  This paper aims to solve the optimal strategy against a well-known adaptive
algorithm, the Hedge algorithm, in a finitely repeated $2\times 2$ zero-sum
game. In the literature, related theoretical results are very rare. To this
end, we make the evolution analysis for the resulting dynamical game system and
build the action recurrence relation based on the Bellman optimality equation.
First, we define the state and the State Transition Triangle Graph (STTG);
then, we prove that the game system will behave in a periodic-like way when the
opponent adopts the myopic best response. Further, based on the myopic path and
the recurrence relation between the optimal actions at time-adjacent states, we
can solve the optimal strategy of the opponent, which is proved to be periodic
on the time interval truncated by a tiny segment and has the same period as the
myopic path. Results in this paper are rigorous and inspiring, and the method
might help solve the optimal strategy for general games and general algorithms.
