  We study optimal equilibria in multi-player games. An equilibrium is optimal
for a player, if her payoff is maximal. A tempting approach to solving this
problem is to seek optimal Nash equilibria, the standard form of equilibria
where no player has an incentive to deviate from her strategy. We argue that a
player with the power to define an equilibrium is in a position, where she
should not be interested in the symmetry of a Nash equilibrium, and ignore the
question of whether or not her outcome can be improved if the other strategies
are fixed. That is, she would only have to make sure that the other players
have no incentive to deviate. This defines a greater class of equilibria, which
may have better (and cannot have worse) optimal equilibria for the designated
powerful player. We apply this strategy to concurrent bimatrix games and to
turn based multi-player mean-payoff games. For the latter, we show that such
political equilibria as well as Nash equilibria always exist, and provide
simple examples where the political equilibrium is superior. We show that
constructing political and Nash equilibria are NP-complete problems. We also
show that, for a fixed number of players, the hardest part is to solve the
underlying two-player mean-payoff games: using an MPG oracle, the problem is
solvable in polynomial time. It is therefore in UP and CoUP, and can be solved
in pseudo polynomial and expected subexponential time.
