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A core tension in the study of plurality elections is the clash between the classic Hotelling-Downs model, which predicts that two office-seeking candidates should cater to the median voter, and the empirical observation that democracies often have two major parties with divergent policies. Motivated in part by this tension, we introduce a dynamic model of candidate positioning based on a simple bounded rationality heuristic: candidates imitate the policy of previous winners. The resulting model is closely connected to evolutionary replicator dynamics. For uniformly-distributed voters, we prove in our model that with k = 2, 3, or 4 candidates per election, any symmetric candidate distribution converges over time to the center. With k ≥ 5 candidates per election, however, we prove that the candidate distribution does not converge to the center and provide an even stronger non-convergence result in a special case with no extreme candidates. Our conclusions are qualitatively unchanged if a small fraction of candidates are not winner-copiers and are instead positioned uniformly at random in each election. Beyond our theoretical analysis, we illustrate our results in extensive simulations; for five or more candidates, we find a tendency towards the emergence of two clusters, a mechanism suggestive of Duverger's Law, the empirical finding that plurality leads to two-party systems. Our simulations also explore several variations of the model, where we find the same general pattern: convergence to the center with four or fewer candidates, but not with five or more. Finally, we discuss the relationship between our replicator dynamics model and prior work on strategic equilibria of candidate positioning games. 

Instant runoff voting (IRV) has recently gained popularity as an alternative to plurality voting for political elections, with advocates claiming a range of advantages, including that it produces more moderate winners than plurality and could thus help address polarization. However, there is little theoretical backing for this claim, with existing evidence focused on case studies and simulations. In this work, we prove that IRV has a moderating effect relative to plurality voting in a precise sense, developed in a 1-dimensional Euclidean model of voter preferences. We develop a theory of exclusion zones, derived from properties of the voter distribution, which serve to show how moderate and extreme candidates interact during IRV vote tabulation. The theory allows us to prove that if voters are symmetrically distributed and not too concentrated at the extremes, IRV cannot elect an extreme candidate over a moderate. In contrast, we show plurality can and validate our results computationally. Our methods provide new frameworks for the analysis of voting systems, deriving exact winner distributions geometrically and establishing a connection between plurality voting and stick-breaking processes. 

Abstract Choices made by individuals have widespread impacts—for instance, people choose between political candidates to vote for, between social media posts to share, and between brands to purchase—moreover, data on these choices are increasingly abundant.Discrete choice modelsare a key tool for learning individual preferences from such data. Additionally, social factors like conformity and contagion influence individual choice. Traditional methods for incorporating these factors into choice models do not account for the entire social network and require hand-crafted features. To overcome these limitations, we use graph learning to study choice in networked contexts. We identify three ways in which graph learning techniques can be used for discrete choice: learning chooser representations, regularizing choice model parameters, and directly constructing predictions from a network. We design methods in each category and test them on real-world choice datasets, including county-level 2016 US election results and Android app installation and usage data. We show that incorporating social network structure can improve the predictions of the standard econometric choice model, the multinomial logit. We provide evidence that app installations are influenced by social context, but we find no such effect on app usage among the same participants, which instead is habit-driven. In the election data, we highlight the additional insights a discrete choice framework provides over classification or regression, the typical approaches. On synthetic data, we demonstrate the sample complexity benefit of using social information in choice models. 

Instant runoff voting (IRV) is an increasingly-popular alternative to traditional plurality voting in which voters submit rankings over the candidates rather than single votes. In practice, elections using IRV often restrict the ballot length, the number of candidates a voter is allowed to rank on their ballot. We theoretically and empirically analyze how ballot length can influence the outcome of an election, given fixed voter preferences. We show that there exist preference profiles over k candidates such that up to k-1 different candidates win at different ballot lengths. We derive exact lower bounds on the number of voters required for such profiles and provide a construction matching the lower bound for unrestricted voter preferences. Additionally, we characterize which sequences of winners are possible over ballot lengths and provide explicit profile constructions achieving any feasible winner sequence. We also examine how classic preference restrictions influence our results—for instance, single-peakedness makes k-1 different winners impossible but still allows at least Ω(√k). Finally, we analyze a collection of 168 real-world elections, where we truncate rankings to simulate shorter ballots. We find that shorter ballots could have changed the outcome in one quarter of these elections. Our results highlight ballot length as a consequential degree of freedom in the design of IRV elections. 

Abstract Motivation There has been recent increased interest in using algorithmic methods to infer the evolutionary tree underlying the developmental history of a tumor. Quantitative measures that compare such trees are vital to a number of different applications including benchmarking tree inference methods and evaluating common inheritance patterns across patients. However, few appropriate distance measures exist, and those that do have low resolution for differentiating trees or do not fully account for the complex relationship between tree topology and the inheritance of the mutations labeling that topology. Results Here we present two novel distance measures, Common Ancestor Set distance (CASet) and Distinctly Inherited Set Comparison distance (DISC), that are specifically designed to account for the subclonal mutation inheritance patterns characteristic of tumor evolutionary trees. We apply CASet and DISC to multiple simulated datasets and two breast cancer datasets and show that our distance measures allow for more nuanced and accurate delineation between tumor evolutionary trees than existing distance measures. Availability and implementation Implementations of CASet and DISC are freely available at: https://bitbucket.org/oesperlab/stereodist. Supplementary information Supplementary data are available at Bioinformatics online. 

A number of methods have recently been proposed to reconstruct the evolutionary history of a tumor from noisy DNA sequencing data. We investigate when and how well these histories can be reconstructed from multi-sample bulk sequencing data when considering only single nucleotide variants (SNVs). We formalize this as the Enumeration Variant Allele Frequency Factorization Problem and provide a novel proof for an upper bound on the number of possible phylogenies consistent with a given dataset. In addition, we propose and assess two methods for increasing the robustness and performance of an existing graph based phylogenetic inference method. We apply our approaches to noisy simulated data and find that low coverage and high noise make it more difficult to identify phylogenies. We also apply our methods to both chronic lymphocytic leukemia and clear cell renal cell carcinoma datasets.