Centralized Allocation Module

• Centralized allocation module is a top-level component that assigns resources to agents based on global state and fairness criteria.
• It leverages logic-driven optimization and utility functions that incorporate factors like waiting time and urgency to balance efficiency and safety.
• The module integrates within a hierarchical control structure, separating global decision-making from decentralized real-time execution.

A Centralized Allocation Module refers to a top-level, logic-driven or optimization-based component in hierarchical control frameworks that, through global state access or consensus, makes discrete resource-assignment or access-control decisions for a set of agents, vehicles, or computational entities. Its primary function is to enforce fairness, efficiency, or other system-level objectives that cannot be robustly or efficiently achieved through purely decentralized or local methods. Centralized allocation modules appear in a range of multi-agent systems, from real-time traffic control to federated learning over wide-area networks, and commonly form the apex of a layered architecture, with lower levels dedicated to tracking, execution, or detailed constraint satisfaction.

1. Core Functions and Mathematical Structure

The centralized allocation module is responsible for determining, at regular intervals, an assignment of access rights or resources to one or more agents from a global candidate set. Let $\mathcal{V}$ denote the set of agents (e.g., vehicles in an intersection or participants in federated learning). At time $t$, the module computes for each $i \in \mathcal{V}$ a utility or payoff $U_i(t)$ according to a function reflecting priority, access history, urgency, or system objectives. The allocation problem is then cast as

$A_t = \arg\max_{i \in \mathcal{E}_t} U_i(t)$

where $\mathcal{E}_t$ is a set of eligible candidates, possibly subject to exclusion rules encoding recent access history or minimum separation requirements. Utility functions can incorporate convex combinations of factors such as waiting time, recent allocations, urgency, and domain-specific fairness or efficiency terms (e.g., speed-deviation bonuses for vehicles).

A representative form, seen in fairness-aware traffic control (Shi et al., 8 Nov 2025), is the "inequity-aversion utility": $$U_i(t) = p_i(t) - \frac{\beta_1}{N-1} \sum_{j \neq i} \max(p_j - p_i, 0) - \frac{\beta_2}{N-1} \sum_{j \neq i} \max(p_i - p_j, 0) + \delta v_i(t)$$ with $p_i(t)$ a convex combination of recent-control ratio, waiting-time factor, and urgency. This form explicitly penalizes both disadvantageous and advantageous inequity, and can be evaluated in $O(N)$ time per step.

2. Hierarchical Control Architecture

The centralized allocation module occupies the top layer of a hierarchical system. Its output—a discrete assignment (e.g., which agent may proceed at a traffic intersection, or which tasks receive bandwidth in a federated learning round)—is passed down to lower layers which focus on trajectory tracking, constraint enforcement, or fine-grained scheduling.

This structure decouples global, combinatorial or logic-based objectives (e.g., fairness under complex historical, social, or operational constraints) from the continuous, real-time task of control or execution. For example, in intersection management, the authorized vehicle executes a reference trajectory computed by a lower-level controller, which may use LQR and high-order control barrier functions to guarantee tracking accuracy and enforce formal safety (Shi et al., 8 Nov 2025). In federated edge learning, the central allocation may determine resource slices or aggregation weights, while lower layers execute communication protocols and update models (Huang et al., 5 Aug 2024).

The separation ensures that non-differentiable or global fairness rules reside in a tractable optimization layer, while real-time responsiveness is achieved by lighter-weight, decentralized, or soft-constrained methods.

3. Fairness, Efficiency, and Tradeoffs

Centralized allocation modules provide a direct mechanism for encoding and enforcing sophisticated group-level fairness or efficiency-reliability tradeoffs. Key mathematical measures include:

• Jain’s Fairness Index: $JFI(t)=\frac{(\sum c_i)^2}{N\sum c_i^2}$; $JFI\to 1$ is perfect fairness.
• Gini Coefficient: quantifies pairwise inequity (0 = perfect equity).
• Statistical Parity Gap: $$|\mathbb{P}(\hat{y}=1 \mid s=1) - \mathbb{P}(\hat{y}=1 \mid s=0)|$$ in GNN fairness-aware systems (Yang et al., 27 Oct 2025).

