Variance strikes back: sub-game--perfect Nash equilibria in time-inconsistent $N$-player games, and their mean-field sequel
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914190156890112 |
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| author | Possamaï, Dylan Rossato, Chiara |
| author_facet | Possamaï, Dylan Rossato, Chiara |
| contents | We investigate a time-inconsistent, non-Markovian finite-player game in continuous time, where each player's objective functional depends non-linearly on the expected value of the state process. As a result, the classical Bellman optimality principle no longer applies. To address this, we adopt a two-layer game-theoretic framework and seek sub-game--perfect Nash equilibria both at the intra-personal level, which accounts for time inconsistency, and at the inter-personal level, which captures strategic interactions among players. We first characterise sub-game--perfect Nash equilibria and the corresponding value processes of all players through a system of coupled backward stochastic differential equations. We then analyse the mean-field counterpart and its sub-game--perfect mean-field equilibria, described by a system of McKean-Vlasov backward stochastic differential equations. Building on this representation, we finally prove the convergence of sub-game--perfect Nash equilibria and their corresponding value processes in the $N$-player game to their mean-field counterparts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08745 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variance strikes back: sub-game--perfect Nash equilibria in time-inconsistent $N$-player games, and their mean-field sequel Possamaï, Dylan Rossato, Chiara Probability Theoretical Economics Optimization and Control We investigate a time-inconsistent, non-Markovian finite-player game in continuous time, where each player's objective functional depends non-linearly on the expected value of the state process. As a result, the classical Bellman optimality principle no longer applies. To address this, we adopt a two-layer game-theoretic framework and seek sub-game--perfect Nash equilibria both at the intra-personal level, which accounts for time inconsistency, and at the inter-personal level, which captures strategic interactions among players. We first characterise sub-game--perfect Nash equilibria and the corresponding value processes of all players through a system of coupled backward stochastic differential equations. We then analyse the mean-field counterpart and its sub-game--perfect mean-field equilibria, described by a system of McKean-Vlasov backward stochastic differential equations. Building on this representation, we finally prove the convergence of sub-game--perfect Nash equilibria and their corresponding value processes in the $N$-player game to their mean-field counterparts. |
| title | Variance strikes back: sub-game--perfect Nash equilibria in time-inconsistent $N$-player games, and their mean-field sequel |
| topic | Probability Theoretical Economics Optimization and Control |
| url | https://arxiv.org/abs/2512.08745 |