  Stochastic games with discounted payoff, introduced by Shapley, model
adversarial interactions in stochastic environments where two players try to
optimize a discounted sum of rewards. In this model, long-term weights are
geometrically attenuated based on the delay in their occurrence. We propose a
temporally dual notion -- called past-discounting -- where agents have
geometrically decaying memory of the rewards encountered during a play of the
game. We study objective functions based on past-discounted weight sequences
and examine the corresponding stochastic games with liminf, discounted, and
mean payoffs. For objectives specified as the limit inferior of past-discounted
reward sequences, we show that positional determinacy fails and that optimal
strategies may require unbounded memory. To overcome this obstacle, we study an
approximate windowed objective based on the idea of using sliding windows of
finite length to examine infinite plays. On the other hand, for objectives
specified as the discounted and average limits of past-discounted reward
sequences we establish determinacy in mixed stationary strategies in the
setting of concurrent stochastic games and show how the values of these games
may be computed via reductions to standard discounted and mean-payoff games.
