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

Martin Bichler, Alok Gupta, Matthias Oberlechner

TL;DR

The paper investigates how revenue in display advertising auctions changes when moving from second-price to first-price auctions in markets where bidding agents learn. It combines analytical modeling for payoff, ROI, and ROS utilities with extensive numerical experiments, including complete- and incomplete-information settings, and applies learning algorithms (Exp3, Q-learning, and SODA) to study convergence to equilibrium and the resulting revenue differences. A key finding is that learning agents converge to Bayes-Nash equilibria across settings, but revenue equivalence breaks for ROI/ROS objectives, with first-price auctions yielding lower revenue than second-price auctions as competition rises. These results imply that algorithmic collusion is not a universal driver of lower FP revenue and that the auction-format switch has substantial, non-obvious consequences for ad exchanges and advertisers. The work provides practical insights for ad-platform design and DSP budgeting, highlighting the robustness of equilibrium convergence and the nuanced impact of bidder objectives on revenue.

Abstract

The transition of display ad exchanges from second-price auctions (SPA) to first-price auctions (FPA) has raised questions about its impact on revenue. Auction theory predicts the revenue equivalence between these two auction formats. However, display ad auctions are different from standard models in auction theory. First, automated bidding agents cannot easily derive equilibrium strategies in FPA because information regarding competitors is not readily available. Second, due to principal-agent problems, bidding agents typically maximize return-on-investment (ROI), not payoff. The literature on learning agents for real-time bidding is growing because of the practical relevance of this area; most research has found that learning agents do not converge to an equilibrium. Specifically, research on algorithmic collusion in display ad auctions has argued that FPA can induce symmetric Q-learning agents to tacitly collude, resulting in bids below equilibrium, leading to lower revenue compared to the SPA. Whether bids are in equilibrium cannot easily be determined from field data since the underlying values of bidders are unknown. In this paper, we draw on analytical modeling and numerical experiments and explore the convergence behavior of widespread online learning algorithms in both complete and incomplete information models. Contrary to prior results, we show that there are no systematic deviations from equilibrium behavior. We also explore the differences in revenue of the FPA and SPA, which have not been done for utility functions relevant to this domain, such as ROI. We show that learning algorithms also converge to equilibrium. Still, revenue equivalence does not hold, indicating that collusion may not be the explanation for lower revenue with FPA, and the change in auction format might have had substantial and non-obvious consequences for ad exchanges and advertisers.

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

TL;DR

The paper investigates how revenue in display advertising auctions changes when moving from second-price to first-price auctions in markets where bidding agents learn. It combines analytical modeling for payoff, ROI, and ROS utilities with extensive numerical experiments, including complete- and incomplete-information settings, and applies learning algorithms (Exp3, Q-learning, and SODA) to study convergence to equilibrium and the resulting revenue differences. A key finding is that learning agents converge to Bayes-Nash equilibria across settings, but revenue equivalence breaks for ROI/ROS objectives, with first-price auctions yielding lower revenue than second-price auctions as competition rises. These results imply that algorithmic collusion is not a universal driver of lower FP revenue and that the auction-format switch has substantial, non-obvious consequences for ad exchanges and advertisers. The work provides practical insights for ad-platform design and DSP budgeting, highlighting the robustness of equilibrium convergence and the nuanced impact of bidder objectives on revenue.

Abstract

The transition of display ad exchanges from second-price auctions (SPA) to first-price auctions (FPA) has raised questions about its impact on revenue. Auction theory predicts the revenue equivalence between these two auction formats. However, display ad auctions are different from standard models in auction theory. First, automated bidding agents cannot easily derive equilibrium strategies in FPA because information regarding competitors is not readily available. Second, due to principal-agent problems, bidding agents typically maximize return-on-investment (ROI), not payoff. The literature on learning agents for real-time bidding is growing because of the practical relevance of this area; most research has found that learning agents do not converge to an equilibrium. Specifically, research on algorithmic collusion in display ad auctions has argued that FPA can induce symmetric Q-learning agents to tacitly collude, resulting in bids below equilibrium, leading to lower revenue compared to the SPA. Whether bids are in equilibrium cannot easily be determined from field data since the underlying values of bidders are unknown. In this paper, we draw on analytical modeling and numerical experiments and explore the convergence behavior of widespread online learning algorithms in both complete and incomplete information models. Contrary to prior results, we show that there are no systematic deviations from equilibrium behavior. We also explore the differences in revenue of the FPA and SPA, which have not been done for utility functions relevant to this domain, such as ROI. We show that learning algorithms also converge to equilibrium. Still, revenue equivalence does not hold, indicating that collusion may not be the explanation for lower revenue with FPA, and the change in auction format might have had substantial and non-obvious consequences for ad exchanges and advertisers.
Paper Structure (36 sections, 2 theorems, 23 equations, 12 figures, 3 tables, 3 algorithms)

This paper contains 36 sections, 2 theorems, 23 equations, 12 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Learning agents and algorithmic collusion
  3. Return-on-investment
  4. Contributions
  5. Related Literature
  6. Utility Models of Bidders in Display Ad Auctions
  7. Collusion in Auctions and Pricing
  8. Equilibrium Learning
  9. Equilibrium Strategies
  10. Model
  11. Payoff-Maximizing Bidders
  12. ROI-Maximizing Bidders
  13. ROS-Maximizing Bidders
  14. Learning Algorithms
  15. Exp3-Algorithm
  16. ...and 21 more sections

Key Result

Theorem 3.1

The second-price sealed-bid single-object auction is strategyproof for bidders with an ex-post utility function of $u_i(b, v_i) = x_i(b) \tfrac{ v_i - p_i(b)}{p_i(b)}$.

Figures (12)

  • Figure 1: Analytical BNE for Payoff- and ROI-Maximizing Agents in a First-Price Auction with Uniform Prior.
  • Figure 2: Median Bids in the FPSB Auction with Two Agents using Different Q-Learners
  • Figure 3: Revenue in the FPSB Auction with Two Agents Using Different Learning Algorithms
  • Figure 4: Expected Revenue in the FPSB and SPSB Auction with Different Utility Models.
  • Figure 5: Expected Revenue in the FPSB and SPSB Auction with Two Asymmetric Agents.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Proposition 3.1
  • proof