Quantitative convergence for displacement monotone Mean Field Games of control
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918102562766848 |
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| author | Jackson, Joe Mészáros, Alpár R. |
| author_facet | Jackson, Joe Mészáros, Alpár R. |
| contents | In this paper we establish quantitative convergence results for both open and closed-loop Nash equilibria of N-player stochastic differential games in the setting of Mean Field Games of Controls (MFGC), a class of models where interactions among agents occur through both states and controls. Our analysis covers a general class of non-separable Hamiltonians satisfying a displacement monotonicity condition, along with mild regularity and growth conditions at infinity. A major novelty of our work is the rigorous treatment of a nontrivial fixed-point problem on a space of measures, which arises naturally in the MFGC formulation. Unlike prior works that either restrict to separable Hamiltonians - rendering the fixed-point map trivial - or assume convergence or regularity properties of the fixed point map, we develop a detailed structural analysis of this equation and its N-player analogue. This leads to new regularity results for the fixed-point maps and, in turn, to quantitative convergence of open-loop equilibria. We further derive sharp a priori estimates for the N-player Nash system, enabling us to control the discrepancy between open and closed-loop strategies, and thus to conclude the convergence of closed-loop equilibria. Our framework also accommodates common noise in a natural way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17014 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative convergence for displacement monotone Mean Field Games of control Jackson, Joe Mészáros, Alpár R. Probability Analysis of PDEs Optimization and Control 91A16, 49N80, 60H30 In this paper we establish quantitative convergence results for both open and closed-loop Nash equilibria of N-player stochastic differential games in the setting of Mean Field Games of Controls (MFGC), a class of models where interactions among agents occur through both states and controls. Our analysis covers a general class of non-separable Hamiltonians satisfying a displacement monotonicity condition, along with mild regularity and growth conditions at infinity. A major novelty of our work is the rigorous treatment of a nontrivial fixed-point problem on a space of measures, which arises naturally in the MFGC formulation. Unlike prior works that either restrict to separable Hamiltonians - rendering the fixed-point map trivial - or assume convergence or regularity properties of the fixed point map, we develop a detailed structural analysis of this equation and its N-player analogue. This leads to new regularity results for the fixed-point maps and, in turn, to quantitative convergence of open-loop equilibria. We further derive sharp a priori estimates for the N-player Nash system, enabling us to control the discrepancy between open and closed-loop strategies, and thus to conclude the convergence of closed-loop equilibria. Our framework also accommodates common noise in a natural way. |
| title | Quantitative convergence for displacement monotone Mean Field Games of control |
| topic | Probability Analysis of PDEs Optimization and Control 91A16, 49N80, 60H30 |
| url | https://arxiv.org/abs/2507.17014 |