  Direct reciprocity is a mechanism for sustaining mutual cooperation in
repeated social dilemma games, where a player would keep cooperation to avoid
being retaliated by a co-player in the future. So-called zero-determinant (ZD)
strategies enable a player to unilaterally set a linear relationship between
the player's own payoff and the co-player's payoff regardless of the strategy
of the co-player. In the present study, we analytically study zero-determinant
strategies in finitely repeated (two-person) prisoner's dilemma games with a
general payoff matrix. Our results are as follows. First, we present the forms
of solutions that extend the known results for infinitely repeated games (with
a discount factor w of unity) to the case of finitely repeated games (0 < w <
1). Second, for the three most prominent ZD strategies, the equalizers,
extortioners, and generous strategies, we derive the threshold value of w above
which the ZD strategies exist. Third, we show that the only strategies that
enforce a linear relationship between the two players' payoffs are either the
ZD strategies or unconditional strategies, where the latter independently
cooperates with a fixed probability in each round of the game, proving a
conjecture previously made for infinitely repeated games.
