{"ID":2879750,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.15356","arxiv_id":"2508.15356","title":"ε-Stationary Nash Equilibria in Multi-player Stochastic Graph Games","abstract":"A strategy profile in a multi-player game is a Nash equilibrium if no player can unilaterally deviate to achieve a strictly better payoff. A profile is an $ε$-Nash equilibrium if no player can gain more than $ε$ by unilaterally deviating from their strategy. In this work, we use $ε$-Nash equilibria to approximate the computation of Nash equilibria. Specifically, we focus on turn-based, multiplayer stochastic games played on graphs, where players are restricted to stationary strategies -- strategies that use randomness but not memory. The problem of deciding the constrained existence of stationary Nash equilibria -- where each player's payoff must lie within a given interval -- is known to be $\\exists\\mathbb{R}$-complete in such a setting (Hansen and Sølvsten, 2020). We extend this line of work to stationary $ε$-Nash equilibria and present an algorithm that solves the following promise problem: given a game with a Nash equilibrium satisfying the constraints, compute an $ε$-Nash equilibrium that $ε$-satisfies those same constraints -- satisfies the constraints up to an $ε$ additive error. Our algorithm runs in FNP^NP time. To achieve this, we first show that if a constrained Nash equilibrium exists, then one exists where the non-zero probabilities are at least an inverse of a double-exponential in the input. We further prove that such a strategy can be encoded using floating-point representations, as in the work of Frederiksen and Miltersen (2013), which finally gives us our FNP^NP algorithm. We further show that the decision version of the promise problem is NP-hard. Finally, we show a partial tightness result by proving a lower bound for such techniques: if a constrained Nash equilibrium exists, then there must be one that where the probabilities in the strategies are double-exponentially small.","short_abstract":"A strategy profile in a multi-player game is a Nash equilibrium if no player can unilaterally deviate to achieve a strictly better payoff. A profile is an $ε$-Nash equilibrium if no player can gain more than $ε$ by unilaterally deviating from their strategy. In this work, we use $ε$-Nash equilibria to approximate the c...","url_abs":"https://arxiv.org/abs/2508.15356","url_pdf":"https://arxiv.org/pdf/2508.15356v2","authors":"[\"Ali Asadi\",\"Léonard Brice\",\"Krishnendu Chatterjee\",\"K. S. Thejaswini\"]","published":"2025-08-21T08:34:45Z","proceeding":"cs.GT","tasks":"[\"cs.GT\"]","methods":"[]","has_code":false}