Exercise 2. Consider the following game. Left Center Right Up 350, 350 350, 250 1000,0 Middle...

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Exercise 2. Consider the following game. Left Center Right Up 350, 350 350, 250 1000,0 Middle 250, 350 550,550 0,0 Down 0,1000 0,0 500, 500 (i) Find all pure-strategy Nash equilibria of the game. (ii) Find all mixed-strategy Nash equilibria of the game. Solution: (i) There are two pure-strategy Nash equilibria: (Up, Left) and (Middle, Center). (ii) First, note that "Down" and "Right" can never be played in a mixed-strategy Nash equi- librium because they are strictly dominated by "Up" and "Left". Furthermore, there is no mixed-strategy Nash equilibrium where one player chooses one of the actions with certainty and the other one randomizes because given that one player chooses one of the actions with certainty there is always a single best response for the other player. Hence, there is only one possibility for a mixed-strategy Nash equilibrium: Player 1 random- izes between "Up and Middle" while player 2 randomizes between "Left" and "Center". Suppose that player 2 chooses "Left" with probability p and Center with probability 1 p. Then player 1 is indifferent between Up" and "Middle" if 2 350=250p+550(1 - p) or p 3' The game is symmetric and so player 2 is indifferent between Left and Center" when player = 1 picks "Up" with probability q and "Middle" with probability 1 2 3 1 q = Thus, there 3 is a unique mixed-strategy Nash equilibrium in which player 1 chooses Up" with probability 2 1 2 and "Middle" with probability and player 2 chooses "Left" with probability and 3 3' 3 Center with probability In this equilibrium both players expect to earn 350. Exercise 2. Consider the following game. Left Center Right Up 350, 350 350, 250 1000,0 Middle 250, 350 550,550 0,0 Down 0,1000 0,0 500, 500 (i) Find all pure-strategy Nash equilibria of the game. (ii) Find all mixed-strategy Nash equilibria of the game. Solution: (i) There are two pure-strategy Nash equilibria: (Up, Left) and (Middle, Center). (ii) First, note that "Down" and "Right" can never be played in a mixed-strategy Nash equi- librium because they are strictly dominated by "Up" and "Left". Furthermore, there is no mixed-strategy Nash equilibrium where one player chooses one of the actions with certainty and the other one randomizes because given that one player chooses one of the actions with certainty there is always a single best response for the other player. Hence, there is only one possibility for a mixed-strategy Nash equilibrium: Player 1 random- izes between "Up and Middle" while player 2 randomizes between "Left" and "Center". Suppose that player 2 chooses "Left" with probability p and Center with probability 1 p. Then player 1 is indifferent between Up" and "Middle" if 2 350=250p+550(1 - p) or p 3' The game is symmetric and so player 2 is indifferent between Left and Center" when player = 1 picks "Up" with probability q and "Middle" with probability 1 2 3 1 q = Thus, there 3 is a unique mixed-strategy Nash equilibrium in which player 1 chooses Up" with probability 2 1 2 and "Middle" with probability and player 2 chooses "Left" with probability and 3 3' 3 Center with probability In this equilibrium both players expect to earn 350.