  Zero-determinant strategies are a class of strategies in repeated games which
unilaterally control payoffs. Zero-determinant strategies have attracted much
attention in studies of social dilemma, particularly in the context of
evolution of cooperation. So far, not only general properties of
zero-determinant strategies have been investigated, but zero-determinant
strategies have been applied to control in the fields of information and
communications technology and analysis of imitation. Here, we further deepen
our understanding on general mathematical properties of zero-determinant
strategies. We first prove that zero-determinant strategies, if exist, can be
implemented by some one-dimensional transition probability. Next, we prove
that, if a two-player game has a non-trivial potential function, a
zero-determinant strategy exists in its repeated version. These results assist
us to implement zero-determinant strategies in broader situations.
