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.
Efficiency of non-truthful auctions in auto-bidding with budget constraints
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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.