Triadic Consensus: A Randomized Algorithm for Voting in a Crowd

Typical voting rules do not work well in settings with many candidates. If there are just several hundred candidates, then even a simple task such as choosing a top candidate becomes impractical. Motivated by the hope of developing group consensus mechanisms over the internet, where the numbers of candidates could easily number in the thousands, we study an urn-based voting rule where each participant acts as a voter and a candidate. We prove that when participants lie in a one-dimensional space, this voting protocol finds a $(1-\epsilon/sqrt{n})$ approximation of the Condorcet winner with high probability while only requiring an expected $O(\frac{1}{\epsilon^2}\log^2 \frac{n}{\epsilon^2})$ comparisons on average per voter. Moreover, this voting protocol is shown to have a quasi-truthful Nash equilibrium: namely, a Nash equilibrium exists which may not be truthful, but produces a winner with the same probability distribution as that of the truthful strategy.

💡 Research Summary

The paper tackles the fundamental scalability problem of collective decision‑making when the number of candidates runs into the thousands—a regime where traditional voting rules (plurality, Borda, Condorcet methods) become infeasible because each voter would need to evaluate or rank an overwhelming set of alternatives. To address this, the authors introduce Triadic Consensus, an urn‑based randomized protocol in which every participant simultaneously plays the role of voter and candidate.

Algorithmic description. All candidates are placed as balls in a single urn. In each round three distinct candidates are drawn uniformly at random. Every voter observes this triple, orders the three according to his/her underlying preference (which is assumed to be derived from a one‑dimensional ideological space), and declares the most preferred candidate as the “winner” of that comparison. The winning candidate is returned to the urn, while the other two are permanently removed. The process repeats until only one candidate remains; that candidate is declared the election winner. The protocol requires only local information (the three‑way comparison) and no global ranking, making it naturally suited for large‑scale online settings where communication bandwidth is limited.

Theoretical guarantees. The authors assume that candidates lie on a real line and that each voter’s utility decreases with Euclidean distance from his/her ideal point. Under this model, the Condorcet winner (the candidate that would beat every other candidate in a head‑to‑head majority contest) exists and is unique. The main analytical result shows that after a number of rounds proportional to (\log n) (with (n) the number of candidates), the algorithm outputs a candidate that is a ((1-\epsilon/\sqrt{n}))‑approximation of the Condorcet winner with high probability. In other words, the probability that the true Condorcet winner is selected differs from one by at most (\epsilon/\sqrt{n}).

The expected number of three‑way comparisons each voter must perform is bounded by

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