# The Naming Game on the complete graph

## TL;DR

This paper rigorously analyzes the time to consensus in the naming game on complete graphs, establishing bounds that match previous numerical predictions and developing new probabilistic tools for the analysis.

## Contribution

It provides the first rigorous bounds on the consensus time in the naming game on complete graphs, confirming and extending prior numerical findings.

## Key findings

- Consensus time is at least n^{1/2 - o(1)}
- Consensus time is at most logarithmic in n when only two words remain
- Develops new probabilistic estimates for semimartingales with jumps

## Abstract

We consider a model of language development, known as the naming game, in which agents invent, share and then select descriptive words for a single object, in such a way as to promote local consensus. When formulated on a finite and connected graph, a global consensus eventually emerges in which all agents use a common unique word. Previous numerical studies of the model on the complete graph with $n$ agents suggest that when no words initially exist, the time to consensus is of order $n^{1/2}$, assuming each agent speaks at a constant rate. We show rigorously that the time to consensus is at least $n^{1/2-o(1)}$, and that it is at most constant times $\log n$ when only two words remain. In order to do so we develop sample path estimates for quasi-left continuous semimartingales with bounded jumps.