By E. Arthur Robinson Jr., Daniel H. Ullman
What Ralph Nader's spoiler function within the 2000 presidential election tells us concerning the American political process. Why Montana went to court docket to change the 1990 apportionment to Dean's process. How the united states attempted to take advantage of video game idea to win the chilly warfare, and why it didn't paintings. whilst scholars observe that mathematical pondering can handle those types of urgent matters of the political international it evidently sparks their curiosity within the underlying arithmetic.
A Mathematical examine Politics is designed instead to the standard arithmetic texts for college students in quantitative reasoning classes. It applies the facility of mathematical considering to difficulties in politics and public coverage. strategies are accurately outlined. Hypotheses are laid out. Propositions, lemmas, theorems, and corollaries are said and proved. Counterexamples are provided to refute conjectures. scholars are anticipated not just to make computations but additionally to kingdom effects, turn out them, and draw conclusions approximately particular examples.
Tying the liberal arts school room to real-world mathematical functions, this article is extra deeply enticing than a standard common schooling booklet that surveys the mathematical panorama. It goals to instill a passion for arithmetic in a inhabitants now not continually confident that arithmetic is proper to them.
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Additional info for A Mathematical Look at Politics
By monotonicity, A is the unique winner whenever A receives at least t/2 + 1/2 votes. But by neutrality, it is also true that B is the unique winner whenever B receives at least t/2 + 1/2. So, any candidate who has more votes than the other is the unique winner. Again, the method must be the simple majority method. We leave as an exercise the proof of the following extension of May’s Theorem. 24. In an election with two candidates, a voting method that is anonymous, neutral, and monotone and that isn’t the all-ties method must be a either the simple majority method or a supermajority method.
3, then the Borda count method gives a candidate 4, 3, 2, 1, or 0 points for every voter who ranks them first, second, third, fourth, or fifth (last), respectively. 3 gets 4 points from each of the 5 voters represented by the first column but gets no points at all from the other voters, so candidate A tallies 20 points. By contrast, candidate B obtains 3 points from each of the 5 voters represented by the first column, but 4 additional points for the next 4 voters, 3 points for the next 4, 3 for the subsequent 4, and 2 points for the final 3, for a total of 3×5+4×4+3×4+3×4+2×3 = 61, easily defeating candidate A.
In effect, then, the antiplurality winner is the candidate with the fewest last-place votes. The positional method P (1, 1, 0, . . , 0, 0) is called the vote-for-two method. We examine a number of other interesting social choice functions in the exercises. Most of these are variants on the methods explored in this section. 4 Some Methods on the Edge The following methods mark the outlying areas of the theory. They are all familiar from the two-candidate case. 13. Let v be one voter. The dictatorship method with voter v as dictator is the social choice function in which the candidate ranked first by v is the unique winner, regardless of how anyone else votes.
A Mathematical Look at Politics by E. Arthur Robinson Jr., Daniel H. Ullman