Zero-Determinant strategies in finitely repeated n-player games

In this paper we study the existence of zero-determinant (ZD) strategies in finitely repeated n-player games. Necessary and sufficient conditions are derived for a linear relation to be enforceable by a ZD strategy in finitely repeated n-player social dilemmas. The finite number of repetitions is modeled by a so-called discount factor (0 < δ < 1) that discounts future payoffs. The novelty of this work is the extension of results for finitely repeated two-player, two-strategy games to finitely repeated n-player, two-strategy games. Our results show that depending on the group size and the focal player’s initial probability to cooperate, for finitely repeated n-player social dilemmas, it is possible for extortionate, generous and equalizer ZD strategies to exist. This differs from infinitely repeated games in which fair strategies exist for any group size and the existence of ZD strategies is independent of the initial condition. Specifically, in the finitely repeated games in order for generous (resp. extortionate) strategies to exist, the player who employs the ZD strategy must initially cooperate (resp. defect).