A town consists of only one street in the form of a circle. The town authorities
ID: 2986145 • Letter: A
Question
A town consists of only one street in the form of a circle. The town authorities give out four licenses for a particular kind of business. The inhabitants of the town live in equal density along the circle and will always go to the closest business for what they need. Business A gets to choose a location first, then business B, then C, and finally D. Each business desires to carve out as much business for themselves as possible but each knows the others all have the same motive. Assume that if a business is indifferent between locating in two different sections of the circle it will choose a section at random. Also assume that the business that goes last will choose a location in the middle of the largest (or one of the largest) sections. Where should business B choose relative to the location of A?Explanation / Answer
This is only a quick overview of the solution, the details are left up to you. Let the circumference of the circle be 1 and that A chooses a location at point 0. Business D will choose a location in the middle of the largest section.
Business C will also choose the midpoint of the larger of the two gaps between the first two businesses.
If business B chooses a point before 1/3 then C will choose a point halfway between B and 1. Business D will choose randomly between the halfway point between A and C or B and C. If x is the location of business B then the area which B will carve out of the circle will be either (1+3x)/8 if D goes between B and C or (1+x)/4 if D goes between A and C. The average of these is (3+5x)/16. The same logic applies if B chooses a point after 2/3.
If business B chooses a point after 1/3 (but before 1/2) then C will choose a point halfway between A and B going the long way and D will choose the halfway point between A and B the short way. This will leave B exactly a 1/4 share of the business. The same logic applies if A chooses a point between 1/2 and just before 2/3.
If B should choose a location at exactly 1/3 then C would choose at 2/3 and D would be indifferent between 1/6, 1/2, and 5/6. B would have a 2/3 chance of having 1/4 of the business share and 1/3 chance of having 1/3, the average being 5/18 =~ 0.27778 .
Thus B should try to maximize (3+5x)/16 without choosing x equal or greater to 1/3. The optimal choice of location would be just a hair short of 1/3 (or just a hair after 2/3). At this point B will have a 50/50 chance at having either 1/4 or 1/3 of the market share for an average of 7/24=~ 0.29166667 of the market share.
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