Nash Equilibria in a High-Dimensional Duopoly Location Game

We investigate a two-player location-price game in the unit ball [Formula: see text] of Euclidean space [Formula: see text], where firms simultaneously choose locations and prices to attract a continuum of customers uniformly distributed over the ball. Customers minimize a utility function that accounts for both transportation costs and prices. We first analyze the case where both firms have fixed locations and derive the equilibrium prices using first-order conditions. We then examine the location decision of one firm in a competitive setting with the other firm’s location fixed. Symmetric cases are also explored, revealing notable insights into economic behavior.