Characterization of incentive compatible single-parameter mechanisms revisited

We reexamine the characterization of incentive compatible single-parameter mechanisms introduced in Archer and Tardos (2001). We argue that the claimed uniqueness result, called 'Myerson's Lemma' was not well established. We provide an elementary proof of uniqueness that unifies the presentation for two classes of allocation functions used in the literature and show that the general case is a consequence of a little known result from the theory of real functions. We also clarify that our proof of uniqueness is more elementary than the previous one. Finally, by generalizing our characterization result to more dimensions, we provide alternative proofs of revenue equivalence results for multiunit auctions and combinatorial auctions.