Discrete Incremental Voting

Authors Colin Cooper, Tomasz Radzik , Takeharu Shiraga

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Colin Cooper
  • Department of Informatics, King’s College, University of London, UK
Tomasz Radzik
  • Department of Informatics, King’s College, University of London, UK
Takeharu Shiraga
  • Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, Tokyo, Japan

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Colin Cooper, Tomasz Radzik, and Takeharu Shiraga. Discrete Incremental Voting. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 ("disagree strongly"), 2 ("disagree"), …, 5 ("agree strongly"). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbour’s opinion, if different. For opinions drawn from a set {1,2,…,k}, the opinion of the vertex would change by +1 if the opinion of the neighbour is larger, or by -1, if it is smaller. It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as complete graphs K_n and random graphs G_{n,p} for suitably large p. If the average of the original opinions satisfies i ≤ c ≤ i+1 for some integer i, then the asymptotic probability that opinion i wins is i+1-c, and the probability that opinion i+1 wins is c-i. With high probability, the winning opinion cannot be other than i or i+1. To contrast this, we show that for a path and opinions 0,1,2 arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions 0 and 2 is initially supported by a constant fraction of vertices.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Random distributed processes
  • Pull voting


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