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A non-parametric approach for estimating consumer valuation distributions using second price auctions

Sourav Mukherjee, Ziqian Yang, Rohit K Patra, Kshitij Khare

TL;DR

This work introduces a fully non-parametric framework to estimate the consumer valuation distribution $F$ from second-price auctions by leveraging the entire standing-price sequence rather than only final prices. It develops a likelihood-based approach, recasts $F$ via a constraint-free parameterization $\boldsymbol{\theta}$, and optimizes the augmented likelihood $Lik_{A}$ with a coordinate ascent algorithm to obtain $\widehat{F}_{MLE}$. An explicit initial estimator $\widehat{F}_{init}$ is constructed from final selling prices and first observed bids to seed the algorithm, and performance is validated through extensive simulations and an Xbox eBay dataset. Results show substantial accuracy gains over traditional final-price-based methods, with practical implications for sellers aiming to set profit-maximizing prices under incomplete bidding information.

Abstract

We focus on online second price auctions, where bids are made sequentially, and the winning bidder pays the maximum of the second-highest bid and a seller specified reserve price. For many such auctions, the seller does not see all the bids or the total number of bidders accessing the auction, and only observes the current selling prices throughout the course of the auction. We develop a novel non-parametric approach to estimate the underlying consumer valuation distribution based on this data. Previous non-parametric approaches in the literature only use the final selling price and assume knowledge of the total number of bidders. The resulting estimate, in particular, can be used by the seller to compute the optimal profit-maximizing price for the product. Our approach is free of tuning parameters, and we demonstrate its computational and statistical efficiency in a variety of simulation settings, and also on an Xbox 7-day auction dataset on eBay.

A non-parametric approach for estimating consumer valuation distributions using second price auctions

TL;DR

This work introduces a fully non-parametric framework to estimate the consumer valuation distribution from second-price auctions by leveraging the entire standing-price sequence rather than only final prices. It develops a likelihood-based approach, recasts via a constraint-free parameterization , and optimizes the augmented likelihood with a coordinate ascent algorithm to obtain . An explicit initial estimator is constructed from final selling prices and first observed bids to seed the algorithm, and performance is validated through extensive simulations and an Xbox eBay dataset. Results show substantial accuracy gains over traditional final-price-based methods, with practical implications for sellers aiming to set profit-maximizing prices under incomplete bidding information.

Abstract

We focus on online second price auctions, where bids are made sequentially, and the winning bidder pays the maximum of the second-highest bid and a seller specified reserve price. For many such auctions, the seller does not see all the bids or the total number of bidders accessing the auction, and only observes the current selling prices throughout the course of the auction. We develop a novel non-parametric approach to estimate the underlying consumer valuation distribution based on this data. Previous non-parametric approaches in the literature only use the final selling price and assume knowledge of the total number of bidders. The resulting estimate, in particular, can be used by the seller to compute the optimal profit-maximizing price for the product. Our approach is free of tuning parameters, and we demonstrate its computational and statistical efficiency in a variety of simulation settings, and also on an Xbox 7-day auction dataset on eBay.
Paper Structure (18 sections, 2 theorems, 54 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 18 sections, 2 theorems, 54 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology for learning the consumer valuation distribution $F$
  3. Joint density of the observed data in a single second price auction
  4. Likelihood based on the observed data from multiple identical, non-overlapping second price auctions
  5. Estimation of $F$ given $\lambda$: Some new notation based on pooled standing prices across all auctions
  6. Estimation of $F$ given $\lambda$: A constraint-free reparametrization
  7. Maximizing $Lik_{A} ( \lambda, \boldsymbol{\theta} )$: Coordinate ascent algorithm
  8. Coordinate-wise maximizers for $Lik_{A} ( \lambda, \boldsymbol{\theta} )$
  9. Initial estimator of $\lambda$: Generalized method of moments
  10. Constructing an initial estimate $\widehat{F}_{init}$ of $F$ (and $\boldsymbol{\theta}^{(0)}$ of $\boldsymbol{\theta}$) based exclusively on final selling prices and first observed bids
  11. The Coordinate ascent algorithm for maximizing $Lik_{A}$
  12. Simulation study
  13. Data generation
  14. Simulation Results
  15. Empirical application
  16. ...and 3 more sections

Key Result

Lemma 2.1

For the second price auction described above, the joint density of $M$, $O$, $\{X_i\}_{i=1}^{M}$, $\{T_i\}_{i=0}^{M-1}$ at values $m$, $o$, $\{x_i\}_{i=1}^{m}$, $\{t_i\}_{i=0}^{m-1}$ is given by where $C_1$ does not depend on $F$, $x_0$ represents the reserve price ($r$), $\lambda$ denotes the constant rate of arrival of the bidders throughout the course of the auction, and $f$ represents the den

Figures (10)

  • Figure 1: An illustration of a single second price auction. True bid values are generated from a Pareto distribution and reserve price is at $\$2$. Blue vertical lines are the bid values, black horizontal lines are the current selling prices, and black dots are the time points when the bids are made.
  • Figure 2: An illustration of 4 separate second price auctions for a given product. For each auction, true bid values are generated from a discrete Uniform distribution and reserve price is respectively set to $\$10,\$5,\$13,\$17$. Blue vertical lines are the bid values, black horizontal lines are the current selling prices, and black dots are the time points when the bids are made.
  • Figure 3: Plot of $g$ over the interval $\left[0,5\right]$.
  • Figure 4: "True" underlying valuation distribution functions used in the simulation studies.
  • Figure 5: Plot of the MLE $\widehat{F}_{MLE}$ (red), initial estimator $\widehat{F}_{init}$ (blue), Polya Tree estimator (green) and the true valuation distribution $F$ (taken to be piece-wise Uniform) for a random chosen replicate with $K=100$ (left) and $K=1000$ independent auctions (right). $90\%$-HulC confidence intervals or credible interval (for Polya Tree) are also provided for both estimators (dotted lines, matching colors).
  • ...and 5 more figures

Theorems & Definitions (2)

  • Lemma 2.1
  • Lemma 2.2