A county consists, for all intents and purposes, of two towns, Right and Left. Each has one- third of the population; the remaining one-third lives uniformly scattered on the one straight road that connects the two towns. Everyone is an imbiber. Jim Beam and Jack Daniels own the only two liquor licenses. Each drinker will go regularly to whichever bar is closest to him. Unfortunately for Jim and Jack, Antie Temperance has convinced the powers that be that if these two gentlemen were unregulated, they would conspire to raise prices; the resulting hardship would be intolerable! Hence, the price of a drink in each bar is equal and determined by the government. The only variable that Jack and Jim have control over is their location.
(a) What is the appropriate definition of a Nash equilibrium for this location game?
(b) Find the Nash equilibrium locations of the two drinking establishments.
(c) If we reinterpret "location" as "type of product," how would you expect the two drinks to taste vis-a-vis each other? What does this say about the market's ability to provide a variety of products?
(d) Are the Nash equilibrium locations likely to be the "socially optimal" ones?
(e) Suppose instead that 1/8 of the population lives in each of the two towns, with the remaining 3/4 distributed uniformly between the two towns. If there are four taverns, what is the symmetric Nash equilibrium in locations?