[Paper Review] A note on the voting problem

M.A. Fiol|arXiv (Cornell University)|Jun 4, 2012

Game Theory and Voting Systems2 references3 citations

TL;DR

This paper provides an improved upper bound on the minimum number of voters with transitive preferences needed to generate any strong tournament (i.e., complete, asymmetric preference pattern) on $ n $ candidates. Using an inductive construction based on transitive subtournaments and parity considerations, it proves $ v(n) \leq n - k $ if $ n $ and $ k = \lfloor \log_2 n \rfloor $ have different parity, and $ v(n) \leq n - k + 1 $ otherwise, refining prior bounds by Sterns and McGarvey.

ABSTRACT

Let $v(n)$ be the minimum number of voters with transitive preferences which are needed to generate any strong preference pattern (ties not allowed) on $n$ candidates. Let $k=\lfloor \log_2 n floor$. We show that $v(n)\le n-k$ if $n$ and $k$ have different parity, and $v(n)\le n-k+1$ otherwise.

Motivation & Objective

• To determine the minimal number of voters with transitive preferences required to realize any strong preference pattern (tournament) on $ n $ candidates.
• To improve upon existing upper bounds for $ v(n) $, the maximum, over all tournaments on $ n $ vertices, of the minimum number of voters needed to generate it.
• To provide an explicit inductive construction method that generates any desired tournament using a minimal set of voters.
• To refine earlier results by Sterns and McGarvey by incorporating the parity of $ n $ and $ k = \lfloor \log_2 n \rfloor $, leading to tighter bounds.

Proposed method

• The method relies on recursively building voter sets for larger tournaments from smaller ones using Theorem 2.2, which shows $ v(T_{n+2}) \leq v(T_n) + 2 $.
• It uses the existence of large transitive subtournaments—guaranteed by Theorem 2.3—where $ f(n) \geq \lfloor \log_2 n \rfloor + 1 $, to start the inductive construction.
• The construction defines two new voters per step: one placing a before all others and one placing b after all others, with intermediate orderings adjusted via reverse sequences to preserve majority preferences.
• The voter permutations are carefully structured using sequences $ \gamma, \delta, \sigma, \mu $ and their reverses to maintain correct pairwise majority outcomes.
• Parity is used to determine whether $ n - k $ or $ n - k + 1 $ voters suffice, based on whether $ n $ and $ k $ have the same parity.
• The proof leverages Lemma 2.1, which establishes that $ v(n) $ must be odd, ensuring consistency in the inductive step.

Experimental results

Research questions

• RQ1 What is the smallest number of voters with transitive preferences needed to generate any strong preference pattern on $ n $ candidates?
• RQ2 How does the parity of $ n $ and $ k = \lfloor \log_2 n \rfloor $ affect the minimal voter count?
• RQ3 Can an inductive construction method be used to systematically build voter sets that realize any tournament with minimal size?
• RQ4 How do the bounds from Sterns and McGarvey compare to tighter bounds when considering transitive subtournaments and parity?
• RQ5 To what extent can the Erdős–Moser $ O(n / \log_2 n) $ bound be improved for strong preference patterns?

Key findings

• The paper establishes that $ v(n) \leq n - k $ when $ n $ and $ k = \lfloor \log_2 n \rfloor $ have different parity.
• When $ n $ and $ k $ have the same parity, the bound is $ v(n) \leq n - k + 1 $, which is tighter than previous results.
• The bound improves upon Sterns’ result of $ v(n) \leq n+1 $ for even $ n $ and $ v(n) \leq n $ for odd $ n $, especially for larger $ n $.
• The construction is explicit and inductive, starting from a transitive subtournament of size $ k+1 $, and extending it step-by-step.
• The method ensures that all pairwise majority preferences are preserved through symmetric voter pairings and reverse-order permutations.
• The result confirms that $ v(n) $ is always odd, as shown in Lemma 2.1, which underpins the consistency of the inductive construction.