Side-by-side first-price auctions with imperfect bidders

TL;DR

This paper analyzes a procurement model with two imperfect bidders in side-by-side first-price auctions, proving convergence of best response dynamics and offering a numerical method for equilibrium analysis.

Contribution

It introduces a theoretical framework for side-by-side first-price auctions with imperfect bidders, including convergence proofs and conditions for uniqueness.

Findings

Iterated best response algorithm converges to equilibrium.

Provided sufficient conditions for equilibrium uniqueness.

Developed a numerical method for analyzing side-by-side procurement.

Abstract

We model a procurement scenario in which two \textit{imperfect} bidders act simultaneously on behalf of a single buyer, a configuration common in display advertising and referred to as \textit{side-by-side bidding} but largely unexplored in theory. We prove that the iterated best response algorithm converges to an equilibrium under standard distributional assumptions and provide sufficient condition for uniqueness. Beyond establishing existence and convergence, our analysis provides a tractable numerical method for quantitative studies of side-by-side procurement.