# How Much Does Correctness Cost? Budgeted Placement of Strong Correctors in a Weak Multi-Agent Swarm

arXiv: https://arxiv.org/abs/2607.09765 · HTML: https://mlsanity.com/paper/2607.09765/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 21 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [Correlated Errors in Large Language Models](https://mlsanity.com/paper/2506.07962/) (2025)
- [When Models Lie, We Learn: Multilingual Span-Level Hallucination Detection with PsiloQA](https://mlsanity.com/paper/2510.04849/) (2025)
- [Automated Composition of Agents: A Knapsack Approach for Agentic Component Selection](https://mlsanity.com/paper/2510.16499/) (2025)
- [Phase Transition for Budgeted Multi-Agent Synergy](https://mlsanity.com/paper/2601.17311/) (2026)
- [Delayed Verification Destabilizes Multi-Agent LLM Belief: Instability Thresholds and Optimal Corrector Placement](https://mlsanity.com/paper/2606.27409/) (2026)
- [Submodular Optimization for Consensus Networks with Noise-Corrupted Leaders](https://mlsanity.com/paper/1712.08212/) (2017)
- [Algorithms for leader selection in stochastically forced consensus networks](https://mlsanity.com/paper/1302.0450/) (2013)

## Abstract (arXiv)

A cheap swarm of unreliable agents can be steered to a correct consensus by a few strong, expensive "oracle" correctors. We ask how much one must spend, and where to place the oracles. We model the swarm as a consensus on a graph in which each oracle pins one node toward the truth at a cost-coupled, concave strength, and measure quality by the coherence H(R)=tr M(R)^{-1}. Our first result is that H stays submodular (each added oracle helps less than the last) even when the oracles differ in strength, so a cost-benefit greedy comes within 1-1/e of the best placement at any budget. Inverting the budget gives the budget-correctness frontier B*(eps), the least spend that guarantees an eps-correct consensus: closed-form on the complete graph, and a minimal oracle count k* when oracles cost the same. Whether a budget then buys a few strong oracles or many medium onese curvature of the cost-quality law: diminishing returns favour spreadsharply increasion. Measured onthe Qwen3 ladder (0.6-32B), the law is concave for math verificatio convex foremergent code tracing, so the verdict is genuinely task-dependent.https://github.com/YehudaItkin/budgeted-oracle-placemen