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Microscopic dynamics of consensus formation in multi-agent LLM Naming Games
Cristiano De Nobili, Vijayasri Iyer, Alessandro Codello, Raffaella Burioni
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
This paper investigates the microscopic dynamics of consensus formation in decentralized populations of Large Language Model (LLM) agents using a modified Naming Game. By replacing deterministic inventory checks with single-token LLM calls at varying decoding temperatures, the study identifies two key conditional rates: consolidation (pi) and repaint (phi). The research demonstrates that decoding temperature acts as an architecture-dependent control parameter, leading to three distinct listener regimes (permissive, near-deterministic, and conservative) across different LLM architectures. Mean-field theory and finite-size scaling analysis reveal how these microscopic stochasticities influence macroscopic ordering and convergence times.
Entities (10)
Relation Signals (7)
LLM Naming Game → usesparameter → Decoding Temperature
confidence 95% · We study a minimal LLM Naming Game in which the listener's decision is a single-token LLM call at decoding temperature T
llama3.1:8b → exhibitsregime → Permissive Regime
confidence 92% · llama3.1:8b (permissive listener)... repaint-noise dominated
phi3:14b → exhibitsregime → Conservative Regime
confidence 92% · phi3:14b (conservative listener)... missed-collapse dominated
Decoding Temperature → influences → Consolidation Rate
confidence 90% · pi(T) is the rate at which a collapse is correctly triggered... shifts with temperature
Decoding Temperature → influences → Repaint Rate
confidence 90% · phi(T) is the rate at which one is triggered erroneously... shifts with temperature
Mean-Field Theory → predicts → Repaint Rate
confidence 85% · A mean-field theory of the two-rate dynamics yields an analytical ordering condition
Mean-Field Theory → predicts → Consolidation Rate
confidence 85% · A mean-field theory of the two-rate dynamics yields an analytical ordering condition
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Abstract
Abstract:Decentralized populations of Large Language Model (LLM) agents can spontaneously reach consensus on shared conventions, yet the microscopic mechanisms by which their internal stochasticity shapes macroscopic ordering remain unexplored. We study a minimal LLM Naming Game in which the listener's decision is a single-token LLM call at decoding temperature $T$, replacing the inventory check of the deterministic Naming Game. Each interaction decomposes into an in-inventory and an out-inventory channel with conditional rates $\pi(T)\!\equiv\!P(\text{YES}\mid w\in P_j)$ and $\phi(T)\!\equiv\!P(\text{YES}\mid w\notin P_j)$, whose balance controls an ordering-disordering drift. A mean-field theory of the two-rate dynamics yields an analytical ordering condition that generalizes the consensus threshold of the stochastic Naming Game to a critical line in the $(\pi,\phi)$ plane. Across three open-weight architectures, consensus is always reached, but through three distinct listener regimes: permissive (repaint-noise dominated), near-deterministic, and conservative (missed-collapse dominated). The effective finite-size exponent $\beta(T)$ in $t_{\rm conv}\!\sim\!N^{\beta}$ shifts with temperature, and the temperature-sensitivity $\alpha$ in $t_c\!\sim\!e^{\alpha T}$ ranges from ${\approx}\,0.67$ to ${\approx}\,0$ across architectures. Decoding temperature thus emerges as an architecture-dependent control parameter for decentralized LLM populations, quantitatively characterized by the statistical-physics toolkit.
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- Source: https://arxiv.org/abs/2608.02178v1
- Canonical: https://arxiv.org/abs/2608.02178v1
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email: ]cristiano@critiqality.ai Microscopic dynamics of consensus formation in multi-agent LLM Naming Games Cristiano De Nobili [ Critiqality, Via Pinturicchio 21, 20133, Milan, Italy Vijayasri Iyer Independent Researcher Alessandro Codello DSMN, Ca’ Foscari University of Venice, Via Torino 155, 30172 Venice, Italy IFFI, Universidad de la República, J.H.y Reissig 565, 11300 Montevideo, Uruguay Raffaella Burioni Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 7/A 43124 Parma, Italy INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy Abstract Decentralized populations of Large Language Model (LLM) agents can spontaneously reach consensus on shared conventions, yet the microscopic mechanisms by which their internal stochasticity shapes macroscopic ordering remain unexplored. We study a minimal LLM Naming Game in which the listener’s decision is a single-token LLM call at decoding temperature T, replacing the inventory check of the deterministic Naming Game. Each interaction decomposes into an in-inventory and an out-inventory channel with conditional rates π(T)≡P(YES∣w∈Pj)π(T)\!≡\!P(YES w∈ P_j) and ϕ(T)≡P(YES∣w∉Pj)φ(T)\!≡\!P(YES w∉ P_j), whose balance controls an ordering-disordering drift. A mean-field theory of the two-rate dynamics yields an analytical ordering condition that generalizes the consensus threshold of the stochastic Naming Game to a critical line in the (π,ϕ)(π,φ) plane. Across three open-weight architectures, consensus is always reached, but through three distinct listener regimes: permissive (repaint-noise dominated), near-deterministic, and conservative (missed-collapse dominated). The effective finite-size exponent β(T)β(T) in tconv∼Nβt_ conv\! \!N^β shifts with temperature, and the temperature-sensitivity α in tc∼eαTt_c\! \!e^α T ranges from ≈ 0.67≈\,0.67 to ≈ 0≈\,0 across architectures. Decoding temperature thus emerges as an architecture-dependent control parameter for decentralized LLM populations, quantitatively characterized by the statistical-physics toolkit. I Introduction Consensus formation, that is the spontaneous emergence of a shared convention from purely local interactions, is a central problem at the intersection of statistical physics, complex systems, and multi-agent intelligence [6]. Classical agent-based models such as the Voter model, Axelrod dynamics, and bounded-confidence frameworks have shown that macroscopic order can arise from simple microscopic stochastic rules, with the nature of the resulting phase transitions, coarsening dynamics, and scaling laws depending sensitively on the interaction topology and on the