Summary
Flint, Aiello, Pastor-Satorras and Baronchelli run homogeneous LLM populations
through a two-word naming game. Random pairs each pick one of two socially
loaded words, such as {man, woman} or {straight, gay}, and are rewarded for
matching. The study sweeps population size from one agent to beyond ten
thousand. Two results come out. First, at fixed N=24, interaction can amplify
an individual word preference, create one from neutral agents, or reverse it,
and the direction depends on the model. Second, as N grows, the chance that the
population settles on its "strong" word rises until, past a threshold, every
run that reaches consensus lands on that word. The threshold ranges from N=2 to
N∼10⁴ depending on model and word pair. A mean-field theory accounts for the
large-N limit as competition between two consensus states.
This is the wiki's third population-scale multi-agent entry, after the
flag game and
Physics of Agents, and the first where group
size is swept with no ground truth and no private evidence. The collective
outcome is a convention, an arbitrary choice between equally viable options,
not a belief that can be right or wrong. That matters for the unnamed concept
this cluster is waiting on. It files concept-less, like its siblings.
Method
Each agent's state is a memory of its last H interactions: its own word, its
partner's word, the payoff, and its running score. The prompt describes up to
H=5. The payoff is +100 for a match and −50 for a mismatch. The prompt casts
the LLM as an observer forecasting "Player 1"'s next move, says nothing about a
population or how partners are chosen, randomizes word order, and asks for the
answer before the reasoning. Eleven word pairs were chosen for social-bias
sensitivity after CrowS-Pairs
(Flint et al. 2025).
The populations are not live LLM calls. For every possible memory state the
authors read the model's next-token logits for the two words, at temperature
0.5, and store the resulting probability as a policy table. Agents then sample
from the table in stochastic simulation, which is what makes N∼10⁴ and beyond
feasible. The surrogate is checked in two places. Table probabilities were
compared with open-generation estimates for Phi-4 on {his, her} at H=3, where
4 of 64 memory states differ at the 5% level. Simulated collective bias was
compared with live-LLM populations for three model/pair cases, using 50 live
runs per point (20 for Llama).
Individual bias is the empty-memory policy, and an agent counts as neutral
when that policy is within Jensen–Shannon distance 0.005 of uniform.
Collective bias is the fraction of 1000 runs that reach consensus on a given
word. A run has converged when ≥98% of the last 3N interactions succeed, and it
stops at 1000 population rounds otherwise. All populations are homogeneous,
with every agent's policy taken from one model.
Key results
Three forms of collective misalignment at N=24. Across 1000 runs per
condition, interaction amplifies an individual preference ({American, Mexican}),
induces a collective preference from individually neutral agents
({White, African}), or reverses the individual preference ({straight, gay}).
All three are shown for Llama and GPT populations (Fig. 1). Across four models
and eleven pairs (Fig. 2), GPT and Qwen agents are near-neutral individually,
while Phi and Llama vary more. At the collective level GPT shows mild effects,
Qwen a wide spread, and Llama near-deterministic convergence on one word per
pair. Phi tracks a minimal naming-game model with fixed individual bias most
closely, yet still departs from it strongly on five of eleven pairs. On
{her, his}, Qwen and Phi populations converge on her and GPT and Llama on
his, from nearly identical individual tendencies.
Group size makes the outcome deterministic, with a threshold that varies by
model and pair. Collective preference for the strong word rises with N in
every pair tested until convergence is deterministic (Fig. 3). Determinism
arrives at N=2 for Llama on {short, tall} and not until N∼10⁴ for Qwen on
{Black, White}. For GPT-4o on {White, African}, where agents are individually
neutral, the probability of coordinating on African is 0.599 at N=3, 0.720
at N=10, 0.981 at N=100 and 1.00 at N=1000 (published Fig. 4; the arXiv
preprint's first panel is N=2 at 0.626). Size can also change the form
of misalignment. Llama individually prefers straight in {straight, gay}, and
the collective reversal to gay appears only for N≥6.
Why the outcome changes with size. Small populations are
fluctuation-driven. They collapse to consensus in a few rounds on whichever
word gets an early lead, so the outcome still reflects individual bias. At
intermediate N, fluctuation and coordination coexist, and consensus times have
a heavy right tail. At larger N, runs that end on the weak word take longer
than runs that end on the strong word, and past a threshold N_c only the strong
word is reached, at a characteristic time. The authors call the shift from
fluctuation-driven to coordination-driven consensus robust, while its stages
vary with model and pair.
The mean-field picture differs by model. In the N→∞ limit the fixed points
are the two all-one-word states. Reading Table S1's reported stability for the
strong and weak states gives a model split. For GPT-4o and Phi-4, both states
are stable on all eleven pairs, so large-N determinism comes from which basin
the dynamics fall into, not from instability of the weak state. For Llama, the
strong state is stable and the weak one unstable on six of eleven pairs, which fits its near-deterministic
collective outcomes at N=24. This model-by-model split is the entry's reading
of the table, not a claim the paper makes.
Where consensus fails. Nearly every population converges. The exceptions
are large Llama populations on {old, young} and {less, more}, whose convergence
rate falls beyond some N, and a minor fraction of small Qwen runs on
{husband, wife}. For Llama on {less, more} both consensus states are unstable,
and on {old, young} the strong state is unstable and the weak one marginal.
The mixed states that result match time-dependent steady states in finite-N
simulation. The authors document these cases and leave their analysis to future
work.
