Revenue in First- and Second-Price Display Advertising Auctions: Understanding Markets with Learning Agents

The transition of display ad exchanges from second-price to first-price auctions has raised questions about its impact on revenue. Auction theory predicts revenue equivalence between these two auction formats under standard assumptions. However, display ad auctions differ from standard auction models in at least two important ways. First, automated bidding agents cannot easily derive equilibrium strategies in first-price auctions because distributional information about competitors' values, or even the number of competitors, is often unavailable. Second, due to principal-agent problems, bidding agents often optimize return on investment (ROI) rather than quasilinear payoff. The literature on learning agents for real-time bidding is growing because of the practical relevance of this setting. However, whether such learning agents converge to equilibrium is an open question: learning dynamics in games can cycle, become chaotic, or generate off-equilibrium outcomes. Recent experiments suggest that learning agents may also converge to collusive low-price outcomes. Since bidders' underlying values are typically unobserved, it is difficult to determine from field data alone whether observed bids are consistent with equilibrium behavior. In this paper, we derive equilibrium predictions and study the convergence behavior of widely used online learning algorithms in a stationary benchmark model of display-advertising auctions. We also leverage recent developments in equilibrium computation to obtain equilibrium predictions in settings where analytical solutions to the governing differential equations are unavailable. In the stationary benchmark environments we study, the learning algorithms do not exhibit systematic bid suppression and instead move toward the computed equilibrium. The benchmark identifies a theoretically important channel through which auction-format changes can affect revenues when bidders optimize ROI rather than quasilinear payoff: lower first-price revenues can arise from non-collusive ROI-based equilibrium behavior in a canonical auction model. We show that, in equilibrium, second-price auctions achieve higher expected revenue than first-price auctions with ROI-maximizing bidders. These results do not imply that algorithmic collusion is unlikely in real display-advertising markets, but they show that lower first-price revenues are not uniquely diagnostic of collusion and may also reflect bidder objectives.