agents’ decision rule [6]. Among these, the Naming Game (NG) [26, 4, 19] occupies a distinguished position: it provides a minimal, exactly solvable model of how a population bootstraps a shared vocabulary through pairwise negotiation, exhibiting symmetry breaking and a convergence time tconv∼N3/2t_ conv N^3/2 on fully connected graphs [4]. The deterministic NG has been studied on regular lattices [2], complex networks [10], and generalized to include irresolute agents with a tunable commitment parameter, revealing a non-equilibrium phase transition from consensus to fragmentation [3]. The rapid deployment of Large Language Models (LLMs) as autonomous, interacting agents has opened a new arena for these questions. LLM-based multi-agent systems are now used in negotiation, planning, and tool-use [15, 17], and populations of such agents interact with one another and with humans at increasing scale [18]. Whether and how decentralized populations of LLMs can self-organize on shared conventions is therefore not only a foundational scientific question but also a prerequisite for the safe and reliable deployment of agentic AI. Recent empirical studies have begun to answer it: Ashery, Aiello, and Baronchelli [1] demonstrated that populations of LLM agents playing a minimal coordination game, of the type used in human experiments on convention formation [7], spontaneously reach group-wide linguistic conventions and exhibit emergent collective biases not present at the single-agent level. De Marzo, Castellano, and Garcia [12] showed that LLM agents can self-organize on arbitrary binary choices but only below a model-dependent critical group size, and Flint et al. [14] developed a mean-field analytical framework relating group size to the structure of basins of attraction. Related work has investigated conformity and social influence in LLM agents [11, 5], detailed balance in LLM-driven dynamics [25], cultural attractors in transmission chains [22], opinion dynamics in networks of LLMs [8, 23], agreement protocols among reasoning agents [24], and Ising-like collective alignment on lattices [13]. Mean-field methods from statistical mechanics have also proven effective in large-population multi-agent reinforcement learning [28], and differentiable surrogates of agent-based models are being developed to bridge microscopic rules and emergent collective phenomena [9]. A consistent picture is emerging: LLM populations do reach consensus on shared conventions, the resulting macroscopic dynamics can be modelled with statistical-physics tools, and the outcome depends in non-trivial ways on the underlying architecture. What is still missing, however, is a microscopic theory linking the stochasticity of LLM inference to the macroscopic dynamics of consensus. In particular, the decoding temperature T, the primary hyperparameter governing the randomness of each agent’s output [16], has not been studied as an effective control parameter in the statistical-physics sense; existing LLM-NG studies fix T to a single value [1, 14]. This gap is especially relevant as LLM agents move toward decentralized deployment, where no central coordinator exists and collective outcomes emerge bottom-up from local interactions [18, 9]. In this work we fill this gap. We implement the minimal NG with LLM agents, retaining the original topology and update rule of the deterministic NG but replacing the listener’s deterministic inventory check with a single-token LLM call at temperature T. We then introduce a microscopic decomposition of each interaction into four channels, parametrized by two conditional rates, π(T)π(T) and ϕ(T)φ(T), and show that this pair suffices to organize the model-dependent phenomenology into three qualitatively distinct regimes. The setup is deliberately minimal: it isolates the effect of LLM-generated stochasticity on a well-understood ordering dynamics, free from prompt engineering, multi-turn memory, or strategic play. Our main contributions are: (i) the (π,ϕ)(π,φ) decomposition itself, which compresses the stochasticity of LLM inference into two measurable conditional rates and thereby exposes the microscopic origin of collective order; (i) a drift analysis that links these microscopic rates to the net ordering tendency; (i) the identification of three qualitatively distinct listener regimes (permissive, near-deterministic, and conservative) that emerge from the interplay of architecture and temperature, with qualitatively different macroscopic signatures including an inverted temperature ordering for the conservative regime; (iv) finite-size scaling and temperature-response measurements over the accessible range N∈[50,150]N\!∈\![50,150] indicating that the effective exponent β(T)β(T) in tconv∼Nβt_ conv\! \!N^β varies with decoding temperature and, for the most noise-dominated architecture, reaches values above the canonical mean-field 3/23/2, and that the exponential rate α governing tc(T)t_c(T) is itself architecture-dependent, ranging from essentially zero to ≈ 0.7≈\,0.7 across the three models tested. (v) a complete-graph mean-field theory of the two-rate dynamics whose two-word sector yields the analytical ordering condition 3π−2ϕ−1>03π-2φ-1>0, generalizing the known threshold of the stochastic NG and rationalizing the three regimes. Together these results promote decoding temperature to a bona-fide effective control parameter for decentralized LLM populations, and show that the statistical-physics toolkit of coarsening, drift analysis, and scaling provides a natural language for characterizing and ultimately designing the emergent behaviour of multi-agent AI systems. I Model Figure 1: Schematic of the Naming Game interaction. In the deterministic NG, the decision step is an inventory check (w∈Pjw∈ P_j); in the LLM-NG it is replaced by the listener’s LLM call at temperature T. In the deterministic NG, N agents are placed on a fully connected graph. Each agent i carries an inventory Pi(t)⊆P_i(t) , where V is a pool of words. All inventories start empty. At each step t, an ordered pair (i,j)(i,j) is drawn uniformly (speaker i, listener j). If an agent has an empty inventory it invents by sampling from V. The speaker picks w uniformly from PiP_i and transmits it. The interaction succeeds if w∈Pjw∈ P_j: both agents collapse to w\w\; otherwise j appends w (Fig. 1). On the fully connected graph, the deterministic NG exhibits a characteristic three-stage dynamics: an initial phase in which agents accumulate distinct words in their inventories, a coarsening stage in which pairwise agreements build up correlations between inventories, and a final rapid collapse to a single conventional name [4, 19]. The resulting consensus time scales as tc∼N3/2t_c N^3/2, a benchmark against which we compare the LLM-driven dynamics throughout this work. Figure 2: The two new interaction channels of the LLM-NG that do not exist in the deterministic NG (Fig. 1). Top: a missed collapse occurs when w∈Pjw\!