  In this paper, we consider the $n \times n$ two-payer zero-sum repeated game
in which one player (player X) employs the popular Hedge (also called
multiplicative weights update) learning algorithm while the other player
(player Y) adopts the myopic best response. We investigate the dynamics of such
Hedge-myopic system by defining a metric $Q(\textbf{x}_t)$, which measures the
distance between the stage strategy $\textbf{x}_t$ and Nash Equilibrium (NE)
strategy of player X. We analyze the trend of $Q(\textbf{x}_t)$ and prove that
it is bounded and can only take finite values on the evolutionary path when the
payoff matrix is rational and the game has an interior NE. Based on this, we
prove that the stage strategy sequence of both players are periodic after
finite stages and the time-averaged strategy of player Y within one period is
an exact NE strategy. Accordingly, we propose an asymmetric paradigm for
solving two-player zero-sum games. For the special game with rational payoff
matrix and an interior NE, the paradigm can output the precise NE strategy; for
any general games we prove that the time-averaged strategy can converge to an
approximate NE. In comparison to the NE-solving method via Hedge self-play,
this HBR paradigm exhibits faster computation/convergence, better stability and
can attain precise NE convergence in most real cases.
