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Bertrand & Cournot Mean Field Games (2014)
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BibTeX
@MISC{Chan14bertrand&,
author = {Patrick Chan and Ronnie Sircar},
title = {Bertrand & Cournot Mean Field Games},
year = {2014}
}
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Abstract
We study how continuous time Bertrand and Cournot competitions, in which firms producing similar goods compete with one another by setting prices or quantities respectively, can be analyzed as continuum dynamic mean field games. Interactions are of mean field type in the sense that the demand faced by a producer is affected by the others through their average price or quantity. Motivated by energy or consumer goods markets, we consider the setting of a dynamic game with uncertain market demand, and under the constraint of finite supplies (or exhaustible resources). The continuum game is characterized by a coupled system of partial differential equations: a backward HJB PDE for the value function, and a forward Kolmogorov PDE for the density of players. Asymptotic approximation enables us to deduce certain qualitative features of the game in the limit of small competition. The equilibrium of the game is further studied using numerical solutions, which become very tractable by considering the tail distribution function instead of the density itself. This also allows us to consider Dirac delta distributions to use the continuum game to mimic finite N-player nonzero-sum differential games, the advantage being having to deal with two coupled PDEs instead of N. We find that, in accordance with the two-player game, a large degree of competitive interaction causes firms to slow down production. Consumers benefit from market competition in the sense that high product substitutability drives average price down. The continuum system can therefore be used as an effective approximation to even small player dynamic games. 1
Keyphrases
bertrand cournot mean field game average price continuum game backward hjb pde partial differential equation market competition exhaustible resource mean field type high product substitutability value function numerical solution continuum system asymptotic approximation continuum dynamic mean field game cournot competition dynamic game similar good compete dirac delta distribution coupled system large degree finite supply two-player game consumer good market effective approximation competitive interaction small player dynamic game tail distribution function finite n-player nonzero-sum differential game certain qualitative feature uncertain market demand continuous time bertrand small competition kolmogorov pde
