Zero-Sum Fictitious Play Cannot Converge to a Point

Abstract

Fictitious play is a history-based learning process in which players best respond to the empirical distribution of their opponents' past play. Although classical results show that fictitious play converges to the set of Nash equilibria in zero-sum games, this set convergence does not imply convergence to a single equilibrium when the equilibrium set is non-singleton. This paper shows that pointwise convergence fails in a strong sense. Suppose the equilibrium set of a player has positive measure and consists only of fully mixed strategies. Then, under any tie-breaking rule, the empirical strategy of that player cannot converge to any equilibrium point, provided it is initialized outside the equilibrium set -- in particular, when the player starts with no prior beliefs. The proof identifies two mechanisms behind this instability. In the interior of the equilibrium set, the dynamics retain inertia that prevents settling. At the boundary, the opponent's deviations from its unique equilibrium action steadily unbalance the cumulative payoffs that drive best responses, so that not all actions can remain competitive, as a fully mixed limit would require. The results clarify the gap between convergence to an equilibrium set and convergence to an equilibrium point in fictitious play.