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.