[Paper Review] Autobidding Equilibria in Sponsored Shopping

Paul Dütting, Li, Yuhao+3 more|arXiv (Cornell University)|Feb 25, 2026

Auction Theory and Applications0 citations

TL;DR

The paper proves existence of autobidding equilibria in multi-item sponsored shopping auctions under GSP and VCG, and tightens the Price of Anarchy to 2.

ABSTRACT

As commerce shifts to digital marketplaces, platforms increasingly monetize traffic through Sponsored Shopping auctions. Unlike classic ``Sponsored Search", where an advertiser typically bids for a single link, these settings involve advertisers with broad catalogs of distinct products. In these auctions, a single advertiser can secure multiple slots simultaneously to promote different items within the same query. This creates a fundamental complexity: the allocation is combinatorial, as advertisers simultaneously win a bundle of slots rather than a single position. We study this setting through the lens of autobidding, where value-maximizing agents employ uniform bidding strategies to optimize total value subject to Return-on-Investment (ROI) constraints. We analyze two prevalent auction formats: Generalized Second-Price (GSP) and Vickrey-Clarke-Groves (VCG). Our first main contribution is establishing the universal existence of an Autobidding Equilibrium for both settings. Second, we prove a tight Price of Anarchy (PoA) of 2 for both mechanisms.

Motivation & Objective

• Motivate the study of Sponsored Shopping where advertisers can win multiple items in a single query.
• Model autobidding with uniform multipliers to respect ROI constraints in a multi-item, multi-slot setting.
• Establish existence of Autobidding Equilibria for GSP and VCG mechanisms in this complex setting.
• Derive tight welfare guarantees (PoA = 2) for autobidding in both mechanisms.

Proposed method

• Model bidders with multi-item valuations and slot-dependent click-through rates.
• Use a uniform-bidding (autobidding) strategy where each bidder’s bid is their value times a common multiplier across items.
• Define autobidding equilibria as a pair (α, π) with ROI feasibility and maximal autobidding conditions.
• Introduce a smoothed input framework perturbing each item bid by independent Uniform[0, ε] noise to ensure unique allocations.
• Prove existence of equilibria in the smoothed game via Brouwer’s fixed point theorem and take ε → 0 to recover original equilibrium.
• Apply a smoothness-style argument to show PoA ≤ 2 and establish a matching lower bound.

Experimental results

Research questions

• RQ1 Do autobidding equilibria exist for multi-item, multi-slot auctions under GSP and VCG when bidders use uniform multipliers?
• RQ2 What is the welfare efficiency (PoA) of autobidding equilibria in these settings, and is it tight?
• RQ3 Can a general input-smoothed framework yield existence results beyond single-item or simpler auction formats?
• RQ4 How do GSP and VCG payments interact with uniform autobidders under ROI constraints to bound welfare loss?

Key findings

• There exists an autobidding equilibrium for both GSP and VCG in the sponsored shopping setting.
• The Price of Anarchy for autobidding equilibria is tight at 2 for both GSP and VCG.
• A novel input-smoothing framework yields equilibrium existence by perturbing bids and applying Brouwer’s fixed point theorem.
• A disjoint-ownership argument underpins the PoA bound by ensuring displaced optimal items serve as price-setters in payments.
• The results extend known PoA bounds from multi-slot auctions to the combinatorial, multi-item sponsored shopping context.