Abstract
This article is devoted to various methods (optimal transport, fixed-point, ordinary differential equations) to obtain existence and/or uniqueness of Cournot–Nash equilibria for games with a continuum of players with both attractive and repulsive effects. We mainly address separable situations but for which the game does not have a potential, contrary to the variational framework of Blanchet and Carlier (Optimal transport and Cournot–Nash equilibria, 2012). We also present several numerical simulations which illustrate the applicability of our approach to compute Cournot–Nash equilibria.
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Notes
This is the case, for instance, if \({\mathcal {V}}[\nu ]\) has the form
$$\begin{aligned} {\mathcal {V}}[\nu ](y):=\int _Y \phi (y,z) \;\mathrm{d}\nu (z) \end{aligned}$$with \(\phi \) smooth and convex with respect to its first argument.
This is the case as soon as \({\mathcal {V}}[\nu ]\) fulfils some coercivity assumption and \(U\) is chosen large enough.
By definition the \(1\)-Wasserstein distance \({\mathcal {W}}_1\) between probability measures \(\nu _1\) and \(\nu _2\) is the least average distance for transporting \(\nu _1\) into \(\nu _2\):
$$\begin{aligned} {\mathcal {W}}_1(\nu _1, \nu _2):=\inf _{\eta \in \Pi (\nu _1, \nu _2)} \int _{Y\times Y} \vert y_1-y_2\vert \;\mathrm{d}\eta (y_1, y_2). \end{aligned}$$
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Acknowledgments
The authors gratefully acknowledge the support of INRIA and the ANR through the Projects ISOTACE (ANR-12-MONU-0013) and OPTIFORM (ANR-12-BS01-0007).
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Blanchet, A., Carlier, G. Remarks on existence and uniqueness of Cournot–Nash equilibria in the non-potential case. Math Finan Econ 8, 417–433 (2014). https://doi.org/10.1007/s11579-014-0127-z
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DOI: https://doi.org/10.1007/s11579-014-0127-z