[Paper Review] Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed Logit Demand: A Technical Framework for Analysis and Efficient Computational Methods

William R. Morrow, Steven J. Skerlos|arXiv (Cornell University)|Dec 28, 2010

Economic theories and models41 references3 citations

TL;DR

This paper proposes a fixed-point iteration method based on a novel $ζ$-markup equation to compute Bertrand-Nash equilibrium prices under mixed logit demand, offering greater robustness and computational efficiency than Newton’s method or other standard approaches. The method ensures superlinear local convergence and avoids solving linear systems, significantly reducing computational burden while maintaining theoretical convergence guarantees.

ABSTRACT

This article presents a detailed technical framework for modeling with Bertrand-Nash equilibrium prices under Mixed Logit demand. Two coercive fixed-point equations provide more stable computational methods than those obtained from the literal first-order conditions. Assumptions to justify derivation and use of these equations are provided. A brief discussion of a GMRES-Newton method with hookstep globalization strategy originally due to Viswanath is also given. This article can be considered a supplement to an article by the authors forthcoming in the journal {\em Operations Research}.

Motivation & Objective

• To address the lack of systematic treatment of numerical computation methods for Bertrand-Nash equilibrium prices in differentiated product markets.
• To improve the reliability and efficiency of computing equilibrium prices when exogenous variables change significantly, especially in the absence of observed price initialization.
• To develop a computationally efficient alternative to Newton’s method that avoids solving linear systems and ensures better convergence behavior.
• To establish theoretical and computational advantages of fixed-point iteration on the $ζ$-markup equation over existing methods like Newton’s method, tatonnement, and Gauss-Newton.
• To provide a technical framework for analyzing and implementing efficient computational methods in mixed logit demand models with Bertrand competition.

Proposed method

• Introduces the $ζ$-markup equation as a novel fixed-point reformulation of the first-order conditions for Bertrand-Nash equilibrium, equivalent to the BLP-markup equation but with superior convergence properties.
• Employs fixed-point iteration on the $ζ$-markup equation, which is superlinearly locally convergent and does not require solving linear systems, reducing computational cost per iteration.
• Derives theoretical conditions under which the $ζ$-markup fixed-point iteration converges, showing that the limit of the derivative ratio of the profit function ensures positive marginal adjustment.
• Proposes the GMRES-Newton hookstep method as a robust inexact Newton method for cases requiring higher-order convergence, using Krylov subspace methods and preconditioning to improve numerical stability.
• Compares the $ζ$-markup approach to alternative methods including Newton’s method, tatonnement, variational formulations, and Gauss-Newton minimization, highlighting computational and convergence trade-offs.
• Uses directional finite differences for Jacobian approximation in Newton-type methods and provides practical implementation guidelines, including truncation of low-probability products and termination conditions.

Experimental results

Research questions

• RQ1 Can fixed-point iteration on the $ζ$-markup equation provide more reliable and efficient convergence than Newton’s method when computing Bertrand-Nash equilibrium prices under mixed logit demand?
• RQ2 Why does the BLP-markup equation fail to ensure local convergence in some cases, while the $ζ$-markup equation does?
• RQ3 What theoretical conditions guarantee the superlinear local convergence of the $ζ$-markup fixed-point iteration?
• RQ4 How does the computational burden of fixed-point iteration on the $ζ$-markup equation compare to Newton’s method and Gauss-Newton methods in terms of Jacobian and linear system requirements?
• RQ5 In what scenarios does fixed-point iteration on the $ζ$-markup equation outperform tatonnement or variational formulations for equilibrium computation?

Key findings

• The $ζ$-markup fixed-point iteration is superlinearly locally convergent, while the BLP-markup equation may fail to converge locally, as demonstrated in Example 7 of the paper.
• The $ζ$-markup fixed-point method avoids solving linear systems, making each iteration computationally cheaper than Newton’s method, which requires solving a system at each step.
• Theoretical analysis proves that under standard assumptions on the profit function, the limit of the ratio of second to first derivatives of the profit function is less than one, ensuring positive marginal adjustment and convergence of the fixed-point iteration.
• The GMRES-Newton hookstep method provides a robust inexact Newton approach with improved conditioning and reduced error accumulation compared to standard Newton or Gauss-Newton methods.
• Variational formulations and Gauss-Newton methods are less favorable due to poor conditioning, higher computational burden, and lack of convergence guarantees, especially when third-order derivatives are required.
• Tatonnement lacks convergence guarantees and is inefficient when firms’ optimal prices are highly interdependent, making it less suitable than fixed-point or Newton-based methods.