Generalizing Condorcet's Jury Theorem to Social Networks
Dan Braha, Marcus A. M. de Aguiar
TL;DR
This work extends Condorcet's jury theorem to socially connected populations by modeling peer imitation on complete graphs with zealots. It derives exact stationary distributions for the number of voters supporting the correct option under binary and multi-alternative settings, revealing Beta-Binomial and Dirichlet-Multinomial laws and Beta/Dirichlet limits for vote shares. The authors show that, under an absolute-majority condition, collective accuracy under both majority and plurality rules strictly exceeds the performance of any single voter, demonstrating amplification of informative signals through social interaction. These results provide closed-form benchmarks for networked decision-making and deepen understanding of how dependence and topology influence wisdom of crowds in social, biological, and engineered systems.
Abstract
We generalize Condorcet's jury theorem (CJT) to socially connected populations in which agents revise discrete choices on a network in the presence of zealots. Free agents receive privately informative signals about the correct alternative and, at each update, either retain their state or imitate a uniformly chosen neighbor (free or zealot). For finite networks, we derive closed-form stationary laws for vote counts, and we characterize the corresponding vote-share limits as the number of free voters tends to infinity. For majority rule -- both in binary and multi-alternative settings -- we obtain an exact accuracy limit in closed form via the regularized incomplete beta function. For plurality rule, we establish sharp closed-form lower bounds on accuracy, expressed in terms of regularized incomplete beta functions. Under an absolute-majority condition for the correct alternative, both majority and plurality accuracies strictly exceed the accuracy of any single voter, showing that informative signals, coupled through social interaction, are amplified at the group level. These results extend CJT beyond independence and provide closed-form accuracy benchmarks for networked decision systems in social, biological, and engineered settings.