A modified proportional mechanism for online auto-bidding achieves a price of anarchy of 1 + O(1)/(n-1) for liquid welfare, improving on the standard bound of 2.
Vcg-kelly mechanisms for allocation of divisible goods: Adapting vcg mechanisms to one-dimensional signals.IEEE Journal on Selected Areas in Communica- tions, 25(6):1237–1243
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Efficiency of Proportional Mechanisms in Online Auto-Bidding Advertising
A modified proportional mechanism for online auto-bidding achieves a price of anarchy of 1 + O(1)/(n-1) for liquid welfare, improving on the standard bound of 2.