Question

Find all Nash equilibria in the following three simultaneous-move games:

Game 1 :

Column

Left Center Right

Up 3,1 4,2 3,3

High 5,7 1,3 2,4

Low 6,1 2,5 3,4

Down 1,1 4,6 5,2

Game 2:

Left Right

Up 2,4 7,7

Down 3,12 5,8

Game 3:

Left Right

Up 2,4 3,3

Down . 6,0 1,5

Answer 1

Game 1:
Let player 1 choose Up, Player 2's best response is Right (3).
Let player 1 choose High, Player 2's best response is Left (7).
Let player 1 choose Low, Player 2's best response is Center (5).
Let player 1 choose Down, Player 2's best response is Center (6).
Let player 2 choose Left, Player 1's best response is Low (6).
Let player 2 choose Center, Player 1's best response is Up and Down (4).
Let player 2 choose Right, Player 1's best response is Down (5).
so, the Nash equilibrium is (Down, Center) = (4,6) as best response of both players occur simultaneously at this set.

Game 2:
Let player 1 choose Up, Player 2's best response is Right (7).
Let player 1 choose Down, Player 2's best response is Left (12).
Let player 2 choose Left, player 1's best response is Down (3).
Let player 2 choose Right, player 1's best response is Up (7).
So, there are 2 Nash equilibria. They are (Up, Right) = (7,7) and (Down, Right) = (3,12) as best response of both players occur simultaneously at these sets.

Game 3:
Let player 1 choose Up, Player 2's best response is Left (4).
Let player 1 choose Down, Player 2's best response is Right (5).
Let player 2 choose Left, player 1's best response is Down (6).
Let player 2 choose Right, player 1's best response is Up (3).
So, there is no Nash equilibria as best response of both players never occur simultaneously.