Question

Q1. Consider the following game. A spy (row player) is trying to get away from the villain (column player) by skiing down one of three routes. The villain can choose to explode a bomb (which is costly) and cause an avalanche or not explode a bomb. Payo↵s are given by the following matrix:

Don't explode Explode

1 (12,0) (0.6)

2 (7,1) (1,5)

3 (9,3) (6,0)

a. Are there any routes that the spy should never to choose?

b. Let q be the probability with which the villain chooses ‘Don’t Explode’. What should

the spy do if q > 2/3? What about if q < 2/3?

c. Derive the mixed strategy Nash equilibrium of this game. Prove that it is an equilib- rium.

(Remember to show all working)

Answer 1

In above game we can see that Spy has 3 routes to choose

Among these 3 routes route 3 is strcitly dominating route 2 hence Spy will never play route 2 when route 3 is available to him.

Answer for B)

If Mixed strategy profile for villain we can calculate as below

if Villain plays DE with probability q and E with probability 1-q

Spy to be indifferent between DE and E

Expected payoff (Route 1)=12q+0*(1-q)=12q

Expected payoff (Route 3)=9q+6(1-q)=3q+6

Expected payoff (Route 1)=Expected payoff (Route 3)

12q=3q+6

9q=6

q=2/3

Hence when q<2/3

Expected Payoff(Route 1)=12(2/3)=8 and Expected Payoff(Route 3)=3(2/3)+6=8 Hence Equillibruim

hence if q=1/3<2/3

Expected Payoff(Route 1)=4 and Expected Payoff(Route 2)=7

hence Spy will play Route 2 when q<2/3 and will play Route 1 when q>2/3