  In this paper, we study the dynamic behavior of Hedge, a well-known algorithm
in theoretical machine learning and algorithmic game theory. The empirical
average (arithmetic mean) of the iterates Hedge generates is known to converge
to a minimax equilibrium in zero-sum games. We generalize that result to show
convergence of the empirical average to Nash equilibrium in symmetric bimatrix
games (that is bimatrix games where the payoff matrix of each player is the
transpose of that of the other) in the sense that every limit point of the
sequence of averages is an $\epsilon$-approximate symmetric equilibrium
strategy for any desirable $\epsilon$. Our analysis gives rise to a symmetric
equilibrium fully polynomial-time approximation scheme, implying P = PPAD.
