Smith-Consistent Budgeting Algorithm

• The paper demonstrates that SBA is a polynomial-time algorithm achieving Smith-consistency by selecting budgets from the minimal majority-supported set.
• It employs a two-phase method—ranking via a majority graph and greedy pruning—to handle complex quantitative and hierarchical voter inputs.
• The SBA framework generalizes Condorcet principles, ensuring transparent, democratic budget selection adaptable to real-world organizations.

A Smith-Consistent Budgeting Algorithm (SBA) is a polynomial-time procedure for participatory democratic budgeting that produces complete, quantitative (i.e., items with arbitrary units and non-linear costs), and hierarchical budgets based on ranked voter input, budgetary constraints, and a previous-year reference ("reality-aware budgeting"). SBA provably satisfies Smith-consistency: its output budget lies in the minimal set (the Smith set) of feasible budgets such that, for any other feasible alternative, there exists a majority-supported "path" of dominance or non-domination leading to the output. When a Condorcet-winning (majority-dominant) budget exists, SBA returns it; in cases of cycles, it returns a Smith-set member. SBA structurally generalizes Condorcet-consistency to budgeting problems, making it uniquely suited for transparent, participatory allocation in real-world organizations, cities, and states (Shapiro et al., 2017).

1. Problem Setting and Input Specification

SBA addresses the construction of democratic budgets where voter preferences may be complex, quantitative, and hierarchical. The essential inputs are:

• Budget Proposal ($\mathcal{P}$): A collection of distinct budget items $b_i$, each possibly available in multiple quantities, with cost functions $F_{b_i}(q)$ that may be non-linear (e.g., exhibiting discounts or thresholds).
• Votes ($\mathcal{V}$): Each participant ranks (partially or totally) the proposed budget items—including, in the quantitative setting, specifying priority classes for item quantities.
• Budget Limit ($\ell$): The total spending constraint for the selected budget.
• Previous Budget ($B_{-1}$): Used as a tie-breaker and for introducing inertia/hysteresis to prevent radical oscillations.

Budgets can have hierarchical structures: sub-budgets are ranked and allocated individually within sections, followed by top-level aggregation.

2. Preference Aggregation: Majority Graph and Condorcet Principle

SBA extends the Condorcet winner concept to complex budget selection by defining budget dominance:

• Voter Preference over Budgets is specified using the minmax set extension. For voter $v$, with budgets $B$ and $B'$,

$$v \text{ prefers } B \text{ over } B' \iff \max(\text{pos}_v(B \setminus B')) < \min(\text{pos}_v(B' \setminus B))$$

where $b_i$0 indicates the set of ranking positions by $b_i$1.

• Budget Dominance: Budget $b_i$2 dominates $b_i$3 if a majority of voters prefer $b_i$4 over $b_i$5.
• Condorcet-Winning Budget: A feasible budget that dominates all others; however, cycles may prevent existence.

3. SBA Algorithm Structure

The SBA is divided into ranking and pruning phases:

1. Ranking Phase:
   • Construct the majority graph over items: an arc $b_i$6 denotes a majority for $b_i$7 over $b_i$8.
   • Extract ordered partitions using the Schwartz set: each block represents a maximal undominated set. Iteration yields a partitioned ranking $b_i$9.
2. Pruning Phase:
   • Sequentially, for each block $F_{b_i}(q)$0, greedily add a maximal subset to the budget such that cumulative costs remain under $F_{b_i}(q)$1.
   • Ties are resolved by choosing the subset closest (minimal symmetric difference of total cost) to last year’s budget $F_{b_i}(q)$2.
   • Output is exhaustive: no further items can be added within the limit.
3. Hierarchical Aggregation:
   • Local rankings and pruning are run within each section/sub-budget.
   • Global aggregation treats section outputs as meta-items for allocation, producing multi-level, organization-reflective budgets.

Pseudocode (from Algorithm 1):

$\mathcal{V}$8

4. Mathematical Properties and Smith-Consistency

Majority Graph: Vertices are items, arcs encode majority preference.

• Schwartz set: Each extracted block is a minimal undominated subset.
• Smith set: Minimal set of items dominating all others; contains Condorcet winners if they exist.

Smith-Consistency Theorem: SBA outputs a budget $F_{b_i}(q)$3 such that for any feasible alternative $F_{b_i}(q)$4, a weak dominance path (via majority or non-domination) leads from $F_{b_i}(q)$5 to $F_{b_i}(q)$6. If a Condorcet winner exists, SBA finds it; otherwise, a Smith-set member is chosen.

• The proof utilizes relations in the majority graph to construct induction over dominance paths, ensuring exhaustive but feasible assembly of the budget.

Quantitative Extension: When items have quantities and arbitrary cost functions, voting and graph construction are extended via item splitting and block index calculation, maintaining computational efficiency and Smith-consistency.

Complexity: SBA is polynomial-time, given construction of majority graphs and splitting domain as per voting distinctions.

5. Condorcet and Smith-Consistency Analysis

SBA guarantees both Condorcet- and Smith-consistency:

• Condorcet-consistency: If a Condorcet-winning budget exists, SBA returns it.
• Smith-consistency: For all cases, including cycles, SBA returns a member of the Smith set (the minimal mutually dominating set), closely analogous to the Schwartz set in voting theory.

This ensures optimal majority-reflective outcomes and robustness to voting paradoxes.

6. Illustrative Example

Consider items $F_{b_i}(q)$7, $F_{b_i}(q)$8; budget limit $F_{b_i}(q)$9, votes $\mathcal{V}$0, $\mathcal{V}$1, previous budget $\mathcal{V}$2:

• Feasible exhaustive budget: $\mathcal{V}$3.
• Majority graph: derived pairwise from votes.
• Ranking: e.g., $\mathcal{V}$4.
• Pruning: selects $\mathcal{V}$5 to satisfy exhaustiveness and proximity to $\mathcal{V}$6.

This process yields a majority-reflecting, exhaustive budget.

7. Democratic and Practical Properties

SBA supports flexible participation and real-world democratic needs:

• Majority Protection: No feasible change to the output budget has majority support.
• Vote Flexibility: Accepts ranked, partially ordered, or amendment-based votes.
• Quantitative and Hierarchical Budgeting: Supports complex allocations with quantity decisions and multi-level budget sections.
• Practical Scalability: Applicable in organizations ranging from cooperatives to governments.

Summary Table

Property	SBA Algorithm
Input	$\mathcal{V}$7
Budget Structure	Quantitative, Hierarchical
Preference Model	Rankings (linear, partial, with quantities)
Aggregation	Majority graph → Schwartz sets → Ordered partition → Greedy pruning
Democratic Guarantee	Condorcet-consistent and Smith-consistent
Complexity	Polynomial time

SBA transforms participatory budgeting by ensuring computationally efficient, majority-reflective, and exhaustive selection compatible with hierarchical and quantitative organizational realities. Its Smith-consistency secures robust democratic outcomes regardless of voting paradoxes or complex preferences (Shapiro et al., 2017).