∈\!P_j but the LLM answers NO, at rate 1−π1-π; the inventories are left unchanged. Bottom: a repaint occurs when w∉Pjw\!∉\!P_j but the LLM answers YES, at rate ϕφ; the listener discards its inventory and adopts the alien word w. Together with Fig. 1, these panels enumerate the four channels (TP, FN, FP, TN) into which the LLM-NG interaction decomposes (Sec. I). In the LLM-NG, the listener receives a prompt containing its inventory and the proposed word, and returns a single YES/NO token at temperature T. If YES, both collapse to w\w\; if NO, j appends w. The prompt used throughout our simulations is: System: “You are an agent with your own language and vocabulary. You can and must reply with yes or no.” User: “Your words are: PjP_j. Do we add w to the list?”. The only control parameter varied is T. In this work, we chose ||=104|V|\!=\!10^4 English words. This number is much larger than any explored population size, so inventory collisions during invention are negligible and V is effectively unlimited, as in the deterministic NG [4]. We track: the number of distinct words Nd(t)=|⋃iPi(t)|N_d(t)=| _iP_i(t)| and the success rate S(t)S(t). Consensus is reached at tct_c defined by Nd(tc)=1N_d(t_c)=1. A comment on the prompt. Any wording choice can induce a listener bias beyond the architecture- and temperature-dependent one under study. Alternative phrasings such as “Is the proposed word already in your list? Answer only YES or NO.” are equally legitimate and could shift the numerical values of (π,ϕ)(π,φ). A parallel statistical-physics study of LLM populations on 2D lattices [13] observes that prompt rewordings can move the effective microscopic parameters quantitatively while leaving the overall collective behaviour and its physical description qualitatively similar. Given the compute budget of the present study we did not perform a systematic prompt-variation scan and defer this robustness check to future work. I Microscopic decomposition At each interaction, the proposed word w is either already present in the listener’s inventory, or absent from it. We refer to these as the in-inventory and out-inventory channels, respectively. The LLM listener can answer YES or NO in either case, producing four microscopic outcomes: true positive (TP: w∈Pjw∈ P_j, YES), false negative (FN: w∈Pjw∈ P_j, NO), false positive (FP: w∉Pjw∉ P_j, YES), and true negative (TN: w∉Pjw∉ P_j, NO). We define two conditional acceptance rates (Fig. 2): π(T) π(T) ≡P(YES∣w∈Pj)=TPTP+FN, ≡ P(YES w∈ P_j)= TPTP+FN\,, (1) ϕ(T) φ(T) ≡P(YES∣w∉Pj)=FPFP+TN. ≡ P(YES w∉ P_j)= FPFP+TN\,. (2) The deterministic NG corresponds to π=1π\!=\!1, ϕ=0φ\!=\!0: π(T)π(T) is the rate at which a collapse is correctly triggered, ϕ(T)φ(T) the rate at which one is triggered erroneously. To weight these rates by how often each situation occurs, we introduce the in-inventory fraction m(t)≡P(w∈Pj(t)),m(t)\;≡\;P (w\!∈\!P_j(t) )\,, (3) that is, the probability that at time t the transmitted word w already belongs to the listener’s inventory, averaged over uniformly random speaker–listener pairs (i,j)(i,j) and over words w drawn uniformly from Pi(t)P_i(t). Empirically, m(t)m(t) is estimated from the running fraction of in-inventory interactions, m^(t)=TP(t)+FN(t)TP(t)+FN(t)+FP(t)+TN(t), m(t)\;=\; TP(t)+FN(t)TP(t)+FN(t)+FP(t)+TN(t)\,, (4) evaluated in a small window around t. By construction m(t)m(t) starts near zero (empty or disparate inventories) and approaches unity at consensus. The two rates then control competing tendencies: an in-inventory–YES event (rate mπm\,π) is a legitimate collapse, the ordering channel; an out-inventory–YES event (rate (1−m)ϕ(1-m)\,φ) is a repaint, in which the listener discards its inventory and adopts an alien word, the disordering channel. Two clarifications on this labelling are in order. First, “disordering” is an ensemble-averaged statement, not a property of every event: a single repaint does commit both interacting agents to the singleton w\w\, but because w is by construction not the word the listener held, repeated repaints on average redirect the population toward random attractors rather than toward the name emerging as the global consensus. Second, repaints are not always harmful: near consensus, when only two or three names coexist in a metastable configuration, a rare repaint can break the deadlock by knocking an agent out of one attractor so that the majority absorbs it. This is how a small residual ϕφ can accelerate late-time convergence, and why the drift proxy introduced below predicts the sign of ordering only on average. In what follows, “ordering” and “disordering” should be read as ensemble labels valid at first order in a mean-field description. We define a drift proxy Δ(t)=m(t)π(t)−λ(1−m(t))ϕ(t), (t)=m(t)\,π(t)-λ\,(1-m(t))\,φ(t)\,, (5) with λ=1λ=1. Positive drift implies net ordering; negative drift signals that repaint noise overwhelms consolidation. The choice λ=1λ\!=\!1 deserves a brief comment. A true-positive collapse consolidates two agents around a word they already share, leaving the number of w-holders unchanged; a false-positive collapse instead recruits the listener into the population of w-holders from scratch, forcing it to abandon its entire prior inventory. The consequences for NdN_d, for individual inventory sizes, and for the overlap distribution therefore differ in magnitude between the two events, and a rigorous mean-field treatment would yield λ≠1λ≠ 1, possibly state-dependent. We adopt λ=1λ\!=\!1 as the simplest phenomenological choice: since it is the sign of Δ , and not its magnitude, that determines the ordering versus disordering balance and hence the three regimes of Sec. IV.1, this suffices for the qualitative claims made here. A mean-field derivation of λ from the microscopic rates, along the lines of Ref. [3], is a natural extension. The (π,ϕ)(π,φ) plane organizes several known limits: the deterministic NG (π=1,ϕ=0π\!=\!1,φ\!=\!0) [4], lazy consolidation (π<1,ϕ=0π\!<\!1,φ\!