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Statistical Inference for Fisher Market Equilibrium

Luofeng Liao, Yuan Gao, Christian Kroer

TL;DR

This work develops a statistical inference framework for Fisher market equilibria, connecting finite observed markets to the long-run, infinite-item Fisher market via a dual Eisenberg-Gale formulation. Using a sample average approximation approach, it proves consistency and finite-sample bounds for key equilibrium quantities such as individual utilities, pacing multipliers, and Nash social welfare, and derives central limit theorems under mild smoothness assumptions. The paper also provides practical methods for constructing confidence intervals and variance estimators for NSW, pacing multipliers, and utilities, and extends the framework to revenue inference in quasilinear Fisher markets. Collectively, the results supply principled uncertainty quantification for equilibrium-based resource allocations and ad-auction revenue in large-scale, data-driven settings.

Abstract

Statistical inference under market equilibrium effects has attracted increasing attention recently. In this paper we focus on the specific case of linear Fisher markets. They have been widely use in fair resource allocation of food/blood donations and budget management in large-scale Internet ad auctions. In resource allocation, it is crucial to quantify the variability of the resource received by the agents (such as blood banks and food banks) in addition to fairness and efficiency properties of the systems. For ad auction markets, it is important to establish statistical properties of the platform's revenues in addition to their expected values. To this end, we propose a statistical framework based on the concept of infinite-dimensional Fisher markets. In our framework, we observe a market formed by a finite number of items sampled from an underlying distribution (the "observed market") and aim to infer several important equilibrium quantities of the underlying long-run market. These equilibrium quantities include individual utilities, social welfare, and pacing multipliers. Through the lens of sample average approximation (SSA), we derive a collection of statistical results and show that the observed market provides useful statistical information of the long-run market. In other words, the equilibrium quantities of the observed market converge to the true ones of the long-run market with strong statistical guarantees. These include consistency, finite sample bounds, asymptotics, and confidence. As an extension, we discuss revenue inference in quasilinear Fisher markets.

Statistical Inference for Fisher Market Equilibrium

TL;DR

This work develops a statistical inference framework for Fisher market equilibria, connecting finite observed markets to the long-run, infinite-item Fisher market via a dual Eisenberg-Gale formulation. Using a sample average approximation approach, it proves consistency and finite-sample bounds for key equilibrium quantities such as individual utilities, pacing multipliers, and Nash social welfare, and derives central limit theorems under mild smoothness assumptions. The paper also provides practical methods for constructing confidence intervals and variance estimators for NSW, pacing multipliers, and utilities, and extends the framework to revenue inference in quasilinear Fisher markets. Collectively, the results supply principled uncertainty quantification for equilibrium-based resource allocations and ad-auction revenue in large-scale, data-driven settings.

Abstract

Statistical inference under market equilibrium effects has attracted increasing attention recently. In this paper we focus on the specific case of linear Fisher markets. They have been widely use in fair resource allocation of food/blood donations and budget management in large-scale Internet ad auctions. In resource allocation, it is crucial to quantify the variability of the resource received by the agents (such as blood banks and food banks) in addition to fairness and efficiency properties of the systems. For ad auction markets, it is important to establish statistical properties of the platform's revenues in addition to their expected values. To this end, we propose a statistical framework based on the concept of infinite-dimensional Fisher markets. In our framework, we observe a market formed by a finite number of items sampled from an underlying distribution (the "observed market") and aim to infer several important equilibrium quantities of the underlying long-run market. These equilibrium quantities include individual utilities, social welfare, and pacing multipliers. Through the lens of sample average approximation (SSA), we derive a collection of statistical results and show that the observed market provides useful statistical information of the long-run market. In other words, the equilibrium quantities of the observed market converge to the true ones of the long-run market with strong statistical guarantees. These include consistency, finite sample bounds, asymptotics, and confidence. As an extension, we discuss revenue inference in quasilinear Fisher markets.
Paper Structure (28 sections, 24 theorems, 110 equations, 5 figures)

