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 within polynomial time if the number of peaks is polynomial in the number of nodes.