  Repeated games are useful models to analyze long term interactions of living
species and complex social phenomena. Zero-determinant (ZD) strategies in
repeated games discovered by Press and Dyson in 2012 enforce a linear payoff
relationship between a focal player and the opponent. This linear relationship
can be set arbitrarily by a ZD player. Hence, a subclass of ZD strategies can
fix the opponent's expected payoff and another subclass of the strategies can
exceed the opponent for the expected payoff. Since this discovery, theories for
ZD strategies are extended to cope with various natural situations. It is
especially important to consider the theory of ZD strategies for repeated games
with a discount factor and observation errors because it allows the theory to
be applicable in the real world. Recent studies revealed their existence of ZD
strategies even in repeated games with both factors. However, the conditions
for the existence has not been sufficiently analyzed. Here, we mathematically
analyzed the conditions in repeated games with both factors. First, we derived
the thresholds of a discount factor and observation errors which ensure the
existence of Equalizer and positively correlated ZD (pcZD) strategies, which
are well-known subclasses of ZD strategies. We found that ZD strategies exist
only when a discount factor remains high as the error rates increase. Next, we
derived the conditions for the expected payoff of the opponent enforced by
Equalizer as well as the conditions for the slope and base line payoff of
linear lines enforced by pcZD. As a result, we found that, as error rates
increase or a discount factor decreases, the conditions for the linear line
that Equalizer or pcZD can enforce become strict.
