Zero-determinant strategies in finitely repeated games

Genki Ichinose, Naoki Masuda*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)61-77
Number of pages17
JournalJournal of Theoretical Biology
Publication statusPublished - 2018 Feb 7
Externally publishedYes


  • Cooperation
  • Direct reciprocity
  • Discount factor
  • Prisoner's dilemma game

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Biochemistry, Genetics and Molecular Biology(all)
  • Immunology and Microbiology(all)
  • Agricultural and Biological Sciences(all)
  • Applied Mathematics


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