Sharpness of the quantitative voting-rule bound

Determine whether the quantitative bound asserting that a voting rule based on a monotone Boolean function with every voter’s Shapley value at most e^{-C\sqrt{m\log m}} can generate every asymmetric preference relation is sharp, where m is the number of candidates and C is an absolute constant.

Background

The paper applies Theorem 1.2, together with an argument from Kalai and a result of Alon, to obtain a quantitative version of a voting-theoretic result: for some absolute constant C, when there are m candidates, a voting rule derived from a monotone Boolean function whose every voter has Shapley value at most e{-C\sqrt{m\log m}} can realize every asymmetric social preference relation. The authors explicitly state that the sharpness of this quantitative estimate is unresolved.

References

Theorem 1.2 together with an argument from Kalai [12] and a result of Alon [1], implies that for some absolute constant C, when there are m candidates, a voting rule based on f, for which the Shapley values of every voter is at most e-Cvmlogm, leads to every asymmetric preference relation. This provides a quantitative version of a result from [12]; however, its sharpness remains an open question.

— On Shapley Values and Threshold Intervals  (2502.05990 - Kalai et al., 9 Feb 2025) in Section 1.7, “Voting paradoxes, Condorcet and McGarvey”