Two equally competitive pet shops want to locate stores in lake Tahoe, where the

Question

Two equally competitive pet shops want to locate stores in lake Tahoe, where there are currently none. There are three main business centers. Tahoe city serves 25% of the population, incline village 44% and south lake tahoe 31%. If both shops locate in the same center they split all the business equally, if they locate in different centers, they get all the business in the center in which they locate plus half the business in the third center. Where should the pet shops locate? Set up a game matrix and solve. Firstly I need to set up the matrix, that is the first question, and then obviously solve the problem.

Explanation / Answer

Tahoe City Incline Village South Lake Tahoe Tahoe City 50% _____% _____% Incline Village _____% 50% _____% South Lake Tahoe _____% _____% 50% The completed game matrix is: 50 25.5 39.5 74.5 50 64 60.5 36 50 ? ? ? ? ? ? ? ? ? ? The rows are the locations for the first shop (Tahoe City, Incline Village, and South Lake Tahoe, going down vertically). The columns are the locations for the second shop ((Tahoe City, Incline Village, and South Lake Tahoe, going across from left to right. The values come from the following calculations: Second column of first row: Shop 1 sets up in Tahoe City and Shop 2 sets up in Incline Village. Shop 1 gets all Tahoe City business, and half of the South Lake Tahoe business: 10% + (1/2)(31%) = 25.5% Third column of the first row: Shop 1 sets up in Tahoe City and Shop 2 sets up in South Lake Tahoe. Shop 1 gets all Tahoe City business, and half of the Incline Village business: 10% + (1/2)(59%) = 39.5% The remaining values in the game matrix are calculated in similar fashion. To determine the solution, first determine if there is a saddle point in the matrix. If there is, then this is the solution for the game. To determine the saddle point, identify the smallest value in each row. Then identify the largest value in each column. A value which is in both sets represents a saddle point. The smallest values in each row are identified below: 50 25.5 39.5 74.5 50 64 60.5 36 50 ? ? ? ? ? ? ? ? ? ? The largest values in each column are identified below: 50 25.5 39.5 74.5 50 64 60.5 36 50 ? ? ? ? ? ? ? ? ? ? Note that the middle value of 50 is marked in both cases. This is the saddle point. The solution then is for both shops to set up in Incline Village

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