Computing Nash equilibria for product design based on hierarchical Bayesian mixed logit models

TL;DR

This paper evaluates the effectiveness of hierarchical Bayesian mixed logit models in accurately identifying Nash equilibria in competitive product design scenarios, highlighting the importance of choice behavior type and Bayesian uncertainty.

Contribution

It introduces a comprehensive analysis of Nash equilibria using fully Bayesian choice models, revealing their dependence on choice behavior and uncertainty in competitive markets.

Findings

Deterministic choice models better recover true equilibria.

Bayesian uncertainty improves equilibrium detection.

Choice behavior type significantly affects equilibrium outcomes.

Abstract

Despite a substantial body of theoretical and empirical research in the fields of conjoint and discrete choice analysis as well as product line optimization, relatively few papers focused on the simulation of subsequent competitive dynamics employing non-cooperative game theory. Only a fraction of the existing frameworks explored competition on both product price and design, none of which used fully Bayesian choice models for simulation. Most crucially, no one has yet assessed the choice models' ability to uncover the true equilibria, let alone under different types of choice behavior. Our analysis of thousands of Nash equilibria, derived in full and numerically exact on the basis of real prices and costs, provides evidence that the capability of state-of-the-art mixed logit models to reveal the true Nash equilibria seems to be primarily contingent upon the type of choice behavior (probabilistic versus deterministic), regardless of the number of competing firms, offered products and features in the market, as well as the degree of preference heterogeneity and disturbance. Generally, the highest equilibrium recovery is achieved when applying a deterministic choice rule to estimated preferences given deterministic choice behavior in reality. It is especially in the latter setting that incorporating Bayesian (hyper)parameter uncertainty further enhances the detection rate compared to posterior means. Additionally, we investigate the influence of the above factors on other equilibrium characteristics such as product (line) differentiation.