We provide a new characterisation of the set of stationary equilibriafor finite discounted N-player stochastic games, based on the defini-tion of an auxiliary one-shot game with the same set of equilibria. Thisresult is the extension, to the N-player case, of a similar characterisa-tion for two-player zero-sum stochastic games (Attia and Oliu-Barton,Proceedings of the National Academy of Sciences of the USA, 2019)which led to a tractable formula for the limit value. Though the generalcase presents additional challenges, our characterisation may have fur-ther applications, notably in terms of the description and computationof stationary equilibria, and of their limit as the discount rates vanish.