Naming Game

A population invents private names for one shared thing, then prunes them in pairwise talk until—often—a single convention remains. Colours and labels are arbitrary identifiers. The drama is agreement without a central dictionary, not language in the full linguistic sense.

The minimal Naming Game asks how agents can bootstrap a shared word for a referent without global coordination. Baronchelli, Felici, Loreto, Caglioti and Steels analysed a microscopic model inspired by Luc Steels’s Naming Game and Talking Heads experiments: agents hold inventories of candidate words; two agents meet as speaker and hearer; the speaker utters a word for the object (inventing one if its inventory is empty); on success both keep only that word; on failure the hearer adds it. Homonymy is neglected so that, without loss of generality, the environment can be reduced to a single object. The system can then be studied as pure synonym competition around one referent.

In the well-mixed (mean-field) case the population typically accumulates many distinct words, reaches a memory peak, then undergoes a sharp disorder-to-order transition toward a shared convention. Baronchelli et al. showed that peak memory and convergence times scale as N^{3/2} for population size N, and that the final consensus is an absorbing state: once every agent holds the same unique word, the rules cannot invent diversity again. Finite-population proofs guarantee absorption with probability one when innovation is absent; the interesting scientific question is the scaling and the sharpness of the transition.

Topology changes the story. On low-dimensional lattices, local success creates domains that coarsen slowly; memory stays O(N) rather than O(N^{3/2}), and consensus times stretch dramatically. Small-world shortcuts can restore faster, mean-field-like late stages after an early coarsening regime. Community structure with sparse bridges prolongs multiple conventions. These are still games about one referent—the network only changes who can speak to whom.

AL.Floys ships three topologies: well-mixed, periodic lattice, and communities with adjustable bridge density. An optional innovation rate lets speakers invent even with non-empty inventories, destroying the absorbing character of consensus. COIN and FORGET are visitor interventions on inventories. Word identities are reclaimed from a free list; colours and on-canvas labels carry no semantics.

Steels’s artificial-intelligence programme asked whether embodied agents could negotiate lexicons through local games. The statistical-physics minimal model distilled that programme into inventory rules simple enough for scaling analysis and network embedding. The 2005–2006 papers established the mean-field peak, the absorbing consensus, and the lattice/small-world corrections. Social-tagging systems and robot lexicon experiments sit in the same conceptual neighbourhood; none of them turn this sphere into a model of grammar or child language acquisition.

Aspect	Current implementation
Referents	One shared object (M = 1).
Interaction	Speaker–hearer pair; success contracts both inventories to the uttered word; failure adds the word to the hearer.
Topologies	Well mixed; lattice (periodic 4-neighbour); communities + bridges.
Innovation	Optional per-interaction invent probability (0 in most presets).
Populations	Showcase desktop N = 1200; mobile N = 800; Perpetual Slang ≤ 400.
Telemetry	Active words, leading share, inventory totals, success rate, entropy, consensus flag.
Tools	COIN invents and paints a word; FORGET clears inventories in radius.

First Word (well-mixed, innovation 0) should show vocabulary growth, an inventory peak, rising success rate, then collapse toward one leading word and a consensus flag. Local Tongues on the lattice shows coloured domains that coarsen; do not expect on-screen global consensus. Dialect Islands keeps several conventions alive longer across sparse bridges; Open Borders raises bridge density for faster mixing in community mode. Perpetual Slang sets innovation to 0.008 so fresh words keep appearing—leading share may rise locally while active words refuse to fall to one. Word labels, when enabled, name majority single-word conventions; the strings are identifiers, not meanings.

Pick a topology-first preset, then change only Innovation or Bridge density (community mode) after you understand the baseline. Interactions/s·agent sets tempo. Agents changes finite-size scaling—larger well-mixed populations make the late transition look sharper on collective timescales, matching the published size trend. Use COIN to inject a competitor word; use FORGET to clear a region and watch re-invention or invasion. Restart after topology or population changes.

• Topology. Well-mixed versus lattice versus communities is the main structural lesson.
• Bridge density. Only meaningful for communities; sparse bridges protect dialect islands.
• Innovation. Zero → consensus can absorb; nonzero → perpetual lexical diversity is possible.
• Interactions/s·agent. Clock rate, not a change of update rule.
• Word labels / Trails. Visual explanations only.

First Word

Well-mixed, N = 1200, innovation 0. Showcase consensus path; population chosen from sweep evidence that N ≤ 1200 still reaches consensus in the harness window at 20 interactions/agent/s.

Local Tongues

Lattice. Domain coarsening; consensus not expected on screen.

Dialect Islands

Communities with bridge fraction 0.02. Multiple conventions persist longer.

Open Borders

Communities with bridge fraction 0.12. Faster cross-group mixing.

Perpetual Slang

Innovation 0.008, smaller N. Diversity remains above one by design.

First Word (Mobile)

Well-mixed N = 800 for mobile showcase budgets.

Recommended order: First Word → Local Tongues → Dialect Islands → Perpetual Slang.

• COIN invents one word per gesture and writes it onto agents in the brush—an exogenous competitor.
• FORGET clears inventories locally so invention and invasion can restart.
• Interaction chords and success-rate telemetry track actual game outcomes, not decorative links.

This is not a model of grammar, semantics, phonology, comprehension, bilingualism, or human language acquisition. Agents do not parse sentences or ground rich meaning; they compete synonyms for one referent. Calling the sphere “Language” as a public title would overclaim—the atlas keeps Naming Game and states the one-referent scope in the HUD and Field Note.

Consensus with innovation zero is an absorbing convention, not proof that real communities must converge. Lattice presets that remain multi-coloured are faithful to slow coarsening, not bugs. Colours are category codes. Innovation, COIN, FORGET and community bridges are implementation extensions beyond the strictest mean-field minimal game.

Naming Game closes the Mind trio with cultural agreement: Associative Memory repairs a pattern inside one network; Belief Dynamics moves continuous opinions under a hearing threshold; Naming Game shows how discrete invented signals can win a population. Together they sketch three complementary collective computations—attractor, compromise, and convention.

Foundational and primary sources

1. Andrea Baronchelli, Maddalena Felici, Vittorio Loreto, Emanuele Caglioti and Luc Steels, “Sharp transition towards shared vocabularies in multi-agent systems,” Journal of Statistical Mechanics: Theory and Experiment (2006), P06014. Preprint: arXiv:physics/0509075.
2. Luca Dall’Asta, Andrea Baronchelli, Alain Barrat and Vittorio Loreto, “Agreement dynamics on small-world networks,” Europhysics Letters 73, 969–975 (2006). Preprint: arXiv:cond-mat/0603205.

Implementation evidence

3. AL.Floys naming-game population sweep under docs/evidence/mind-expansion/naming-game/ (pack @uridolan/al-floys-engines@0.10.1): N = 200…1200 reached consensus in the measured harness at 20 interactions/agent/s with innovation 0.