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This book provides a comprehensive mathematical description and analysis of the delegate allocation processes in the US Democratic and Republican presidential primaries, focusing on the role of apportionment methods and the effect of thresholds—the minimum levels of support required to receive delegates.  The analysis involves a variety of techniques, including theoretical arguments, simplicial geometry, Monte Carlo simulation, and examination of presidential primary data from 2004 to 2020.The book is divided into two parts: Part I defines the classical apportionment problem and explains...