Zero-determinant (ZD) strategies are a class of strategies in repeated games, which unilaterally control payoffs. It has been shown that several ZD strategies promote cooperation in social dilemma games. It has been widely believed that the performance of ZD strategies is expressed as ``unilaterally enforce linear relations between payoffs''. However, in order to interpret properties of ZD strategies in repeated games without discounting, previous studies assumed that the limit of the Cesaro average of the probability distribution of the action profile exists. Here, we explain the performance of ZD strategies without this assumption, which leads to a stronger result than previous ones.

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