give player 1’s payoffs and the second numbers give player 2’s payoffs. Find all pure strategy Nash equilibria         Player 1 Player 2 L R   T 1, 6 1, 2   B 0, 6 2, 1       

Problem set #1

 

 

 

• Consider the 2-player strategic game represented by the following payoff matrix. Player 1’s actions are denoted by T and B and player 2’s actions are denoted by L and R, respectively. Each cell shows the payoff outcomes for the respective pair of actions chosen by the two players. The first numbers in the cells give player 1’s payoffs and the second numbers give player 2’s payoffs.

 

Find all pure strategy Nash equilibria

 

 

	Player 1		Player 2
			L	R
		T	1, 6	1, 2
		B	0, 6	2, 1

 

 

 

• Consider the 2-player strategic game represented by the following payoff matrix. Player 1’s actions are denoted by T and B and player 2’s actions are denoted by L and R, respectively. Each cell shows the payoff outcomes for the respective pair of actions chosen by the two players. The first numbers in the cells give player 1’s payoffs and the second numbers give player 2’s payoffs.

 

Find all pure strategy Nash equilibria

 

 

	Player 1		Player 2
			L	R
		T	1, 2	2, 3
		B	2, 3	-5, 1

 

 

 

 

3)    Consider the 2-player strategic game represented by the following payoff matrix. Player 1’s actions are denoted by T, M, and B and player 2’s actions are denoted by L, C, and R, respectively. Each cell shows the payoff outcomes for the respective pair of actions chosen by the two players. The first numbers in the cells give player 1’s payoffs and the second numbers give player 2’s payoffs.

 

Player 1		Player 2
		L	C	R
	T	0, 4	4, 0	5, 3
	M	4, 0	0, 4	5, 3
	B	3, 5	3, 5	6, 6

 

 

 

1. Find all pure strategy Nash equilibria of the game. Are these strict or weak Nash equilibria?

 

1. Does any action strictly or weakly dominate another action? Check for each player!

 

4)    Consider the 2-player strategic game represented by the following payoff matrix. Player 1’s actions are denoted by T, M, and B and player 2’s actions are denoted by L, C, and R, respectively. Each cell shows the payoff outcomes for the respective pair of actions chosen by the two players. The first numbers in the cells give player 1’s payoffs and the second numbers give player 2’s payoffs.

 

 

 

Player 1		Player 2
		L	C	R
	T	3, 3	8, 0	0, 0
	M	0, 8	5, 5	0, 0
	B	0, 0	0, 0	4, 4

 

1. Find all pure strategy Nash equilibria of the game! Are these strict or weak Nash equilibria?

 

1. Does any action strictly or weakly dominate another action? Check for each player!

 

 

 

 

5)  Consider the 2-player strategic game represented by the following payoff matrix. Player 1’s actions are denoted by T, M, and B and player 2’s actions are denoted by L, C, and R, respectively. Each cell shows the payoff outcomes for the respective pair of actions chosen by the two players. The first numbers in the cells give player 1’s payoffs and the second numbers give player 2’s payoffs.

 

 

Player 1		Player 2
		L	C	R
	T	2, 0	1, 1	4, 2
	M	3, 4	1, 2	2, 3
	B	1, 3	0, 2	3, 0

 

1. Which actions survive iterated elimination of strictly dominated strategies?

 

 

1. Find all pure strategy Nash equilibria of the game that remain after iterated elimination of strictly dominated strategies. Are these strict or weak Nash equilibria?

 

 

• In a Hotelling model of electoral competition two candidates, A and B, vie for office by simultaneously and independently announcing binding policy positions p_A and p_B, respectively, on a one-dimensional policy space [0, 100]. Each candidate prefers winning the simple majority election to a tie, and she prefers a tie to a defeat. There are i = 1,2, …, 21 citizens with rather ‘polarized’ ideal points , which are shown in the figure below (each black ball represents an individual citizen and the position of the ball indicates this citizen’s ideal position). All citizens have the same payoff function, which is given by where  denotes the winning candidate’s policy position. We assume that citizens vote for their preferred candidate (after the policy positions of the two candidates are announced).

 

 

1. What is the modal, median, and mean ideal voter position?

 

1. What are the policy positions that the two candidates take in the unique Nash equilibrium? Show that these positions do indeed constitute a Nash equilibrium!

 

1. Why is p_A = 25 and p_B = 75 not a Nash equilibrium?

 

 

1. What is the equilibrium payoff of a citizen with an ideal position at 50? And what is the equilibrium payoff of a citizen with an ideal position at 25?

 

 

 

 

Problem+set+1

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The post Consider the 2-player strategic game represented by the following payoff matrix. Player 1’s actions are denoted by T and B and player 2’s actions are denoted by L and R, respectively. Each cell shows the payoff outcomes for the respective pair of actions chosen by the two players. The first numbers in the cells give player 1’s payoffs and the second numbers give player 2’s payoffs. Find all pure strategy Nash equilibria         Player 1 Player 2 L R   T 1, 6 1, 2   B 0, 6 2, 1        appeared first on Apax Researchers.