Error Bounds of Statistical Estimators for Utility Elicitation

We consider preference elicitation in Bradley-Terry-Luce (BTL) model with possibly nonlinear parametric multivariate utility function. The set of selected pairwise questionnaires is non-uniform, deterministic, and otherwise arbitrary over a collection of d alternatives, provided that it satisfies a joint identifiability condition. We presume minimax lower bounds under the standard bounded dynamic range condition, and would like to understand how Fisher information geometry, as in the classic non-asymptotic theory, underpins the intrinsic difficulty of the estimation problem in finite sample regime. We further wish to identify a design-dependent critical sample-size threshold above which the unconstrained canonical maximum likelihood estimator exists and is unique with high probability. The overall target of the project aims at providing a unified non-asymptotic theory for parametric utility elicitation and revealing how the questionnaire structure determines the statistical efficiency under the BTL model.