A note on the efficiency of position mechanisms with budget constraints

TL;DR

This paper analyzes the efficiency of well-known position auction mechanisms under budget constraints, establishing a tight bound of 2 on their liquid price of anarchy and stability for pure Nash equilibria.

Contribution

It provides the first tight bounds on the liquid price of anarchy and stability for GSP, VCG, and EGFP mechanisms with budget constraints.

Findings

Liquid price of anarchy bound of 2 for pure Nash equilibria

Liquid price of stability bound of 2 for pure Nash equilibria

Results apply to GSP, VCG, and EGFP mechanisms

Abstract

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 the 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.