=\!0), voter-like (π=1,ϕ=1π\!=\!1,φ\!=\!1), and maximal chaos (π=0,ϕ=1π\!=\!0,φ\!=\!1). The lazy-consolidation edge is the stochastic negotiation model of Baronchelli, Dall’Asta, Barrat, and Loreto [3], whose hand-tuned commitment probability β plays the role of π, with ϕφ identically zero by construction because out-inventory interactions cannot trigger a collapse in their update rule. The decisive difference, developed in Sec. VIII, is that their β is imposed externally on the update rule, whereas our π(T)π(T) and ϕ(T)φ(T) are emergent outputs of the LLM listener, measured a posteriori from actual multi-agent simulations. IV Results We simulate the LLM-NG for three open-weight models served locally via Ollama [21]: llama3.1:8b (Meta), mistral:7b (Mistral AI), and phi3:14b (Microsoft). Unless stated otherwise, N=150N=150 agents interact for up to 10510^5 steps (1.75×1051.75× 10^5 for phi3:14b), with T∈0.05,0.2,0.4,0.6,0.8,1.0,1.2,1.4,1.6,1.8,2.0T∈\0.05,0.2,0.4,0.6,0.8,1.0,1.2,1.4,1.6,1.8,2.0\. Each configuration is averaged over 1010 to 1515 seeds. Throughout, consensus is defined by strict 11-consensus (Nd(tc)=1N_d(t_c)\!=\!1); central lines in all consensus-time plots show the median across seeds, and shaded bands span the interquartile range (p25, p75). This robust statistic was preferred over the mean and standard deviation because of the heavy-tailed seed distributions characteristic of the conservative listener regime. IV.1 Microscopic rates Figure 3 summarizes the microscopic behaviour of the three models through the conditional rates π(t)π(t) and ϕ(t)φ(t). llama3.1:8b (permissive listener). The consolidation rate π(t)π(t) settles at a temperature-dependent plateau: ≈ 1.0≈\,1.0 at T=0.05T\!=\!0.05, decreasing to ≈ 0.55≈\,0.55 at T=2.0T\!=\!2.0 (Fig. 3a). The repaint rate ϕ(t)φ(t) starts high and decays as the system orders (Fig. 3d): at T=2.0T\!=\!2.0, ϕφ reaches ≈ 0.50≈\,0.50 at early times, remaining elevated for tens of thousands of steps. Both π and ϕφ show clear, monotonic temperature ordering. This places llama in a repaint-noise dominated regime where ϕφ is the main lever through which temperature controls the dynamics. mistral:7b (near-deterministic listener). π(t)π(t) saturates at ≈ 1.0≈\,1.0 for all tested temperatures (Fig. 3b), and ϕ(t)φ(t) shows only a small early-time hump (≲ 0.15 \,0.15) that decays to zero by t≈5×103t\!≈\!5× 10^3 (Fig. 3e). A subtle but robust inverted temperature ordering appears in the transient: the T=0.05T\!=\!0.05 curve has the slowest π approach and the highest early-time ϕφ peak, while T=2.0T\!=\!2.0 converges fastest. At low T, the listener is slightly more conservative and makes errors in both directions; higher T increases overall permissiveness, paradoxically improving consistency. Since ϕφ is already small, this cost is negligible and the net effect is beneficial. phi3:14b (conservative listener). π(t)π(t) settles at dramatically lower plateaux: ≈ 0.75≈\,0.75 at T=0.05T\!=\!0.05, falling to ≈ 0.30≈\,0.30 at T=2.0T\!=\!2.0 (Fig. 3c). At T=2.0T\!=\!2.0, 70% of in-inventory interactions fail to trigger the collapse they should. Meanwhile ϕ(t)φ(t) is near zero at all temperatures (Fig. 3f), with only a faint persistent level ≈ 0.03≈\,0.03–0.050.05 at T=2.0T\!=\!2.0. This places phi3 in the lazy-consolidation limit (π<1π<1, ϕ≈0φ≈ 0): the disordering channel is shut off, but the ordering channel is heavily attenuated. Figure 3: Microscopic conditional rates for N=150N\!=\!150 agents at selected temperatures. Top row: consolidation rate π(t)=P(YES∣w∈Pj)π(t)\!=\!P(YES w∈ P_j) for (a) llama3.1:8b, (b) mistral:7b, (c) phi3:14b. Dashed line: deterministic NG (π=1π\!=\!1). Bottom row: repaint rate ϕ(t)=P(YES∣w∉Pj)φ(t)\!=\!P(YES w∉ P_j) for (d) llama3.1:8b, (e) mistral:7b, (f) phi3:14b. Dashed line: deterministic NG (ϕ=0φ\!=\!0). Shaded bands indicate seed-to-seed variance. Note the different y-axis scales in the bottom row, reflecting the order-of-magnitude difference in repaint noise across models. Figure 4: Drift proxy Δ(t)=mπ−λ(1−m)ϕ (t)=mπ-λ(1-m)φ (λ=1λ\!=\!1) for (a) llama3.1:8b, (b) mistral:7b, (c) phi3:14b. Positive drift implies net ordering. Figure 5: Logarithm of the number of distinct words lnNd(t) N_d(t) for N=150N\!=\!150 agents at all explored temperatures (colour-coded from dark=\,=\,low T to light=\,=\,high T). Solid black: deterministic NG baseline. (a) llama3.1:8b: standard ordering (high T = slow). (b) mistral:7b: near-deterministic, all T bunched. (c) phi3:14b: inverted ordering (low T = slowest). Figure 6: Average inventory size k¯(t)=N−1∑i|Pi(t)| k(t)=N^-1 _i|P_i(t)| for N=150N\!=\!150 agents at all explored temperatures for (a) llama3.1:8b, (b) mistral:7b, (c) phi3:14b. Central lines are seed averages; shaded bands are one standard error. Note the different vertical scales: the phi3 panel spans roughly an order of magnitude more than the other two. Figure 7: Finite-size scaling of the consensus time tct_c vs. N (log-log) for (a) llama3.1:8b, (b) mistral:7b, (c) phi3:14b. Each colour is a different temperature. Central lines are medians across seeds, shaded bands are the interquartile range. Dashed black: average power-law guide N⟨β⟩N β . Upward arrows indicate lower bounds: at least one seed did not reach consensus within the simulation horizon, so the true tct_c is at least as large as the plotted value. IV.2 Drift and macroscopic convergence The drift proxy Δ(t) (t), Eq. (5), translates the microscopic rates into a net ordering tendency (Fig. 4). For llama (Fig. 4a), Δ is strongly temperature-dependent: at T=0.05T\!=\!0.05 it reaches unity (indistinguishable from deterministic NG), while at T=2.0T\!=\!2.0 it starts weakly negative (repaint noise overwhelms consolidation) and climbs only slowly to ≈ 0.55≈\,0.55. The time spent at low or negative drift maps directly onto the long plateaux in Nd(t)N_d(t) (Fig. 5a). For mistral (Fig. 4b), all temperatures converge to Δ=1 \!=\!1 within ∼ 104 \,10^4 steps. Surprisingly, T=0.05T\!