  The paper studies the highly prototypical Fictitious Play (FP) algorithm, as
well as a broad class of learning processes based on best-response dynamics,
that we refer to as FP-type algorithms. A well-known shortcoming of FP is that,
while players may learn an equilibrium strategy in some abstract sense, there
are no guarantees that the period-by-period strategies generated by the
algorithm actually converge to equilibrium themselves. This issue is
fundamentally related to the discontinuous nature of the best response
correspondence and is inherited by many FP-type algorithms. Not only does it
cause problems in the interpretation of such algorithms as a mechanism for
economic and social learning, but it also greatly diminishes the practical
value of these algorithms for use in distributed control. We refer to forms of
learning in which players learn equilibria in some abstract sense only (to be
defined more precisely in the paper) as weak learning, and we refer to forms of
learning where players' period-by-period strategies converge to equilibrium as
strong learning. An approach is presented for modifying an FP-type algorithm
that achieves weak learning in order to construct a variant that achieves
strong learning. Theoretical convergence results are proved.
