Question

SUBJECT; GAME THEORY

Consider the following pricing game

James	Dean	Swerve (q)	Straight (1-q)
Swerve (p)		0,0	-1,1
Straight (1-p)		2,-1	-2,-2

PLEASE EXPLAIN IN DETAIL.

Find the mixed-strategy equilibrium in this game, including the expected payoffs for the players.

Answer 1

Mixed strategy Nash equilibrium.

Let James choose swerve with a probability of p and Dean choose swerve with a probability of q

In order for James to be indifferent between choosing swerve and straight, the expected payoff from choosing both the strategies should be equal

=> E(swerve) = E(straight)

=> q*0 + (1-q)*(-1) = q*2 + (1-q)*(-2)

=> q -1 = 2q -2 +2q

=> 3q = 1 => q = 1/3

In order for Dean to be indifferent between choosing swerve and straight, the expected payoff from choosing both the strategies should be equal

=> E(swerve) = E(straight)

=> p*0 + (1-p)*(-1) = p*1 + (1-p)*(-2)

=> p - 1 = p + 2p -2

=>2p = 1 => p =1/2

Expected utility for James = q*0 + (1-q)*(-1) = q*2 + (1-q)*(-2) = -2/3

Expected utility for Dean = p*0 + (1-p)*(-1) = p*1 + (1-p)*(-2) = -1/2

Thsu, in the moxed strategy nash equilibrium James choose to swerve with a probability of 1/2 and Dean chooses to swerve with a probability of 1/3