Deployment of AI-Assisted Interventions: Capacity Constraints and Noisy Compliance

Carri W. Chan; Yi Han; Hannah Li; Benjamin L. Ranard

stat.MEApr 15, 2026

Deployment of AI-Assisted Interventions: Capacity Constraints and Noisy Compliance

Authors: Carri W. Chan, Yi Han, Hannah Li, Benjamin L. Ranard

Organizations: Decision, Risk, and Operations, Columbia Business School · Department of Statistics, Columbia University · Division of Pulmonary, Allergy, and Critical Care Medicine, Department of Medicine, Vagelos College of Physicians and Surgeons, Columbia University

Abstract

AI tools increasingly drive targeted interventions in various service settings, including healthcare, education, and public services. Algorithms score individuals, trigger outreach to those above a threshold (e.g., high-risk or high-value), and encourage them to request service; then providers deliver service to those who request. Much of the work in this area has focused on improving predictive accuracy, implicitly assuming that better predictions lead to better outcomes. We show that predictions are only one component of a larger service system: when service capacity is limited and behavioral responses to outreach are probabilistic, system efficacy depends on operational forces that predictive accuracy does not capture. In such settings, the optimal score threshold must balance two effects: ensuring all capacity is filled (utilization) and, when capacity is constrained, preventing low-value requests from crowding out high-value ones (cannibalization). We characterize the optimal threshold and prove that thresholds based solely on predictive accuracy are generally suboptimal. Further, algorithm selection metrics such as AUC can be misaligned with operational performance: they weight all thresholds equally, while optimal deployment uses a subset of thresholds that depends on both capacity and compliance behavior. We introduce a new metric, Operational AUC (OpAUC), and show that it identifies the efficacy-optimal algorithm. Finally, we conduct a case study on sepsis early warning data that illustrates the magnitude of the improvement available from better threshold selection and shows that a predictor with lower AUC can achieve higher system efficacy under optimal deployment.

Figures & tables

Figure 1 : AI-assisted intervention with noisy compliance and capacity constraints.

Figure 2 : Decision tree of optimal thresholding policies across operational regimes. For simplicity, we write

\tau^{*A}_{\mathrm{score}}

for

\tau^{*A}_{\mathrm{score}}(p_{0},\Delta_{P})

\tau_{c}

for

\tau_{c}(M/N\mid p_{0},\Delta_{P})

\tau_{c,I}

for

\tau_{c,I}(M_{I}/N)

and

\tau^{*A}

for

\tau^{*A}(M/N\mid p_{0},\Delta_{P})

. Each leaf is read in the parameter context defined by its branch.

Figure 3 : Visualizations of fluid approximation accuracy. The fluid approximation consistently upper bounds the true objective (left panel), and the error converges to 0 as $M$ and $N$ scale up (right panel). Parameters: $M/N=0.2$ , $p_{0}=0.1$ , $\Delta_{P}=0.5$ and $\hat{r}=r\sim 0.7\mathrm{Beta}(2,10)+0.3\mathrm{Beta}(8,2)$ .

Figure 4 : Requests decomposition under varying outreach thresholds. Each row corresponds to a setting from the motivating example: Setting 1 (

\tau=0.9

) shows underutilization; Setting 2 (

\tau=0.8

) matches capacity exactly; Setting 3 (

\tau=0.7

) over-expands outreach, shifting capacity toward targeted requests; Setting 4 (

\tau=0.6

) further expands outreach but increases cannibalization from lower-value individuals.

Figure 5 : Optimal thresholding policy as the capacity ratio $\tfrac{M}{N}$ or the baseline request $p_{0}$ varies. Parameters: $N=1000,M=200,\Delta_{P}=0.5,p_{0}=0.1$ and $\hat{r}=r\sim 0.7\mathrm{Beta}(2,10)+0.3\mathrm{Beta}(8,2).$

Figure 6 : Thresholding behavior and performance under varying capacity and baseline request probability. The top row varies the capacity ratio $M/N$ with the baseline request probability fixed at $p_{0}=0.1$ , while the bottom row varies the baseline request probability $p_{0}$ with capacity fixed at $M=200$ . Each column reports (from left to right) the threshold policy $\tau$ , the resulting objective value $\tilde{W}(\tau)$ , and the relative gap $(\tilde{W}(\tau^{*})-\tilde{W}(\tau))/\tilde{W}(\tau^{*})$ . Curves compare the optimal threshold $\tau^{*}$ , the capacity-matching threshold $\tau_{c}$ , and several prediction-based thresholds. Parameters: $N=1000$ , $\Delta_{P}=0.5$ , and $\hat{r}=r\sim 0.7\,\mathrm{Beta}(2,10)+0.3\,\mathrm{Beta}(8,2)$ .

