Fictitious play (FP) is a canonical game-theoretic learning algorithm which has been deployed extensively in decentralized control scenarios. However standard treatments of FP, and of many other game-theoretic models, assume rather idealistic conditions which rarely hold in realistic control scenarios. This paper considers a broad class of best response learning algorithms that we refer to as FP-type algorithms. In such an algorithm, given some (possibly limited) information about the history of actions, each individual forecasts the future play and chooses a (myopic) best response strategy given their forecast. We provide a unified analysis of the behavior of FP-type algorithms under an important class of perturbations, thus demonstrating robustness to deviations from the idealistic operating conditions that have been previously assumed. This robustness result is then used to derive convergence results for two control-relevant relaxations of standard game-theoretic applications: distributed (network-based) implementation without full observability and asynchronous deployment (including in continuous time). In each case the results follow as a direct consequence of the main robustness result.