Question

Pure strategy Nash equilibrium

3. In the following games, use the underline method to find all pure strategy Nash equilibrium.

(B ) [0, 4, 4 0, 5, 3]

[4, 0 0 4, 5, 3]

[3, 5, 3, 5 6, 6]

(C) [2, -1 0,0]

[0,0 1,2]

(D) [4,8 2,0]

[6,2 0,8]

(E) [3,3 2,4]

[4,2 1,1]

4. In the following 3-player game, use the underline method to find all pure strategy Nash equilibria. Player 1 picks the row, Player 2 picks the column, and Player 3 picks the page. For each outcome of the game, the three players’ payoffs are listed in order: 1’s payoff, 2’s payoff, 3’s payoff. For example, when the outcome of the game is (row 1, column 1, page 2), the payoffs are (3, 3, 6) for Players 1, 2, and 3, respectively.

page 1

[5, 5, 5 3, 6, 3]

[6, 3, 3 4, 4, 1]

page 2

[3, 3, 6 1, 4, 4]

[4, 1, 4 2, 2, 2]

I will get you started, so you can see the idea. The best response to (row 1, column 1) is page 2, so I underlined the 6. The best response to (column 2, page 1) is row 2, so I underlined the 4. Each player will have 4 situations to best respond to, so there should be 12 things underlined in total.

5. Is the above 3-player game dominance solvable? If not, explain why not. If yes, write down an order of deletion that solves the game.

Answer 1

In this game we have following Pure Nash Equilibrias

Row is Player 1 & Column is Player 2

(0,4)	(4,0)	(5,3)
(4,0)	(0,4)	(5,3)
(3,5)	(3,5)	(6,6)

Hence Best Responses for plater 1 for Player 2's repsonses from column 1 to column 3 are Row 2, Row 1 and Row 3 respectively (Row2,Col1);(Row1,Col2);(Row3,Col3)

SImilarly for player 2 we have Col1,Col2 & Col3 respectiely for Row1, Row2& Row3 Hence (Row1,Col1);(Row2,Col2);(Row3,Col3)

Intersection between these two players is (Row3,Col3)=(6,6) Pure Nash Equilibriuim

Game C)

(2,-1)	(0,0)
(0,0)	(1,2)

In the above game we have Nash Equilibria equals to (1,2)

Game D)

(4,8)	(2,0)
(6,2)	(0,8)

Hence Nash Equilibria for this game is (2,0)

Game E)

(3,3)	(2,4)
(4,2)	(1,1)

Above Game has NE at (2,4)