From Nash to Cournot--Nash equilibria via $Γ$-convergence
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866911561387343872 |
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| author | Machado, João Miguel Mazanti, Guilherme Pfeiffer, Laurent |
| author_facet | Machado, João Miguel Mazanti, Guilherme Pfeiffer, Laurent |
| contents | This work addresses the issue of the convergence of an $N$-player game towards a limit model involving a continuum of players, as the number of agents $N$ goes to infinity. More precisely, we investigate the convergence of Nash equilibria to a Cournot--Nash equilibrium of the limit model. When the cost function of the players is the first variation of some potential function, equilibria can be characterized by a stationarity condition, satisfied in particular by the minimizers of the potential. We demonstrate such a characterization under low regularity assumptions. Then we focus on the case where the players interact in a pairwise fashion; in this case we show that the original sequence of $N$-player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of $Γ$-convergence to the potential function of the limit game. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Nash to Cournot--Nash equilibria via $Γ$-convergence Machado, João Miguel Mazanti, Guilherme Pfeiffer, Laurent Analysis of PDEs This work addresses the issue of the convergence of an $N$-player game towards a limit model involving a continuum of players, as the number of agents $N$ goes to infinity. More precisely, we investigate the convergence of Nash equilibria to a Cournot--Nash equilibrium of the limit model. When the cost function of the players is the first variation of some potential function, equilibria can be characterized by a stationarity condition, satisfied in particular by the minimizers of the potential. We demonstrate such a characterization under low regularity assumptions. Then we focus on the case where the players interact in a pairwise fashion; in this case we show that the original sequence of $N$-player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of $Γ$-convergence to the potential function of the limit game. |
| title | From Nash to Cournot--Nash equilibria via $Γ$-convergence |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.02205 |