  Evolutionary game theory is a powerful mathematical framework to study how
intelligent individuals adjust their strategies in collective interactions. It
has been widely believed that it is impossible to unilaterally control players'
payoffs in games, since payoffs are jointly determined by all players. Until
recently, a class of so-called zero-determinant strategies are revealed, which
enables a player to make a unilateral payoff control over her partners in
two-action repeated games with a constant continuation probability. The
existing methods, however, lead to the curse of dimensionality when the
complexity of games increases. In this paper, we propose a new mathematical
framework to study ruling strategies (with which a player unilaterally makes a
linear relation rule on players' payoffs) in repeated games with an arbitrary
number of actions or players, and arbitrary continuation probability. We
establish an existence theorem of ruling strategies and develop an algorithm to
find them. In particular, we prove that strict Markov ruling strategy exists
only if either the repeated game proceeds for an infinite number of rounds, or
every round is repeated with the same probability. The proposed mathematical
framework also enables the search of collaborative ruling strategies for an
alliance to control outsiders. Our method provides novel theoretical insights
into payoff control in complex repeated games, which overcomes the curse of
dimensionality.
