{\epsilon}-Stationary Nash Equilibria in Multi-player Stochastic Graph Games

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 $\epsilon$-Nash equilibrium if no player can gain more than $\epsilon$ by unilaterally deviating from their strategy. In this work, we use $\epsilon$-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 S{\o}lvsten, 2020). We extend this line of work to stationary $\epsilon$-Nash equilibria and present an algorithm that solves the following promise problem: given a game with a Nash equilibrium satisfying the constraints, compute an $\epsilon$-Nash equilibrium that $\epsilon$-satisfies those same constraints -- satisfies the constraints up to an $\epsilon$ 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.