This comprehensive text, primarily excerpts from a book, investigates the role of mathematics in optimizing democratic processes such as voting, apportionment, and redistricting. It argues that quantitative methods provide a politically impartial path to improving representation, although democracy's inherent "messiness" often introduces non-mathematical considerations. The discussion critically analyzes various voting systems, contrasting the limitations of simple majority (plurality) with more sophisticated methods like Instant Runoff Voting (IRV), the Borda count, and Condorcet methods, often in the context of impossibility theorems like May’s and Arrow’s. Furthermore, the source examines the mathematics of apportionment discussing historical and current divisor methods (like Huntington-Hill and Webster) used to allocate legislative seats. Finally, it explores methods for quantifying and combating gerrymandering, introducing measures like the efficiency gap and Polsby-Popper scores to assess partisan bias and district compactness.

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