Question

TWO STORE CHAIN A and B plan to open stores in some of the following three town X, Y Z. In total chain A plan to open one store while chain B plans to open two. The market in each of the town is worth million dollars and it will be split between the chains according to the following rules: If in a town there are store of both chains present (it does not matter how many) then the market in this town will be split exactly in half between the chain If in a town there are stores of just one chain present ( it does not matter how many ) then the market in this town will go to entirely to the chain present in the town If in a town there are no stores of any of the chains presents ,, then the entire market of the town will go to the chain whose store is located in a town that is closer to the town considered. If in that closer town there are stores of both chain present (it does not matter how many) then the market of the town considered will be split equally between the two chain The distance between towns are following X Y 5 MILES X Z 10 MILES Y Z 13 MILES List the strategies for both chains; Write down a pay-off matrix for chain A; Find all saddle points. No explanation necessary for this question. (15 points)

Answer 1

Part 1 and 2

The best strategy for both chains would be to keep symmetry between the towns in positioning the stores.

Since A is going to open only one store it makes sense that A will open the store in Town X

Pay off Matrix for Store Chain A ( Game theory)

Town X Town Y Town Z Profit for A

A opens B opens B opens 1 million

A opens B opens    B opens   1 million

A opens B opens B opens 1 million

A opens B opens both 2 million ( Saddle point )

A opens    B opens both     2 million ( Saddle Point)

A opens

B opens both 1.5 million

B opens both A opens 1 million

B opens both A opens 1 million

Bopens A opens Bopens 1 million

A opens

B opens both 1.5 million

B opens both A opens 2 million ( Saddle Point)

Few more repetive cases

Case 1 A opens store in X.

He will land up in one of scenarios depending on placement of B

1) He will 50% market from town X and again 50% from either Y or Z (1 million total)

2) Hew will get 100% from town X and remote possibility of 100% from from Y or Z ( 2 million total)

3) He will get 50% from all three towns (1.5 million total)

So the Best Option for A is to open in City X

Now strategy for B starts with realization that Knowing the best strategy for A is to open The shop in Town X

By that Logic the B would be the most profitable if it tries to fit in the Case 1 of A

i,e B opens one Shop in X and other Shop either in Y or Z. By doing this B will ensure that it will recieve 2 million

So the best strategy for Both the players is A will open its shop in X and B will open its shop in X and either Y or Z

Part 3 : There are total 3 Saddle points. Please refer to the pay off matrix for the same

Town X Town Y Town Z Profit for A

A opens B opens both 2 million ( Saddle point )

A opens    B opens both     2 million ( Saddle Point)

B opens both A opens 2 million ( Saddle Point)