The initial algorithm may run in a huge polynomial, but further improvements may yield practical algorithms. The result is due to Evangelos Markakis, Aranyak Mehta, and myself: it proves that every such game has a strategy that is an Nash solution, and it finds this solution in time at most
Note, this is clearly a galactic algorithm: even for modest approximations the exponent is huge, and the algorithm cannot be used. In any event, David Johnson famously once said,
Matrix Product: One of the most beautiful two results are the initial brilliant result of Volker Strassen on fast matrix product, and the subsequent deep work of Don Coppersmith and Shmuel Winograd on an even faster matrix product algorithm.