Progressive Budgeting Strategy

• Progressive budgeting strategies are adaptive, iterative approaches for reallocating resources as budgets increase, ensuring efficient updates without full recomputation.
• They incorporate methodologies like AMES, EES, and ordered budget games to achieve proportional fairness and computational scalability in participatory budgeting and strategic games.
• These strategies also extend to multi-period asset management using hierarchical reinforcement learning, balancing total and incremental allocations under dynamic constraints.

A progressive budgeting strategy is an adaptive, iterative, or staged approach to the allocation and reallocation of resources under budget constraints. It enables either (1) efficient, proportional expansion of funded projects as budgets grow incrementally—central to participatory budgeting and utility games—or (2) optimized, budget-conforming decision-making over multiple periods, as in infrastructure asset management. Modern progressive budgeting strategies are algorithmically rigorous, ensuring computational scalability, proportional fairness, and, where applicable, exact feasibility with respect to budget and representational constraints.

1. Foundational Principles

Progressive budgeting strategies aim to update funding decisions or resource allocations efficiently as the available budget evolves, typically without recomputing solutions ab initio. This paradigm is instantiated in several domains:

• Approval-based participatory budgeting: Budget is distributed among projects based on voters’ approvals, with methods such as Adaptive Method of Equal Shares (AMES) and Exact Equal Shares (EES) algorithmically ensuring that increases in budget result in proportionate, certifiably fair expansions of the winning project set (Kraiczy et al., 2023, Kraiczy et al., 17 Feb 2025).
• Budget games with ordered strategic decisions: Players select subsets of tasks competing for budget-limited resources, with the allocation adapting as players sequentially join or update their strategies, influencing both efficiency and equity (Drees et al., 2014).
• Multi-period resource planning under uncertainty: Budget decisions are staged over time in response to observed system evolution, with hierarchical reinforcement learning frameworks optimizing both total and incremental annual allocations to balance multiple objectives and dynamic uncertainties (Fard et al., 25 Jul 2025).

Progressive budgeting is distinguished by its ability to maintain rigorous guarantees—such as Extended Justified Representation (EJR), proportionality, or strict constraint feasibility—throughout incremental or hierarchical budget expansion.

2. Participatory Budgeting: Adaptive and Exact Equal Shares

The state-of-the-art coalition-based progressive budgeting methodology in participatory budgeting is realized through the MES, AMES, and EES frameworks:

• AMES (Kraiczy et al., 2023): Given a stable outcome $(W, X)$ for budget $b$, AMES efficiently updates to a new budget $b' > b$ by recalculating each voter’s capacity and greedily performing update steps. These steps involve either adding a new project to $W$ or increasing contributors for an existing project, possibly removing current projects with higher per-voter cost. Each update is executed in time $O(n\log n + mn)$, culminating in a stable, EJR-satisfying solution for $b'$.
• EES + Add-Opt (Kraiczy et al., 17 Feb 2025): EES requires equal per-voter contributions and relies on the add-opt heuristic to compute, in $O(mn)$ or $O(m^2n)$ time (for cardinal or uniform utilities, respectively), the minimal next budget increment $d^*$ that will alter the outcome. Progressive budgeting is realized as a sequence $(b^{(i)})$ where at each step, outcomes are updated only at budgets where the solution changes. This approach avoids redundant computation characteristic of naïve, serial budget increment techniques.
• Empirical performance: Add-opt-skip achieves high spending efficiency (e.g., $0.853$ for cardinal utilities), with $>10\times$ fewer calls than baseline add-one completion and with near-exhaustive project funding (Kraiczy et al., 17 Feb 2025).

