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

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.