  Zero-determinant (ZD) strategies, a recently found novel class of strategies
in repeated games, has attracted much attention in evolutionary game theory. A
ZD strategy unilaterally enforces a linear relation between average payoffs of
players. Although existence and evolutional stability of ZD strategies have
been studied in simple games, their mathematical properties have not been
well-known yet. For example, what happens when more than one players employ ZD
strategies have not been clarified. In this paper, we provide a general
framework for investigating situations where more than one players employ ZD
strategies in terms of linear algebra. First, we theoretically prove that a set
of linear relations of average payoffs enforced by ZD strategies always has
solutions, which implies that incompatible linear relations are impossible.
Second, we prove that linear payoff relations are independent of each other
under some conditions. These results hold for general games with public
monitoring including perfect-monitoring games. Furthermore, we provide a simple
example of a two-player game in which one player can simultaneously enforce two
linear relations, that is, simultaneously control her and her opponent's
average payoffs. All of these results elucidate general mathematical properties
of ZD strategies.
