Alternative polynomial-time algorithm for Bipartite Matching

If $G$ is a bipartite graph, Hall's theorem \cite{H35} gives a condition for the existence of a matching of $G$ covering one side of the bipartition. This theorem admits a well-known algorithmic proof involving the repeated search of augmenting paths. We present here an alternative algorithm, using a game-theoretic formulation of the problem. We also show how to extend this formulation to the setting of balanced hypergraphs.