Question: Suppose that there are four towns connected by a highway as in the following diagram:
Assume that 15% of the total populations of the four towns are nearest to town 1, 30% nearest to town two, 20% nearest to town three, and 35% nearest to town four. There are two superstores, say, I and II, thinking about building a store to serve these four towns. If both stores are in the same town or in two different towns but with the same distance to a town, then I will get a 65% market share of business. Each store gets 90% of the business of the town in which they put the store if they are in two different towns. If store I is closer to a town than II is, then I gets 90% of the business of that town. If store I is farther than store II from a town, store I still gets 40% of that town's business, except for the town II is in. Find the payoff matrix and solve the game.