Eliciting Von Neumann-Morgenstern utility from discrete choices with response error

Abstract

We develop a preference elicitation method for a Von Neumann-Morgenstern (VNM)-type decision-maker from pairwise comparison data in the presence of response errors. We apply the maximum likelihood estimation (MLE) method to jointly elicit the non-parametric systematic VNM utility function and the scale parameter of the response error, assuming a Gumbel distribution. We incorporate structural preference information known in advance about the decision-maker's risk attitude through linear constraints on the utility function, including monotonicity, concavity, and Lipschitz continuity. Under discretely distributed lotteries, the resulting MLE problem can be reformulated as a convex program. We derive finite-sample error bounds between the MLE and the true parameters, and establish quantitative convergence of the MLE-based VNM utility function to the true utility function in the sense of the Kolmogorov distance under some conditions on the lotteries. These conditions may have potential applications in the design of efficient lotteries for preference elicitation. We further show that the optimization problem maximizing the expected MLE-based VNM utility is robust against the response error and estimation error in a probabilistic sense. Numerical experiments in a portfolio optimization application illustrate and support the theoretical results.