Question

Suppose that two players are playing the following game. Player 1 can choose either Top or Bottom, and Player 2 can choose either Left or Right. The payoffs are given in the following table:

player 2:

left right  

player 1 : Top: (1,2) (2,4)

bottom: (3,4) (0,3)

where the number on the left is the payoff to Player 1, and the number on the right is the payoff to Player 2.

A) (2 points) Does Player 1 have a dominant strategy, and if so what is it?

B) (2 points) Does Player 2 have a dominant strategy and if so what is it?

C) (1 point each) For each of the following strategy combinations, write TRUE if it is a Nash Equilibrium, and FALSE if it is not:

• i) Top/Left
• ii) Top/Right
• iii) Bottom/Left
• iv) Bottom Right

D) (2 points) What is Player 1’s maximin strategy?

E) (2 points) What is Player 2’s maximin strategy?

F) (2 points) If the game were played with Player 1 moving first and Player 2 moving second, using the backward induction method we went over in class, what strategy will each player choose?

Answer 1

PLAYER 1	PLAYER 2
	LEFT	RIGHT
TOP	1,2	2,4
BOTTOM	3,4	0,3

A) DOMINANT STRATEGY IS A STRATEGY OF A PLAYER THAT HE CHOOSES IRRESPECTIVE OF WHAT STRATEGY OTHER PLAYER CHOOSES.

IF PLAYER 2 CHOOSES LEFT, PLAYER 1 CHOOSES BOTTOM. IF PLAYER 2 CHOOSES RIGHT, PLAYER 1 CHOOSES TOP.  

PLAYER1 DOES NOT HAVE A DOMINANT STRATEGY.

B)

IF PLAYER 1 CHOOSES TOP, PLAYER 2 CHOOSES RIGHT. IF PLAYER 1 CHOOSES BOTTOM, PLAYER 2 CHOOSES LEFT.  

PLAYER 2 DOES NOT HAVE A DOMINANT STRATEGY.

C) WE WILL UNDERLINE THE BEST RESPONSE OF EACH PLAYER, GIVEN ALL THE STRATEGIES OF THE OTHER PLAYER. THE INTERSECTION OF BOTH PLAYER'S BEST RESPONSE IS THE NASH EQUILIBRIUM.

IF PLAYER 2 CHOOSES LEFT, PLAYER 1 CHOOSES BOTTOM.

PLAYER 1	PLAYER 2
	LEFT	RIGHT
TOP	1,2	2,4
BOTTOM	3,4	0,3

IF PLAYER 2 CHOOSES RIGHT, PLAYER 1 CHOOSES TOP.  

PLAYER 1	PLAYER 2
	LEFT	RIGHT
TOP	1,2	2,4
BOTTOM	3,4	0,3

IF PLAYER 1 CHOOSES TOP, PLAYER 2 CHOOSES RIGHT.

PLAYER 1	PLAYER 2
	LEFT	RIGHT
TOP	1,2	2,4
BOTTOM	3,4	0,3

IF PLAYER 1 CHOOSES BOTTOM, PLAYER 2 CHOOSES LEFT.

PLAYER 1	PLAYER 2
	LEFT	RIGHT
TOP	1,2	2,4
BOTTOM	3,4	0,3

TOP/LEFT - FALSE

TOP/RIGHT- TRUE

BOTTOM/ LEFT- TRUE

BOTTOM/RIGHT- FALSE

D) Maximin strategy maximizes the minimum gain that can be achieved.

THE LOWEST PAYOFF THAT PLAYER1 CAN GET IS 0. HE GETS 0, BY CHOOSING BOTTOM. HE WILL MINIMIZE THE RISK OF GETTING THE WORST PAYOFF BY CHOOSING TOP. THUS PLAYER 1'S MAXIMIN STRATEGY IS TOP.

E) THE LOWEST PAYOFF THAT PLAYER 2 CAN GET IS 2. HE GETS 2, BY CHOOSING LEFT. HE WILL MINIMIZE THE RISK OF GETTING THE WORST PAYOFF BY CHOOSING RIGHT. THUS PLAYER 2'S MAXIMIN STRATEGY IS RIGHT.

F)