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## Location Location Location!...Makes Cents

Four fast food stands are to be opened on a circular highway. These are the rules or conditions known to all four buisnesses:

• The four businesses are "undifferentiated," that is they will serve exactly the same "cuisine" and have similar decor.
• They will be open the same hours and have the same prices.
• Every owner of a stand wants to maximize their income via placement relative to the other stands.
• Because all features of the fast food stands will be alike, all shoppers will decide which business to patronize only on the basis of how close it is to where they live.
• As they are built, the businesses will open in this order: Stand A, Stand B, Stand C, and Stand D.
• Relocations are not possible, that is once a stand is opened, it must remain at that location
Where on the circular highway should Stand B be located in order to get the most business and income?

Source: B.B. (former student)

Hint: Draw a circle and place Stand A anywhere (WLOG assume it is at 12 o'clock position). Ask: what happens if Stand B is placed here...e.g. at the 6 o'clock position or the 3 c'clock position. That is, what will owners of Stands C and D probably do given possible placements of Stand A and B?

Solution Commentary: Draw pictures as you follow this solution commentary from person B.B. who submitted this problem: "The placement of Stand B is the one that really controls the arrangement of the stands. It wants as much of the business possible with the least risk; that is, when Stand C and Stand D open up, Stand B wants to control more than a quarter of the circle's business.

If after A opens any place on the circle and B opens at the 1/2 mark directly opposite A, then C and D would probably place themselves at the respective midpoints of the half-arcs, and each of the four stands would control 1/4 of the total distance. [Note: If C and D wanted to hurt B, they could place themselves very close to B and still ensure themselves of 1/4 the business but definitely hurt B.]

If B places itself at the 1/3 position from A, stand C would pick the remaining 2/3 position, hoping to get one-half of the business between Stand's A and B (i.e. equal to 1/3). Stand D would then have to locate at the 5/6, 1/2, or 1/6 mark (viewed clockwise).

If D located at the 1/6 or 1/2 mark, B would gain 1/4 of the business (i.e. 1/12 + 1/6). But, if D located at the 5/6 mark, B would get 1/3 of the business. Note: In all cases, D gets the same business. Thus, as D's three possible positions are random, the average business expected for B is (1/4+1/4+1/3)/3 = 5/18, which is greater than the eralier case of 1/4.

A more creative answer would be to place B slightly closer to A than the 1/3 mark, making arc AB slightly smaller than arcs AC or CB. Then, to maximize its business, D would avoid arc AB and locate at the midpoint of arc AC or CB. If D locates at the midpoint of AC then B gets 1/3 of the business, while if D locates at the midpoint of BC, then B gets 1/4 of the business. Thus, again the average is (1/4+1/3)/2 = 7/24 of the business, which is greater than the previous values of 5/18 or 1/4.

Do you agree with this reasoning?