In 2012, Press and Dyson discovered a strategy set, called Zero-determinant (ZD) strategies, which enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy in the repeated prisoner's dilemma (RPD) game. In the RPD game, a discount factor and observation errors are both important because they often happen in society. Here, we examined strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first revealed that the payoffs of two players can be represented by the form of determinants even with these two factors. Then, we searched for all possible strategies that enforce linear payoff relationships and found that both ZD strategies and unconditional strategies are the only strategy sets which satisfy the condition.

The repeated prisoner’s dilemma (RPD) game has revealed how cooperation and competition arise among competitive players in long-run relationships. In the RPD game with no errors, zero-determinant (ZD) strategies allow a player to unilaterally set a linear relationship between the player’s own payoff and the opponent’s payoff regardless of the strategy of the opponent. On the other hand, unconditional strategies such as ALLD and ALLC also unilaterally set a linear relationship. However, little is known about the existence of such strategies in the RPD game with errors. Here, we analytically search for the strategies that enforce a linear payoff relationship under observation error in the RPD game. As a result, we found that, even in the case with observation errors, the only strategy sets that enforce a linear payoff relationship are either ZD strategies or unconditional strategies.