## Chaotic Elections!: A Mathematician Looks at VotingWhat does the 2000 U.S. presidential election have in common with selecting a textbook for a calculus course in your department? Was Ralph Nader's influence on the election of George W. Bush greater than the now-famous chads? In Chaotic Elections!, Don Saari analyzes these questions, placing them in the larger context of voting systems in general. His analysis shows that the fundamental problems with the 2000 presidential election are not with the courts, recounts, or defective ballots, but are caused by the very way Americans vote for president. This expository book shows how mathematics can help to identify and characterize a disturbingly large number of paradoxical situations that result from the choice of a voting procedure. Moreover, rather than being able to dismiss them as anomalies, the likelihood of a dubious election result is surprisingly large. These consequences indicate that election outcomes--whether for president, the site of the next Olympics, the chair of a university department, or a prize winner--can differ from what the voters really wanted. They show that by using an inadequate voting procedure, we can, inadvertently, choose badly. To add to the difficulties, it turns out that the mathematical structures of voting admit several strategic opportunities, which are described. Finally, mathematics also helps identify positive results: By using mathematical symmetries, we can identify what the phrase ``what the voters really want'' might mean and obtain a unique voting method that satisfies these conditions. Saari's book should be required reading for anyone who wants to understand not only what happened in the presidential election of 2000, but also how we can avoid similar problems from appearing anytime any group is making a choice using a voting procedure. Reading this book requires little more than high school mathematics and an interest in how the apparently simple situation of voting can lead to surprising paradoxes. |

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#### LibraryThing Review

User Review - dougb56586 - LibraryThingIn this short book (152 pages), a Mathematics Professor explains how elections can often yield drastically different outcomes depending on which voting procedure is used, and further, that if a low ... Read full review

#### LibraryThing Review

User Review - FPdC - LibraryThingOf the two expository books by Saari on the mathematics of voting systems, this is clearly the more mathematically oriented , although it is not exactly a mathematical text, containing no proofs of ... Read full review

### Contents

A Mess of an Election | 1 |

1 Electoral College | 4 |

2 Other procedures | 17 |

Voter Preferences or the Procedure? | 33 |

1 Some examples | 34 |

2 Representation triangle and profiles | 40 |

3 Procedure lines and elections | 45 |

4 Approval or Cumulative voting? | 53 |

2 Strategic voting | 94 |

3 Debate and selecting amendments | 100 |

4 Any relief? | 102 |

5 Changing the outcome | 103 |

What Do the Voters Want? | 109 |

1 Breaking ties and cycles | 110 |

2 Reversal effects | 123 |

3 A profile coordinate system | 129 |

5 More candidates toward Lincolns election | 60 |

Chaotic Election Outcomes | 69 |

1 Deanna had to withdraw | 70 |

2 General results | 72 |

3 Consequences | 79 |

4 Chaotic notions for chaotic results | 84 |

How to Be Strategic | 91 |

1 Choice of a procedure | 92 |

Other Procedures Other Assumptions | 137 |

1 Beyond voting other aggregation methods | 138 |

2 Apportioning congressional seats on a torus | 143 |

3 Other procedures and other assumptions | 148 |

4 Concluding comment | 152 |

153 | |

157 | |

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### Common terms and phrases

apportionment approach Approval Voting Arrow's Arrow's Theorem ballots behavior Borda Count bottom ranked Bush Chapter Chen Lu choice coalition component computations Condorcet terms Condorcet winner construction create cue ball cumulative voting cyclic define described determine dropped election outcomes election procedure election rankings election tallies Electoral College electoral votes example four candidates geometric Gibbard-Satterthwaite Theorem Gore happen illustrate indicated instance Kruskal-Wallis test Lincoln mathematical mathematician Nader number of candidates number of voters occur one-ball pairwise outcomes pairwise rankings pairwise tallies pairwise vote paradoxes party players plurality vote points positional election positional methods positional procedures possible President problems procedure line ranking regions ranking wheel represent result reversal terms Saari SC&W specified strategic voting subsets of candidates suppose symmetry target region tetrahedron Theorem three candidates top ranked triangle U.S. Presidential election values vector vertex victory voter preferences voters vote voting methods voting paradoxes voting procedures wins