Suppose an arbitrator needs to allocate an asset among two players, whose claims on the asset are incompatible. The allocation outcome is said to be fair if the arbitrator awards an outcome that brings the same utility payoff to the two players whenever the two players' claims are symmetric and the allocation set is symmetric. In conjunction with other natural axioms, this fairness requirement implies a unique allocation outcome for any claims problem. We propose a mechanism which can be used by the arbitrator to implement this allocation outcome, even when the players' preferences are unknown to the arbitrator. (JEL C78, D63, J52)