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Who Is the Influencer? Consensus Dynamics in Multi-Agent Systems

Gouki Minegishi, Jingyuan Feng, Hiroki Furuta, Takeshi Kojima, Yusuke Iwasawa, Yutaka Matsuo

arXiv:2610.31105v1cs.AI

TL;DR

Multi-agent systems can improve performance, yet their behavior is difficult to understand, and prior consensus studies leave influence in heterogeneous populations open. This paper uses a meaningless-label naming game and a particle-based account, finding that the stronger model’s label most often becomes consensus across seven families.

  • Problem

    Prior work on consensus considered homogeneous populations or two-agent conversations, leaving open which model steers consensus in heterogeneous systems.

    Source: §1 · p. 1

  • Method

    The paper mixes models in a naming game with meaningless labels and models the agents as weighted particles moving on a simplex under an energy landscape.

    Source: §1 · p. 2

  • Results

    Across seven model families, populations most often converge to the stronger model’s label even though the labels have no correct answer.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Consensus can establish conventions or formats without indicating their quality, making influence relevant even when agreement says nothing about correctness.

    Source: §5 · p. 8

    The study uses well-mixed populations exchanging single labels and does not test structured networks, free-form text, or settings with objective ground-truth answers.

    Source: §5 · p. 9

Abstract

from arXiv · show

Multi-agent LLM systems are increasingly used in practice, but their behavior is complex and hard to understand. We design a simple task to ask which model has the most influence on the final consensus when several models are mixed in one system. Experimentally, across seven model families, we find that the opinions of the other models are absorbed into the opinion of the stronger model, the one with more parameters. This happens even though the task requires no pretraining knowledge and has no correct answer. To understand this phenomenon, we interpret the agents as weighted particles moving on a simplex. Larger models carry a larger weight and therefore occupy a deeper well in the energy landscape. A large weight means strong in-context learning: a model with a large weight amplifies the majority of its context. Larger models are generally better at in-context learning, and this is what makes the larger model the influencer. We hope that this work helps to interpret multi-agent systems, whose behavior is becoming too complex for humans to understand.

1 INTRODUCTION

The paper asks which model steers consensus in heterogeneous multi-agent systems, where behavior is difficult to understand and prior work has focused on homogeneous populations or two-agent conversations. It introduces a meaningless-label task and a particle-based account, finding that stronger models tend to shape the final consensus.

1 INTRODUCTION

  • Prior consensus studies considered homogeneous populations or two-agent conversations, leaving open whose view prevails when systems combine different models.Such systems often mix a strong model with cheaper ones.Source: §1 · p. 1
  • The naming game assigns meaningless labels to models and seeds each model’s agents with its own label, so the adopted label measures influence without relying on pretrained knowledge.The task separates consensus dynamics from knowledge acquired during pretraining.Source: §1 · pp. 1–2
  • Across seven model families, populations most often converge to the stronger model’s label, despite labels having no correct answer.For API models, stronger means higher-performing; for open-weight models, it means larger.Source: §1 · p. 2
  • The paper’s contributions include a task designed to remove the role of pretrained knowledge and evidence that stronger and larger models exert greater influence.The paper presents these as contributions to studying heterogeneous multi-agent systems.Source: §1 · p. 2
  • The proposed particle framework gives models different weights and interprets consensus as motion down an energy landscape, aligning the account with the empirical results.A model’s weight represents how strongly it pulls consensus toward its label.Source: §1 · p. 2

2 A SYNTHETIC TASK FOR CONSENSUS FORMATION

The paper operationalizes consensus as a naming game in which agents exchange meaningless labels through limited memories. It measures influence by which model’s seeded label the population adopts, and represents agents’ states on a simplex using memory composition.

