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

