Third-party conflict management, particularly legal dispute resolution (arbitration and adjudication) and mediation, can help improve the willingness of disputants to make asymmetric concessions by ameliorating commitment problems and providing political cover. In both regards, and especially pertaining to commitment problems, mediation has substantial limitations when compared to legal dispute resolution. We develop these arguments and test the observable implications on the Issue Correlates of War data. To get traction on the mechanisms at work, we distinguish between challenger concessions and defender concessions, positing that challenger concessions face the primary hurdle of political cover while defender concessions face the primary hurdle of commitment problems. We find that legal dispute resolution strongly increases the propensity for concessions by both challengers and targets, even major asymmetric concessions. Mediation, on the other hand, only helps increase minor challenger concessions. Also consistent with expectations, mediation best enables asymmetric challenger concessions in the highly salient cases that need the most political cover, and legal dispute resolution best enables asymmetric concessions when there has been a history of failed conflict management attempts that perpetuate mistrust.