=\!0.05 shows the deepest early negative dip (≈−0.25≈\,-0.25), reflecting the inverted-transient physics: the conservative low-T listener produces both missed collapses (lower π) and a slightly elevated ϕφ in the early disordered phase. For phi3 (Fig. 4c), Δ is always positive (no repaint noise) but never exceeds ≈ 0.75≈\,0.75. The drift mirrors π: weak but directed ordering without disordering competition. The macroscopic convergence (Fig. 5) reveals three qualitatively different responses of Nd(t)N_d(t) to temperature: llama (Fig. 5a) shows standard temperature ordering: higher T produces longer plateaux and slower convergence, reflecting the monotonic growth of repaint noise ϕ(T)φ(T). mistral (Fig. 5b) shows all temperature curves bunched tightly together and decaying faster than the deterministic baseline during the initial phase. The mild late-time splitting is consistent with the inverted transient: low-T curves have slightly longer tails. phi3 (Fig. 5c) shows a strongly inverted temperature ordering: the lowest temperature (T=0.05T\!=\!0.05, highest π) is the slowest to converge, with NdN_d plateauing at ≈ 7≈\,7 distinct words through 9×1049× 10^4 steps without reaching consensus. Higher temperatures converge faster despite having lower π. The mechanism is directly visible in the average inventory size k¯(t)=N−1∑i|Pi(t)| k(t)=N^-1 _i|P_i(t)| (Fig. 6). For phi3 at T=0.05T\!=\!0.05, agents accumulate on average up to ≈ 34≈\,34 words each around t≈7×104t\!≈\!7× 10^4 and remain at a large-inventory plateau k¯≳30 k\! \!30 throughout the entire 1.75×1051.75× 10^5-step simulation window without reaching consensus; at T=2.0T\!=\!2.0, by contrast, k¯ k peaks briefly below ≈ 4≈\,4 and decays to k¯≈1 k\!≈\!1 within ∼ 104 \,10^4 steps. llama and mistral peak at k¯≲5 k\! \!5 at every temperature and collapse to k¯=1 k\!=\!1 within (104)O(10^4) steps. This order-of-magnitude architecture-dependent difference in per-agent inventory size is the microscopic origin of phi3’s inverted temperature ordering: the narrow π≈0.75π\!≈\!0.75 channel must consolidate a much larger accumulated vocabulary at low T than at high T. A faint persistent ϕ≈0.04φ\!≈\!0.04 at T=2.0T\!=\!2.0 may further help break late-time deadlocks. This inverted ordering demonstrates that the macroscopic convergence speed is not determined by π alone, but by the interplay of π with the inventory structure, a genuinely new feature of the LLM-NG that emerges only when temperature is varied systematically. V Finite-size scaling In the deterministic NG on a fully connected graph, tconv∼N3/2t_ conv N^3/2 [4]. This scaling results from the interplay between the initial phase of word accumulation in the inventories of individual agents, the increase in correlations among the inventories of different agents, and the final coarsening collapse. We measure tconv(N,T)t_ conv(N,T) for N∈50,70,90,110,150N∈\50,70,90,110,150\ and fit tconv∼Nβ(T)t_ conv N^β(T) (Fig. 7). The explored range spans only about half a decade in N, so the fitted exponents should be interpreted as effective quantities over this window rather than as asymptotic critical exponents; a definitive determination of the true asymptotic scaling would require sizes at least an order of magnitude larger, beyond the reach of the present LLM-inference budget. For llama (Fig. 7a), the effective exponent varies from β≈1.3β\!≈\!1.3 at low T to β≈2.0β\!≈\!2.0 at T=2.0T\!=\!2.0, with a temperature-averaged value ⟨β⟩=1.61±0.28 β \!=\!1.61± 0.28. Over the accessible size range, β(T)β(T) reaches values above the canonical 3/23/2 at high temperatures, consistent with persistent repaint noise slowing the ordering drift. Whether this reflects a genuine change of universality class or a slow crossover to the deterministic 3/23/2 asymptote at larger N cannot be decided from the current data. For mistral (Fig. 7b) we obtain ⟨β⟩=1.28±0.13 β \!=\!1.28± 0.13, broadly consistent with an exponent near the canonical 3/23/2 and tighter than for the other two architectures. This is the expected behaviour for a near-deterministic listener whose microscopic rates barely depend on temperature. For phi3 (Fig. 7c) we find ⟨β⟩=1.59±0.47 β \!=\!1.59± 0.47, consistent with 3/23/2 within the sizable uncertainty. The wide IQR bands and the persistent lower-bound arrows in Fig. 7c are not statistical undersampling: with 1515 seeds at horizon 1.75×1051.75× 10^5 steps they reflect the intrinsic path-dependence of the conservative listener regime, where low-π dynamics make consensus trajectories strongly history dependent. The reported value should be interpreted as a lower bound on the true exponent, since seeds that fail to reach strict consensus within the simulation horizon would, on average, raise it. It is worth noting that phi3 is the empirical LLM realization of the lazy-consolidation model of Ref. [3]: to a very good approximation (ϕ≈0φ\!≈\!0 at every T), phi3’s dynamics coincides with theirs under the identification β↔π(T)β\! \!π(T). Their analytical prediction of a consensus–fragmentation transition at βc=1/3 _c\!=\!1/3 is therefore a direct, testable expectation for the phi3 regime (the same threshold emerges from the two-word mean field of Sec. VII as the ϕ=0φ\!=\!0 limit of the critical line): since phi3’s measured π ranges from ≈ 0.75≈\,0.75 at T=0.05T\!=\!0.05 to ≈ 0.30≈\,0.30 at T=2.0T\!=\!2.0 (Sec. IV.1), the threshold is nominally crossed inside the accessible temperature window. We nevertheless observe strict consensus at every temperature (Sec. VIII), so testing the prediction properly would require system sizes larger than those explored here. VI Temperature response of the consensus time While Fig. 7 examines how tct_c scales with N at fixed temperature, the complementary question is how tct_c depends on T at fixed N. We address it in Fig. 8, which displays tc(T)t_c(T) for N∈50,70,90,110,150N∈\50,70,90,110,150\ across the three architectures. Because tct_c varies multiplicatively with T, we fit each curve in log-linear form, lntc(T)=lnA+αT, t_c(T)\,=\, A\,+\,α\,T\,, (6) so that tc(T)=AeαTt_c(T)\!=\!A\,e^α T and the slope α has units of inverse temperature. The sign and magnitude of α provide a direct, single-number summary of the temperature sensitivity of the macroscopic consensus dynamics: α>0α\!>\!0 means tct_c grows with T, α≈0α\!≈\!0 means T-independence, and α<0α\!<\!0 would indicate accelerated convergence with T. The exponential form mirrors the power-law guide of Fig. 7 (a straight line on log-y) and produces a clean dashed-line overlay on each panel of Fig. 8. Figure 8: Temperature response of the consensus time tct_c vs. T (log-y, linear-x) for (a) llama3.1:8b, (b) mistral:7b, (c) phi3:14b. Each colour is a different agent count N∈50,70,90,110,150N\!∈\!\50,70,90,110,150\; central lines are medians across seeds and shaded bands span the interquartile range. Dashed black: average exponential guide tc∼e⟨α⟩Tt_c\! \!e α \,T. The fitted rates are ⟨α⟩=0.67±0.14 α \!