Determine the Optimal Welfare-Comparison Constant

Determine the smallest constant c such that, for every fractional assignment profile P, the egalitarian overlap welfare produced by the WF mechanism is at most c times the egalitarian overlap welfare produced by the QP mechanism.

Background

The paper compares the water filling (WF) and quadratic programming (QP) mechanisms using egalitarian overlap welfare, defined as the minimum overlap utility among all agents. The authors prove that the QP mechanism can outperform the WF mechanism by a factor of order n, while the WF mechanism can exceed the QP mechanism by at most a factor of 2.

The paper provides an example showing that the ratio of WF to QP egalitarian overlap welfare can be 7/6, and establishes the bounds 7/6 ≤ c ≤ 2 for the smallest universal comparison constant. Determining the exact value of this constant remains unresolved.