  Self-serving, rational agents sometimes cooperate to their mutual benefit.
The two-player iterated prisoner's dilemma game is a model for including the
emergence of cooperation. It is generally believed that there is no simple
ultimatum strategy which a player can control the return of the other
participants. The recent discovery of the powerful class of zero-determinant
strategies in the iterated prisoner's dilemma dramatically expands our
understanding of the classic game by uncovering strategies that provide a
unilateral advantage to sentient players pitted against unwitting opponents.
However, strategies in the prisoner's dilemma game are only two strategies. Are
there these results for general multi-strategy games? To address this question,
the paper develops a theory for zero-determinant strategies for multi-strategy
games, with any number of strategies. The analytical results exhibit a similar
yet different scenario to the case of two-strategy games. Zero-determinant
strategies in iterated prisoner's dilemma can be seen as degenerate case of our
results. The results are also applied to the snowdrift game, the hawk-dove game
and the chicken game.
