Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand

Abstract

This article describes numerical methods that exploit fixed-point equations equivalent to the first-order condition for Bertrand-Nash equilibrium prices in a class of differentiated product market models based on the mixed-logit model of demand. One fixed-point equation is already prevalent in the literature, and one is novel. Equilibrium prices are computed for the calendar year 2005 new-vehicle market under two mixed-logit models using (i) a state-of-the-art variant of Newton's method applied to the first-order conditions as well as the two fixed-point equations and (ii) a fixed-point iteration generated by our novel fixed-point equation. A comparison of the performance of these methods for a simple model with multiple equilibria is also provided. The analysis and trials illustrate the importance of using fixed-point forms of the first-order conditions for efficient and reliable computations of equilibrium prices.