Pasting of Equilibria and Donsker-type Results for Mean Field Games
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912100267327488 |
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| author | Dianetti, Jodi Nendel, Max Tangpi, Ludovic Wang, Shichun |
| author_facet | Dianetti, Jodi Nendel, Max Tangpi, Ludovic Wang, Shichun |
| contents | This paper studies the relation between equilibria in single-period, discrete-time and continuous-time mean field game models. First, for single-period mean field games, we establish the existence of equilibria and then prove the propagation of the Lasry-Lions monotonicity to the optimal equilibrium value, as a function of the realization of the initial condition and its distribution. Secondly, we prove a pasting property for equilibria; that is, we construct equilibria to multi-period discrete-time mean field games by recursively pasting the equilibria of suitably initialized single-period games. Then, we show that any sequence of equilibria of discrete-time mean field games with discretized noise converges (up to a subsequence) to some equilibrium of the continuous-time mean field game as the mesh size of the discretization tends to zero. When the cost functions of the game satisfy the Lasry-Lions monotonicity property, we strengthen this convergence result by providing a sharp convergence rate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pasting of Equilibria and Donsker-type Results for Mean Field Games Dianetti, Jodi Nendel, Max Tangpi, Ludovic Wang, Shichun Optimization and Control Probability This paper studies the relation between equilibria in single-period, discrete-time and continuous-time mean field game models. First, for single-period mean field games, we establish the existence of equilibria and then prove the propagation of the Lasry-Lions monotonicity to the optimal equilibrium value, as a function of the realization of the initial condition and its distribution. Secondly, we prove a pasting property for equilibria; that is, we construct equilibria to multi-period discrete-time mean field games by recursively pasting the equilibria of suitably initialized single-period games. Then, we show that any sequence of equilibria of discrete-time mean field games with discretized noise converges (up to a subsequence) to some equilibrium of the continuous-time mean field game as the mesh size of the discretization tends to zero. When the cost functions of the game satisfy the Lasry-Lions monotonicity property, we strengthen this convergence result by providing a sharp convergence rate. |
| title | Pasting of Equilibria and Donsker-type Results for Mean Field Games |
| topic | Optimization and Control Probability |
| url | https://arxiv.org/abs/2411.00633 |