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

1. Problem Statement

The paper addresses the challenge of estimating the consumer valuation distribution ($F$) in online second-price auctions (e.g., eBay).

• Context: In these auctions, a product is sold to the highest bidder if their bid exceeds a seller-specified starting price ($r$). The winner pays the maximum of the starting price and the second-highest bid.
• Data Limitation: Sellers typically do not observe the total number of bidders ($N$) or the specific bid values of all participants. They only observe the sequence of current selling prices (standing prices) and the times at which these prices change.
• Gap in Literature: Existing semi-parametric or non-parametric methods (e.g., George and Hui, 2012) rely solely on the final selling price and require knowledge of the total number of bidders ($N$). Using only the final price discards significant information contained in the intermediate price changes and leads to identifiability issues when $N$ is unknown.
• Goal: Develop a method to estimate $F$ using the entire sequence of standing prices without requiring knowledge of the total number of bidders.

2. Methodology

The authors propose a novel semi-parametric maximum likelihood estimation (MLE) framework.

2.1. Assumptions

To make the problem tractable, the authors assume:

1. Independent Private Values (IPV): Bidders have independent valuations drawn from a common distribution $F$.
2. Truthful Bidding: Bidders bid their true valuation (dominant strategy in second-price auctions). This implies no multiple bidding by a single user (or that proxy bidding effectively acts as a single bid at the true valuation).
3. Poisson Arrival: Bidders arrive according to a Poisson process with a constant rate $\lambda$.
4. No Knowledge of $N$: The total number of bidders is unobserved.

2.2. Likelihood Construction

• Joint Density: The authors derive the joint density of the observed data ($M$, the number of price changes; $O$, the sale indicator; $\{X_i\}$, the sequence of standing prices; $\{T_i\}$, the inter-change times) for a single auction.
• Multiple Auctions: They extend this to $K$ independent auctions with potentially different starting prices.
• Intractability: Direct maximization of the likelihood is intractable because $F$ is an infinite-dimensional parameter.
• Discretization: Following standard non-parametric conventions, the parameter space for $F$ is restricted to discrete distributions with jumps only at the observed standing prices (and starting prices). This transforms the problem into a finite-dimensional optimization.

2.3. Reparametrization and Optimization

• Constraint-Free Transformation: To handle the monotonicity constraint of the CDF ($F$), the authors reparametrize the problem using a vector $\theta$. If $z_1 < z_2 < \dots < z_{L+K}$ are the pooled observed prices, they define $\theta_i = G(z_i)/G(z_{i-1})$, where $G(z) = 1 - F(z)$. This ensures $0 \le \theta_i \le 1$.
• Coordinate Ascent Algorithm: They develop a coordinate-wise ascent algorithm to maximize the log-likelihood with respect to $\lambda$ and $\theta$.
  • Closed-form Updates: The maximizers for each coordinate $\theta_i$ and the rate $\lambda$ are derived in closed form.
  • Initialization: A crucial step involves constructing a robust initial estimator ($\hat{F}_{init}$) using only the final selling prices and the first observed bid, combined with a Generalized Method of Moments (GMM) estimator for $\lambda$. This ensures the coordinate ascent algorithm converges to a high-quality solution.

3. Key Contributions

1. Utilization of Full Data: Unlike previous methods that use only the final selling price, this approach utilizes the entire sequence of standing prices and their timing.
2. No Need for Bidder Counts: The method does not require knowledge of the total number of bidders ($N$), a significant practical advantage as this data is rarely available to sellers.
3. Semi-Parametric Efficiency: The method is free of tuning parameters (unlike kernel-based non-parametric methods) and provides a consistent estimator under the stated assumptions.
4. Algorithmic Innovation: The development of a coordinate ascent algorithm with closed-form updates and a principled initialization strategy for a complex, non-convex likelihood surface.

4. Results

4.1. Simulation Studies

The authors tested the method against various true distributions (Uniform, Pareto, Gamma, Beta, and Piecewise Uniform) and compared it with:

• Initial Estimator ($\hat{F}_{init}$): Uses only final prices and first bids.
• Pólya Tree Estimator (George & Hui, 2012): Uses only final prices (requires $N$).
• Parametric Methods: Gamma-F and Truncated Normal-F.

Findings:

• Accuracy: The proposed MLE ($\hat{F}_{MLE}$) consistently outperformed all other methods in terms of Kolmogorov-Smirnov (KS) and Total Variation (TV) distances.
• Sample Size: Performance improved significantly as the number of auctions ($K$) increased from 100 to 1000.
• Profit Maximization: The MLE provided the most accurate estimates of the profit-maximizing price and the resulting maximum profit, with lower bias and variance compared to competitors.
• Uncertainty Quantification: The HulC (Hold-out Likelihood Cross-validation) confidence intervals for the MLE showed superior coverage rates (closer to the nominal 90%) and narrower band areas compared to the initial estimator and the Pólya Tree approach.

4.2. Empirical Application (eBay Xbox Data)

The method was applied to a dataset of 93 seven-day Xbox auctions on eBay.

• Data Preprocessing: Multiple bids by the same user were filtered to align with the single-bid assumption; small noise was added to handle ties.
• Results:
  • The MLE estimate of the valuation distribution was distinct from the initial estimator, suggesting that the initial estimator (based on limited data) stochastically dominated the MLE (implying the initial estimator overestimated valuations).
  • Validation: Using a train-test split, the MLE-based model generated simulated final selling prices that were significantly closer to the observed test data (lower Mean Squared Error) than the initial estimator.

5. Significance and Implications

• Managerial Value: Accurate estimation of the consumer valuation distribution allows sellers to determine the optimal starting price and reserve price to maximize profit. The proposed method offers a more precise tool for this than existing techniques.
• Data Efficiency: By leveraging the "standing price sequence," the method extracts more information from the same auction data, making it highly valuable for platforms where bidder counts are hidden.
• Robustness: While the method relies on assumptions (constant arrival rate, single bidding), the authors note that deviations in real-world data (like eBay) are often minor (e.g., <10% multiple bidding), making the approach practically viable.
• Future Directions: The paper suggests relaxing the constant arrival rate assumption (e.g., allowing for "sniping" behavior with a rate change) and incorporating covariates (item condition, seller reputation) into the valuation distribution.

In summary, this paper presents a rigorous, computationally efficient, and statistically superior framework for recovering consumer valuation distributions from incomplete second-price auction data, filling a critical gap in the econometric literature on online markets.