The max-product “belief propagation” algorithm is an iterative, local, message passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding and computer vision which involve graphs with many cycles, theoretical convergence results are only known for graphs which are tree-like or have a single cycle. In this paper, we consider a weighted complete bipartite graph and define a probability distribution on it whose MAP assignment corresponds to the maximum weight matching (MWM) in that graph. We analyze the fixed points of the max-product algorithm when run on this graph and prove the surprising result that even though the underlying graph has many short cycles, the maxproduct assignment converges to the correct MAP assignment. We also provide a bound on the number of iterations required by the algorithm