How Much Does Correctness Cost? Budgeted Placement of Strong Correctors in a Weak Multi-Agent Swarm
Quick Answer
This study explores the cost-effectiveness of placing strong 'oracle' correctors in a swarm of unreliable agents to achieve consensus.
Quick Take
It finds that a cost-benefit greedy algorithm can yield placements within 1-1/e of the optimal solution, with diminishing returns on quality as budget increases. The budget-correctness frontier is established, providing a framework for determining the least expenditure for guaranteed consensus accuracy.
Key Points
- Strong correctors can steer unreliable agent swarms to consensus at a cost.
- The coherence measure H remains submodular, indicating diminishing returns.
- A greedy algorithm achieves placements close to optimal within budget constraints.
- The budget-correctness frontier B*(eps) quantifies minimum spending for consensus accuracy.
- Findings are validated through mathematical verification and emergent code tracing.
Paper Resources
📖 Reader Mode
~2 min readAbstract: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 this http URL://github.com/YehudaItkin/budgeted-oracle-placemen
| Comments: | 30 pages, 9 figures, 1 table.. Code and data: this https URL |
| Subjects: | Artificial Intelligence (cs.AI); Computation and Language (cs.CL); Multiagent Systems (cs.MA); Systems and Control (eess.SY); Optimization and Control (math.OC) |
| MSC classes: | 90C27, 05C50, 93A16 |
| ACM classes: | I.2.11; G.1.6; G.2.2 |
| Cite as: | arXiv:2607.09765 [cs.AI] |
| (or arXiv:2607.09765v1 [cs.AI] for this version) | |
| https://doi.org/10.48550/arXiv.2607.09765 arXiv-issued DOI via DataCite |
Submission history
From: Igor Itkin [view email]
[v1]
Tue, 7 Jul 2026 05:48:17 UTC (887 KB)
— Originally published at arxiv.org
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