SERVER
Forehand Backhand
Forehand 9,1 0,10
RECEIVER
Backhand 0,10 6,4
This game is like the Tennis Game in Dixit and Nalebuff, except that if the Receiver is surprised, he never succeeds in returning the serve.
1_2 What is the Nash equilibrium probability of Forehand for the Receiver?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7
A. Try again. If the Receiver chooses such a low probability of Forehand, the Server can take advantage of that by serving to his Forehand all the time.
If the Receiver chooses the a probability of .1, for example, then
Server's Payoff (Forehand) = .1 (1) + .9(10) = 9.1 > Server's Payoff (Backhand) =.1(10) + .9 (4) = 4.6,
so the Server will serve to his Forehand all the time.
To get the correct answer, you need to choose a mixing probability Y for the Receiver such that the Server does no better from Forehand than from Backhand. To do that, you solve
Server's Payoff (Forehand) = Y (1) + (1-Y)(10) = Server's Payoff (Backhand) =Y(10) + (1-Y)(4).
Return to Self Test 2.