The Distributed Multiple Voting Problem

A networked set of agents holding binary opinions does not seem to be able to compute its majority opinion by means of local binary interactions only. However, the majority problem can be solved using two or more bits, instead of one (F. Bénézit , “Interval consensus: From quantized gossip to voting”, Apr. 2009, pp. 3661-3664). Pairs of agents asynchronously exchange their states and update them according to a voting automaton. This paper presents binary voting automata as well as solutions to the multiple voting problem, where agents can vote for one candidate among |C| ≥ 2 candidates and need to determine the majority vote. The voting automata are derived from the pairwise gossip algorithm, which computes averages. In the binary case (|C|=2), we focus on averages in dimension 1, but in the multiple case (|C| ≥ 2) we quantize gossip in dimension |C| - 1 , which is larger than or equal to 1. We show in particular that a consensus on majority can be reached using 15 possible states (4 bits) for the ternary voting problem, and using 100 possible states (7 bits) for the quaternary voting problem.