The only Dance in Town: Unique Equilibrium in a Generalized Model of Price Competition*

Abstract

We study a model of simultaneous price competition that subsumes many employed in the literature over the last several decades. Firms sell a homogeneous good to consumers characterized by the number of prices they (exogenously) consider. We show there is a unique equilibrium if and only if some consumers consider exactly two prices. The equilibrium is in symmetric mixed strategies. When no consumer considers exactly two prices, there is, in addition, an uncountable infinity of asymmetric equilibria. Our results show that the paradigm generically produces a unique equilibrium, and only the commonly sought symmetric equilibrium is robust to perturbations in consumer behavior.