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

1. Problem Statement

The paper addresses a critical gap in the deployment of AI-assisted interventions in service systems (e.g., healthcare, education, recruiting). Standard practice assumes that maximizing predictive accuracy (e.g., AUC, sensitivity) automatically leads to optimal system outcomes. However, this assumption fails in real-world scenarios characterized by two specific constraints:

1. Capacity Constraints: Service providers have a limited number of slots ($M$) to serve a population ($N$).
2. Noisy Compliance: AI tools act as "nudges" rather than direct assignments. Flagged individuals request service with a higher probability ($p_0 + \Delta P$) than unflagged individuals ($p_0$), but compliance is probabilistic.

The Core Conflict:
When capacity is limited, a "funnel" exists between algorithmic flagging and actual service delivery. Two competing forces arise:

• Underutilization: If the flagging threshold is too high (conservative), demand is lower than capacity, leaving service slots empty.
• Cannibalization: If the threshold is too low (aggressive), demand exceeds capacity. Under random allocation (common in practice due to fairness or logistical constraints), low-value requests from unflagged or low-scoring flagged individuals crowd out high-value requests, reducing the average efficacy of served cases.

The paper asks: How should one set the classification threshold and select the algorithm to maximize total system efficacy (sum of values of served individuals) given these constraints?

2. Methodology

A. Mathematical Model

The authors develop a stochastic model for $N$ individuals where:

• Request Behavior: An individual $i$ requests service with probability $p_0$ if unflagged and $p_0 + \Delta P$ if flagged.
• True Scores: Each individual has a latent true score $r_i$ representing the value of serving them.
• Algorithm: An algorithm $A$ predicts scores $\hat{r}_i$.
• Allocation: If total requests exceed capacity $M$, slots are allocated uniformly at random among requesters.

B. Fluid Approximation

To derive analytical insights, the authors introduce a Fluid Model that replaces stochastic variables with their expectations.

• Total expected requests: $N[p_0 + \Delta P(1-\tau)]$.
• Expected served requests: $\min(N[p_0 + \Delta P(1-\tau)], M)$.
• The objective function $\tilde{W}$ (System Efficacy) is the product of the number of served requests and the average efficacy per slot.
• Proposition 1: The fluid objective provides an upper bound on the stochastic objective and converges to it as $N, M \to \infty$.

C. Threshold Characterization

The authors analyze the trade-off between maximizing the number of served requests (utilization) and maximizing the average value of those requests (efficacy per slot). They define:

1. Score-Optimal Threshold ($\tau^*_{score}$): The threshold that maximizes the average efficacy per served slot (ignoring capacity limits).
2. Capacity-Matching Threshold ($\tau_c$): The threshold that generates exactly enough expected requests to fill capacity $M$.

3. Key Contributions

A. Characterization of the Optimal Threshold (Theorem 1)

The paper proves that the optimal threshold $\tau^*$ has a remarkably simple two-point structure:
$$\tau^* = \min(\tau^*_{score}, \tau_c)$$

• Interpretation: The optimal policy is the minimum of the threshold that maximizes per-slot value and the threshold that ensures full capacity utilization.
• Regimes:
  • Scarce Capacity: When capacity is tight relative to baseline demand, $\tau^* = \tau^*_{score}$. The system prioritizes serving high-value individuals, accepting that some capacity might be wasted if the optimal score threshold doesn't generate enough volume (though usually, it does).
  • Abundant Capacity: When capacity is high, $\tau^* = \tau_c$. The system lowers the threshold to ensure all slots are filled, even if it means serving slightly lower-value individuals.

B. Suboptimality of Standard Policies

• Prediction-Based Thresholds: Policies based solely on metrics like sensitivity or specificity (Definition 1) are generally suboptimal. They ignore operational parameters ($M, p_0, \Delta P$) and fail to adapt to capacity changes, leading to significant efficacy losses (up to 40% in simulations).
• Capacity-Matching Thresholds: While better than prediction-based thresholds, $\tau_c$ is also suboptimal when cannibalization is strong (i.e., when $p_0 > 0$ and capacity is scarce). It ignores the quality of the marginal requesters, leading to the crowding out of high-value cases.

C. Algorithm Selection: Operational AUC (OpAUC)

The paper demonstrates that AUC is misaligned with system efficacy in capacity-constrained settings.

• The Problem: AUC weights performance across all possible thresholds equally. However, an optimal deployment only operates within a specific range of thresholds (determined by capacity). An algorithm with a higher AUC might perform worse in the specific operational range where the system actually functions.
• The Solution: The authors propose Operational AUC (OpAUC).
  $$\text{OpAUC}(A) = \int \text{Efficacy at } \tau^*(\rho) \, d\mu(\rho)$$
  OpAUC weights the algorithm's performance only over the thresholds it will actually be deployed at, given the distribution of capacity constraints. It is proven to select the efficacy-optimal algorithm.

D. Robustness to Prioritization

The authors extend the analysis to scenarios where providers partially prioritize high-scoring requesters (mixture mechanism). They show that the two-point threshold derived under random allocation remains robust and effective across a wide range of prioritization levels, whereas fixed thresholds fail.

4. Results

A. Simulation Study (Sepsis Early Warning)

Using data from 8 hospitals in the NewYork-Presbyterian system (85,160 patient encounters):

• Threshold Optimization: Adjusting the threshold based on capacity and behavioral parameters (using the two-point rule) improved system efficacy by up to 40% compared to standard prediction-based thresholds.
• Algorithm Selection: The study compared the Epic Sepsis Model (AUC = 0.826) and a custom XGBoost model (AUC = 0.806).
  • Despite the lower AUC, the XGBoost model outperformed the Epic model by up to 22.8% in system efficacy when capacity was limited ($M/N \in [0.05, 0.15]$).
  • This occurred because XGBoost had a higher True Positive Rate (TPR) specifically in the high-risk threshold range relevant to the capacity constraints, a nuance missed by the aggregate AUC metric.
  • OpAUC correctly identified XGBoost as the superior choice in this regime.

B. Sensitivity Analysis

• The gap between optimal and naive thresholds is non-monotonic with respect to capacity.
• As baseline request probability ($p_0$) increases, cannibalization becomes more severe, making the capacity-matching threshold increasingly suboptimal.

5. Significance and Implications

1. Shift from Development to Implementation: The paper argues that the bottleneck in AI-assisted interventions is often not model accuracy, but deployment strategy. Practitioners can achieve massive gains (up to 40%) without retraining models, simply by adjusting thresholds based on operational constraints.
2. New Metric for Selection: It challenges the dominance of AUC in capacity-constrained environments. OpAUC provides a theoretically grounded, actionable metric for selecting algorithms that align with specific operational realities.
3. Practical Applicability: The framework requires only estimable operational parameters (capacity distribution, baseline request rates, and nudge lift), making it immediately applicable to existing systems in healthcare, education, and labor markets.
4. Theoretical Insight: It formalizes the "cannibalization" effect in AI deployment, showing that the optimal policy is a simple minimum of two interpretable thresholds, unifying the trade-off between utilization and quality.

In summary, the paper provides a rigorous framework proving that operational constraints fundamentally alter the definition of "optimal" AI deployment, necessitating a move away from pure predictive accuracy toward operationally-aware thresholding and selection metrics.