Location games Two competing coffee house chains, Pete's Coffee and Caribou Coffee, are seeking locations for new branch stores in Cambridge. The town is comprised of only one street, along which all the residents live.

Each of the two chains therefore needs to choose a single point within the interval [0, 1], which represents the exact location of the branch store along the road. It is assumed that each resident will go to the coffee house that is nearest to his place of residence.

If the two chains choose the exact same location, they will each attract an equal number of customers. Each chain, of course, seeks to maximize its number of customers. To simplify the analysis required here, suppose that each point along the interval [0, 1] represents a town resident, and that the fraction of residents who frequent each coffee house is the fraction of points closer to one store than to the other.

(a) Describe this situation as a two-player strategic-form game.

(b) Prove that the only equilibrium in this game is that given by both chains selecting the location x =1/2 .

(c) Prove that if three chains were to compete for a location in Cambridge, the resulting game would have no equilibrium. (Under this scenario, if two or three of the chains choose the same location, they will split the points closest to them equally between them.)