Question

Frank and Nancy met at a sorority sock hop. They agreed to meet for a date at a local bar the next week. Regrettably, they were so fraught with passion that they forgot to agree on which bar would be the site of their rendezvous. Luckily, the town has only two bars, Rizotti’s and the Oasis. Having discussed their tastes in bars at the sock hop, both are aware that Frank prefers Rizotti’s to the Oasis and Nancy prefer the Oasis to Rizotti’s. In fact, the payoffs are as follows. If both go to the Oasis, Nancy’s utility is 3 and Frank’s utility is 2. If both go to Rizotti’s, Frank’s utility is 3 and Nancy’s utility is 2. If they don’t both go to the same bar, both have a utility of 0.

(1) Show the game in the matrix form. (Denote players, strategies and payoffs clearly)

(2) Is there any pure Nash equilibrium for the game? If any, find out the pure Nash equilibrium(s)

(3) Find out the mixed Nash equilibrium. (Show your calculation in details)

(4) What is the probability that Frank and Nancy go to the same bar?

Answer 1

Q1)

Payoff matrix

Frank/Nancy	R	O
Rizotti R	(3*,2•)	(0,0)
Oasis O	(0,0)	(2*,3•)

2 players : F Frank, N Nancy

Two strategies R & O

Q2) pure strategy NE

(R,R) & (O,O)

Q3) MSNE

let F plays R & O with probability p & (1-p)

N plays R & O with q & (1-q)

then EU_F = 3pq + 0 + 0 + 2(1-p)(1-q)

= 3pq + 2 -2p -2q + 2pq

= 5pq -2p -2q + 2

so, dEU_F/dp =5q -2 = 0

q = 2/5

EU_N = 2pq + 3(1-p)(1-q)

= 2pq + 3 -3p -3q + 3pq

= 5pq - 3p -3q +3

dEU_N/dq= 5p - 3= 0

p = 3/5

MSNE : (p,q) = (3/5, 2/5)

4) both go to same bar, probability

= (3/5)*(2/5)

= 6/25