Question

Brett and Carl are playing chicken in their cars. If they both stay straight they will be horribly disfigured in a car accident, but both are hoping the other guy swerves so they can be the stronger man who stays straight. They have gotten so close that it is effectively a simultaneous move game. The table shows their possible payoffs, where the contents of each cell are (Brett’s payoff, Carl’s payoff).

		Carl's option 1	Carl's option 2
		Keep Straight	Swerve
Brett's option 1	Keep Straight	(-10 , -10)	( 1, -1)
Brett's option 2	Swerve	(-1 , 1)	(0 , 0)

a. Do either/both Brett and Carl have dominant strategies?
b. Does this game have one or more pure strategy Nash equilibriums? If so, what is/are it/they?
c. Is there a mixed strategy Nash equilibrium? If so, what is it?

Answer 1

Ans for a)

No neither of the players have any dominant strategy here in this game

Ans for b)

Yes this game has 2 Pure Nash Equilibrium and they are as follows

(Keep Straight,Swerve) & (Swerve, Keep Straight)

Becasue these are PSNE from which no player has profitable deviation to play away from it

Ans for c)

We have 2 NE then we cn find Mixed strategy Nash Equilibrium

Lets assume Brett plays Swerve with prob (1-p) and Carl plays swerve with prob (1-q)

then we need to find value of p and q for which Brett and Carl is indifferent between any of his two strategies

therefore we have

-10q+1*(1-q)=-1*q+0(1-q)

-10q+1-q=q

1/12=q similarly for Brett we can find values of p

-10p+1(1-p)=-1*p

p=1/12

Therefore MSNE are (11/12,1/12) and (11/12,1/12) respectivelt for Brett and Carl

Hence this game has 2 PSNE and 1 MSNE