STATISTICAL INFERENCE FOR FISHER MARKET EQUI-LIBRIUM

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 infinitedimensional 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 (SAA), 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.

1. INTRODUCTION

In a Fisher market there is a set of n buyers that are interested in buying goods from a distinct seller. A market equilibrium (ME) is then a set of prices for the goods, along with a corresponding allocation, such that demand equals supply. One important application of market equilibrium (ME) is fair allocation using the competitive equilibrium from equal incomes (CEEI) mechanism (Varian, 1974) . In CEEI, each individual is given an endowment of faux currency and reports her valuations for items; then, a market equilibrium is computed, and the items are allocated accordingly. The resulting allocation has many desirable properties such as Pareto optimality, envy-freeness and proportionality. For example, Fisher market equilibrium has been used for fair work allocation, impressions allocation in certain recommender systems, course seat allocation and scarce computing resources allocation; see Appendix A for an extensive overview. Despite numerous algorithmic results available for computing Fisher market equilibria, to the best of our knowledge, no statistical results were available for quantifying the randomness of market equilibrium. Given that CEEI is a fair and efficient mechanism, such statistical results are useful for quantifying variability in CEEI-based resource allocation. For example, for systems that assign blood donation to hospitals and blood banks (McElfresh et al., 2020) , or donated food to charities in different neighborhoods (Aleksandrov et al., 2015; Sinclair et al., 2022) , it is crucial to quantify the variability of the amount of resources (blood or food donation) received by the participants (hospitals or charities) of these systems as well as the variability of fairness and efficiency metrics of interest in the long run. Making statistical statements about these metrics is crucial for both evaluating and improving these systems. In addition to fair resource allocation, statistical results for Fisher markets can also be used in revenue inference in Internet ad auction markets. While much of the existing literature uses expected revenue as performance metrics, statistical inference on revenue is challenging due to the complex interaction among bidders under coupled supply constraints and common price signals. As shown by Conitzer et al. (2022a) , in budget management through repeated first-price auctions with pacing, the optimal pacing multipliers correspond to the "prices-per-utility" of buyers in a quasilinear Fisher market at equilibrium. Given the close connection between various solution concepts in Fisher market models and first-price auctions, a statistical framework enables us to quantify the variability in long-run revenue of an advertising platform. Furthermore, a statistical framework would also help answer other statistical questions such as the study of counterfactuals and theoretical guarantees for A/B testing in Internet ad auction markets. For a detailed survey on related work in the areas of statistical inference, applications of Fisher market models, and equilibrium computation algorithms, see Appendix A. Our contributions are as follows. A statistical Fisher market model. We formulate a statistical estimation problem for Fisher markets based on the continuous-item model of Gao and Kroer (2022) . We show that when a finite set of goods are sampled from the continuous model, the observed ME is a good approximation of the long-run market. In particular, we develop consistency results, finite-sample bounds, central limit theorems, and asymptotically valid confidence interval for various quantities of interests, such as individual utility, Nash social welfare, pacing multipliers, and revenue (for quasilinear Fisher markets). Technical challenges. In developing central limit theorems for pacing multipliers and utilities in Fisher markets (Theorem 5), we note that the dual objective is potentially not twice differentiable. This is a required condition, which is common in the sample average approximation or M-estimation literature. We discover three types of market where such differentiability is guaranteed. Moreover, the sample function is not differentiable, which requires us to verify a set of stochastic differentiability conditions in the proofs for central limit theorems. Finally, we achieve a fast statistical rate of the empirical pacing multiplier to the population pacing multiplier measured in the dual objective by exploiting the local strong convexity of the sample function. Notation. For a sequence of events A n we define the set limit by lim inf n→∞ A n = n≥1 j≥n A j = {A t eventually} and lim sup n→∞ A n = n≥1 j≥n A j = {A t i.o.}. Let [n] = {1, . . . , n}. We use 1 t to denote the vector of ones of length t and e j to denote the vector with one in the j-th entry and zeros in the others. For a sequence of random variables {X n }, we say X n = O p (1) if for any > there exists a finite M and a finite N such that P(|X n | > M ) < for all n ≥ N . We say X n = O p (a n ) if X n /a n = O p (1). We use subscript for indexing buyers



Our contributions. Left panel: a Fisher market with a finite number of items. Right panel: a 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.