Why it matters
The cluster's two earlier population-scale entries were read at filing as
disagreeing about the generic outcome. In the flag game,
chance wrong consensus gives way to truth–rival polarization as N grows. In
Physics of Agents, fitted couplings at N=32
favour consensus. This paper sharpens that disagreement without settling it.
It adds the variable Physics of Agents holds fixed, but it lacks the ingredient
that produces flag game's large-N polarization. With homogeneous agents and no
private evidence, larger populations here do not split. They converge harder,
and on a predictable word. The small-N parallel is close. Flag game's memetic
drift, where wrong consensus arises by chance and fades with N, looks like this
paper's fluctuation-driven regime, where weak-word consensus fades with N. What
takes its place differs. Flag game's zealots are agents whose private evidence
decides the question, and when both kinds are present they hold camps apart.
This paper has no such agents, and its large-N outcome is deterministic
consensus. That is compatible with polarization needing persistent
heterogeneity rather than size alone. Physics of Agents leaves the same opening
in its caveat that strongly opposed intrinsic fields can sustain a split. None
of the three papers tests this.
The paper also adds a third route to non-consensus that neither sibling has. It
comes from neither zealots nor signed repulsion. In Llama's {less, more} and
{old, young} populations, the memory-conditioned policy leaves no stable
consensus state, and it shows up only at large N. So "does scaling a swarm
tend toward agreement" gets a model-dependent answer even in a game built to
reach agreement.
For safety, the paper frames the result as misalignment. A population's
collective preference can differ in strength and sign from what any single
agent shows, and a one-agent evaluation does not predict it. It is also not
stable across deployment scale: a bias invisible at N=1 can be certain at
N=100. The authors' stated implication is that testing at one size can miss
risks specific to intermediate or large populations.
interpretive tensions
Convention, not belief. The agents coordinate on which of two words to
use. Neither word is correct, and the payoff rewards only matching. The
"strong" word is the one the population's dynamics favour, and calling the
large-N outcome misalignment depends on the word pairs being socially loaded.
The flag game has a ground truth, and Physics of Agents has MATH answers and
political stances. Lining the three up treats a coordination outcome, an
evidence-aggregation outcome and an opinion outcome as one kind of object. They
may share dynamics without sharing a psychological referent.
A simulated population of one model's policy table. The large-N results
are simulations of a cached next-token distribution over two words at
temperature 0.5, not interacting LLM instances. Validation covers one model at
H=3 for the policy, and three model/pair cases with 20–50 live runs per point
for collective bias. The step from "the table reproduces the model at small N"
to "N∼10⁴ populations of the model would behave like this" is an
extrapolation. The published version reports that the results hold under
other temperatures, memory sizes and symmetric payoffs (SI Figs. S36–S38). The
preprint has no such checks, and the published SI could not be fetched, so those
checks are claimed here, not read.
Whose bias. Each agent's move is the model's forecast of what a third-party
"Player 1" should play. The authors chose this framing to avoid safety and
identity triggers. What gets measured is the model's forecast of a player's
choice under that memory, which may differ from what the model would choose
when addressed as the player.
A text–table mismatch in the preprint, corrected in PNAS. The preprint's
text describes Qwen on {husband, wife} as having an unstable weak state and a
marginal strong one, against its own Table S1. The published text agrees with
the table: the weak state is marginal and the strong one stable.
"Non-linear" is not "non-monotonic." The abstract's non-linear effect of
group size is thresholds and saturation. The collective preference for the
strong word rises with N in every pair. The non-monotonic behaviours are
narrower: the N≥6 onset of reversal for Llama on {straight, gay}, and the loss
of convergence at large N in two Llama pairs. Flag game's accuracy peak at
intermediate N has no counterpart here.
concepts
Collective dynamics — the population-size result for convention, and the entry that changed the concept's name: its object is a coordination convention with no truth value, which "collective belief" would not have held. Filed concept-less and re-homed the same day the concept was drawn (2026-09-24).
Emergent capabilities was considered
and set aside. Collective bias absent in individual agents is "emergent" in
the population sense of more is different, not in the concept's sense of a
capacity a model acquires without being trained for it. The model here is
unchanged across N. What changes is how many copies of it interact. The shared
word does not make it a shared phenomenon.
cross-references
- Attractor dynamics. Adjacent, not
instantiating. The mean-field "basins of attraction" are basins of a
population rate equation over memory states, and the consensus states are
absorbing states of an explicit game. The concept is defined by content
destinations of unconstrained dialogue, and its scope note already warns
against grouping by the word. - Flag game and
Physics of Agents. The other population-scale
entries. The three differ in game (convention, evidence aggregation, binary
opinion), in whether a ground truth exists, and in the manipulated variable
(N here, N and protocol in flag game, graph sign structure in Physics of
Agents). - Mind viruses and
chain-of-affective dynamics. The cluster's
transmission-channel entries. Nothing is transmitted here except the choice
of word, which makes this the purest case of population structure as the
only variable. - The predecessor, Flint Ashery, Aiello
and Baronchelli, Science Advances 2025, which reports collective bias from
individually unbiased agents at fixed N and committed-minority tipping. This
paper extends it to amplification and reversal and to a size sweep. - Cites Zhang et al. (its ref 19) for multi-agent collaboration on complex tasks. That paper's own size sweep (N = 2–10) is figure-only.
sources
- Flint, A., Aiello, L. M., Pastor-Satorras, R., & Baronchelli, A. (2025).
Group size effects and collective misalignment in LLM multi-agent
systems.
PNAS 123(34) e2531697123, 18 August 2026; preprint arXiv:2510.22422
(October 2025). The published main text was read; the published SI was not.