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5. Consumers are uniformly distributed along a boardwalk of length 1 mile. The price of ice-cream is fixed by a regulator, so consumers go to the nearest firm because they dislike walking (assume that at the regulated price all consumers will purchase an ice-cream cone even if they have to walk a full mile). If more than one vendor is at the same location, they evenly split the business Consider a game where two ice-cream vendors pick their locations simultaneously. Show that there exists a unique pure strategy Nash equilibrium and that it involves both vendors locating at the midpoint of the boardwalk. (Hint: show that in any OTHER arrangement of their locations, at least one vendor has an incentive to deviate. In doing this, you must cover all cases, so you should think carefully about this! )