# The Ringelmann Effect in Multi-Agent LLM Systems: A Scaling Law for Effective Team Size

arXiv: https://arxiv.org/abs/2606.02646 · HTML: https://mlsanity.com/paper/2606.02646/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in General LLM agent frameworks and benchmarks, one of 16 directions in LLM agents and tool use.
- Builds on 32 of its 50 references that are in the map.
- First posted on arXiv 31 May 2026.

## Placed in

- LLM agents and tool use / [General LLM agent frameworks and benchmarks](https://mlsanity.com/agents/general-llm-agent-frameworks-and-benchmarks/)

## Builds on (most cited first, 30 of 32)

- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Improving Factuality and Reasoning in Language Models through Multiagent Debate](https://mlsanity.com/paper/2305.14325/) (2023)
- [Towards Understanding Sycophancy in Language Models](https://mlsanity.com/paper/2310.13548/) (2023)
- [Encouraging Divergent Thinking in Large Language Models through Multi-Agent Debate](https://mlsanity.com/paper/2305.19118/) (2023)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Discovering Language Model Behaviors with Model-Written Evaluations](https://mlsanity.com/paper/2212.09251/) (2022)
- [ChatEval: Towards Better LLM-based Evaluators through Multi-Agent Debate](https://mlsanity.com/paper/2308.07201/) (2023)
- [Mixture-of-Agents Enhances Large Language Model Capabilities](https://mlsanity.com/paper/2406.04692/) (2024)
- [AI safety via debate](https://mlsanity.com/paper/1805.00899/) (2018)
- [Debating with More Persuasive LLMs Leads to More Truthful Answers](https://mlsanity.com/paper/2402.06782/) (2024)
- [Scaling Large Language Model-based Multi-Agent Collaboration](https://mlsanity.com/paper/2406.07155/) (2024)
- [More Agents Is All You Need](https://mlsanity.com/paper/2402.05120/) (2024)
- [Rethinking the Bounds of LLM Reasoning: Are Multi-Agent Discussions the Key?](https://mlsanity.com/paper/2402.18272/) (2024)
- [Towards a Science of Scaling Agent Systems](https://mlsanity.com/paper/2512.08296/) (2025)
- [Should we be going MAD? A Look at Multi-Agent Debate Strategies for LLMs](https://mlsanity.com/paper/2311.17371/) (2023)
- [Debate or Vote: Which Yields Better Decisions in Multi-Agent Large Language Models?](https://mlsanity.com/paper/2508.17536/) (2025)
- [Talk Isn't Always Cheap: Understanding Failure Modes in Multi-Agent Debate](https://mlsanity.com/paper/2509.05396/) (2025)
- [Voting or Consensus? Decision-Making in Multi-Agent Debate](https://mlsanity.com/paper/2502.19130/) (2025)
- [Conformity in Large Language Models](https://mlsanity.com/paper/2410.12428/) (2024)
- [Peacemaker or Troublemaker: How Sycophancy Shapes Multi-Agent Debate](https://mlsanity.com/paper/2509.23055/) (2025)
- [Can LLM Agents Really Debate? A Controlled Study of Multi-Agent Debate in Logical Reasoning](https://mlsanity.com/paper/2511.07784/) (2025)
- [Understanding Agent Scaling in LLM-Based Multi-Agent Systems via Diversity](https://mlsanity.com/paper/2602.03794/) (2026)
- [When Identity Skews Debate: Anonymization for Bias-Reduced Multi-Agent Reasoning](https://mlsanity.com/paper/2510.07517/) (2025)

## Built on by (most cited first, 3 of 3)

- [OpenCollab: A Multi-Agent Coding Framework with Programmable Collaboration and Controllable Runtime](https://mlsanity.com/paper/2609.38345/) (2026)
- [How does Adversarial Influence Scale in Multi-Agent Systems?](https://mlsanity.com/paper/2609.30028/) (2026)
- [Multi-agent Scaling Across Disjunctive and Compensatory Tasks](https://mlsanity.com/paper/2609.31563/) (2026)

## Abstract (arXiv)

Inference-time multi-agent LLM scaling lacks a shared unit: counting nominal agents conflates cost with independent evidence. We derive a two-parameter scaling law R(N) = Neff/N = 1/(1+c(N-1)N-β) where the regime exponent β classifies any configuration into one of three asymptotic regimes -- hard-ceiling at 1/c (β = 0), sublinear at Nβ/c (0 < β < 1), or linear (β ≥ 1), and a mean-field theorem predicts that peer count k and rounds τ during agent debate enter the dynamics only through their product kτ. The law applies at two levels: answer diversity and correctness redundancy. Across 44 (model × task × condition) cells spanning peer debate, self-correction, random-noise placebo, self-consistency, three open-weight families (Qwen, Llama, Ministral) at scales from 7B to 32B with a frontier API check (Gemini), thinking models, heterogeneous teams, and sparse communication, the functional form fits every condition at R2 > 0.99; only (c, β) shifts. On free-form math, dense peer influence collapses the answer-level regime from sublinear into hard-ceiling; correctness-level fits remain hard-ceiling throughout. Three findings have practical implications. (i)~Thirty dense debating agents produce no more answer diversity than one on MMLU-Hard. (ii)~A noise placebo tracks self-correction on free-form math and at 4× scale, so within homogeneous teams the gain commonly attributed to ``debate'' comes from re-evaluation, not peer content. (iii)~A single N ≤ 5 pilot predicts the N=30 structural ceiling, and within the configurations tested only architectural diversity (heterogeneous teams) lowers c and escapes the hard-ceiling regime, communication-mode interventions do not.