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Potential Outcome Rankings for Counterfactual Decision Making

Yuta Kawakami, Jin Tian

TL;DR

This work introduces two counterfactual decision rules, PoR and PoB, to rank and select actions based on individualized potential outcomes under uncertainty. It provides formal definitions, identification results under rank invariance, and Fréchet-type bounds without that assumption, along with plug-in estimators and convergence guarantees. Through illustrative examples, numerical experiments, and a real-world coagulation dataset, the paper demonstrates that PoR and PoB can yield different, nuanced recommendations compared to traditional RoE-based decisions, offering insights into personalized policy and treatment selection. The results highlight both the practical potential and the methodological challenges, including non-sharp bounds for general $K$ and increased estimation difficulty as the number of actions grows, pointing to avenues for refinement and broader application in causal decision-making and reinforcement learning contexts.

Abstract

Counterfactual decision-making in the face of uncertainty involves selecting the optimal action from several alternatives using causal reasoning. Decision-makers often rank expected potential outcomes (or their corresponding utility and desirability) to compare the preferences of candidate actions. In this paper, we study new counterfactual decision-making rules by introducing two new metrics: the probabilities of potential outcome ranking (PoR) and the probability of achieving the best potential outcome (PoB). PoR reveals the most probable ranking of potential outcomes for an individual, and PoB indicates the action most likely to yield the top-ranked outcome for an individual. We then establish identification theorems and derive bounds for these metrics, and present estimation methods. Finally, we perform numerical experiments to illustrate the finite-sample properties of the estimators and demonstrate their application to a real-world dataset.

Potential Outcome Rankings for Counterfactual Decision Making

TL;DR

This work introduces two counterfactual decision rules, PoR and PoB, to rank and select actions based on individualized potential outcomes under uncertainty. It provides formal definitions, identification results under rank invariance, and Fréchet-type bounds without that assumption, along with plug-in estimators and convergence guarantees. Through illustrative examples, numerical experiments, and a real-world coagulation dataset, the paper demonstrates that PoR and PoB can yield different, nuanced recommendations compared to traditional RoE-based decisions, offering insights into personalized policy and treatment selection. The results highlight both the practical potential and the methodological challenges, including non-sharp bounds for general and increased estimation difficulty as the number of actions grows, pointing to avenues for refinement and broader application in causal decision-making and reinforcement learning contexts.

Abstract

Counterfactual decision-making in the face of uncertainty involves selecting the optimal action from several alternatives using causal reasoning. Decision-makers often rank expected potential outcomes (or their corresponding utility and desirability) to compare the preferences of candidate actions. In this paper, we study new counterfactual decision-making rules by introducing two new metrics: the probabilities of potential outcome ranking (PoR) and the probability of achieving the best potential outcome (PoB). PoR reveals the most probable ranking of potential outcomes for an individual, and PoB indicates the action most likely to yield the top-ranked outcome for an individual. We then establish identification theorems and derive bounds for these metrics, and present estimation methods. Finally, we perform numerical experiments to illustrate the finite-sample properties of the estimators and demonstrate their application to a real-world dataset.

Paper Structure

This paper contains 19 sections, 9 theorems, 57 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Illustrating Example
  3. Notations and Backgrounds
  4. Parameter $\psi$ in Fay2018.
  5. Decision-Making Using PoR and PoB
  6. Probabilities of potential outcome ranking (PoR).
  7. Probability of achieving the best potential outcome (PoB).
  8. Example 1 (Homogeneous causal effects).
  9. Example 2 (Heterogeneous causal effects).
  10. Identification and Estimation of PoR and PoB
  11. Identification of PoR and PoB.
  12. Bounding PoR and PoB
  13. Estimation of Bounds.
  14. Numerical Experiments
  15. Three Candidate Actions
  16. ...and 4 more sections

Key Result

Proposition 1

Under SCM ${\cal M}$ and Assumption ASEXO2, the expected PO $\mathbb{E}[Y_{x}]$ is identifiable by $\mathbb{E}[Y|X={x}]$.

Figures (2)

  • Figure 1: Absolute errors of PoR estimates with 95% CIs (y-axis) vs. $K$ (x-axis).
  • Figure 2: Absolute errors of PoB estimates with 95% CIs (y-axis) vs. $K$ (x-axis).

Theorems & Definitions (18)

  • Proposition 1: Identifiability of RoE
  • Proposition 2: Identification of the unobserved PO for two POs
  • Definition 1: PoR
  • Definition 2: PoB
  • Lemma 1: Identification of the unobserved POs
  • Theorem 1: Identification of PoR
  • Theorem 2: Identification of PoB
  • Theorem 3
  • Theorem 4: Bounds of PoR
  • Theorem 5: Bounds of PoB
  • ...and 8 more