Abstracts Mathematics

Quantitative Analysis of Strategic Voting in Anonymous Voting Systems

by Tiance Wang

Democratically choosing a single preference from three or more candidate options is not a straightforward matter. There are many competing ideas on how to aggregate rankings of candidates. However, the Gibbard-Satterthwaite theorem implies that no fair voting system (equality among voters and equality among candidates) is immune to strategic voting, also known as manipulation. This dissertation is a quantitative analysis of strategic voting from a geometric perspective. Anonymous voting rules, where all voters are equal, can be viewed as a partition of a high dimensional simplex, where different distributions of votes correspond to different points in the simplex, and each particular way of partitioning the simplex corresponds to a voting rule. It is revealed that the orientation, instead of the location, of the boundary determines manipulability. A boundary that separates two winning candidates is not manipulable if and only if the boundary is parallel to all vote changes that does not switch the order of the candidate pair. We analyze the vulnerability to strategic voting of several popular voting systems, including plurality, Borda count and Kemeny-Young, under various vote distributions. When there are three candidates, we show that the Kemeny-Young method, and Condorcet methods in general, are categorically more resistant to strategic voting than many other common voting systems, due to the existence of non-manipulable boundaries. We verify our results on voting data that we collected through an online survey on the 2012 US President Election. Finally, we explore the collective behaviors of manipulative voters. Assume every voter can change their vote for an infinite number of times. They formulate strategies based on their observations on the preference of the population. The observations, which contain noise, are generated by some distribution conditioned on the current vote status. We show that the plurality rule almost always elects the instant run-off winner, while Borda count almost always elects the Condorcet winner (when one exists) as the number of voters grow. Advisors/Committee Members: Cuff, Paul (advisor), Kulkarni, Sanjeev (advisor).