We are interested in a well-known problem in Computer Science and Economics, the Stable Marriage Problem (SM). Considering two communities in which each member has some preferences on the potential partners, the goal is to make pairs taking into account their preferences. This abstract problem has many applications. From a multiagent approach, the seminal Gale-Shapley algorithm [2] solves the SM problem by distinguishing two agent behaviors: a community of proposers and a community of responders [1]. The negotiations between agents lead to a stable solution which is unfair: the community of proposers is favored. In fact, even if the solution given by the Gale-Shapley algorithm is stable, it is the best one for the community of proposers, but the worst for the community of responders. Therefore, we think that this solution is not socially acceptable for a part of the users. In this paper, we propose the Swing method where agents alternatively play the two roles in many bilateral negotiations. Our approach may lead to the emergence of some stable matchings which cannot be reached by the Gale-Shapley algorithm. These matchings are more fair, because they do not favor one community and more optimal for the whole society viewpoint. Then, it is suitable for real-world applications because the solutions are socially acceptable for the users involved in the process.