Two simplified proofs for Roberts' theorem

Roberts (Aggregation and Revelation of Preferences. Papers presented at the 1st European Summer Workshop of the Econometric Society, pp. 321-349. North-Holland, 1979) showed that every social choice function that is ex-post implementable in private value settings must be weighted VCG, i. e. it maximizes the weighted social welfare. This paper provides two simplified proofs for this. The first proof uses the same underlying key-point, but significantly simplifies the technical construction around it, thus helps to shed light on it. The second proof builds on monotonicity conditions identified by Rochet (J Math Econ 16: 191-200, 1987) and Bikhchandani et al. (Econometrica 74(4): 1109-1132, 2006). This proof is for a weaker statement that assumes an additional condition of "player decisiveness".