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Budget-Constrained Auctions with Unassured Priors: Strategic Equivalence and Structural Properties

Zhaohua Chen, Mingwei Yang, Chang Wang, Jicheng Li, Zheng Cai, Yukun Ren, Zhihua Zhu, Xiaotie Deng

TL;DR

This paper addresses budget-constrained auctions when advertisers’ value priors are unassured, formulating an unassured-prior game between a seller and budgeted buyers. It compares five budget-constrained mechanisms, including the Bayesian revenue-optimal auction (BROA) and budget-extracting variants of first-price and second-price auctions, through a budget-extracting framework and a strategic-equivalence lens. The authors prove strong and broad strategic-equivalence results between BROA and budget-extracting bid-discount first-price auctions, and establish extensive symmetry and dominance relations among the mechanisms, both with and without strategic bidding. The findings show that simple, budget-extracting mechanisms can match the revenue and equilibrium outcomes of the more complex optimal mechanism under unassured priors, with important implications for practical advertising platforms deploying pacing or bid-discount strategies. The work advances understanding of budget-constrained auctions in uncertain environments and suggests robust mechanism choices for online ad markets where priors are imperfect or private.

Abstract

In today's online advertising markets, it is common for advertisers to set long-term budgets. Correspondingly, advertising platforms adopt budget control methods to ensure that advertisers' payments lie within their budgets. Most budget control methods rely on the value distributions of advertisers. However, due to the complex advertising landscape and potential privacy concerns, the platform hardly learns advertisers' true priors. Thus, it is crucial to understand how budget control auction mechanisms perform under unassured priors. This work answers this problem from multiple aspects. We consider the unassured prior game among the seller and all buyers induced by different mechanisms in the stochastic model. We restrict the parameterized mechanisms to satisfy the budget-extracting condition, which maximizes the seller's revenue by extracting buyers' budgets as effectively as possible. Our main result shows that the Bayesian revenue-optimal mechanism and the budget-extracting bid-discount first-price mechanism yield the same set of Nash equilibrium outcomes in the unassured prior game. This implies that simple mechanisms can be as robust as the optimal mechanism under unassured priors in the budget-constrained setting. In the symmetric case, we further show that all these five (budget-extracting) mechanisms share the same set of possible outcomes. We further dig into the structural properties of these mechanisms. We characterize sufficient and necessary conditions on the budget-extracting parameter tuple for bid-discount/pacing first-price auctions. Meanwhile, when buyers do not take strategic behaviors, we exploit the dominance relationships of these mechanisms by revealing their intrinsic structures.

Budget-Constrained Auctions with Unassured Priors: Strategic Equivalence and Structural Properties

TL;DR

This paper addresses budget-constrained auctions when advertisers’ value priors are unassured, formulating an unassured-prior game between a seller and budgeted buyers. It compares five budget-constrained mechanisms, including the Bayesian revenue-optimal auction (BROA) and budget-extracting variants of first-price and second-price auctions, through a budget-extracting framework and a strategic-equivalence lens. The authors prove strong and broad strategic-equivalence results between BROA and budget-extracting bid-discount first-price auctions, and establish extensive symmetry and dominance relations among the mechanisms, both with and without strategic bidding. The findings show that simple, budget-extracting mechanisms can match the revenue and equilibrium outcomes of the more complex optimal mechanism under unassured priors, with important implications for practical advertising platforms deploying pacing or bid-discount strategies. The work advances understanding of budget-constrained auctions in uncertain environments and suggests robust mechanism choices for online ad markets where priors are imperfect or private.

