  In a zero-sum stochastic game, at each stage, two adversary players take
decisions and receive a stage payoff determined by them and by a controlled
random variable representing the state of nature. The total payoff is the
normalized discounted sum of the stage payoffs. In this paper we solve the
"constant payoff" conjecture formulated by Sorin, Vigeral and Venel (2010): if
both players use optimal strategies, then for any alpha>0, the expected
discounted payoff between stage 1 and stage alpha/lambda tends to the limit
discounted value of the game, as the discount rate lambda goes to 0.
