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

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 l...

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Bibliographic Details
Published in:Operations research 2011-03, Vol.59 (2), p.328-345
Main Authors: Morrow, W. Ross, Skerlos, Steven J.
Format: Article
Language:English
Subjects:
Analysis
Applied sciences
Automobile industry
Bertrand-Nash equilibrium prices
Computer simulation
Computer-generated environments
Cost estimation models
differentiated product markets
Economic competition
Economic models
Equilibrium
Equilibrium (Economics)
Equilibrium prices
Exact sciences and technology
fixed-point iteration
Game theory
GMRES-Newton hookstep
Iterative methods
Markets (Economics)
Mathematical vectors
Methods
mixed logit
Newton's method
Newtons method
Operational research and scientific management
Operational research. Management science
Price efficiency
Product introduction
Standard deviation
Studies
Unit costs
Vehicles
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Summary: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.
ISSN:0030-364X
1526-5463
DOI:10.1287/opre.1100.0894