Figure 7 : Performance of thresholding policies under score-based prioritization with weight $\beta_{1}$ . Each column reports (from left to right) the threshold policies, system efficacy, and relative gap with respect to the optimal threshold under prioritization, computed via exhaustive grid search over $\tau\in[0,1]$ . The two-point threshold derived under random allocation ( Theorem 1 ) closely tracks the prioritization optimum across a wide range of $\beta_{1}$ , while fixed thresholds fail at one or both extremes. Parameters: $N=1000$ , $M=200$ , $p_{0}=0.1$ , $\Delta_{P}=0.5$ , $r\sim 0.7\,\mathrm{Beta}(2,10)+0.3\,\mathrm{Beta}(8,2)$ .

Figure 8 : Misalignment between AUC and system efficacy. Algorithm $B$ achieves higher overall AUC, but algorithm $A$ dominates over the operationally relevant threshold range $[0,\tau^{*A}_{\mathrm{score}}]$ (shaded).

Panel of Figure 9 — Figure 9 : Comparison of Epic and XGBoost sepsis prediction models. Panel ( 9(a) ) shows ROC curves, with Epic achieving higher AUC ( $0.826$ vs. $0.806$ ). Panel ( 9(b) ) shows TPR as a function of threshold $\tau$ , illustrating that the curves cross. The score-optimal threshold is $\tau^{*}_{\mathrm{score}}=0.954$ for XGBoost and $\tau^{*}_{\mathrm{score}}=0.920$ for Epic.

Panel of Figure 10 — Figure 10 : Threshold and algorithm comparison under varying capacity. Panels ( 10 ) and ( 10 ) report relative gaps in system efficacy under different thresholding policies, compared to the efficacy-optimal threshold, for Epic and XGBoost respectively; suboptimal thresholds can lead to efficacy losses of up to $40\%$ . Panel ( 10 ) compares algorithms under optimal thresholding: despite having lower overall AUC, XGBoost achieves higher system efficacy than Epic when $M\in[10,30]$ , with differences exceeding one standard error. Parameters: $N=200$ , $p_{0}=0.1$ , $\Delta_{P}=0.5$ . Shaded regions indicate $\pm$ one standard error across trials.

Figure 11 : Robustness of thresholding policies to score-based prioritization in the NYP sepsis case study. The left panel corresponds to $\beta_{1}=0.5$ and the right panel corresponds to $\beta_{1}=1.0$ . The optimal threshold under random allocation remains near-optimal across all capacity levels and both algorithms, where the optimal threshold under prioritization is computed via exhaustive grid search over $\tau\in[0,1]$ . Fixed prediction-based thresholds ( $\tau=0.8$ , $\tau=0.6$ ) incur losses up to 50% at low capacity ratios, irrespective of $\beta_{1}$ . Parameters: $N=200$ , $p_{0}=0.1$ , $\Delta_{P}=0.5$ . Shaded regions indicate $\pm$ one standard error across trials.

Figure 12 : Performance of threshold strategies as a function of the correlation $\rho(p_{0},\mu)$ between baseline request probability and mean risk score across $K=3$ subpopulations. Parameters: $N=1000$ , $M=175$ , $\bar{p}_{0}=0.1$ , $\Delta_{P}=0.5$ , $r\sim 0.7\,\mathrm{Beta}(2,10)+0.3\,\mathrm{Beta}(8,2)$ , $\hat{r}=r+\mathcal{N}(0,0.1^{2})$ .

Figure 13 : Performance of threshold strategies as a function of the HTE magnitude $\eta$ . Parameters: $N=1000$ , $p_{0}=0.1$ , $K+1=3$ equal-mass regions with $(\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3})=(0,1/3,2/3,1)$ , $r\sim 0.7\,\mathrm{Beta}(2,10)+0.3\,\mathrm{Beta}(8,2)$ , $\hat{r}=r+\mathcal{N}(0,0.1^{2})$ .

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 14 : System efficacy as a function of the bilevel thresholds $(\tau_{W},\tau_{S})$ under varying total capacity $M\in\{200,300\}$ and strong-tier capacity $M_{S}\in\{200,\infty\}$ . The color gradient represents system efficacy, with darker shading indicating higher values. The star marks the bilevel optimal $(\tau_{W}^{*},\tau_{S}^{*})$ , and the remaining markers show benchmark policies. Dashed vertical lines indicate the capacity-matching threshold $\tau_{c,S}$ and the score-optimal threshold $\tau^{*}_{\mathrm{score},S}$ ; in the binding regime (right panels), the dot-dashed horizontal line indicates $\tilde{\tau}^{*}_{\mathrm{score},W}$ . Parameters: $N=1000$ , $\Delta_{P}^{W}=0.3$ , $\Delta_{P}^{S}=0.6$ , $r\sim 0.7\,\mathrm{Beta}(2,10)+0.3\,\mathrm{Beta}(8,2)$ , $\hat{r}=r+\mathcal{N}(0,0.1^{2})$ .