3. Utility Games and Staged Equilibrium

Progressive budgeting in strategic budget games entails staged or ordered entry of tasks or agents:

• Ordered budget games (Drees et al., 2014): The progression of the game—i.e., how the budget is allocated among competing agents/tasks—depends critically on the order of participation, with “bootstrapping” phases where agents sequentially join and “local adjustment” phases for equilibrium refinement. Main progressive budgeting guidelines include:
  • Bootstrapping via sequential player insertions with best-responses (O(n) to strong equilibrium).
  • Leveraging randomized reordering or coalition moves to overcome local optima.
  • Enforcing fairness caps so no agent receives strictly less than a pre-specified fraction.
• Complexity/efficiency: Computing an optimal social welfare (maximum utility) is NP-hard, but incremental, ordered insertions find strong equilibria efficiently. Price of Anarchy is at most $2$, and the system is guaranteed to converge (potentially slowly) to a pure equilibrium.

4. Multi-Period and Hierarchical Optimization

In long-horizon asset planning under total and period-wise budget constraints—typical of infrastructure management—a progressive budgeting strategy decomposes macro and micro decisions:

• Hierarchical Deep RL (Fard et al., 25 Jul 2025):
  • High-level Planner allocates annual budgets through a continuous action $a^{(1)}_t$. The budget for year $t$ is set by:

  $$b_t = \max \left( b^l_t + \frac{(a^{(1)}_t + 1)}{2} (b^u_t - b^l_t),\, b^{\rm total} - \sum_{k=1}^{t-1}b_k - \sum_{k=t+1}^h b^l_k \right)$$

  ensuring both local and total budget feasibility. - Low-level Planner produces priorities $a^{(2)}_t \in \mathbb{R}^n$ for maintenance actions, solved via an LP projection ensuring precise compliance with $b_t$. - Training/optimization: Entire hierarchical policy is trained in a Soft Actor-Critic (SAC) framework, ensuring efficient learning and constraint satisfaction at each period.

• Scalability/robustness: Hierarchical separation reduces action space complexity from $O(2^n)$ (monolithic) to $O(n+1)$ (hierarchical), enabling tractable solutions for large $n$.

5. Performance, Complexity, and Trade-Offs

Multiple empirical evaluations across settings demonstrate that progressive budgeting strategies yield computational and allocation efficiencies:

• Participatory budgeting: Add-opt completion for EES reduces computation by an order of magnitude relative to naïve add-one (Kraiczy et al., 17 Feb 2025), while AMES eliminates the need to recompute full solutions from scratch for every incremental budget (Kraiczy et al., 2023).
• Strategic games: Incremental insertion and coalition adjustment reach strong equilibria with low computational cost, though worst-case convergence can be slow ($\Theta(2^n)$ in adversarial orderings) (Drees et al., 2014).
• Asset management: HDRL achieves Pareto-optimal or near-optimal solutions with lower variance and more stable returns than standard DQL or hybrid methods; performance is robust as the network size or planning horizon increases (Fard et al., 25 Jul 2025).

Trade-offs are domain-dependent but include:

• Reduced solution monotonicity if random restarts or dynamic coalitions are used in games.
• Potential budget underspending if completion heuristics (MES) are not exhaustive.
• Increased model complexity (four neural networks in RL frameworks), offset by linear scaling in problem size.

6. Generalization and Domain-Specific Guidelines

Across applications, the following principles are essential for robust progressive budgeting strategies:

• Exploit hierarchy: Decompose high-level resource allocation from granular actions, using continuous parametrization for the macro budget layer and combinatorial solvers (e.g., LP, knapsack) for the micro layer.
• Monotonicity and stability: Use greedy update steps or bootstrapping to maintain monotonic improvements.
• Guarantee proportionality and feasibility: Enforce per-voter or per-agent fairness (EJR, EJR1) and budget precision via LP/projected updates.
• Efficient state/solution updates: Reuse preceding computations to avoid redundant evaluation when budgets increment.
• Empirical tuning: Adjust learning rates, exploration, and tie-breaking logic in RL or game dynamics to ensure convergence and allocate budgets in an equitable and efficient manner.

These strategies extend naturally to domains such as project portfolio selection, multi-user resource allocation, power grid maintenance, and other large-scale, time-dependent decision problems, provided there are clear hierarchical and incremental decision structures present.