Bayesian Nash Equilibrium in price competition under multinomial logit demand

Abstract

In this paper, we propose a Bayesian Nash equilibrium (BNE) model for analyzing price competition under multinomial logit demand where firm’s marginal cost is private information: each firm may predict the range of the marginal cost of its rival but does not know the true marginal cost. Differing from the existing Nash equilibrium models (Aksoy-Pierson et al., 2013; Pang et al., 2015) where the market equilibrium is described as a tuple of prices at which no firm can be better off by unilaterally changing its position, the BNE is a tuple of firm’s optimal price functions each of which depends on their respective marginal costs and the equilibrium is a situation at which no firm can be better off by unilaterally changing its own optimal price function. This kind of equilibrium model may help individual firms set optimal prices strategically for their future products. We derive sufficient conditions for existence and uniqueness of a continuous BNE. Moreover, we propose a computational scheme to calculate an approximate BNE. Specifically, we develop step-like approximation of firm’s optimal price function and then convert the BNE problem into a finite-dimensional stochastic variational inequality problem (SVIP). We demonstrate how the specific structure of the SVIP may be decomposed into scenario-based VIP and solve the latter by the well-known progressive hedging method. Preliminary numerical tests show that the computational scheme works well.