Tighten bounds for robust social inefficiency in object allocation

Determine tighter quantitative bounds on the robust social inefficiency guarantees in the object allocation problem under the paper’s social inefficiency function Î, specifically improving the comparison factor between the worst-case social inefficiency of Random Serial Dictatorship and that of ordinal mechanisms that currently follows from the upper bound sup_{(X,≽)} Î((X,≽), RSD(≽)) ≤ ln 2 and the lower bound inf_{μ} sup_{(X,≽)} Î((X,≽), μ(≽)) ≥ 1/2 − 1/(2n), which together yield the 1/(2 ln 2) factor.

Background

The paper introduces a cardinal social inefficiency function and applies it to object allocation without money. It proves a lower bound on the worst-case social inefficiency attainable by any ordinal mechanism and an upper bound on the worst-case social inefficiency of Random Serial Dictatorship (RSD). Combining these yields that no ordinal mechanism can guarantee a worst-case social inefficiency strictly below a 1/(2 ln 2) fraction of RSD’s worst-case social inefficiency.

The authors explicitly state that they did not attempt to refine these bounds and leave tightening them as an open problem. Improving either the upper bound for RSD or the lower bound for general ordinal mechanisms would sharpen the comparison factor.