Computing Voting Rules with Elicited Incomplete Votes

TL;DR

This paper fully characterizes the set of positional scoring rules that can be computed for any 1 ≤ t < m, which, notably, does not include plurality, and extends this to show a similar impossibility result for single transferable vote (elimination voting).

Abstract

Motivated by the difficulty of specifying complete ordinal preferences over a large set of $m$ candidates, we study voting rules that are computable by querying voters about $t < m$ candidates. Generalizing prior works that focused on specific instances of this problem, our paper fully characterizes the set of positional scoring rules that can be computed for any $1 \leq t < m$, which, notably, does not include plurality. We then extend this to show a similar impossibility result for single transferable vote (elimination voting). These negative results are information-theoretic and agnostic to the number of queries. Finally, for scoring rules that are computable with limited-sized queries, we give parameterized upper and lower bounds on the number of such queries a deterministic or randomized algorithm must make to determine the score-maximizing candidate. While there is no gap between our bounds for deterministic algorithms, identifying the exact query complexity for randomized algorithms is a challenging open problem, of which we solve one special case.

Figures (9)

• Figure 1: A process inducing the distribution over rankings for ${\boldsymbol{\sigma}}$.
• Figure 2: A process inducing the distribution over rankings for ${\boldsymbol{\sigma}}^c$.
• Figure 3: A process inducing the distribution over rankings for ${\boldsymbol{\sigma}}^{\text{unif}}$.
• Figure 4: A process inducing ${\boldsymbol{\sigma}}^i$. This is parameterized by two disjoint sets, $C^1$ and $C^2$ and two candidates $a, b \in C^1$. The profile ${\boldsymbol{\sigma}} \in \Pi(C^1)$ satisfies the conditions of \ref{['lem:construction']} with $C_1, a,$ and $b$. The indexing $c^2_1 \succ \cdots \succ c^2_{|C^2|}$ is an arbitrary order of the candidates in $C^2$.
• Figure 5: The space of positional scoring rules for $m = 4$ candidates. The corners represent rules that give all weight to a single position, with the top being Plurality. The red kite-shaped region is the 2-dimensional subspace $R_{4, 3}$ spanned by the vectors ${\boldsymbol{\alpha}}^1 = (3, 1, 0, 0)$, ${\boldsymbol{\alpha}}^2 = (0, 1, 1, 0)$, and ${\boldsymbol{\alpha}}^3 = (0, 0, 1, 3)$, which are normalized in the simplex as the three red points at the corners of the kite. As we have shown, this subspace does not contain Plurality. Nested within this subspace is $R_{4, 2}$, the green 1-dimensional subspace spanned by the Borda and Anti-Borda scoring vectors. The purple point at the middle of this line is the trivial voting rule that gives every candidate the same score, which is the only element of the 0-dimensional subspace $R_{4, 1}$.

Theorems & Definitions (19)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Lemma 4: Theorem 1 of bentert2020comparing
• proof