This paper contains 28 sections, 24 theorems, 110 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. The Estimands
  4. The Data
  5. Dual Programs: Bridging Data and the Estimands
  6. Consistency and Finite-Sample Bounds
  7. Consistency
  8. High Probability Bounds
  9. Asymptotics and Inference
  10. Asymptotics
  11. Analytical Properties of the Dual Objective
  12. Inference
  13. Extension: Revenue Inference in Quasilinear Fisher Market
  14. Related Work
  15. Extended Analytical Properties of the Dual Objective
  16. ...and 13 more sections

Key Result

Lemma 1

Define the event $A_t = \{ \beta^\gamma \in C \}$ . (i) If $t \geq 2{ \bar{v} ^2 } {\log(2n/\eta)}$, then ${\mathbb P}(A_t) \geq {\mathbb P}(\frac{1}{2} \leq \frac{1}{t} {\sum_{\tau=1}^{t}} { v_i(\theta^\tau) } \leq 2,\forall i ) \geq 1- \eta$. (ii) It holds ${\mathbb P}( A_t \text{ eventually}) =

Figures (5)

  • Figure 1: Our contributions. Left panel: a Fisher market with a finite number of divisible items. Buyer $i$ has value $v_i(\theta)$ for item $\theta$. The goal is to allocate items so that equilibrium conditions are met (\ref{['def:observed_market']}). Right panel: an infinite-dimensional Fisher market with a continuum of items. Middle arrow: this paper provides various forms of statistical guarantees to characterize the convergence of observed finite Fisher market (left) to the long-run market (right) when the items are drawn from a distribution corresponding to the supply function in the long-run market.
  • Figure 2: Mean and standard errors of ${\small \mathrm{{NSW}}}^\gamma$ of observed markets of sizes $t=100, 200, \dots, 5000$ ($k=10$ repeats) sampled from the infinite-dimensional market $\mathcal{M}_1$ with linear valuations $v_i(\theta) = a_i\theta+c_i$.
  • Figure 3: Empirical distribution of $\sqrt{t}({\small \mathrm{{NSW}}}^\gamma - {\small \mathrm{{NSW}}}^*)$ and $N(0, \sigma^2_{ \mathrm{{N}}})$. Kolmogorov-Smirnov test null hypothesis: $\sqrt{t}({\small \mathrm{{NSW}}}^\gamma - {\small \mathrm{{NSW}}}^*)$ values are sampled i.i.d. from $N(0, \sigma^2_{ \mathrm{{N}}})$; alternative hypothesis: they are not sampled i.i.d. from $N(0, \sigma^2_{ \mathrm{{N}}})$; test statistic: $0.1256$; $p$-value: $0.3779$.
  • Figure 4: Q-Q Plot of $\sqrt{t}({\small \mathrm{{NSW}}}^\gamma - {\small \mathrm{{NSW}}}^*)$ values against theoretical quantiles of $N(0, \sigma^2_{ \mathrm{{N}}})$; a (near) straight line indicates that $\sqrt{t}({\small \mathrm{{NSW}}}^\gamma - {\small \mathrm{{NSW}}}^*)$ values appear to be normal.
  • Figure 5: Mean and standard errors of ${\small \mathrm{{NSW}}}^\gamma$ of observed markets of sizes $t=100, 200, \dots, 5000$ ($k=10$ repeats) sampled from the infinite-dimensional market $\mathcal{M}_2$ with linear valuations $v_i(\theta) = a_i^\top \theta+c_i$, $a_i\in {\mathbb R}^{10}$.

Theorems & Definitions (57)

  • Definition 1: The long-run market equilibrium
  • Definition 2: Observed Market Equilibrium
  • Lemma 1
  • Theorem 1: Consistency
  • Theorem 2
  • Theorem 3: Concentration of Approximate Market Equilibrium
  • Corollary 1
  • Corollary 2
  • Theorem 4: Asymptotic Normality of Nash Social Welfare
  • Theorem 5: Asymptotic Normality of Individual Behavior
  • ...and 47 more