  Since Press and Dyson's ingenious discovery of ZD (zero-determinant) strategy
in the repeated Prisoner's Dilemma game, several studies have confirmed the
existence of ZD strategy in repeated multiplayer social dilemmas. However, few
researches study the evolutionary performance of multiplayer ZD strategies,
especially from a theoretical perspective. Here, we use a newly proposed
state-clustering method to theoretically analyze the evolutionary dynamics of
two representative ZD strategies: generous ZD strategies and extortionate ZD
strategies. Apart from the competitions between the two strategies and some
classical strategies, we consider two new settings for multiplayer ZD
strategies: competitions in the whole ZD strategy space and competitions in the
space of all memory-1 strategies. Besides, we investigate the influence of
level of generosity and extortion on the evolutionary dynamics of generous and
extortionate ZD, which was commonly ignored in previous studies. Theoretical
results show players with limited generosity are at an advantageous place and
extortioners extorting more severely hold their ground more readily. Our
results may provide new insights into better understanding the evolutionary
dynamics of ZD strategies in repeated multiplayer games.
