Analytical characterization of a(D) achieving expected budget balance in the α-Reduction double auction

Derive an analytical expression for the mixing parameter a(D) that yields zero expected revenue (ex ante budget balance) in the α-Reduction Double Auction for a given distribution D over agents’ private values.

Background

The authors propose two randomized double auction rules, including the α-Reduction Double Auction, which mixes between the Trade Reduction and VCG rules. For any known value distribution D, there exists a mixing probability a(D) that achieves zero expected revenue while improving expected efficiency over pure trade reduction.

Although they empirically estimate a*(U) for the uniform distribution via simulation, they explicitly state they do not solve for a(D) analytically, leaving open the analytical determination of a(D) for general distributions.