=\!0.67± 0.14 for llama, ⟨α⟩=0.01±0.02 α \!=\!0.01± 0.02 for mistral, and ⟨α⟩=0.43±0.31 α \!=\!0.43± 0.31 for phi3. Upward arrows indicate lower bounds: at least one seed did not reach consensus within the simulation horizon. The three measured rates differ qualitatively across architectures and map cleanly onto the (π,ϕ)(π,φ) regimes identified in Sec. IV.1. llama (permissive listener, α=0.67±0.14α\!=\!0.67± 0.14). The exponent is clearly positive and statistically significant: across the explored range tct_c grows by a factor e0.67×2.0≈4e^0.67× 2.0\!≈\!4. This is the macroscopic fingerprint of the repaint-dominated regime: as T rises, ϕ(T)φ(T) grows and the disordering channel becomes increasingly active; the consensus time inherits this growth at an exponential rate. The exponential, rather than linear, dependence on T is itself informative; it suggests that tct_c is sensitive to the cumulative effect of many repaint events along an ordering trajectory, rather than to any single rate-limiting step. mistral (near-deterministic listener, α=0.01±0.02α\!=\!0.01± 0.02). The exponent is indistinguishable from zero: tct_c is statistically independent of decoding temperature across a 40×40× range in T. This is the macroscopic fingerprint of the near-deterministic regime: with π≈1π\!≈\!1 and ϕ≈0φ\!≈\!0 at every T (Sec. IV.1), neither microscopic rate has room to move, and the resulting consensus dynamics is effectively decoupled from the LLM’s primary stochasticity knob. A similar architecture-dependent insensitivity to decoding temperature has been reported in a related statistical-physics study of LLM populations on lattices [13], suggesting that this “temperature blindness” is a robust feature of certain LLM architectures across distinct collective-dynamics settings. phi3 (conservative listener, α=0.43±0.31α\!=\!0.43± 0.31). The central value is positive and moderate, but the uncertainty band overlaps zero. Together with the wide IQR shading in Fig. 8c, this reflects the genuine path-dependence of the conservative regime: low-π dynamics is intrinsically variable from seed to seed, and the temperature response is more subtle than for the other two architectures. The trend is mediated by the inventory-diversity mechanism described in Sec. IV.2 rather than by a direct change in the microscopic rates, since ϕφ remains near zero at every T. The pair (β,α)(β,α) thus provides a compact two-dimensional fingerprint of each architecture in the multi-agent NG: β measures how the consensus time scales with system size, α how it responds to decoding temperature. Their joint values cleanly separate the three regimes (Table 1). VII Mean-field theory of the two-rate dynamics At fixed decoding temperature, the LLM listener can be approximated by the effective two-rate rule q(YES∣Pj,w)=π,w∈Pj,ϕ,w∉Pj,q(YES P_j,w)\;=\; casesπ,&w∈ P_j,\\[2.0pt] φ,&w∉ P_j, cases (7) with π and ϕφ treated as constants, in practice the plateau values π¯(T) π(T) and ϕ¯(T) φ(T) of Sec. IV.1. This defines a stochastic NG with two acceptance channels that contains the deterministic NG (π=1π\!=\!1, ϕ=0φ\!=\!0) and the stochastic negotiation model of Ref. [3] (π=βπ\!=\!β, ϕ=0φ\!=\!0) as special cases, and for which exact complete-graph mean-field equations can be written for the densities of agents carrying each possible inventory. The resulting (2m−1)(2^m-1)-dimensional hierarchy is not solvable in closed form, but it admits a natural approximate closure for the inventory-size distribution; both are given in Appendix A. The critical line, however, follows analytically from the two-word sector, to which we now turn. The ordering mechanism is exposed by the two-word sector. When only two words A and B compete, each agent is in one of three states, A, B, or ABAB, with fractions x, y, and z=1−x−yz=1-x-y. On the complete graph the mean-field equations read x˙ x =−(1−ϕ)xy+3π−12xz+ϕyz+πz2, =-(1-φ)\,xy+ 3π-12\,xz+φ\,yz+π z^2, (8) y˙ y =−(1−ϕ)xy+3π−12yz+ϕxz+πz2, =-(1-φ)\,xy+ 3π-12\,yz+φ\,xz+π z^2, (9) reducing to the standard NG mean field for π=1π\!=\!1, ϕ=0φ\!=\!0. Introducing the magnetization-like order parameter u=x−yu=x-y and subtracting Eq. (9) from Eq. (8) gives u˙=z2(3π−2ϕ−1)u. u\;=\; z2\, (3π-2φ-1 )\,u\,. (10) The symmetric state is unstable, and one word is amplified over the other, when R≡ 3π−2ϕ−1> 0,i.e.π>πc(ϕ)=1+2ϕ3.R\;≡\;3π-2φ-1\;>\;0, .e. π\;>\; _c(φ)= 1+2φ3\,. (11) For ϕ=0φ\!=\!0 this recovers the known threshold βc=1/3 _c\!=\!1/3 of the stochastic NG [3]; the false-positive channel shifts the threshold upward along the critical line πc(ϕ) _c(φ), quantifying how repaint noise obstructs ordering. The quantity R thus plays the role of the effective distance from the transition, replacing the scalar β−βcβ- _c, and provides an analytical counterpart to the phenomenological drift proxy of Eq. (5). Evaluated on the measured plateau rates, R rationalizes the three regimes. For mistral, π¯≈1 π\!≈\!1 and ϕ¯≈0 φ\!≈\!0 give R≈2R\!≈\!2, the maximum possible value: the dynamics sits deep inside the ordering region at every temperature, which is precisely the temperature blindness of Sec. VI. For llama, increasing T lowers π and raises ϕφ; both changes decrease R, predicting the observed monotonic slowdown, and the elevated early-time ϕφ at T=2.0T\!=\!2.0 transiently drives R negative, consistent with the negative drift transient of Fig. 4a. For phi3, ϕ¯≈0 φ\!≈\!0 reduces the condition to π>1/3π>1/3: the measured plateau falls from π¯≈0.75 π\!≈\!0.75 (R≈1.25R\!≈\!1.25) at T=0.05T\!=\!0.05 to π¯≈0.30 π\!≈\!0.30 (R≈−0.1R\!≈\!-0.1) at T=2.0T\!=\!2.0, nominally crossing the critical line inside the explored temperature window. That strict consensus is nevertheless reached at all temperatures (Sec. VIII) is consistent with the mean-field transition being sharp only as N→∞N→∞ and with the time dependence of the measured rates, and identifies phi3 at high T as the natural setting in which to search for a fragmented phase at larger system sizes. By construction, the two-word reduction does not capture the inventory-size effects that dominate phi3 at low T (Sec. IV.2); the inventory-size closure of Appendix A is the natural starting point to include them. VIII Discussion and outlook The central result of this work is that two conditional rates, π(T)π(T) and ϕ(T)φ(T), provide a compact microscopic parametrisation of the LLM-NG that captures its leading-order macroscopic phenomenology and cleanly organizes the observed architecture-dependent behaviour into three qualitatively distinct regimes (Table 1). The (π,ϕ)(π,φ) pair is not a full microscopic theory: it averages over inventory size, over word identity, and over per-agent heterogeneity, and phi3 already provides an example (Sec. IV.2) in which macroscopic convergence is co-determined by π and by the inventory structure it generates. Nevertheless, the sign of the drift proxy Δ(t) (t) built from (π,ϕ)(π,φ) tracks the qualitative ordering / disordering balance of every architecture we tested, and the pair suffices as a first-order diagnostic and classification tool. Table 1: Classification of the three LLM architectures by their (π,ϕ)(π,φ) response, effective scaling exponent ⟨β⟩ β over N∈[50,150]N\!