Figure 15 : Illustration of bilevel interventions in the NYP sepsis case study. Panels (a) and (b) report results for Epic under strong-tier capacity $M_{S}=20$ (binding) and $M_{S}=\infty$ (non-binding); panels (c) and (d) report the analogous results for XGBoost. Both benchmark policies fix the weak threshold at $\tau_{W}=0$ and differ in the strong threshold: $\tau_{S}=\tau_{c,S}(M/N)$ (capacity-matching) and $\tau_{S}=0.8$ (fixed prediction-based). Relative gap is measured against the bilevel optimum $(\tau_{W}^{*},\tau_{S}^{*})$ characterized in Theorem 5 . Parameters: $N=200$ , $p_{0}=0.1$ , $\Delta_{P}^{W}=0.1$ , $\Delta_{P}^{S}=0.5$ . Shaded regions indicate $\pm$ one standard error across trials.

Figure 16 : Convergence of the moment estimators $\hat{p}_{0}$ (left) and $\hat{\Delta}_{P}$ (right) as the size $N$ of the request log grows. Solid line: mean across $30$ replications; shaded band: $\pm 1$ standard deviation; dashed line: true value. Parameters: $p_{0}=0.1$ , $\Delta_{P}=0.5$ , $r\sim 0.7\,\mathrm{Beta}(2,10)+0.3\,\mathrm{Beta}(8,2)$ .

Figure 17 : Convergence of the plug-in threshold (left) and the relative system efficacy (right) as $N$ grows. Left: mean of $\hat{\tau}^{*A}$ (solid) compared with the population optimum $\tau^{*A}$ (dashed). Right: ratio of the fluid objective at $\hat{\tau}^{*A}$ to its value at $\tau^{*A}$ , which approaches $1$ from below. Parameters: $M/N=0.2$ ; remaining parameters as in Figure 16 .

Figure 18 : ROC curves (left) and TPR curves (right) for three prediction models: XGBoost(1) (AUC $=0.806$ ), Epic (AUC $=0.826$ ), and XGBoost(2) (AUC $=0.854$ ). Dots on the TPR curves mark the optimal score threshold $\tau^{*}_{\mathrm{score}}$ for each model (0.954, 0.920, and 0.957, respectively).

Figure 19 : Threshold and algorithm comparison under varying capacity, extending Figure 10 to include XGBoost(2). Panels (a)–(c) report relative gaps in system efficacy under different thresholding policies, compared to the efficacy-optimal threshold, for Epic, XGBoost(1), and XGBoost(2) respectively; the capacity-matching policy achieves near-zero gap across all capacities, while fixed thresholds incur up to 40% relative loss at low capacity. Panel (d) reports the absolute number of true positives under optimal thresholding for all three models; despite having the highest overall AUC, XGBoost(2) does not outperform XGBoost(1) at any capacity level. Parameters: $N=200$ , $p_{0}=0.1$ , $\Delta_{P}=0.5$ . Shaded regions indicate $\pm$ one standard error across trials.

Figure 20 : Operational AUC curves for the three models given $\mu=\mathrm{Uniform}(M^{L}/N,M^{U}/N)$ . Despite XGBoost(2) having the highest overall AUC, it does not achieve the highest operational AUC. Parameters: $N=200$ , $p_{0}=0.1$ , $\Delta_{P}=0.5$ .

Variable Type	Variables
Demographics	Age, Gender, Race
Encounter Information	Admission Source, Admission Type
Comorbidities	Congestive Heart Failure, Valvular, Pulmonary circulation disorders, Peripheral vascular disease, Hypertension, Paralysis, Other neurological disorders, Chronic pulmonary disease, Diabetes without chronic complications, Diabetes with chronic complications, Hypothyroidism, Renal failure, Liver disease, Chronic peptic ulcer disease, HIV, Lymphoma, Metastatic cancer, Tumor, Rheumatoid arthritis, Coagulation deficiency, Obesity, Weight loss, Blood loss, Anemia, Alcohol abuse, Drug abuse, Psychoses, Depression, Leukemia
Laboratory Values	Albumin, ALT, Ammonia, AST, Bandemia, Bicarbonate, Bilirubin, BUN, CK-MB, Creatine Kinase, CRP, D-Dimer, ESR, Fibrinogen, Glucose, Hematocrit, INR, Lactate, LDH, Magnesium, PCO2, pH, Platelets, PO2, Potassium, Serum Creatinine, Sodium, Troponin, WBC
Vital Signs	BP, Pulse, Pulse Oximetry, Respiratory Rate, Temperature, Weight
Medications	Antibiotics, Benzodiazepines, Chemotherapy, Heparins, Immunosuppressants, Insulins, IV Fluids, Opioids, Steroids, Vasopressors

Appendix

Table 1: Summary of covariates used in the XGBoost model.

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