Precedence-Aware Resource Allocation: Extending the AUTOC Framework to Multi-Level Treatments

Olin E. Gilster; Shengpu Tang

cs.LGOct 26, 2026

Precedence-Aware Resource Allocation: Extending the AUTOC Framework to Multi-Level Treatments

Authors: Olin E. Gilster, Shengpu Tang

Organizations: Emory University, USA

Abstract

Many real-world decision problems require allocating limited resources across individuals to maximize total benefit often through interventions that vary in intensity rather than binary treat-or-not decisions. In these settings, higher levels treatments require first applying lower levels of treatment, inducing a precedence constraint that complicates allocation under a fixed budget. While the causal inference literature has largely emphasized accurate estimation of conditional average treatment effects (CATE), recent work has shown that estimation accuracy alone does not guarantee effective resource allocation, motivating ranking-based evaluation frameworks. The Area Under the Targeting Operating Characteristic (AUTOC) provides an effective way to evaluate prioritization rules across budget levels, but existing formulations are limited to binary treatments. We introduce Multi-Level AUTOC (ML-AUTOC), which is a generalization for AUTOC designed for multi-level treatment settings with precedence. We also propose precedence-aware policies for mapping true marginal treatment effects to feasible treatment assignments under fixed budget constraints. Our results show that in the setting of multi-level treatments, maximizing realized treatment effects may require re-allocation of treatments across individuals.

Figures & tables

$\delta_{i}^{(\ell)}$	$\ell=1$	$\ell=2$
$x_{1}$	1	5
$x_{2}$	2	3

Table 1: True Marginal Benefit table

Figure 1: Toy example ML-TOC Graph

Figure 2: Average TOC curves across budgets for the strictly increasing marginal benefit dataset. Curves are averaged over 100 datasets and plotted relative to a random baseline.

\pm

one standard deviation error bars are plotted but too small to be visually distinguishable.

Policy	ML-AUTOC (mean $\pm$ std)
PGP	1.666352 $\pm$ 0.003744
IFP	0.941827 $\pm$ 0.009646
LFP	-0.154488 $\pm$ 0.003209
PAG	0.940726 $\pm$ 0.009646
PAO	0.941828 $\pm$ 0.009646

Table 2: ML-AUTOC Values for Strictly Increasing Marginals

Figure 3: Average TOC curves across budgets for the random marginal benefit dataset. Curves are averaged over 100 datasets and plotted relative to a random baseline.

\pm

one standard deviation error bars are plotted but too small to be visually distinguishable.

Policy	ML-AUTOC (mean $\pm$ std)
PGP	0.285943 $\pm$ 0.005926
IFP	0.141575 $\pm$ 0.004947
LFP	0.096445 $\pm$ 0.003341
PAG	0.141024 $\pm$ 0.004727
PAO	0.209599 $\pm$ 0.004183

Table 3: ML-AUTOC Values for Random Marginals

Policy	ML-AUTOC (mean $\pm$ std)
PGP	0.350069 $\pm$ 0.003820
IFP	-0.368648 $\pm$ 0.009757
LFP	0.348233 $\pm$ 0.003348
PAG	0.350069 $\pm$ 0.003820
PAO	0.350069 $\pm$ 0.003820

Table 4: ML-AUTOC Values for Strictly Decreasing Marginals

Policy	ML-AUTOC (mean $\pm$ std)
PGP	1.901681 $\pm$ 0.019583
IFP	1.394540 $\pm$ 0.028555
LFP	0.003476 $\pm$ 0.008919
PAG	1.392382 $\pm$ 0.028573
PAO	1.394548 $\pm$ 0.028554

Table 5: ML-AUTOC Values for Heterogeneous Strictly Increasing Marginals

Figure 4: Average TOC curves across budgets for the heterogeneous strictly increasing marginal benefit dataset. Curves are averaged over 100 datasets and plotted relative to a random baseline.

\pm

one standard deviation error bars are plotted but too small to be visually distinguishable.

Dataset	$\%\,$ Budgets PAO $>$ IFP	$\%\,$ Budgets PAO $>$ PAG
Strictly Increasing	$45.41\pm 31.32$	$87.65\pm 7.82$
Random	$96.70\pm 3.15$	$99.55\pm 0.26$
Strictly Decreasing	$99.94\pm 0.02$	$44.44\pm 29.59$
Heterogeneous Increasing	$47.92\pm 26.15$	$94.17\pm 3.83$

Table 6: Frequency with which PAO strictly outperforms baseline policies across budgets.

Figure 5: Empirical runtime scaling of ranking policies as the number of individuals

n

increases, measured on strictly increasing marginal benefit datasets with

L=4

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

$\delta_{i}^{(\ell)}$	$\ell=1$	$\ell=2$
$x_{1}$	1	5
$x_{2}$	2	3

Appendix

Table 7: True marginal benefit matrix (see also Table 1 )

Policy	Runtime (Big-Oh)
PGP	$O(nL\log(nL))$
IFP	$O(nL+n\log n)$
LFP	$O(nL\log n)$
PAG	$O(nL\log n)$
PAO	$O(n^{2}L^{2})$

Appendix

Table 8: Asymptotic runtime of ranking and precedence-aware policies as a function of the number of individuals

n

and treatment levels

L

Figure 6: Average TOC curves across budgets for the strictly decreasing marginal benefit dataset. Curves are averaged over 100 datasets and plotted relative to a random baseline.

\pm

one standard deviation error bars are plotted but too small to be visually distinguishable.

$n$	PGP	IFP	LFP	PAG	PAO
500	0.0008	0.0009	0.0009	0.0041	0.6721
1,000	0.0017	0.0017	0.0017	0.0086	2.6607
2,000	0.0033	0.0033	0.0033	0.0178	10.7443
4,000	0.0069	0.0069	0.0069	0.0483	43.1270
8,000	0.0143	0.0135	0.0135	0.1529	176.8498

Appendix

Table 9: Runtime (seconds) of ranking policies as a function of dataset size

n

on strictly increasing marginal benefit datasets with

L=4

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Open paper AskPDF arXiv

Public service programs often allocate limited resources under uncertainty about their benefits, creating a need for randomization to support credible evaluation. In practice, however, applicants commonly enter waitlists where resources are prioritized toward individuals judged to have higher need through tiered priority queues, making direct randomization difficult. Motivated by this, we develop an experimental design framework for learning treatment effects while treating those most in need where incoming applicants are randomized into priority queues based on their assessed risk scores. Treatments are then provided across queues in priority order and first-in-first-out within queue as budget becomes available. Our contributions are two-fold. First, we characterize what causal effects are identified under this priority-queue allocation. When arrivals are exogenous, treatments are conditionally randomized, and hence standard estimands are identified; when arrivals are endogenous, queue randomization instead provides an instrument for treatment, identifying local treatment effects induced by the queuing process. Second, we develop optimized queue-assignment designs that trade off statistical efficiency against prioritizing higher-need applicants. We show in the process that, despite dependence in treatment assignments induced by the design, usual iid efficiency bounds remain well-justified design objectives. We illustrate the proposed designs using data from a housing allocation program in a large U.S. county.

Abstract

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