Bayes-Nash Equilibria of the Generalized Second-Price Auction

Bayes-Nash Equilibria of the Generalized Second-Price Auction
Renato Gomes

We develop a Bayes–Nash analysis of the generalized second-price (GSP) auction, the multi-unit auction used by search engines to sell sponsored advertising positions. Our main result characterizes the efficient Bayes–Nash equilibrium of the GSP and provides a necessary

and sufficient condition that guarantees existence of such an equilibrium. With only two positions, this condition requires that the click–through rate of the second position is sufficiently smaller than that of the first.

When an efficient equilibrium exists, we provide a necessary and sufficient condition for the auction revenue to decrease as click–through rates increase. Interestingly, under optimal reserve prices, revenue increases with the click–through rates of all positions. Further, we prove that no inefficient equilibrium of the GSP can be symmetric.

Our results are in sharp contrast with the previous literature that studied the GSP under complete information.

Another publication from the same category: Machine Learning and Data Science

IEEE Computing Conference 2018, London, UK

Regularization of the Kernel Matrix via Covariance Matrix Shrinkage Estimation

The kernel trick concept, formulated as an inner product in a feature space, facilitates powerful extensions to many well-known algorithms. While the kernel matrix involves inner products in the feature space, the sample covariance matrix of the data requires outer products. Therefore, their spectral properties are tightly connected. This allows us to examine the kernel matrix through the sample covariance matrix in the feature space and vice versa. The use of kernels often involves a large number of features, compared to the number of observations. In this scenario, the sample covariance matrix is not well-conditioned nor is it necessarily invertible, mandating a solution to the problem of estimating high-dimensional covariance matrices under small sample size conditions. We tackle this problem through the use of a shrinkage estimator that offers a compromise between the sample covariance matrix and a well-conditioned matrix (also known as the "target") with the aim of minimizing the mean-squared error (MSE). We propose a distribution-free kernel matrix regularization approach that is tuned directly from the kernel matrix, avoiding the need to address the feature space explicitly. Numerical simulations demonstrate that the proposed regularization is effective in classification tasks.