Quantization Games on Social Networks and Language Evolution

Motivated by collaboration in human and human-robot groups, we consider designing lossy source codes for agents in networks that are in different statistical environments but also must communicate with one another along network connections. This yields a strategic network quantizer design problem where agents must balance fidelity in representing their local source distributions against their ability to successfully communicate with other connected agents. Using network game theory, we show existence of Bayes Nash equilibrium quantizers. For any agent, under Bayes Nash equilibrium, we prove that 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 further prove that under certain conditions, the agents need not know the source origin and yet still settle on a Bayes Nash equilibrium using only the observed sources. Further, we prove the network may converge to equilibrium through a distributed version of the Lloyd-Max algorithm, rather than centralized design. In contrast to traditional results in language evolution, we demonstrate several vocabularies may coexist in Bayes 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 prove 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 examples demonstrate our findings.