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.
Fixed-point and minimax theorems in locally convex topological linear spaces.Pro- ceedings of the National Academy of Sciences, 38(2):121–126
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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.