Does this game have a Nash equilibrium? If so, describe it. If not, explain why not.

Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.3P
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There is a city, which looks like a trapezoid, as shown below. Citizens live
uniformly distributed all over the city. Two ice-cream vendors, A and B, must independently set up stores in the city. Each citizen buys from the vendor closest to their location and when equidistant from both vendors they choose by
coin toss. Each vendor’s aim is to maximize the expected number of customers. A choice of
location by the two vendors is a Nash equilibrium if no vendor can do better by deviating
unilaterally. Does this game have a Nash equilibrium? If so, describe it. If not, explain why not.

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