The Ringelmann Effect in Multi-Agent LLM Systems: A Scaling Law for Effective Team Size
Blaž Bertalanič, Carolina Fortuna
physics.soc-ph, cs.AI, cs.MA
2026-06-01
Fitted on 44 configs, a two-parameter Ringelmann law finds 30 dense debating agents yield Neff=1.12 on MMLU-Hard; a noise placebo tracks self-correction.
Pretraining scaling has shared units: parameters, tokens, compute. Inference-time multi-agent scaling does not. One paper reports that N=10 agents helped; another reports that N=10 did not. Nominal headcount mixes spend with independent evidence.
LLM agents sycophantically align under interaction, so their answers correlate. Nominal N overstates information, the clustered-sample problem. Researchers at Jozef Stefan Institute graft the Ringelmann effect (per-person output falls as groups grow) onto the Kish design effect (correlated draws convert to a smaller independent sample) and produce a two-parameter law. A pilot at N≤5 reads the structural ceiling at N=30.
Effective team size is Kish's formula: Neff = N / (1 + (N-1) ρN), where ρN is pairwise answer correlation. At ρN=0, Neff=N. At consensus, Neff collapses to 1 and extra agents add no independent evidence. They model the correlation profile as ρN = c N^(-β) and obtain R(N) = Neff/N = 1/(1 + c(N-1)N^(-β)). The floor parameter c sets the asymptote 1/c when β=0. The exponent β is the regime classifier: hard ceiling (β=0, Neff→1/c), sublinear (0<β<1, Neff N^β/c), or linear (β≥1).
A mean-field closure says leftover disagreement contracts by a constant factor each round. The theorem's falsifiable claim is that peer count k and revision rounds τ enter only through the product kτ. A fixed budget B=kτ is split-invariant to first order. Escaping the hard ceiling requires both the conformity rate α and the pre-communication correlation ρ(0) to vanish with N; one of the two vanishing is not enough.
The sweep covers 44 (model × task × condition) cells: a full grid on Qwen2.5-7B, cross-family copies on Llama-3.1-8B and Ministral-8B, 4× scale on Qwen2.5-32B, Qwen3-8B thinking and Gemini 3.1 Flash-Lite, plus heterogeneous teams. Tasks are MMLU-Hard (724 four-choice items), GSM-Hard (1,017 free-form numeric problems), and GPQA Diamond (198 items), with N from 1 to 30. Three controls hold the prompt skeleton fixed and change only the peer block: debate fills true same-item peers; self-correction fills the agent's own previous output; a noise placebo fills length-matched independent text from other items. Answer-level Neff is 2^H(a) from the team's answer entropy; correctness-level Neff uses the ICC of binary right/wrong indicators. Fits use N∈{2,3,5} and extrapolate to 30.
The same functional form fits every condition, median R²=0.999, floor ≥0.97. Only (c, β) moves. Dense debate on Qwen2.5-7B sits in the hard-ceiling regime on all three tasks. At N=30, answer-level Neff is 1.12 on MMLU-Hard, 1.79 on GSM-Hard, and 1.15 on GPQA.
Thirty densely debating agents on MMLU-Hard produce about as much answer diversity as one.
GSM-Hard crosses an answer-level regime boundary. Self-correction and noise become sublinear, with Neff at N=30 of 6.10 and 6.44. Correctness-level fits stay hard-ceiling throughout: distinct wrong numbers inflate entropy, correctness redundancy does not fall.
| Setup | c | β | Regime | Neff at N=30 |
| MMLU debate | 0.85 | 0 | hard ceiling | 1.12 |
| GSM debate | 0.57 | 0.01 | hard ceiling | 1.79 |
| GSM self-correction | 0.34 | 0.25 | sublinear | 6.10 |
| GSM noise | 0.33 | 0.27 | sublinear | 6.44 |
| GPQA debate | 0.81 | 0 | hard ceiling | 1.15 |
A pilot at N≤5 extrapolated to N=7 to 30 with 5.5% mean relative error, against 32.2% for a generic power law and 73.4% for logistic saturation. Across three open-weight families the error stays at 5% to 12%. Matched kτ products agree within Δρ≈0.04 in the high-agreement regime. The noise placebo tracks self-correction: c=0.33 vs 0.34 on GSM, and at 32B both land at c=0.45, β=0.18. A TOST test bounds the debate vs self-correction accuracy gap within ±5 pp (p<0.02). GSM rises from about 10% to 26% to 28%: self-correction +18.2 pp, debate +16.4 pp. Debate costs 10 to 45× more prompt tokens per effective agent.
Inside homogeneous teams, the gain usually credited to debate is re-evaluation. Peer content does not set the regime.
Architectural diversity is the lever that moves the ceiling. Equal-share Qwen+Llama+Ministral teams drop c from 0.85 to 0.54 (MMLU), 0.57 to 0.35 (GSM), 0.81 to 0.56 (GPQA), with β still near 0. Scaling the same family to 32B tightens c to 0.96/0.77/0.97. Qwen3-8B in thinking mode scores 87.2% solo on MMLU-Hard; after debate at N=5, ρ=0.983, Neff=1.02, c=1.00. Individual learning is real: net wrong-to-correct is about +8 pp on GSM. Team productivity still does not beat the matched no-peer control. R(N) at N=30 is about one-fifteenth of R(N) at N=2.
The practical object is a measuring stick. Run N∈{2,3,5}, read (c, β), and you know whether extra agents still add independent evidence. On instruct-class models and bounded tasks, β≈0, so communication mode, peer count, rounds, and aggregator are interchangeable. The one validated lever that raises the ceiling is lowering ρ(0) via heterogeneous ensembles.
If you already run multi-agent debate: 3 to 5 self-correcting agents capture the ceiling on easy tasks (solo p>0.5); the practical knee on harder tasks is about N=10. Sweeping prompts, voting rules, debate scripts, or topology mostly walks a plane the ceiling does not respond to. Lifting GSM from 10% to 26% to 28% is still worth doing. That bill belongs to re-evaluation. This is incremental measurement work that folds protocols previously treated as orthogonal into shifts on one (c, β) plane.
The sweep is intentionally role-symmetric, fully connected, and tool-free, so conformity can be isolated. Asymmetric roles, tool use, MoA and verifier-aggregator pipelines, multimodal tasks, frontier-vs-frontier debate, N>30 and τ>6 are untested. The authors write these as testable extensions: tools and verifiers should lower ρ(0); high solo accuracy should push c toward 1. If a verifier lowers c at constant pre-verifier ρ(0), initial correlation is not the lever. If MoA or frontier heterogeneous teams drive β to 1, the three-regime split breaks.
Decoding is locked at temperature 0.4, top-p 0.95. The noise placebo is shuffled, length-matched answers from other items; padding-only, fluent off-topic, and adversarial peers were not run. Answer-level Neff=2^H is an upper bound on free-form tasks. Coverage is four-choice and unique-integer answers; a HumanEval pilot finds the same form at R²≥0.99, but open-ended generation and multi-step planning are not systematically covered. Gemini 3.1 Pro was evaluated solo only. Confidence-weighted and top-1 aggregators leave β within noise of plurality and shift accuracy by at most 5 pp.
The answer-level regime crossing does not exist at the correctness level. Diversity numbers look generous; the independent right/wrong mass that majority vote actually consumes stays hard-ceiling. Treating answer-level Neff as a license to add agents will overstate the room left on free-form math.