On the consistency of random serial dictatorship

The random serial dictatorship (RSD) can be generalized to indivisible object allocation problems allowing fractional endowments such that symmetry, ex-post efficiency and strategy-proofness are preserved. However, there exists a consistent extension of RSD if and only if the population is less than four. The inconsistency of the generalized RSD is a common feature of strategy-proof rules that satisfy minimal fairness and efficiency properties: symmetry, ex-post efficiency, consistency and strategy-proofness are not compatible.