Evaluating all Bertrand-Nash equilibria in a discrete spatial duopoly model

This paper studies a spatial duopoly model where customers are located at nodes and the demand functions are given for each node. For any fixed location of two firms, we analyze Bertrand–Nash equilibrium and derive a necessary and sufficient condition for the existence of equilibrium. We present an algorithm to compute all equilibria, provided profit functions have a finite number of peaks. The algorithm terminates with polynomial time if the number of peaks is polynomial in the number of nodes.