∈\![50,150], temperature-sensitivity ⟨α⟩ α , and dominant slowdown mechanism. Arrows ↘(T) (T) and ↗(T) (T) denote decrease and increase with temperature. Model π ϕφ ⟨β⟩ β ⟨α⟩ α Slowdown llama3.1:8b ↘(T) (T) ↗(T) (T) 1.611.61 0.670.67 repaints mistral:7b ≈ 1≈\,1 ≈ 0≈\,0 1.281.28 0.010.01 (weak) phi3:14b low, ↘(T) (T) ≈ 0≈\,0 1.591.59 0.430.43 missed collapses These three regimes are not imposed by hand but emerge from the interaction between the LLM architecture and the decoding temperature. The (π,ϕ)(π,φ) decomposition acts as a bridge between the “black box” of LLM inference and the well-understood physics of the NG. Our work is complementary to but distinct from recent LLM-NG studies in four specific ways. First, Ref. [1] and Ref. [14] characterise the macroscopic outcome (does consensus emerge, on which name, with what bias) at a single fixed temperature T=0.5T\!=\!0.5. We instead decompose each microscopic interaction into the conditional rates (π,ϕ)(π,φ) and trace how decoding temperature reshapes them. Second, their analysis treats the LLM as an unresolved black box and infers asymmetries from observed bias; we expose the underlying in-inventory and out-inventory channel structure that produces those asymmetries. Third, we elevate decoding temperature from a fixed hyperparameter to a tunable control parameter and quantify its effect through two complementary scalar diagnostics, β(T)β(T) and α(N)α(N). Fourth, we identify three qualitatively distinct architecture-dependent regimes that produce different consensus dynamics through different microscopic mechanisms. Together, these moves convert LLM-NG from a phenomenological observation that consensus emerges into a microscopically resolved statistical-physics problem. Our results are also complementary to those of Ref. [12], which established the existence of architecture-dependent critical group sizes, and to those of Ref. [11], which decomposes conformity into competing effective forces; in both cases, the (π,ϕ)(π,φ) rates provide a temporally resolved diagnostic that complements those purely macroscopic or aggregate-state characterizations. Two further studies complement our findings by varying different knobs of the LLM-NG. Mehdizadeh and Hilbert [20] study a networked Naming Game among LLM agents and show that agent memory depth interacts with network topology in a sign-flipping way: longer memory slows convergence in decentralized networks but accelerates the fragmented settling of centralized ones. Their work fixes population size and varies topology and memory, providing a natural counterpart to our temperature-based control parameter; whether their effects persist under finite-size scaling, and how they interact with the (π,ϕ)(π,φ) decomposition, remains open. Separately, Zhang et al. [29] show that imposing lightweight schema structure on the communication channel itself, rather than tuning a decoding hyperparameter, can accelerate naming-game convergence by up to 5.8×5.8×. Together with our results, this points to a broader picture in which convention formation in LLM populations can be steered through several largely independent levers, decoding temperature, memory depth, network topology, and communication schema, each acting on a different part of the underlying (π,ϕ)(π,φ) channel structure. Several features are worth emphasising. First, temperature does not universally slow consensus: for mistral, increasing T paradoxically improves consistency by suppressing the low-T conservative transient; for phi3, high T accelerates convergence despite lowering π, because it also reduces inventory diversity. The macroscopic effect of decoding temperature is thus architecture-dependent, mediated by the model-specific balance of the ordering and disordering channels. This cautions against treating temperature as a universal “noise knob” in multi-agent LLM systems. Second, from a statistical-physics perspective, the relationship between our (π,ϕ)(π,φ) framework and the stochastic negotiation model of Baronchelli et al. [3] is more than an analogy: their commitment probability β is mathematically identical to our π, and their model corresponds exactly to the ϕ=0φ\!=\!0 slice of the (π,ϕ)(π,φ) plane. The essential conceptual difference is in the origin of the stochasticity. In their setup, β is an external, hand-tuned scalar, chosen by the modeller and swept across [0,1][0,1] to trace out a phase diagram; the update rule fixes ϕ=0φ\!=\!0 by construction. In ours, both π(T)π(T) and ϕ(T)φ(T) are emergent: they are properties of the LLM listener that we measure a posteriori from multi-agent simulations, and their values, their T-dependence, and even their qualitative shapes are set by the architecture. This shift, from an external stochasticity parameter to an emergent one, is what turns a mathematically clean toy model into a diagnostic tool for real LLM populations. As a corollary, decoding temperature moves both rates along architecture-dependent trajectories in the two-dimensional (π,ϕ)(π,φ) plane, whereas the phase transition of Ref. [3] is inherently a one-dimensional phenomenon along the ϕ=0φ\!=\!0 edge, of which phi3 is the empirical embodiment (Sec. V). Notably, no fragmented phase was observed for any model at the explored temperatures: strict 11-consensus is always reached on the fully connected graph, even for phi3 at T=2.0T\!=\!2.0, where R≈−0.1R\!≈\!-0.1 lies nominally below the critical line (Sec. VII). As discussed there, this is consistent with a transition that sharpens only as N→∞N→∞, and with the group-size scenarios of Refs. [14, 12], where fragmentation emerges only above model-dependent population thresholds, beyond the sizes explored here. Third, over the accessible size range N∈[50,150]N\!