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Published in: Dynamic Games and Applications 3/2021

01-11-2020

Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach

Authors: Régis Chenavaz, Corina Paraschiv, Gabriel Turinici

Published in: Dynamic Games and Applications | Issue 3/2021

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Abstract

Dynamic pricing of new products has been extensively studied in monopolistic and oligopolistic markets. But, the optimal control and differential game tools used to investigate pricing behavior on markets with a number of firms are not well-suited to model competitive markets with a large number of firms. Using a mean-field game approach, this article develops a setting where numerous firms optimize prices for a new product. We analyze a framework à la Bass with product diffusion and experience effects. The analytical contribution of the paper is to prove the existence and uniqueness of a mean-field game equilibrium, further characterized in terms of mean tendencies and market heterogeneity. We also demonstrate the possible emergence of one or more groups of firms with regards to their pricing strategy. Numerical simulations illustrate how differences in firm experience translate into market heterogeneity in sales and profits. We show that, on a market where the absolute price effect is stronger than the relative price effect, we observe the emergence of two groups of firms, characterized by different prices, sales, and profits. Heterogeneity in firms’ prices and profits is thus compatible with competitive markets.

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Appendix
Available only for authorised users
Footnotes
1
Weintraub et al. [82, p. 1377] affirm that “most industries contain more than 20 firms, but it would require more than 20 million gigabytes of computer memory to store the policy function for an industry with just 20 firms and 40 states.”
 
2
Technically, an MFG equilibrium approximates a Nash equilibrium with a finite but large number of players. The approximation process is the following. First is the optimization of the game with a finite number of players; second is the passage to the limit. Note that the steps are not commutative.
 
3
Technically, modeling rational anticipation is achieved with the forward/backward structure of the mean-field game.
 
4
Stochastic analyses are also frequent in the dynamic pricing literature [15, 16, 36, 71, 72]. A difference between the deterministic and stochastic analyses is that the deterministic setting gives information about the mean tendencies, while the stochastic setting also provides information about the deviations from the mean. While the main results are expected to be similar in the two settings, the stochastic approach complexifies the analysis which is already computationally complex with MFG.
 
5
Note that inequality \({\overline{X}}_t \le N_0\) is only true in average and does not imply \(x_t \le N_0\). This means that a particular firm may sell more than \(N_0\) units. However, if some firms sell more than \(N_0\), some other firms sell necessarily less to compensate at market level.
 
6
The reciprocal is also true: given the price distribution \(P_t\), if the price distribution \(p_t\) is optimal and any moment of \(p_t\) and \(P_t\) coincide, then we have an equilibrium.
 
7
Note that the condition of mean equality (i.e., \({\mathbb E}[p_t]={\mathbb E}[P_t]\)) is necessary for the equilibrium, but not sufficient. If only the means coincide one cannot conclude to a MFG equilibrium because this condition alone does not ensure that the distribution \(P_t\) of prices observed on the market coincides with the distribution \(p_t\) of prices announced by the firms.
 
8
A more general formulation of the type \(\mathrm{d}x_t/\mathrm{d}t = h(x_t,p_t;\mu _t ,P_t )\) is also possible. It corresponds to the case where the firms have more information about the mean-field distributions of cumulative demands and prices.
 
10
Indeed, when the price of each firm is a time-dependent curve, the difficulty comes from the fact that the MFG will be a Nash equilibrium between an infinity of curves (whose mean values \({\overline{X}}_t\) and \({\overline{P}}_t\) represent the mean-fields). The following fixed point procedure is used: Starting with the mean-fields \({\overline{X}}_t\) and \({\overline{P}}_t\), we compute the (optimal) price of any firm through the critical point equations. By averaging, we obtain the novel mean-fields which should coincide with the initial datum \({\overline{X}}_t\) and \({\overline{P}}_t\). The result by Kakutani–Glicksberg–Fan guarantees that a fixed point exists if the space of curves is regular enough (the required mathematical concept is the compactness).
 
11
A log-normal (or Gibrat) distribution accounts for the multiplicative product of numerous independent and identically distributed variables, which are additive on a log scale.
 
12
The definition of the mapping \({\mathscr {E}}\) appears in the proof of point 1 from Theorem 5.6 in “Appendix B”.
 
13
Recall that \(\Pi ^\dagger (x_0):=J^*(x_0,Z^\dagger )\), with \(J^*\) defined in Eq. (B.6) in “Appendix B.”
 
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Metadata
Title
Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach
Authors
Régis Chenavaz
Corina Paraschiv
Gabriel Turinici
Publication date
01-11-2020
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 3/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00369-6

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