We use the United States Supreme Court as an illuminative context in which to discuss three different spatial voting preference models: an instance of the widely used single-peaked preferences, and two models that are more novel in which vote outcomes have a strength in addition to a location. We introduce each model from a formal axiomatic perspective, briefly discuss practical motivation for each in terms of judicial behavior, prove mathematical relationships among the voting coalitions compatible with each model, and then study the two-dimensional setting by presenting computational tools for working with the models and by exploring these with judicial voting data from the Supreme Court.
There has been a recent media blitz on a cohort of mathematicians valiantly working to fix America's democratic system by combatting gerrymandering with geometry. While statistics commonly features in the courtroom (forensics, DNA analysis, etc.), the gerrymandering news raises a natural question: in what other ways has pure math, specifically geometry and topology, been involved in court cases and legal scholarship? In this survey article, we collect a few examples with topics ranging from the Pythagorean formula to the Ham Sandwich Theorem, and we discuss some jurists' perspectives on geometric reasoning in the legal realm. One of our goals is to provide math educators with engaging real-world instances of some abstract geometric concepts.
While the U.S. Supreme Court is commonly viewed as comprising a liberal bloc and a conservative bloc, with a possible "swing vote" or "median justice" between them, surprisingly many case decisions are not explained by this simple model. We introduce a pair of spatial methods for conceptualizing many 5-to-4 voting alignments that have occurred on the Court and which defy the usual liberal/conservative dichotomy. These methods, utilizing higher order Voronoi diagrams and halving lines, are based on the metric geometry of the two-dimensional ideal space locations obtained from applying multidimensional scaling to voting data. We also introduce a two-dimensional metric method for determining the crucial "fifth" vote in each 5-to-4 ruling and for determining the median justice in any collection of terms within a natural court.
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