Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: 
The paper proposes a statistical framework to estimate linear Fisher markets with a continuum distribution over items from observed markets with finitely many items drawn from the distribution. A linear Fisher market consists of a continuum set of items with predetermined supplies and a finite set of buyers (each with a potentially different budget and different valuations for items). The market outcome is an allocation of items to buyers along with prices. The goal is to estimate certain quantities at equilibrium in the continuum market: (1) individual utilities received by buyers, (2) pacing multipliers (defined by the budget divided by the utility), and (3) Nash social welfare. The paper examines the statistical properties of equilibrium market quantities from the observed market with finitely many items as estimators for the corresponding quantities of the continuum market. The main results are showing the following: (A) consistency of these estimators and finite sample bounds for the Nash social welfare, the individual utilities, and the pacing multipliers, (B) convergence of approximate market equilibria, and (C)  asymptotic normality of the Nash social welfare. 
strength_and_weaknesses: Strengths
- The paper bridges linear Fisher markets with statistical inference in an interesting manner. The results on fundamental statistical properties—consistency and asymptotic normality—in the context of a classical market are insightful. 
- The technical results in the paper—especially the consistency results and finite sample bounds as well as asymptotic normality—are mathematically clean and clearly explained. 

Weaknesses
- The main weakness of this paper is in the motivation for the setup. In particular, the paper assumes the observed data comes from a market equilibrium from a finite item sample of the underlying continuum market. However, the paper does not provide much motivation for this modeling choice. Why it is not more natural for the learner to have data from a noisy version of the continuum item market, e.g. the equilibria formed under random perturbations in the values of agents? 
- What are the statistical properties of other measures of social welfare besides Nash social welfare? 
clarity,_quality,_novelty_and_reproducibility: The paper builds on an emerging line of work that brings statistical considerations into classical markets. The study of how well observed market quantities approximate the true market quantities is natural and interesting. The main weakness is it is not clear if the setup for the observed market is well-motivated.

The paper is very well-written and the statistical results are clearly explained. The background section (Section 2) is useful for a machine learning audience. 
summary_of_the_review: The paper investigates statistical considerations that arise in linear Fisher markets and contributes to an emerging line of work bridging statistics and market equilibria. The main weakness is the motivation for how the observed quantities are set up in this paper. Despite this weakness, the paper offers a number of informative insights about the convergence guarantees of linear Fisher markets.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The authors have proposed a statistical framework based on the concept of infinite dimensional Fischer markets. This study attempted a marked formed by the finite number of items sampled from an distribution to infer several equilibrium quantities such as individual utilities, social welfare, and pacing multipliers of the underlying long-run market. 
strength_and_weaknesses: Strengths
The article is well written with sufficient support from literature; the proposed method and its limitations are well written and gives good readability. 
Results and conclusion adhere to the style of writing and provides better readability and clarity.
Detailed theoretical and empirical evaluation 

clarity,_quality,_novelty_and_reproducibility: Needs experimental design and publicly available data set to evaluate the proposed methodology or at the least a section on how to leverage the methodology in a real world application.
The authors had provided detailed materials to reproduce the results.

summary_of_the_review: This paper provides good readability but it has gone in depth to analyze/explain the theoretical algorithm in detail on the fisher market equilibrium.
This article is an incremental work of Gao and Kroer (2022) and extended the Fischer market model and proposed a statistical model based on their infinite-dimensional Fischer Market.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: This paper considers statistical inference in linear Fisher markets. In their framework, a market formed by a finite number of items sampled from an underlying distribution is observed and their goal is to infer several important equilibrium quantities (individual utilities, pacing multipliers, and social welfare) of the underlying long-run market. 


strength_and_weaknesses: Strength:

1. Despite numerous algorithmic results available for computing Fisher market equilibria, limited statistical inference result was available for quantifying the randomness of market equilibrium. 

2. These statistical inference results (CLT and confidence intervals) are highly non-trivial. One technical challenge is that the dual objective is potentially not twice differentiable and hence it requires new technical tools. So the technical novelty is strong.


Weaknesses:


1. Writing: The writing of the paper can be further improved. 

(1) The current version assumes that readers have good knowledges of Fisher markets and market equilibrium. However, this may not be the case for general ICLR readers. It would be helpful to add some background on these topics and provide some real examples of Fisher markets. In addition, some parts of Gao and Kroer (2022) can be included in the appendix of this paper. 

(2) The whole paper is very technical and is hard to follow. It would be helpful to check if some of the math notation and math definition can be simplified or moved to the Appendix. The theoretical results include consistency, finite-sample bounds, CLT, and finally variance estimation for constructing confidence intervals. In my personal opinion, the most interesting part is the CLT and confidence interval parts, followed by the finite-sample bounds. The consistency result is of less interests and can be inferred from the finite-sample bounds. It might be helpful to list key theoretical results in the main paper to save space for additional explanations. In addition, the whole Section 5 on the extension to quasilinear fisher market can be moved to the appendix. This section could also be added in a journal version. On the other hand, some of the numerical experiments in Section M of the appendix may be moved to the main paper. 


2. Motivation. It would be helpful to add more justifications on why the statistical inference for linear Fisher market equilibrium is useful. There are two concerns. 

(1) The authors claimed in the Introduction that statistical inference can be used to "quantify the variability of the resources/fairness/efficiency metrics of interest in the long run". But why do we want to quantify the variability of these metrics in the long run? The example of Internet ad auction markets used in the Introduction did not fully answer this question. Why should an internet company care about the statistical inference of these metrics considered in this paper?

(2) This paper considers a linear Fisher markets where the valuation for each buyer is linear. Although this helps the theoretical developments, such assumptions might be restrictive in practice. Interestingly, the ad auction markets example (Conitzer et al., 2022a) provided in the Introduction of this paper actually corresponds to a quasilinear Fisher market. More justifications on the linear fisher markets would be needed. 


3. Real data analysis. This is related to the previous comment. It would be more convincing if a real data study is added. 


4. Constructing confidence intervals for the pacing multipliers and the utilities. In Section 4.3 Inference, Theorem 8 only contains the inference result for Nash social welfare because the estimation of the variance for $\beta$ and $u$ is more complicated. In the later case, the plug-in estimator is no longer suitable. Can we use Bootstrap method to estimate the variance? 

clarity,_quality,_novelty_and_reproducibility: Clarity: Good.
Quality: The writing can be improved to improve the readability for general machine learning researchers.  
Novelty: Strong.
Reproducibility: Good. The code for the numerical experiments was provided. 

summary_of_the_review: This paper considers statistical inference in linear Fisher markets. Finite-sample bounds, CLT, and statistical inference of a few metrics are derived. The technical contribution is strong as the inference for Fisher markets is non-trivial. However, there are a few major concerns listed in the weakness part. If the authors can nicely address these weakness parts, I am willing to increase my rating later.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: This is a good paper on fair resource allocation, which an interesting topic with many potential applications.

The contributions might not be mind-blowing, but they are still of quality, putting this paper above the acceptance bar.
justification_for_why_not_higher_score: The contributions are interesting, but maybe a bit narrow compared to other papers.
justification_for_why_not_lower_score: It is a good paper
Decision: Accept: poster