  Fictitious play is a popular learning algorithm in which players that utilize
the history of actions played by the players and the knowledge of their own
payoff matrix can converge to the Nash equilibrium under certain conditions on
the game. We consider the presence of an intelligent player that has access to
the entire payoff matrix for the game. We show that by not conforming to
fictitious play, such a player can achieve a better payoff than the one at the
Nash Equilibrium. This result can be viewed both as a fragility of the
fictitious play algorithm to a strategic intelligent player and an indication
that players should not throw away additional information they may have, as
suggested by classical fictitious play.
