[Paper Review] Fishing out Winners from Vote Streams

Arnab Bhattacharyya, Palash Dey|arXiv (Cornell University)|Aug 19, 2015

Game Theory and Voting Systems23 references3 citations

TL;DR

This paper introduces space-efficient streaming algorithms for approximate winner determination in online elections, where votes arrive in arbitrary order. It proposes randomized algorithms that identify an $\varepsilon$-winner—a candidate who could win with a change of at most $\varepsilon n$ votes—using sublinear space, achieving tight bounds for multiple voting rules including plurality, $k$-approval, and Copeland.

ABSTRACT

We investigate the problem of winner determination from computational social choice theory in the data stream model. Specifically, we consider the task of summarizing an arbitrarily ordered stream of $n$ votes on $m$ candidates into a small space data structure so as to be able to obtain the winner determined by popular voting rules. As we show, finding the exact winner requires storing essentially all the votes. So, we focus on the problem of finding an {\em $\eps$-winner}, a candidate who could win by a change of at most $\eps$ fraction of the votes. We show non-trivial upper and lower bounds on the space complexity of $\eps$-winner determination for several voting rules, including $k$-approval, $k$-veto, scoring rules, approval, maximin, Bucklin, Copeland, and plurality with run off.

Motivation & Objective

• Address the challenge of determining winners in large-scale online elections where votes arrive in a data stream and full storage is infeasible.
• Relax the exact winner problem to allow approximation via $\varepsilon$-winners, defined as candidates who could win with a small change in votes.
• Analyze the space complexity of streaming algorithms for $(\varepsilon,\delta)$-winner determination under various voting rules.
• Establish tight upper and lower bounds on space usage, showing that approximation enables significant compression compared to exact winner computation.
• Extend results to the sliding window model, where only the most recent $N$ votes are considered for winner determination.

Proposed method

• Formulate the $(\varepsilon,\delta)$-winner determination problem as a randomized streaming task with bounded error probability.
• Use randomized sketching and sampling techniques to summarize vote streams with minimal space, focusing on vote counts and pairwise comparisons.
• Reduce the winner determination problem to known streaming problems like heavy hitters and max-sum, leveraging reductions from communication complexity.
• Construct vote streams that encode instances of hard communication problems (e.g., Greater-than, Max-sum) to derive lower bounds.
• Design rule-specific algorithms that exploit structural properties of voting rules (e.g., approval, Borda, Copeland) to minimize space usage.
• Analyze space complexity in terms of $\varepsilon$, $\delta$, $m$ (number of candidates), and $n$ (number of votes), deriving bounds via probabilistic and combinatorial arguments.

Experimental results

Research questions

• RQ1 Can we design space-efficient streaming algorithms for approximate winner determination in online elections with arbitrary vote order?
• RQ2 What is the optimal space complexity for computing an $\varepsilon$-winner under various common voting rules?
• RQ3 How do space requirements scale with $\varepsilon$, $m$, and $n$ for different voting rules, and are the bounds tight?
• RQ4 Can the sliding window model be supported efficiently, where only the most recent $N$ votes are considered?
• RQ5 What are the fundamental limits (lower bounds) on space usage for $(\varepsilon,\delta)$-winner determination across different voting rules?

Key findings

• For plurality and generalized plurality, the space complexity is $O(\min\{\frac{1}{\varepsilon}\log m, m\log\frac{1}{\varepsilon}\} + \log\log n)$, matching the lower bound up to logarithmic factors.
• For $k$-approval and $k$-veto, the space complexity is $O(\min\{\frac{k}{\varepsilon}\log m, m\log\frac{\log(m-k+1)}{\varepsilon}\} + \log\log n)$, with matching lower bounds under certain parameter regimes.
• For scoring rules and approval voting, the space complexity is $O(m(\log\log m + \log\frac{1}{\varepsilon}) + \log\log n)$, which is tight up to logarithmic factors.
• For maximin, Bucklin, and runoff rules, the space complexity is $O(\min\{m^2(\log\log m + \log\frac{1}{\varepsilon}), \frac{1}{\varepsilon^2}m\log^2 m\} + \log\log n)$, with matching lower bounds in key regimes.
• For Copeland, the space complexity is $O(\min\{m^2(\log\log m + \log\frac{1}{\varepsilon}), \frac{1}{\varepsilon^2}m\log^4 m\} + \log\log n)$, reflecting the higher complexity of pairwise majority comparisons.
• A $\Omega(\log\log n)$ lower bound holds for plurality and related rules, even with only two candidates, showing that logarithmic dependence on $n$ is unavoidable.