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Allocation Requires Prediction Only if Inequality Is Low

Ali Shirali, Rediet Abebe, Moritz Hardt

Abstract

Algorithmic predictions are emerging as a promising solution concept for efficiently allocating societal resources. Fueling their use is an underlying assumption that such systems are necessary to identify individuals for interventions. We propose a principled framework for assessing this assumption: Using a simple mathematical model, we evaluate the efficacy of prediction-based allocations in settings where individuals belong to larger units such as hospitals, neighborhoods, or schools. We find that prediction-based allocations outperform baseline methods using aggregate unit-level statistics only when between-unit inequality is low and the intervention budget is high. Our results hold for a wide range of settings for the price of prediction, treatment effect heterogeneity, and unit-level statistics' learnability. Combined, we highlight the potential limits to improving the efficacy of interventions through prediction.

Allocation Requires Prediction Only if Inequality Is Low

Abstract

Algorithmic predictions are emerging as a promising solution concept for efficiently allocating societal resources. Fueling their use is an underlying assumption that such systems are necessary to identify individuals for interventions. We propose a principled framework for assessing this assumption: Using a simple mathematical model, we evaluate the efficacy of prediction-based allocations in settings where individuals belong to larger units such as hospitals, neighborhoods, or schools. We find that prediction-based allocations outperform baseline methods using aggregate unit-level statistics only when between-unit inequality is low and the intervention budget is high. Our results hold for a wide range of settings for the price of prediction, treatment effect heterogeneity, and unit-level statistics' learnability. Combined, we highlight the potential limits to improving the efficacy of interventions through prediction.
Paper Structure (47 sections, 10 theorems, 139 equations, 8 figures, 4 tables)

This paper contains 47 sections, 10 theorems, 139 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Our Results
  3. Individual-Level versus Unit-Level Allocations.
  4. Extension to Heterogeneous Treatment Effects.
  5. Extension to Learning Unit-Level Statistics.
  6. Summary.
  7. Related Work
  8. Our Model
  9. The Cost of Prediction.
  10. Intervention Effect.
  11. Individual-Level Allocation (ILA).
  12. Unit-Level Allocation (ULA).
  13. Dominance Notion.
  14. Illustrating the Key Idea
  15. Graphical Illustration
  16. ...and 32 more sections

Key Result

Corollary 5.2

For a sublinear budget $B = o(C)$, if the set of feasible unit profiles ${\mathcal{P}}$ does not exclude profiles with a Gini index of $G_\rho$ or less, as long as $G_\rho \ge \frac{B}{(1-q) C} = o(1)$, there exists ${\bm{\rho}}^* \in {\mathcal{P}}$ for which ULA is optimal.

Figures (8)

  • Figure 1: Sufficient conditions for a dominant unit-level allocation (green) and nondominated unit-level allocation (red). We define a high budget in relation to the cost of treating the whole population and a low budget in relation to the cost of prediction. See \ref{['fig:ula_regimes']} for a quantitative version of this figure.
  • Figure 2: ULA outperforms ILA in a simulated setting with high inequality. The treatment effect is $\delta=0.3$, and half of the units are treated, with $20\%$ of the budget required for prediction. Within each unit, individuals have independent beta-distributed welfare. They are sorted based on their welfare and plotted according to their position and welfare. A within-unit allocation with $q=0.3$ and $q'\approx0$ is considered, and individuals among the top $q$ fraction at each unit are faded. The values obtained by ILA and ULA from each unit are depicted with horizontal bars, and their total values compared. For ease of presentation, we assume ULA gets zero value from treating units above welfare $1-\delta.$
  • Figure 3: ULA outperforms ILA in a real-world high inequality setting. Eight school districts from the greater Los Angeles (LA) area are considered. Household income, in $10$ brackets, is used as a proxy for individual welfare. A within-unit allocation with $q=0.3$ and $q'\approx0$ is considered, where individuals among the top $q$ fraction at each unit appear faded. The values obtained by ILA and ULA from each unit are depicted with horizontal bars, and their overall values are compared.
  • Figure 4: In the special case of uniformly distributed units, each unit is represented by a dot. The first $K$ units (the targeted units) are indicated in blue.
  • Figure 5: Sufficient conditions for dominant ULA (green) and nondominated ULA (red) for $q \le \bar{\rho}$.
  • ...and 3 more figures

Theorems & Definitions (25)

  • Definition 2.1: Dominance notion
  • Definition 4.1: Effective within-unit allocation
  • proof
  • Remark 5.1
  • Corollary 5.2
  • Remark 5.3
  • proof
  • Example 6.3
  • Theorem 6.4: Sufficient conditions for a dominant ULA in case of a heterogeneous effect
  • Proposition D.1
  • ...and 15 more