  Long-term cooperation, competition, or exploitation among individuals can be
modeled through repeated games. In repeated games, Press and Dyson discovered
zero-determinant (ZD) strategies that enforce a special relationship between
two players. This special relationship implies that a ZD player can
unilaterally impose a linear payoff relationship to the opponent regardless of
the opponent's strategies. A ZD player also has a property that can lead the
opponent to an unconditional cooperation if the opponent tries to improve its
payoff. This property has been mathematically confirmed by Chen and Zinger.
Humans often underestimate a payoff obtained in the future. However, such
discounting was not considered in their analysis. Here, we mathematically
explored whether a ZD player can lead the opponent to an unconditional
cooperation even if a discount factor is incorporated. Specifically, we
represented the expected payoff with a discount factor as the form of
determinants and calculated whether the values obtained by partially
differentiating each factor in the strategy vector become positive. As a
result, we proved that the strategy vector ends up as an unconditional
cooperation even when starting from any initial strategy. This result was
confirmed through numerical calculations. We extended the applicability of ZD
strategies to real world problems.
