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The Best of Many Robustness Criteria in Decision Making: Formulation and Application to Robust Pricing

Jerry Anunrojwong, Santiago R. Balseiro, Omar Besbes

Abstract

In robust decision-making under non-Bayesian uncertainty, different robust optimization criteria, such as maximin performance, minimax regret, and maximin ratio, have been proposed. In many problems, all three criteria are well-motivated and well-grounded from a decision-theoretic perspective, yet different criteria give different prescriptions. This paper initiates a systematic study of overfitting to robustness criteria. How good is a prescription derived from one criterion when evaluated against another criterion? Does there exist a prescription that performs well against all criteria of interest? We formalize and study these questions through the prototypical problem of robust pricing under various information structures, including support, moments, and percentiles of the distribution of values. We provide a unified analysis of three focal robust criteria across various information structures and evaluate the relative performance of mechanisms optimized for each criterion against the others. We find that mechanisms optimized for one criterion often perform poorly against other criteria, highlighting the risk of overfitting to a particular robustness criterion. Remarkably, we show it is possible to design mechanisms that achieve good performance across all three criteria simultaneously, suggesting that decision-makers need not compromise among criteria.

The Best of Many Robustness Criteria in Decision Making: Formulation and Application to Robust Pricing

Abstract

In robust decision-making under non-Bayesian uncertainty, different robust optimization criteria, such as maximin performance, minimax regret, and maximin ratio, have been proposed. In many problems, all three criteria are well-motivated and well-grounded from a decision-theoretic perspective, yet different criteria give different prescriptions. This paper initiates a systematic study of overfitting to robustness criteria. How good is a prescription derived from one criterion when evaluated against another criterion? Does there exist a prescription that performs well against all criteria of interest? We formalize and study these questions through the prototypical problem of robust pricing under various information structures, including support, moments, and percentiles of the distribution of values. We provide a unified analysis of three focal robust criteria across various information structures and evaluate the relative performance of mechanisms optimized for each criterion against the others. We find that mechanisms optimized for one criterion often perform poorly against other criteria, highlighting the risk of overfitting to a particular robustness criterion. Remarkably, we show it is possible to design mechanisms that achieve good performance across all three criteria simultaneously, suggesting that decision-makers need not compromise among criteria.
Paper Structure (34 sections, 8 theorems, 33 equations, 2 figures, 5 tables)

This paper contains 34 sections, 8 theorems, 33 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Summary of Main Contributions
  3. Unified Analysis.
  4. Performance of Focal Mechanisms.
  5. The best of many criteria.
  6. Related Work
  7. Robust Pricing.
  8. Robust Decision Making Under Uncertainty.
  9. Robust Optimization.
  10. Best of Many Robustness Criteria Formulation: Robust Pricing
  11. Seller's mechanisms.
  12. Robust criteria.
  13. Focal mechanisms.
  14. Performance across robust criteria.
  15. The best of many robustness criteria.
  16. ...and 19 more sections

Key Result

Proposition 1

Fix a class of distributions $\mathcal{F}$ as in equation (eqn:uncertainty-set). Suppose that the mapping $\lambda \mapsto R_{\lambda}^*(\mathcal{F})$ is known. Then one can directly obtain the maximin revenue, minimax regret and maximin ratio as follows: $\textnormal{MinimaxRegret}(\mathcal{F}) = R

Figures (2)

  • Figure 1: Relative performance of different mechanisms $\Phi \in \{\Phi^*_\textnormal{Regret}, \Phi^*_\textnormal{Revenue}, \Phi^*_\textnormal{Ratio}, \Phi^*_\textnormal{All}\}$ over all criteria as a function of the key parameter, in four classes of uncertainty sets: known mean $\mathcal{F}^{\textnormal{mean}}_{\mu}$, known mean and variance $\mathcal{F}^{\textnormal{mean+var}}_{\mu,\sigma}$, known median $\mathcal{F}^{\textnormal{median}}_{\nu}$, and known lower bound $\mathcal{F}^{\textnormal{LB}}_a$.
  • Figure 2: Comparison of different price distributions $\Phi \in \{\Phi^*_\textnormal{Revenue}, \Phi^*_\textnormal{Regret}, \Phi^*_\textnormal{Ratio}, \Phi^*_\textnormal{All}\}$ for prototypical uncertainty sets in each of the four classes. Each number in parenthesis is $\textnormal{RelPerf}(\Phi,\textnormal{All},\mathcal{F})$ for the corresponding $\Phi$ and $\mathcal{F}$.

Theorems & Definitions (10)

  • Proposition 1: our-paper-ec-a-b
  • Proposition 2: $\lambda$-regret Linear Programs
  • Proposition 3: Linear Program for Cross Regret
  • Proposition 4
  • Proposition 5: Mean Information
  • Theorem 1: Robust Pricing: Best of Many Criteria
  • Proposition A-1: Equivalent Definition of Relative Performance Over All Objectives
  • proof : Proof of Proposition \ref{['prop:relperf-equivalent']}
  • proof : Proof of Proposition \ref{['prop:minimax-lmbd-regret-lp']}
  • Proposition C-2: shixin-ratio