A note on the efficiency of position mechanisms with budget constraints - Voudouris, A (Elsevier, 2018)
We study the social efficiency of several well-known mechanisms for the allocation of a set of available (advertising) positions to a set of competing budget-constrained users (advertisers). Specifically, we focus on the Generalized Second Price auction (GSP), the Vickrey–Clarke–Groves mechanism (VCG) and the Expressive Generalized First Price auction (EGFP). Using liquid welfare as our efficiency benchmark, we prove a tight bound of 2 on the liquid price of anarchy and stability of these mechanisms for pure Nash equilibria.