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

Bo Chen, Jia Liu

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.

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.

Paper Structure

This paper contains 20 sections, 17 theorems, 111 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. VNM-type utility modeling with response error
  3. The VNM utility model with response error
  4. Choice probability under pairwise lottery comparisons
  5. Preference elicitation from pairwise comparison data
  6. Statistical inference by MLE
  7. Solution to the MLE problem \ref{['eq:MLE-origin']}
  8. Existence and uniqueness of the MLE solution
  9. Statistical consistency of MLE
  10. Interpretation and implications of the statistical results
  11. MLE-based VNM utility maximization with application to portfolio optimization
  12. Preference-at-Risk against response error
  13. Preference-Robust-at-Risk against response error and estimation error
  14. Sample average approximation
  15. Numerical test
  16. ...and 5 more sections

Key Result

Proposition 1

Given the set of breakpoints $\mathbb{Y}$, the ambiguity set of VNM utility functions $\mathcal{U}_c$, and its subset $\mathcal{U}_N$ of piecewise-linear lower approximation utility functions, the optimal values of the following two problems are equal:

Figures (4)

  • Figure 1: An illustrative example of $\hat{u}_{\rm MLE}$ as the pointwise lower bound of $\mathcal{U}^{*}_c$ when $N=5$ and $\bar{b}=1$.
  • Figure 2: $\ell_2$ and $\ell_{\infty}$statistical errors between $(\hat{\alpha}^{\rm MLE}, \sigma)$ and $(\alpha^*,\sigma^*)$ with optimized $\hat{\sigma}_{\rm MLE}$, or with fixed but misspecified$\sigma = 1$ and $\sigma = 100$, as the number of pairwise comparison queries increases.
  • Figure 3: $\ell_2$ and $\ell_{\infty}$statistical errors of MLE-based utility elicitation for the four models with different levels of structural preference information about the DM's true VNM utility function, as the number of pairwise comparison queries increases.
  • Figure 4: $\ell_2$ and $\ell_\infty$ statistical errors of MLE-based utility elicitation under datasets with $\lambda_{\text{min}}(\Sigma_D)=0$ and $\lambda_{\text{min}}(\Sigma_D)>0$, as the number of pairwise comparison queries increases.

Theorems & Definitions (40)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1: Piecewise-linear Lower Approximation (PLA)
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark 4
  • Proposition 2
  • ...and 30 more