Constant Payoff Property in Zero-Sum Stochastic Games with a Finite Horizon

This paper examines finite zero-sum stochastic games and demonstrates that when the game's duration is sufficiently long, there exists a pair of approximately optimal strategies such that the expected average payoff at any point in the game remains close to the value. This property, known as the \textit{constant payoff property}, was previously established only for absorbing games and discounted stochastic games.