  Evolutionary game dynamics on networks typically consider the competition
among simple strategies such as cooperation and defection in the Prisoner's
Dilemma and summarize the effect of population structure as network
reciprocity. However, it remains largely unknown regarding the evolutionary
dynamics involving multiple powerful strategies typically considered in
repeated games, such as the zero-determinant (ZD) strategies that are able to
enforce a linear payoff relationship between them and their co-players. Here,
we consider the evolutionary dynamics of always cooperate (AllC), extortionate
ZD (extortioners), and unbending players in lattice populations based on the
commonly used death-birth updating. Out of the class of unbending strategies,
we consider a particular candidate, PSO Gambler, a machine-learning-optimized
memory-one strategy, which can foster reciprocal cooperation and fairness among
extortionate players. We derive analytical results under weak selection and
rare mutations, including pairwise fixation probabilities and long-term
frequencies of strategies. In the absence of the third unbending type,
extortioners can achieve a half-half split in equilibrium with unconditional
cooperators for sufficiently large extortion factors. However, the presence of
unbending players fundamentally changes the dynamics and tilts the system to
favor unbending cooperation. Most surprisingly, extortioners cannot dominate at
all regardless of how large their extortion factor is, and the long-term
frequency of unbending players is maintained almost as a constant. Our
analytical method is applicable to studying the evolutionary dynamics of
multiple strategies in structured populations. Our work provides insights into
the interplay between network reciprocity and direct reciprocity, revealing the
role of unbending strategies in enforcing fairness and suppressing extortion.
