Linear Regression from Strategic Data Sources

Linear regression is a fundamental building block of statistical data analysis. It amounts to estimating the parameters of a linear model that maps input features to corresponding output data. In the classical setting where the precision of each data point is fixed, the famous Aitken/Gauss-Markov theorem in statistics states that the best linear unbiased estimator is generalized least square (GLS). In modern data science, however, one often faces strategic data sources, that is data provided by individuals who incur a cost for providing high-precision data. In this paper, we study a setting in which features are public but individuals choose the precision of the output data they provide. Contrary to the existing literature, we do not consider monetary payments but rather assume that individuals benefit from the outcome of the estimation. We model this scenario as a game where individuals minimize a cost comprising two components: (a) a (local) disclosure cost for providing high-precision data; and (b) a (global) estimation cost representing the inaccuracy in the linear model estimate. In this game, the linear model estimate is a public good that benefits all individuals. We establish that this game has a unique non-trivial Nash equilibrium. We study the efficiency of this equilibrium and we prove tight bounds on the price of stability for a large class of disclosure and estimation costs. Finally, we study the estimator accuracy achieved at equilibrium. We show that Aitken's theorem no longer holds with strategic data sources in general. It holds if individuals have identical disclosure costs (up to a multiplicative factor). When individuals have non-identical costs, we derive a bound on the improvement of the equilibrium estimation cost that can be achieved by deviating from GLS, under mild assumptions on the disclosure cost functions.