  Repeated game theory has been one of the most prevailing tools for
understanding the long-run relationships, which are footstones in building
human society. Recent works have revealed a new set of "zero-determinant (ZD)"
strategies, which is an important advance in repeated games. A ZD strategy
player can exert a unilaterally control on two players' payoffs. In particular
he can deterministically set the opponent's payoff, or enforce an unfair linear
relationship between the players' payoffs, thereby always seizing an
advantageous share of payoffs. One of the limitations of the original ZD
strategy, however, is that it does not capture the notion of robustness when
the game is subjected to stochastic errors. In this paper, we propose a general
model of ZD strategies for noisy repeated games, and find that ZD strategies
have high robustness against errors. We further derive the pinning strategy
under noise, by which the ZD strategy player coercively set the opponent's
expected payoff to his desired level, although his payoff control ability
declines with the increase of noise strength. Due to the uncertainty caused by
noise, the ZD strategy player cannot secure his payoff to be higher than the
opponent's, which implies strong extortions do not exist even under low noise.
While we show that the ZD strategy player can still establish a novel kind of
extortions, named weak extortions, where any increase of his own payoff always
exceeds that of the opponent's by a fixed percentage, and the conditions under
which the weak extortions can be realized are more stringent as the noise
becomes stronger.
