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Bootstrapping Fisher Market Equilibrium and First-Price Pacing Equilibrium

Luofeng Liao, Christian Kroer

TL;DR

This work tackles the challenge of statistical inference for market equilibria by developing statistically valid bootstrap procedures for Linear Fisher Markets and First-Price Pacing Equilibrium. It leverages the Eisenberg-Gale convex program structure and epi-convergence to derive bootstrap methods (exchangeable, numerical, and proximal) and characterizes FPPE asymptotics without requiring strict complementarity, including adaptive schemes for degenerate buyers and confidence region construction. The authors demonstrate that standard multinomial bootstrap can fail for FPPE, provide conditions under which bootstrap methods are consistent, and validate the theory with synthetic and semi-real experiments, including iPinYou data, highlighting practical applicability. The results advance reliable inference for equilibrium-based resource allocation, with implications for revenue, budgets, and pacing decisions in ad platforms and related applications.

Abstract

The linear Fisher market (LFM) is a basic equilibrium model from economics, which also has applications in fair and efficient resource allocation. First-price pacing equilibrium (FPPE) is a model capturing budget-management mechanisms in first-price auctions. In certain practical settings such as advertising auctions, there is an interest in performing statistical inference over these models. A popular methodology for general statistical inference is the bootstrap procedure. Yet, for LFM and FPPE there is no existing theory for the valid application of bootstrap procedures. In this paper, we introduce and devise several statistically valid bootstrap inference procedures for LFM and FPPE. The most challenging part is to bootstrap general FPPE, which reduces to bootstrapping constrained M-estimators, a largely unexplored problem. We devise a bootstrap procedure for FPPE under mild degeneracy conditions by using the powerful tool of epi-convergence theory. Experiments with synthetic and semi-real data verify our theory.

Bootstrapping Fisher Market Equilibrium and First-Price Pacing Equilibrium

TL;DR

This work tackles the challenge of statistical inference for market equilibria by developing statistically valid bootstrap procedures for Linear Fisher Markets and First-Price Pacing Equilibrium. It leverages the Eisenberg-Gale convex program structure and epi-convergence to derive bootstrap methods (exchangeable, numerical, and proximal) and characterizes FPPE asymptotics without requiring strict complementarity, including adaptive schemes for degenerate buyers and confidence region construction. The authors demonstrate that standard multinomial bootstrap can fail for FPPE, provide conditions under which bootstrap methods are consistent, and validate the theory with synthetic and semi-real experiments, including iPinYou data, highlighting practical applicability. The results advance reliable inference for equilibrium-based resource allocation, with implications for revenue, budgets, and pacing decisions in ad platforms and related applications.

Abstract

The linear Fisher market (LFM) is a basic equilibrium model from economics, which also has applications in fair and efficient resource allocation. First-price pacing equilibrium (FPPE) is a model capturing budget-management mechanisms in first-price auctions. In certain practical settings such as advertising auctions, there is an interest in performing statistical inference over these models. A popular methodology for general statistical inference is the bootstrap procedure. Yet, for LFM and FPPE there is no existing theory for the valid application of bootstrap procedures. In this paper, we introduce and devise several statistically valid bootstrap inference procedures for LFM and FPPE. The most challenging part is to bootstrap general FPPE, which reduces to bootstrapping constrained M-estimators, a largely unexplored problem. We devise a bootstrap procedure for FPPE under mild degeneracy conditions by using the powerful tool of epi-convergence theory. Experiments with synthetic and semi-real data verify our theory.
Paper Structure (36 sections, 12 theorems, 78 equations, 5 figures, 3 tables)

This paper contains 36 sections, 12 theorems, 78 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Bootstrap Consistency
  3. Review of Fisher Market and FPPE
  4. Linear Fisher Markets (LFM)
  5. First-Price Pacing Equilibrium (FPPE)
  6. Smoothness Assumptions
  7. Bootstrapping Fisher Market Equilibrium
  8. Exchangeable Bootstrap
  9. Bootstrapping FPPE
  10. The Limit Distribution of General FPPE
  11. Failure of Multinomial Bootstrap for FPPE
  12. Bootstrapping FPPE under Full Budget Exhaustion
  13. Bootstrapping FPPE without Degenerate Buyers
  14. Confidence Regions for General FPPE
  15. Experiments
  16. ...and 21 more sections

Key Result

Theorem 1

$\sqrt t ( \beta ^ b_{\mathsf{ex, LFM}} - \beta^\gamma ) \,{\stackrel{p}{\rightsquigarrow}}\, c\cdot {\mathscr{J}}_{ \mathsf{{LFM}}}$ where the constant $c$ is defined in as:exboot. Proof in sec:proof:thm:exboot_fm.

Figures (5)

  • Figure 1: Bootstrap vs finite-sample distribution of an 8-buyer 1000-item FPPE. Values are i.i.d. uniformly distributed, and budgets are generated randomly in a way that the first three buyers have leftover budgets. Displayed are histograms of $\beta_1,\dots, \beta_8$. Purple: 100 samples of $\epsilon_t ^{{-1}} ( \beta ^ b-\beta^\gamma)$ according to \ref{['eq:adboot_fppe']} given one FPPE. Yellow: 100 samples of $\sqrt t (\beta^\gamma - \beta^*)$. Bootstrap distribution is very similar to FPPE distribution. The similarity is significant, because to obtain the distributions of FPPE, we need to observe multiple market equilibria, to which we usually do not have access. The bootstrap distribution, on the other hand, is generated based on just one finite FPPE.
  • Figure 2: Comparison of Bootstrap and FPPE finite item distribution.
  • Figure 3: Comparison of Bootstrap and FPPE finite item distribution.
  • Figure 4: Comparison of Bootstrap and FPPE finite item distribution.
  • Figure 5: Click-through rate (in 0.01%) distributions from logistic regression.

Theorems & Definitions (47)

  • Definition 1: Limit Linear Fisher Market Equilibrium
  • Definition 2: Finite LFM, informal
  • Definition 3: Limit FPPE, gao2022infiniteconitzer2022pacing
  • Definition 4: Finite FPPE, informal
  • Definition 5: Exchangable bootstrap weights
  • Theorem 1
  • Example 1: The case with $|I_{==}| = 1$.
  • Theorem 2: Failure of Multinomial Bootstrap
  • Theorem 3
  • Theorem 4
  • ...and 37 more