Allocation modules are often tuned to maximize system utility (e.g., total throughput, minimal total delay) subject to fairness constraints or regularizers, or to minimize the worst-case agent utility under resource contention. Explicit tradeoff parameters (e.g., $\lambda$ in joint loss functions) allow practitioners to navigate between optimal fairness and maximal efficiency.

Distinct approaches exist:

• In real-time safety-critical systems, the module prioritizes fairness while guaranteeing zero collision, with observed tradeoffs quantifiable in reduced maximum throughput or marginally increased minimal inter-agent separation (Shi et al., 8 Nov 2025).
• In resource-constrained distributed learning, weighted reward shaping penalizes deviations from mean task accuracy, driving tasks to converge at similar rates without sacrificing average accuracy (Huang et al., 5 Aug 2024).

4. Algorithmic Realizations and Computational Properties

Centralized allocation modules are typically implemented as single-step optimizers without iterative solvers, relying on precomputed or lightweight computations to ensure real-time feasibility:

• In intersection control, $O(N)$ arithmetic and comparisons per 50ms step suffice to enforce fairness and select the next agent (Shi et al., 8 Nov 2025).
• In federated edge learning, dynamic reward shaping and hybrid action spaces are handled via neural network inference per step, requiring only forward passes at inference time (Huang et al., 5 Aug 2024).
• Modules receiving non-differentiable input (e.g., explicit logic rules) are implemented as discrete combinatorial solvers, often with eligibility filtering.

Common eligibility rules include:

• Exclusion of agents with recent excessive resource access.
• Minimum waiting time enforcement before eligibility.
• Windowed history counting (e.g., "fairness window" $W$ steps).

This ensures both computational tractability and the ability to impose arbitrary logic constraints without penalty on lower-level control responsiveness.

5. Empirical Performance and Application Domains

Empirical studies demonstrate that frameworks with centralized allocation modules achieve near-perfect fairness, significant throughput or accuracy improvements, and strict safety guarantees, with tested scalability and resilience:

Domain	Fairness Metric (JFI/Gini)	Throughput/Delay	Safety/Quality
Intersection Control	JFI $\geq 0.94$; Gini $\leq 0.12$	Up to $3480\ \mathrm{veh/hr}$ (vs. $1440$ baseline); avg delay $-60\%$	Zero collisions; min inter-veh dist $\geq 0.14$ m (Shi et al., 8 Nov 2025)
Federated Learning	Min/Max accuracy ratio $\to 1$	Avg test accuracy $91.4\%$ (vs. $85.6\%$)	Online control $>100$ Hz (Huang et al., 5 Aug 2024)
Multi-Agent Resource	CV $\sim 0.06$ vs. $0.86$ baseline	Social welfare $+39$	Eliminated "rich-get-richer" (Jiang et al., 2019)

Performance gains rely on:

• Explicit group-level objective control through the allocation module.
• Real-time feasibility for large $N$ through module simplicity.
• Plug-in compatibility with various tracking/execution architectures beneath the module.

6. Extensions and Limitations

Extensions include:

• Generalization beyond single-agent-at-a-time schemes to multi-slot allocations.
• Incorporation of counterfactual or individual-level fairness by modifying the utility or regularization functions (Yang et al., 27 Oct 2025).
• Application to heterogeneous agents or systems by parameterizing utility functions on agent types or task modalities.
• Optimization over hybrid action spaces (discrete + continuous parameters) through decoupling and joint recoupling in RL-based settings (Huang et al., 5 Aug 2024).

Limitations primarily arise from:

• Dependency on global visibility or reliable consensus for state aggregation.
• Sensitivity to parameterization of fairness vs. efficiency tradeoffs, requiring domain-specific tuning or validation.
• The assumption that lower layers can reliably track assigned plans; deviations may require fallback to safety-only strategies or more conservative allocation.

A plausible implication is that, as system scale increases or as task requirements diversify, the simplicity and interpretability of the centralized allocation module become crucial for maintainability and real-time safety, motivating ongoing research in scalable, distributed, yet fairness-preserving allocation logic.