Abstract

In today's online advertising markets, it is common for advertisers to set long-term budgets. Correspondingly, advertising platforms adopt budget control methods to ensure that advertisers' payments lie within their budgets. Most budget control methods rely on the value distributions of advertisers. However, due to the complex advertising landscape and potential privacy concerns, the platform hardly learns advertisers' true priors. Thus, it is crucial to understand how budget control auction mechanisms perform under unassured priors. This work answers this problem from multiple aspects. We consider the unassured prior game among the seller and all buyers induced by different mechanisms in the stochastic model. We restrict the parameterized mechanisms to satisfy the budget-extracting condition, which maximizes the seller's revenue by extracting buyers' budgets as effectively as possible. Our main result shows that the Bayesian revenue-optimal mechanism and the budget-extracting bid-discount first-price mechanism yield the same set of Nash equilibrium outcomes in the unassured prior game. This implies that simple mechanisms can be as robust as the optimal mechanism under unassured priors in the budget-constrained setting. In the symmetric case, we further show that all these five (budget-extracting) mechanisms share the same set of possible outcomes. We further dig into the structural properties of these mechanisms. We characterize sufficient and necessary conditions on the budget-extracting parameter tuple for bid-discount/pacing first-price auctions. Meanwhile, when buyers do not take strategic behaviors, we exploit the dominance relationships of these mechanisms by revealing their intrinsic structures.
Paper Structure (56 sections, 37 theorems, 99 equations, 4 figures, 2 tables)

This paper contains 56 sections, 37 theorems, 99 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Strategic equivalence among budget-constrained auctions in the unassured prior game.
  4. Properties on variants of first-price auctions.
  5. Dominance relationships on the seller's revenue without strategic bidding.
  6. Related Work
  7. Prior manipulation model.
  8. Market equilibrium with budget-constrained buyers.
  9. Model
  10. Parameterized auction mechanisms and monotonicity.
  11. Bidding functions.
  12. Budget-constrained auction mechanisms.
  13. Unassured prior game with budget constraints among the seller and buyers.
  14. Other terminologies.
  15. Discussions on the model.
  16. ...and 41 more sections

Key Result

Lemma 2.1

For any buyer and bidding function, there exists an increasing bidding function that yields at least the same utility for the buyer under any monotone parameterized auction mechanism and other bidders' strategies.

Figures (4)

  • Figure 1: Summary of the results in \ref{['sec:equivalence']} on the strategic equivalence among different auction types. Two auction forms are strategic-equivalent if they are connected by a bidirectional arrow. Different line types indicate the restrictions. ER: Each buyer's virtual bidding quantile function is strictly increasing and differentiable. IL: Each buyer's bidding quantile function is inverse Lipschitz continuous. $\text{IL}^2$: Each buyer's bidding quantile function and virtual bidding quantile function are both inverse Lipschitz continuous. Sym: Buyers and budget-extracting parameters are both symmetric. Strong (S) and weak (W) strategic equivalence are defined in \ref{['def:strategic-equivalence']}. The "e" at the front of mechanisms stands for budget-extracting, which is defined in \ref{['def:budget-extracting']}.
  • Figure 2: Summary of the results in \ref{['thm:dominance-relationships-seller-revenue']} on the dominance relationships among different auction types when buyers truthful bid. A dominates B if there is an arrow from A to B. Different line types indicate the assumptions. SI: Each buyer's bidding quantile function is strictly increasing. SR: Each buyer's virtual bidding quantile function is strictly increasing. $\text{IL}^2$: Each buyer's bidding quantile function and virtual bidding quantile function are both inverse Lipschitz continuous.
  • Figure 3: An illustration of \ref{['thm:equivalence-BROA-eBDFPA']}. ER: Each buyer's virtual bidding qf is strictly increasing and differentiable; $\text{IL}^2$: each buyer's bidding qf and virtual bidding qf are both inverse Lipschitz continuous.
  • Figure 4: The "lifting" process, which is adopted to construct $\widetilde{v}_i^{(2)}$ when $\widetilde{\psi}_i^{(1)}$ has negative parts.

Theorems & Definitions (67)

  • Definition 2.1: Monotonicity
  • Lemma 2.1
  • Definition 2.2: Budget feasibility
  • Definition 2.3: Individual rationality
  • Definition 2.4: Inverse Lipschitz continuity
  • Definition 3.1: Budget-extracting
  • Theorem 3.1
  • Definition 4.1: Strategic equivalence
  • Lemma 4.1
  • Theorem 4.2
  • ...and 57 more