Voting with Limited Information and Many Alternatives

TL;DR

This paper investigates whether simple plurality voting can reliably select the best option when voters have probabilistic signals, showing it can but requires a large number of voters proportional to the number of options and the inverse of the accuracy.

Contribution

It demonstrates that plurality voting can identify the best option with high probability using voters' signals, but the required voter count grows cubically with options and inversely with accuracy.

Findings

Plurality voting can select the correct option with high probability.

Achieving high accuracy requires a number of voters proportional to $n^3 imes ext{(inverse of } ext{epsilon}^2)$.

The process of correctly identifying the best option is inherently costly in terms of voter number.

Abstract

The traditional axiomatic approach to voting is motivated by the problem of reconciling differences in subjective preferences. In contrast, a dominant line of work in the theory of voting over the past 15 years has considered a different kind of scenario, also fundamental to voting, in which there is a genuinely "best" outcome that voters would agree on if they only had enough information. This type of scenario has its roots in the classical Condorcet Jury Theorem; it includes cases such as jurors in a criminal trial who all want to reach the correct verdict but disagree in their inferences from the available evidence, or a corporate board of directors who all want to improve the company's revenue, but who have different information that favors different options. This style of voting leads to a natural set of questions: each voter has a {\em private signal} that provides probabilistic information about which option is best, and a central question is whether a simple plurality voting system, which tabulates votes for different options, can cause the group decision to arrive at the correct option. We show that plurality voting is powerful enough to achieve this: there is a way for voters to map their signals into votes for options in such a way that --- with sufficiently many voters --- the correct option receives the greatest number of votes with high probability. We show further, however, that any process for achieving this is inherently expensive in the number of voters it requires: succeeding in identifying the correct option with probability at least $1 - \eta$ requires $\Omega(n^3 \epsilon^{-2} \log \eta^{-1})$ voters, where $n$ is the number of options and $\epsilon$ is a distributional measure of the minimum difference between the options.