This work introduces an innovative Access Point (AP) allocation algorithm for dense Wi-Fi networks, which relies on a centralised potential game developed in a Software-Defined Wireless Networking (SDWN)-based framework. The proposed strategy optimises the allocation of the Wi-Fi stations (STAs) to APs and allows their dynamic reallocation according to possible changes in the capacity of the Wi-Fi network. This paper illustrates the design of the proposed framework based on SDWN and the implementation of the potential game-based algorithm, which includes two possible strategies. The main novel contribution of this work is that the algorithm allows us to efficiently reallocate the STAs by considering external interference, which can negatively affect the capacities of the APs handled by the SDWN controller. Moreover, the paper provides a detailed performance analysis of the algorithm, which describes the significant improvements achieved with respect to the state of the art. Specifically, the results have been compared against the AP selection considered by the IEEE 802.11 standards and another centralised algorithm dealing with the same problem, in terms of the data bit rate provided to the STAs, their dissatisfaction and Quality of Experience (QoE). Finally, the paper analyses the trade-off between efficient performance and the computational complexity achieved by the strategies implemented in the proposed algorithm.