2.1 AGENT PROTOCOL

  • The protocol presents candidate labels in shuffled order and the newest memory item first, without mentioning agreement, rewards, or other agents.The labels are meaningless five-letter strings, so their meanings do not favor any label.Source: §2.1 · p. 3

2.2 SPEAKER–LISTENER INTERACTIONS

  • Each interaction selects a speaker and listener; the speaker emits one label, and only the listener appends it while dropping its oldest memory entry.Defaults are K = 3, H = 20, N = 24, and T = 60N interactions.Source: §2.2 · p. 3
  • Because an agent’s label-output probabilities form a simplex state, the method approximates that state with the fractions of labels in its memory.Memory composition rises monotonically with output probability (Pearson r = 0.83–0.93 across 18 models).Source: §2.2 · p. 3

2.3 HETEROGENEOUS POPULATIONS

  • Each model’s agents begin with memories filled by that model’s label, creating initial disagreement; the adopted label identifies the most influential model.The population is partitioned into model-specific subpopulations.Source: §2.3 · pp. 3–4
  • Figure 3 reports, for each family, the fraction of 108 trials in which each model class’s label became the convention, against chance at 1/3.Whiskers show one standard error from a cluster bootstrap over 18 cells.Source: Figure 3 · p. 4
  • In one representative trial per family, agents’ simplex trajectories are absorbed at the large model’s vertex (N = 24, H = 20).The plotted cloud contracts and drifts before reaching that vertex.Source: Figure 4 · p. 4
  • Figure 4 traces agents’ paths on the simplex from class-specific starting vertices through a contracting cloud to the large model’s vertex.Each plotted point represents an agent’s 20-slot memory composition.Source: Figure 4 · p. 4
  • The experiments cover seven model families from four developers, each with large, medium, and small models.Within a family, stronger models are defined by higher scores on standard benchmarks such as MMLU-Pro.Source: §2.3 · p. 4

3 EXPERIMENTAL RESULTS

Across seven model families, the population most often converges to the more capable model’s label; larger populations and longer memories take more interactions to reach consensus.

3.1 STRONGER MODELS DRIVE CONSENSUS

  • In every family, the most capable model’s label most often wins; open-weight families always favor the largest model, and the full large > medium > small ordering holds.The labels are meaningless random strings, so there is no correct label to find.Source: §3.1 · pp. 4–5
  • On the simplex, agents move from their separate starting vertices toward the center before the whole population is absorbed at the strongest model’s vertex.Even the model that ultimately drives consensus first moves toward the center with the other agents.Source: §3.1 · p. 5
  • In two-label trials, the larger model wins in all four tested families, while smaller-model agents shift farther toward its label than larger-model agents do.Each population had 12 agents per model, and Figure 5 tracks trial wins and each agent’s share of the larger model’s label in memory.Source: §3.1 · p. 5, Figure 5 · p. 5
  • Among three nearly equal-sized models from different families, win fractions were 0.39, 0.36, and 0.25, with no definitively dominant influencer.The authors suggest the influencer effect is most prominent when model size, and thus capability, differs substantially.Source: §3.1 · p. 5, Figure 6 · p. 5

3.2 EFFECTS OF POPULATION SIZE AND MEMORY LENGTH

  • Consensus interactions grow roughly as N^1.3–N^1.4 with population size and H^0.6–H^0.9 with memory length, while the larger-model convergence trend remains robust.From N = 3 to N = 24, interactions to consensus rise 14- to 19-fold; Figure 7 reports median interaction counts with interquartile-range bars.Source: §3.2 · p. 6, Figure 7 · p. 6

4 WHY THE STRONGER MODEL WINS

The paper models mixed-model naming-game consensus as particles descending an energy landscape, then tests whether measured assertion strengths explain which label wins and how the population gets there.

4.1 AGENTS AS PARTICLES ON THE SIMPLEX

  • Each agent is a simplex particle, and only the listener moves when it replaces a memory slot with the speaker’s sampled label.An agent’s output probabilities locate it on the simplex; each interaction changes its position by a step of order 1/H.Source: §4.1 · p. 6, Figure 8 · p. 6

4.3 MEASURING AND TESTING THE ASSERTION STRENGTH

  • 78% of Gemma 4 trials reached the 31B model’s vertex, the deepest well, though noisy trials did not always end there.The 31B well has depth −1.72, compared with −1.59 and −1.26 for the other models.Source: Figure 9 · p. 7, §4.3 · p. 7

4.2 A CONSENSUS ENERGY

  • Model-specific assertion strengths tilt the energy toward labels with larger weighted shares, placing the strongest model’s label in the deepest well.The mean output follows the energy’s downhill slope, with labels growing when their weighted share exceeds the population average.Source: §4.2 · pp. 6–7

