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Revenue Guarantees in Autobidding Platforms

Ioannis Caragiannis, Anders Bo Ipsen, Stratis Skoulakis

TL;DR

This work studies revenue maximization for selling multiple divisible goods to budget-constrained buyers with linear valuations, focusing on fixed-unit-price mechanisms that are transparent and implementable in autobidding platforms. It proves that First-Price Pacing Equilibrium (FPPE) guarantees at least $1/2$ of the optimal revenue achievable with variable unit prices, while RMFUP remains APX-hard; FPPE thus provides the best-known polynomial-time approximation in this setting. The authors extend FPPE to online arrival models with a $1/4$-competitive guarantee and generalize to concave valuations via an Eisenberg–Gale convex program, with revenue bounds scaling with the valuations’ curvature. Overall, FPPE offers a robust, tractable approach to revenue-robust autobidding design with strong offline guarantees, practical online adaptability, and principled extensions to nonlinear buyer utilities.

Abstract

Motivated by autobidding systems in online advertising, we study revenue maximization in markets with divisible goods and budget-constrained buyers with linear valuations. Our aim is to compute a single price for each good and an allocation that maximizes total revenue. We show that the First-Price Pacing Equilibrium (FPPE) guarantees at least half of the optimal revenue, even when compared to the maximal revenue of buyer-specific prices. This guarantee is particularly striking in light of our hardness result: we prove that revenue maximization under individual rationality and single-price-per-good constraints is APX-hard. We further extend our analysis in two directions: first, we introduce an online analogue of FPPE and show that it achieves a constant-factor revenue guarantee, specifically a $1/4$-approximation; second, we consider buyers with concave valuation functions, characterizing an FPPE-type outcome as the solution to an Eisenberg-Gale-style convex program and showing that the revenue approximation degrades gracefully with the degree of nonlinearity of the valuations.

Revenue Guarantees in Autobidding Platforms

TL;DR

This work studies revenue maximization for selling multiple divisible goods to budget-constrained buyers with linear valuations, focusing on fixed-unit-price mechanisms that are transparent and implementable in autobidding platforms. It proves that First-Price Pacing Equilibrium (FPPE) guarantees at least of the optimal revenue achievable with variable unit prices, while RMFUP remains APX-hard; FPPE thus provides the best-known polynomial-time approximation in this setting. The authors extend FPPE to online arrival models with a -competitive guarantee and generalize to concave valuations via an Eisenberg–Gale convex program, with revenue bounds scaling with the valuations’ curvature. Overall, FPPE offers a robust, tractable approach to revenue-robust autobidding design with strong offline guarantees, practical online adaptability, and principled extensions to nonlinear buyer utilities.

Abstract

Motivated by autobidding systems in online advertising, we study revenue maximization in markets with divisible goods and budget-constrained buyers with linear valuations. Our aim is to compute a single price for each good and an allocation that maximizes total revenue. We show that the First-Price Pacing Equilibrium (FPPE) guarantees at least half of the optimal revenue, even when compared to the maximal revenue of buyer-specific prices. This guarantee is particularly striking in light of our hardness result: we prove that revenue maximization under individual rationality and single-price-per-good constraints is APX-hard. We further extend our analysis in two directions: first, we introduce an online analogue of FPPE and show that it achieves a constant-factor revenue guarantee, specifically a -approximation; second, we consider buyers with concave valuation functions, characterizing an FPPE-type outcome as the solution to an Eisenberg-Gale-style convex program and showing that the revenue approximation degrades gracefully with the degree of nonlinearity of the valuations.
Paper Structure (18 sections, 19 theorems, 65 equations, 4 figures, 2 algorithms)

This paper contains 18 sections, 19 theorems, 65 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions and Results.
  3. Related Work
  4. Preliminaries and Paper Organization
  5. Revenue Maximization via Fixed Unit Prices
  6. First-Price Pacing Equilibrium
  7. Results and Paper Organization.
  8. Inapproximability of Revenue Maximization with Fixed Unit Prices
  9. Constructing the RMFUP instance
  10. Revenue Guarantees of First-Price Pacing Equilibrium
  11. First-Price Pacing Equilibrium as an Online Algorithm
  12. FPPE as an Online Algorithm
  13. Revenue Guarantees for Concave Valuations
  14. Conclusion
  15. Rounding Algorithm and Proof of Lemma \ref{['l:1']}
  16. ...and 3 more sections

Key Result

Theorem 1

There is no polynomial-time approximation algorithm for $\mathrm{3D}$-2-$\mathrm{M}$ with approximation ratio $\rho \geq 94/95$ unless $\mathrm{P = NP}$.

Figures (4)

  • Figure 1: A 3D-Matching instance with $E_1=\{a_1,a_2\}$, $E_2=\{b_1,b_2\}$, and $E_3=\{c_1,c_2\}$ arranged at the vertices of an abstract triangle. Each filled colored triangle represents a triplet in $S=\{(a_1,b_1,c_1),(a_1,b_2,c_2),(a_2,b_1,c_2)\}$. All three edges of every triplet are explicitly drawn. A valid 3D-Matching corresponds to selecting vertex-disjoint triangles; here, the maximum matching has size 1.
  • Figure 2: Visualization of the conversion of example from Figure \ref{['fig:matching-example']}. Recall that it is defined by $E_1=\{a_1,a_2\}$, $E_2=\{b_1,b_2\}$, and $E_3=\{c_1,c_2\}$ and triplets $S=\{(a_1,b_1,c_1),(a_1,b_2,c_2),(a_2,b_1,c_2)\}$. Notice that element buyers $e$ have budget $B_e = \frac{1}{3}$ and special buyers $s$ have budget $B_s = \frac{2}{3}$. The colors of the special buyers and goods correspond to the color of the triplet from Figure \ref{['fig:matching-example']}.
  • Figure 3: Diagram showing an example of an online FPPE problem. Notice that buyer 1 and 2 are fully revealed at time-step 1 and buyer 3 is only revealed to at time-step 3.
  • Figure 5: Adversarial instance which is difficult to solve well in an online manner.

Theorems & Definitions (45)

  • Example 1: Optimal revenue with variable unit prices
  • Example 2: Optimal revenue with fixed unit prices
  • Definition 1
  • Theorem 1: CC06
  • Lemma 1
  • Claim 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 35 more