A statistical-physics look at how a population of large language models invents, argues over, and settles on a shared name, and why the answer depends on decoding temperature.
⚠️ WORK IN PROGRESS!!
Introduction and context
Do you know how the first group of our ancestors developed a shared language? Have you ever wondered how a crowd spontaneously adopts the same slang word? Collective order in nature rarely needs a conductor. It emerges from local interactions, repeated many times, with no one in charge. In other words in a decentralized manner. Statistical physics has a long tradition of extracting the simplest possible model that still produces this kind of order, then asking exactly which knob controls the transition from disorder to consensus.
For the emergence of shared conventions (a common name, a common word, a common rule), that simplest model is the Naming Game (NG), introduced by Steels and made exactly solvable by Baronchelli, Felici, Loreto, Caglioti, and Steels in a classic 2006 paper. Two agents meet, one speaks a word from its private vocabulary, the other checks whether it already knows that word, and the pair either converges or the listener learns something new. Repeated across a whole population, this elementary rule produces full population-wide consensus, with a precisely known convergence time, , on a fully connected graph. Human populations do this too: a well-known 2015 experiment by Centola and Baronchelli showed that people playing an analogous naming game spontaneously converge on shared conventions, exactly as the model predicts.
Now a new kind of population is running this same game: populations of LLM agents. Recent work has shown that groups of LLMs playing minimal coordination games do spontaneously converge on shared conventions, and can even develop collective biases that no single agent shows in isolation. This matters beyond curiosity. As multi-agent LLM systems move toward genuinely decentralized and self-organizing deployments, with no central coordinator and outcomes emerging bottom-up from purely local interactions, understanding whether and how fast such a population can bootstrap a shared convention becomes a practical engineering question, not just a theoretical one. What none of these studies have done is treat the LLM's own decoding temperature, the parameter that controls how random each token sample is, as a genuine statistical-physics control parameter, the way physicists treat thermal temperature in a magnet.
Can decoding temperature be treated as an effective physical control parameter for the collective dynamics of a population of LLM agents negotiating a shared convention?
This is the question Critiqality new paper sets out to answer.
The Research Question
In the classical, deterministic Naming Game, the listener's decision is a hard rule: if the speaker's word is already in the listener's inventory, they agree; otherwise, the listener just appends it to memory. There is no room for error. The LLM-NG replaces that hard rule with a single-token LLM call: the listener is shown its own inventory and the proposed word, and must reply YES or NO at some sampling temperature . Nothing else about the game changes.
This is a small modification with a large consequence: the LLM can now be wrong in two different ways. It can say NO to a word it already has (a missed opportunity to agree), or it can say YES to a word it has never seen before (an accidental, ungrounded agreement). Neither error exists in the classical game. Both are entirely new physics.
The practical stakes are real. As LLM agents move toward fully decentralized deployments, federated learning, autonomous negotiation, swarms of agents with no central coordinator, knowing whether and how fast they will bootstrap a shared convention is an operational question. If temperature turns out to be a reliable dial for tuning consensus speed, that's a useful design lever. If it doesn't, if different architectures respond to temperature in completely different, even opposite, ways, that's arguably more important to know before deploying anything.