4.3 MEASURING AND TESTING THE ASSERTION STRENGTH

  • Across three open-weight families, assertion-strength ordering followed model size, and strength ordering matched win-rate ordering in all nine class pairs.The measured Gemma 4 strengths were 3.44 (31B), 3.19 (26B-A4B), and 2.52 (E4B).Source: §4.3 · pp. 7–8
  • The energy fell during interaction, and its predicted mean-output flow matched measured motion at r = 0.65.Qwen 3.5 and Qwen 2.5 showed the same pattern.Source: §4.3 · p. 8
  • Large assertion strength amplifies the majority in a model’s memory; the authors conjecture that stronger in-context learning helps larger models resist incoming labels and steer consensus.The energy account is developed for Gemma 4, Qwen 3.5, and Qwen 2.5.Source: §4 · p. 6, §4.3 · p. 8

5 DISCUSSION AND RELATED WORK

The discussion distinguishes influence from accuracy: in the naming game, where no answer is correct, mixed populations converge on the strongest member’s view, while several real-world settings remain outside the study’s scope.

Multi-agent systems; Conventions in agent populations

  • Prior work studies multi-agent performance, risks, or homogeneous populations, while this paper measures whose view wins when heterogeneous agents disagree.The authors position their study as addressing a question left open by work on mixed-model systems.Source: §5 · pp. 8–9

Influence without correctness

  • When correctness cannot be recognized, mixed populations converge on the strongest member’s view, qualifying advice to mix models and mattering for conventions and formats.The paper frames the naming game as a setting where only influence remains, rather than accuracy.Source: §5 · p. 8

Susceptibility and influence

  • Earlier susceptibility studies include majority conformity, confidence-based herding, and influence by committed minorities; here unscripted agents differ only by model.The closest heterogeneous results cited involve two-agent attractors or communities with correct answers.Source: §5 · p. 9

Limitations

  • The experiments use well-mixed populations exchanging single labels; structured networks, free-form text, topology effects, and settings with objective answers remain untested.Source: §5 · p. 9

6 CONCLUSION

Across seven model families, mixed-model naming games converge on the stronger model’s convention, and a particle-and-energy account links that influence to stronger amplification of memory majorities.

6 CONCLUSION

  • In mixed-model naming games across seven families, the stronger model’s convention takes over, with model ranking within a family matching influence ranking.The paper leaves open whether assertion can be measured outside the population to predict influence before a system runs.Source: §6 · p. 9

A EXPERIMENTAL CONFIGURATION

The appendix identifies the model ladders and reports developers’ MMLU-Pro accuracy for open-weight ladders, then specifies label generation and prompt behavior.

A EXPERIMENTAL CONFIGURATION

  • Table 1 lists model identifiers for the seven main ladders, and Table 2 reports developers’ MMLU-Pro accuracy for open-weight ladders.Source: Table 1 · p. 16, Table 2 · p. 16
  • Labels are deterministic lowercase five-character strings, with tokens already appearing as standalone strings elsewhere in the codebase rejected to avoid incidental associations.Source: A EXPERIMENTAL CONFIGURATION · p. 16

B PROMPT TEMPLATE

  • The prompt asks agents to choose among allowed labels using recent memory, shuffles label order per call, and retries invalid outputs before recording misses.The prompt-facing memory order can be changed without mutating the stored FIFO; misses do not push the listener and are excluded from U probe statistics.Source: B PROMPT TEMPLATE · p. 17

C TWO-LABEL POPULATIONS

In two-class populations, the larger model wins at every tested memory length in the reported comparisons. Separately, memory composition tracks output probability across models, supporting its use as a proxy for agents’ positions.

C TWO-LABEL POPULATIONS

  • At H = 6, larger models won all four latest-generation pairings: GPT-5/mini 0.67, Opus 5/Haiku 4.5 0.81, Gemma 4 31B/E4B 0.67, and Qwen 3.5 122B-A10B/35B-A3B 0.83.No trial was unresolved; the larger model also won at every memory length tested, and longer memory helped the larger model in open-weight families.Source: C TWO-LABEL POPULATIONS · p. 17

D STATISTICAL PROCEDURE

  • Trials were grouped by label set and assignment for cluster-bootstrap intervals, with unresolved trials counted as losses for every class.The 95% intervals use 3,000 bootstrap replicates over cells, not individual trials.Source: D STATISTICAL PROCEDURE · p. 17

E MEMORY SHARE AND OUTPUT PROBABILITY

F POPULATION SIZE AND MEMORY LENGTH

In the three-class Gemma 4 population, the large model won the most trials across all tested population sizes and memory lengths. The paper models consensus as a flow on a simplex that descends an energy landscape, while noting that the derivation relies on heuristic approximations.

