A Mathematical Look at Politics
Format: PDF / Kindle (mobi) / ePub
What Ralph Nader's spoiler role in the 2000 presidential election tells us about the American political system. Why Montana went to court to switch the 1990 apportionment to Dean’s method. How the US tried to use game theory to win the Cold War, and why it didn’t work. When students realize that mathematical thinking can address these sorts of pressing concerns of the political world it naturally sparks their interest in the underlying mathematics.
A Mathematical Look at Politics is designed as an alternative to the usual mathematics texts for students in quantitative reasoning courses. It applies the power of mathematical thinking to problems in politics and public policy. Concepts are precisely defined. Hypotheses are laid out. Propositions, lemmas, theorems, and corollaries are stated and proved. Counterexamples are offered to refute conjectures. Students are expected not only to make computations but also to state results, prove them, and draw conclusions about specific examples.
Tying the liberal arts classroom to real-world mathematical applications, this text is more deeply engaging than a traditional general education book that surveys the mathematical landscape. It aims to instill a fondness for mathematics in a population not always convinced that mathematics is relevant to them.
elimination grid. A team keeps playing in such a tournament as long as it keeps winning its individual contests, and once it loses it is eliminated for good. The team that wins its final contest is the tournament winner. In the theory of elections, the schedule of head-tohead match-ups is known as the agenda. One possibility for an agenda comes from a list of the candidates in a particular order. In the first round, the first two candidates on the list compete in a head-to-head match. Then the
no dependence on high school mathematics or government courses. What is really required from the reader is the ability to reason precisely within a carefully prescribed environment and to express that reasoning in clear persuasive writing. This comes easily to some readers but not to others, and this seems to be quite independent of whether those readers have been told in the past that they are good in math. Each chapter begins with a scenario, which is a puzzle or problem meant to engage
many first-place votes a candidate will get. As Condorcet himself was well aware, voters who are queried about their opinions, one pair of alternatives at a time, may not always give “rational” responses. That is, their response may not come from a transitive preference ranking. Consequently, not every Condorcet profile comes from a preference profile. For example, the three-voter three-candidate Profile 6.3 cannot be obtained from any preference profile, since the data for A versus B and for A
the Pareto criterion implies unanimity. We can now state the version of Arrow’s Theorem that applies in the context of social ranking functions. For a proof, see . Theorem 6.33. (Arrow) With three or more candidates, the only social ranking functions that are unanimous and independent are dictatorships. Because the unanimity criterion follows from the Pareto criterion, we have the following weaker result. Corollary 6.34. With three or more candidates, the only social ranking functions that
what point does state k become entitled to a second seat? The Divisor Methods 161 √ answer is: Just when the modified quota pk /d reaches 2, which is the cut-off between being rounded down to 1 and being rounded up to 2. To put it another way, √ state k earns its second seat when the modified divisor d reaches pk / 2. At what point does √ √ state k obtain its third seat? When the modified divisor pk /d reaches 6, that is, when d = pk / 6. In general, state k will get its (m + 1)st seat when d