  Direct reciprocity and conditional cooperation are important mechanisms to
prevent free riding in social dilemmas. But in large groups these mechanisms
may become ineffective, because they require single individuals to have a
substantial influence on their peers. However, the recent discovery of the
powerful class of zero-determinant strategies in the iterated prisoner's
dilemma suggests that we may have underestimated the degree of control that a
single player can exert. Here, we develop a theory for zero-determinant
strategies for multiplayer social dilemmas, with any number of involved
players. We distinguish several particularly interesting subclasses of
strategies: fair strategies ensure that the own payoff matches the average
payoff of the group; extortionate strategies allow a player to perform above
average; and generous strategies let a player perform below average. We use
this theory to explore how individuals can enhance their strategic options by
forming alliances. The effects of an alliance depend on the size of the
alliance, the type of the social dilemma, and on the strategy of the allies:
fair alliances reduce the inequality within their group; extortionate alliances
outperform the remaining group members; but generous alliances increase
welfare. Our results highlight the critical interplay of individual control and
alliance formation to succeed in large groups.