F POPULATION SIZE AND MEMORY LENGTH

  • The large Gemma 4 model won the most trials in all 16 population-size and memory-length cells; its win rate rose from about 0.4–0.5 at H = 6 to 0.7–0.8 at H = 40 and changed little with population size.Table 3 reports the large model’s win fraction, with chance at 1/3.Source: F POPULATION SIZE AND MEMORY LENGTH · p. 19, Table 3 · p. 19

G THE CONSENSUS ENERGY: FLOW, DESCENT, AND WELLS

  • The proposed flow is gradient descent of energy E under the Shahshahani metric, with time scaled so one round moves mean memory by 1/H of mean speech departure from memory.The flow is motivated by interaction dynamics but uses two heuristic closure approximations that do not hold throughout a run.Source: G THE CONSENSUS ENERGY: FLOW, DESCENT,… · p. 19
  • At a label vertex, energy is −w_c(k)/2 and a small share of another label shrinks when w_c(k) > 0; from a balanced start, the label with largest w grows fastest.A vertex is above the balanced start when w_c(k) < w̄/3; equal assertion strengths give equal-depth wells and leave the winner to fluctuations.Source: G THE CONSENSUS ENERGY: FLOW, DESCENT,… · p. 20

H ESTIMATING THE ASSERTION STRENGTH

The paper estimates assertion strength from how strongly a model favors labels that are majorities in its memory. In open-weight families, the estimated strength increases with model size.

H ESTIMATING THE ASSERTION STRENGTH

  • Assertion strength w measures how much more often an agent says a label when it is the memory majority; w = 0 follows memory shares, while w > 0 amplifies the majority.For a two-label memory with 60% of one label, the model’s output probability for it is 60% at w = 0, 72% at w = 2.52, and 77% at w = 3.44.Source: H ESTIMATING THE ASSERTION STRENGTH · p. 20
  • The authors replayed utterance memories and fit each class’s response by least squares, including an echo term for repeating the most recently heard label.The echo term prevents part of that repetition effect from being absorbed into w.Source: H ESTIMATING THE ASSERTION STRENGTH · p. 20
  • In all three open-weight families, assertion strength w grows with model size; the energy analysis is restricted to these families because the API-model regression does not separate large and medium models.Source: H ESTIMATING THE ASSERTION STRENGTH · p. 20
  • The diagnostic figures depict energy landscapes and consensus outputs, predicted versus measured local agreement, energy descent, and assertion strength against win share.Source: Figure 11 · p. 21, Figure 12 · p. 21, Figure 13 · p. 21

I ENERGY RESULTS FOR ALL OPEN-WEIGHT FAMILIES

Across the three open-weight families, the energy landscape favors the large model: its well is deepest, its basin largest, and the start lies inside that basin.

I ENERGY RESULTS FOR ALL OPEN-WEIGHT FAMILIES

J WHERE THE ENERGY ACCOUNT FAILS

Small-model wins expose a limitation in the energy account: under the second closure approximation, the population can climb to a shallow vertex. A modified assertion-strength rule restores descent but removes the energy landscape’s ability to identify the winning label.

J WHERE THE ENERGY ACCOUNT FAILS

  • 7–9% of trials are won by the small model, and energy falls in only 72% (Gemma 4), 42% (Qwen 3.5), and 54% (Qwen 2.5) of rounds.In these trials, the population climbs to a shallow vertex under the second closure approximation.Source: J WHERE THE ENERGY ACCOUNT FAILS · p. 23
  • Letting each agent’s assertion strength act on the labels it says restores descent in 78–82% of rounds and keeps local agreement (r = 0.65–0.66), but equal-depth vertices no longer identify the winner.Source: J WHERE THE ENERGY ACCOUNT FAILS · p. 23
  • Because assertion strengths and outputs are estimated from the same runs used to compare energy with outcomes, the agreement tests consistency, not out-of-sample prediction.Source: J WHERE THE ENERGY ACCOUNT FAILS · p. 23

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