{"ID":2831260,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.08745","arxiv_id":"2512.08745","title":"Variance strikes back: sub-game--perfect Nash equilibria in time-inconsistent $N$-player games, and their mean-field sequel","abstract":"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.","short_abstract":"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-theoret...","url_abs":"https://arxiv.org/abs/2512.08745","url_pdf":"https://arxiv.org/pdf/2512.08745v1","authors":"[\"Dylan Possamaï\",\"Chiara Rossato\"]","published":"2025-12-09T15:53:41Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"econ.TH\",\"math.OC\"]","methods":"[\"Large Language Model\"]","has_code":false}