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_1 What is the Nash equilibrium probability of Forehand for the Server?
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 Server chooses such a low probability of Forehand, the Receiver will take advantage of that by choosing a high probability of Backhand.
For example, if the server chooses a probability of .1, then
Receiver's Payoff (Forehand) = .1 (9) + .9(0) = .09 < Receiver's Payoff (Backhand) =.1(0) + .9 (6) = .54.
To get the correct answer, you need to choose a mixing probability X for the Server such that the Receiver does no better from Forehand than from Backhand. To do that, you solve
Receiver's Payoff (Forehand) =X (9) + (1-X)(0) = Receiver's Payoff (Backhand) =X(0) + (1-X) (6).
Return to Self Test 2.