∈\![50,150] the effective scaling exponent β(T)β(T) varies with temperature, reaching values above 3/23/2 for llama at high T. Discriminating a genuine change of universality class from a slow crossover to the deterministic asymptote would require substantially larger systems, a natural target for future work. More broadly, the (π,ϕ)(π,φ) framework is not specific to the naming game; any binary decision made by an LLM agent in the presence of a ground-truth state can be decomposed analogously. Recent statistical-physics analyses of LLM populations on lattices [13] and in conformity-driven opinion dynamics [11] have similarly relied on decompositions of the LLM response into competing effective parameters (cooperative coupling vs. intrinsic bias; majority force vs. individual preference). Asch-type conformity experiments on multimodal LLM agents [5], where a binary judgement with a known ground truth is flipped by social pressure, provide a particularly close setting: the probability of yielding to the group plays the same role as our false-positive rate ϕφ. Our (π,ϕ)(π,φ) rates fit naturally into this emerging framework as a temporally-resolved diagnostic. These findings carry direct implications for decentralized and self-organizing LLM populations. As multi-agent LLM systems are increasingly deployed without central coordination, for example in federated learning, autonomous negotiation, distributed scientific discovery, and collective decision making, the question of whether and how fast they converge on shared conventions becomes operational. Our results provide three quantitative answers. First, the (π,ϕ)(π,φ) rates can be measured offline through a small set of probing prompts, so that a system designer can determine which regime a given LLM occupies before deploying it. Second, the rate α tells the designer whether decoding temperature is a useful design knob: for mistral-like architectures (α≈0α\!≈\!0) tuning T has essentially no effect on consensus speed, while for llama-like architectures (α≈0.7α\!≈\!0.7) T changes tct_c by a factor of about four across the accessible range. Third, the temperature-dependence of β(T)β(T) means that the effective collective dynamics of an LLM swarm can be shaped through the joint choice of architecture and decoding temperature, opening the door to physics-informed selection of LLMs for specific multi-agent tasks. We view the identification of distinct listener archetypes and their mapping onto macroscopic consensus dynamics as a first step toward a statistical-physics taxonomy of LLM-agent behaviour, of which the (π,ϕ,β,α)(π,φ,β,α) quadruple is a candidate compact “datasheet” summarising an architecture’s collective properties, much as critical exponents summarise a universality class. Natural extensions include: heterogeneous temperatures (quenched disorder), structured interaction topologies, multi-object naming, the inclusion of committed minorities [27, 1], and the exploration of model-native commitment signals (e.g. confidence scores) as additional control parameters. The mean-field theory of Sec. VII opens several analytical directions: the analysis of the inventory-size hierarchy of Appendix A, which would capture the large-inventory effects that dominate the phi3 regime at low T; the structure of the possible fragmented states of the two-rate dynamics; the scaling of the convergence time with the distance R from the critical line; and the finite-N rounding of the transition, which the phi3 high-T regime is best positioned to probe. Appendix A Mean-field equations for arbitrary vocabulary and inventory-size closure Let the active vocabulary contain m words, Ωm=1,…,m _m=\1,…,m\, and let nS(t)n_S(t) be the fraction of agents with inventory S⊆ΩmS _m, S≠∅S≠ , normalized as ∑SnS=1 _Sn_S=1. On the complete graph, in the limit N→∞N→∞, the two-rate rule of Eq. (7) induces the exact mean-field dynamics n˙S=∑A,B≠∅nAnB1|A|∑w∈A[qB(w)(2δS,w−δS,A−δS,B)+(1−qB(w))(δS,B∪w−δS,B)], n_S\;=\; _A,B≠ n_An_B\, 1|A| _w∈ A [\,q_B(w)\, (2\, _S,\w\- _S,A- _S,B )\;+\; (1-q_B(w) ) ( _S,B∪\w\- _S,B ) ], (12) where A and B are the speaker and listener inventories, the speaker chooses w∈Aw∈ A uniformly, qB(w)=πq_B(w)=π if w∈Bw∈ B and qB(w)=ϕq_B(w)=φ otherwise, and δS,X _S,X is the Kronecker delta on inventories. The first bracket describes YES events, in which both agents collapse to w\w\; the second describes NO events, in which the speaker is unchanged and the listener learns w if absent. For ϕ=0φ=0, π=βπ=β, Eq. (12) reduces to the stochastic NG of Ref. [3]; for m=2m=2 it closes on the three densities x=nAx=n_\A\, y=nBy=n_\B\, z=nA,Bz=n_\A,B\ and yields Eqs. (8)–(9). For general m the hierarchy is (2m−1)(2^m-1)-dimensional and not solvable in closed form. A tractable closure is obtained by assuming that all active words are statistically equivalent, so that nSn_S depends on S only through its size: nS=ρk/(mk)n_S= _k/ mk for |S|=k|S|=k, where ρk(t)=∑|S|=knS(t) _k(t)= _|S|=kn_S(t) is the inventory-size distribution. This closure is the inventory analogue of the heterogeneous (degree-based) mean-field approach to dynamical processes on complex networks [6], with the inventory size k playing the role of the node degree. Under this ansatz, a listener with inventory size k contains the transmitted word with probability hk=k/mh_k=k/m, answers YES with probability Qk=πhk+ϕ(1−hk)=ϕ+(π−ϕ)km,Q_k\;=\;π\,h_k+φ\,(1-h_k)\;=\;φ+(π-φ)\, km\,, (13) and learns a new word (out-inventory NO event) with probability Lk=(1−ϕ)(1−km).L_k\;=\;(1-φ) (1- km ). (14) Writing Q=∑kρkQk=ϕ+(π−ϕ)μ/mQ= _k _kQ_k=φ+(π-φ)\,μ/m for the listener-averaged YES probability, with μ=∑kρkμ= _kk _k the mean inventory size, the closed equations read ρ˙1 ρ_1 =Q(1−ρ1)+∑ℓ≥2ρℓQℓ−ρ1L1, =Q\,(1- _1)+ _ ≥ 2 _ \,Q_ - _1L_1\,, (15) ρ˙k ρ_k =−ρk(Q+Qk)+ρk−1Lk−1−ρkLk,2≤k≤m. =- _k\,(Q+Q_k)+ _k-1L_k-1- _kL_k\,, 2≤ k≤ m. (16) In Eq. (16), −ρkQ- _kQ is the loss of size-k speakers that collapse to singletons after a YES interaction, −ρkQk- _kQ_k the loss of size-k listeners that collapse, ρk−1Lk−1 _k-1L_k-1 the gain of size-k listeners by learning one new word, and −ρkLk- _kL_k the corresponding loss towards size k+1k+1. At stationarity, Eq. (16) gives the recursion ρk _k =ρk−1Lk−1Q+Qk+Lk, = _k-1\, L_k-1Q+Q_k+L_k\,, (17) i.e.ρk .e. _k =ρ1∏r=1k−1LrQ+Qr+1+Lr+1, = _1 _r=1^k-1 L_rQ+Q_r+1+L_r+1\,, (18) subject to the normalization ∑kρk=1 _k _k=1 and the self-consistency condition Q=∑kρkQkQ= _k _kQ_k. 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