[Paper Review] Common Voting Rules as Maximum Likelihood Estimators

Vincent Conitzer, Tüomas Sandholm|arXiv (Cornell University)|Jul 4, 2012

Game Theory and Voting SystemsEconomics, Econometrics and Finance11 references37 citations

TL;DR

This paper investigates which common voting rules can be interpreted as maximum likelihood estimators (MLEs) under a probabilistic noise model where voters' rankings are noisy perceptions of a true underlying outcome. It shows that only specific rules—such as scoring rules (plurality, Borda, veto), Bucklin, and STV—can be derived as MLEs under independent voting assumptions, while others like Copeland, maximin, and ranked pairs cannot. The key contribution is a formal characterization of which rules are statistically principled under this model.

ABSTRACT

Voting is a very general method of preference aggregation. A voting rule takes as input every voter's vote (typically, a ranking of the alternatives), and produces as output either just the winning alternative or a ranking of the alternatives. One potential view of voting is the following. There exists a \correct" outcome (winner/ranking), and each voter's vote corresponds to a noisy perception of this correct outcome. If we are given the noise model, then for any vector of votes, we can compute the maximum likelihood estimate of<br>the correct outcome. This maximum likelihood estimate constitutes a voting rule. In this paper, we ask the following question: For which common voting rules does there exist a noise model such that the rule is the maximum likelihood estimate for that noise model? We require that the votes are drawn<br>independently given the correct outcome (we show that without this restriction, all voting rules have the property). We study the question both for the case where outcomes are winners and for the case where outcomes<br>are rankings. In either case, only some of the common voting rules have the property. Moreover, the sets of rules that satisfy the property are incomparable between the two cases (satisfying the property in the one case<br>does not imply satisfying it in the other case).

Motivation & Objective

• . The paper aims to determine which common voting rules can be interpreted as maximum likelihood estimators (MLEs) under a probabilistic model of voter behavior.
• It investigates whether a noise model exists such that a given voting rule is the MLE of the true outcome.
• The study focuses on two outcome types: winner-only (MLEWIV) and full ranking (MLERIV) rules.
• It seeks to distinguish between rules that are statistically principled under the MLE framework and those that are not.
• The research aims to provide insight into the interpretability and rationality of voting rules through statistical estimation theory.

Proposed method

• . The authors define voting rules based on rankings and outcomes, distinguishing between winner-only and full-ranking outputs.
• They introduce a noise model where each voter's ranking is conditionally independent given the true outcome.
• A voting rule is an MLE if it maximizes the likelihood of the observed votes given a true outcome.
• The paper uses a technique based on pairwise election graphs and vote aggregation to test whether a rule can be an MLE.
• It applies Lemma 1 to show that if a rule produces different outcomes on two vote sets but the same outcome on their sum, it cannot be an MLE.
• The method relies on constructing counterexamples using specific vote configurations to disprove MLE status for rules like Copeland, maximin, and ranked pairs.

Experimental results

Research questions

• RQ1 . Which common voting rules can be represented as maximum likelihood estimators under a noise model where votes are independent given the true outcome?
• RQ2 . Do the same rules qualify as MLEs when the outcome is a full ranking versus just a winner?
• RQ3 . Can the Copeland, maximin, and ranked pairs rules be derived as MLEs under any independent noise model?
• RQ4 . Is there a structural difference between rules that are MLEWIV (winner-only) and MLERIV (ranking) rules?
• RQ5 . Can the MLE property be preserved or modified under relaxed independence assumptions?

Key findings

• . The plurality, Borda, and veto rules are MLEWIV and MLERIV rules under specific noise models, with the Borda rule being the MLE under a uniform noise model.
• . The STV rule is an MLEWIV rule but not an MLERIV rule, indicating that winner selection and full ranking are not always consistent under MLE interpretation.
• . The Bucklin rule is an MLEWIV rule, showing that it too can be derived from a probabilistic noise model.
• . The Copeland rule is neither an MLEWIV nor an MLERIV rule, as demonstrated by vote configurations that violate the MLE consistency condition.
• . The maximin rule fails to be an MLEWIV or MLERIV rule, as shown through counterexamples where the sum of two vote sets produces a different ranking than expected.
• . The ranked pairs rule is neither an MLEWIV nor an MLERIV rule, as it fails the MLE consistency test under the proposed method.