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The naming game in language dynamics revisited

Part of: Mathematical sociology Special processes

Published online by Cambridge University Press:  30 March 2016

Nicolas Lanchier
Nicolas Lanchier*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA. Email address: [email protected].
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Abstract

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In this article we study a biased version of the naming game in which players are located on a connected graph and interact through successive conversations in order to select a common name for a given object. Initially, all the players use the same word B except for one bilingual individual who also uses word A. Both words are attributed a fitness, which measures how often players speak depending on the words they use and how often each word is spoken by bilingual individuals. The limiting behavior depends on a single parameter, ϕ, denoting the ratio of the fitness of word A to the fitness of word B. The main objective is to determine whether word A can invade the system and become the new linguistic convention. From the point of view of the mean-field approximation, invasion of word A is successful if and only if ϕ > 3, a result that we also prove for the process on complete graphs relying on the optimal stopping theorem for supermartingales and random walk estimates. In contrast, for the process on the one-dimensional lattice, word A can invade the system whenever ϕ > 1.053, indicating that the probability of invasion and the critical value for ϕ strongly depend on the degree of the graph. The system on regular lattices in higher dimensions is also studied by comparing the process with percolation models.

Keywords

Interacting particle systemnaming gamelanguage dynamicssemiotic dynamics

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 91D25: Spatial models
Type
Part 4. Random graphs and particle systems
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 139 - 158
Copyright
Copyright © Applied Probability Trust 2014 

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