Quantization Games on Social Networks and Language Evolution

We consider a strategic network quantizer design setting where agents must balance fidelity in representing their local source distributions against their ability to successfully communicate with other connected agents. We study the problem as a network game and show existence of Nash equilibrium quantizers. For any agent, under Nash equilibrium, the word representing a given partition region is the conditional expectation of the mixture of local and social source probability distributions within the region. Since having knowledge of the original source of information in the network may not be realistic, we show that under certain conditions, the agents need not know the source origin and yet still settle on a Nash equilibrium using only the observed sources. Further, the network may converge to equilibrium through a distributed version of the Lloyd-Max algorithm. In contrast to traditional results in the evolution of language, we find several vocabularies may coexist in the Nash equilibrium, with each individual having exactly one of these vocabularies. The overlap between vocabularies is high for individuals that communicate frequently and have similar local sources. Finally, we argue that error in translation along a chain of communication does not grow if and only if the chain consists of agents with shared vocabulary. Numerical results are given.