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Dynamic Games and Applications
Published in: Dynamic Games and Applications 2/2020

24-07-2019

Evolution of a Collusive Price in a Networked Market

Author: Yasuhiro Shirata

Published in: Dynamic Games and Applications | Issue 2/2020

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Abstract

This paper studies evolution of firms’ behavior in a networked Bertrand oligopoly market, in which firms who are located on vertices of a network compete in price with their neighbors. This network model is also applied to a market with multi-dimensionally differentiated products. In a non-networked market, it is known that the Bertrand–Nash equilibrium pricing is evolutionarily stable. We show, however, that in our large networked market, the Bertrand–Nash equilibrium price is not stable but a collusive price is evolutionarily stable under weak selection. As the magnitude of transportation cost increases, firms charge a more collusive price in the long run. The results suggest that collusive pricing prevails in a large market if and only if it is networked.
previous article Nonzero-Sum Stochastic Games with Probability Criteria
next article Limit Optimal Trajectories in Zero-Sum Stochastic Games

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Appendix
Available only for authorised users
Footnotes
1
Those papers consider the sequential location-price game. In this paper, however, we only investigate pricing game given the locations of firms, and consider no location choice game.
 
2
Remark that since there are many regular networks for \(k\ge 3\), the interpretation is applied to not all regular networks.
 
3
Salop [42, p.142] states that “By eliminating technical problems, this model allows a focus on the essential interactions of firms in an industry.”
 
4
This paper and those papers assume that consumers are perfectly informed about the distribution of prices, so that they always choose the lowest price. By contrast, Hehenkamp [20] find that firms charge the same identical price above the marginal cost if consumers are imperfectly informed or sluggish.
 
5
The pair approximation is shown to be mathematically correct if a network is a Bethe lattice, which has no loops [39]. The approximation works well for a large random regular network because the probability of short loops becomes negligible, as the number of vertices increases. Ohtsuki and Nowak [38] further examine the performance by running computer simulation.
 
6
The weak selection matters only on a large network. In the Hotelling model, in which two firms located on a line, the evolutionary analysis is irrelevant to intensity w in quality because all firms link with all firms.
 
7
See Mailath and Samuelson [33, Section 2.7] for the detailed discussion.
 
8
Eshel et al. [15] show that the efficient dominated strategy can survive on a circular network, on which players play a prisoners’ dilemma game without transportation cost, using the deterministic imitation. Since we assume imitation is probabilistic, the details of imitation would also matter.
 
9
The circular network is the unique regular network of \(k=2\). In general, however, there are multiple regular networks of k when \(k\ge 3\).
 
10
We further discuss the interpretation and implication of degree k in Sect. 4.2.
 
11
Without the assumption, a monopoly interval can arise for some firm i, in which consumers never choose j for some \(p_j<{\bar{p}}\). Since firm i under-supplies products by monopoly pricing if i’s monopoly interval exits, each firm’s demand is more complicated, which causes the analysis of the dynamics to be difficult. We avoid such a difficulty by the assumption.
 
12
Here, each firm i observes the total profits of adjacent firms. However, it is indifferent to consider the average profit and the total profit in symmetric equilibria.
 
13
By the self-matching, we can ignore finite-population effects.
 
14
The pair approximation is the most common approach to analyze a large complex network. It is well-known that the pair approximation gives good results for large random regular graphs [39].
 
15
For further detailed argument, see Ohtsuki and Nowak [40] and Ohtsuki et al. [38].
 
16
An intuitive interpretation of the form of b is provided by Ohtsuki and Nowak [38].
 
17
To show the theorems below, it suffices to assume that \(N'\ge 1\) and that N is so large that \(\frac{e}{N-1} < \frac{d}{k+1}\).
 
18
Here, we take the large population double limit \(\lim _{u\rightarrow 0} \lim _{|I|\rightarrow \infty } \mu _{u,I}\). Sandholm [43] argues that the large population double limit is favored in economic applications. He further shows that when there is a committed agent who stick to a predetermined strategy for each strategy, the results agree with that of the small noise double limit \( \lim _{|I|\rightarrow \infty } \lim _{u\rightarrow 0} \mu _{u,I}\).
 
19
A strategy \(p_n\) is called risk-dominant if \(U(p_n,p_n)+U(p_n,p_m)-U(p_m,p_n)-U(p_m,p_m)>0\) (\(m\ne n\)).
 
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Metadata
Title
Evolution of a Collusive Price in a Networked Market
Author
Yasuhiro Shirata
Publication date
24-07-2019
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 2/2020
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-019-00322-2

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