  Stochastic games, introduced by Shapley, model adversarial interactions in
stochastic environments where two players choose their moves to optimize a
discounted-sum of rewards. In the traditional discounted reward semantics,
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 past-discounted weight sequences as rewards
on stochastic game arenas and examine the corresponding stochastic games with
discounted and mean payoff objectives. We dub these games forgetful discounted
games and forgetful mean payoff games, respectively. We establish positional
determinacy of these games and recover classical complexity results and a
Tauberian theorem in the context of past discounted reward sequences.
