This week in Occupy Math, we proudly announce a book published by Dr. Andrew McEachern, GAME THEORY: A Classical Introduction, Mathematical Games, and the Tournament. Game theory is a formal structure for studying and resolving conflict and encouraging cooperation that rephrases cooperation and conflict as a game. Andrew developed and taught a course in game theory taken by advanced students from many programs while he was at Queens University. This book is a text based on the course he taught and it is part of an effort to bring textbook prices under control. The book introduces the classical analysis used in game theory — his exposition of The Lady or the Tiger is wonderful — but Andrew also introduces material outside of the standard game theory fare. These include the math behind the fraction teaching game that Dr. McEachern and Occupy Math are developing and techniques for designing fair, balanced tournaments for anything from Prisoner’s Dilemma to Basketball. The book is a text for a course for non-majors that nevertheless has a solid mathematical foundation. We now ask Dr. McEachern a few questions about his book.
Occupy Math is going to look at a simple piece of math that is ignored or, worse, abused by researchers in many fields. It amounts to an example of ignorance of statistics that leads to publishing results that are bogus and so impossible to replicate. This problem is called the replication crisis because many important results seem to disappear when other researchers try and reproduce them. Occasionally this is the result of actual fraud — but more often ignorance of simple facts about statistics can let you publish a paper whose results cannot be replicated (because its results are actually wrong) without even noticing you’re doing it. There is also a separate problem — it is very difficult to completely describe an experiment, which means that the people trying to reproduce your results may not be doing quite the same experiment. That last is a big problem, but not what Occupy Math is looking at today.
The core message of today’s post is that peer-reviewed results in a top journal are sometimes wrong because we don’t teach statistics properly.
This week Occupy Math takes a trip to the land of Freedonia which is beset by a vile dragon that menaces its democracy. A small African nation with a diverse population and the magnificent port city of Great Haven, Freedonia is a constitutional democracy modeled on the American experiment, an active, participatory, free democracy and, until recently, with a vibrant and open economy. Founded in the early 20th century by a largely peaceful revolution organized by tribal leaders — advised by the famous explorer and polymath, Captain Spaulding — Freedonia has been synonymous with hope for generations. Recently, however, the economy has been experiencing problems with corruption. Nepotistic awards of government contracts to incompetent nephews and corrupt back-room deals have taken the economy away from the hard-working farmers, shopkeepers, and craftsmen who have been the backbone of Freedonia society. UN monitors certify each of the biannual elections as free and fair but, somehow, in spite of public outrage, the Lucarian party ekes out a bare majority and restores the corrupt Prime Minister, Joseph Cagliostro to power. What dark force is subverting democracy in Freedonia? Let’s ask no lesser authority that the Governator himself!
Gerrymandering is a subtle way of subverting democracy — and the vorpal sword that can slay it is edged with mathematics.
With all the commotion in the news lately, a lot of people are saying “Oh, yeah? Prove it!” (often in an angry voice). This helped Occupy Math to select his topic for this week which is about proof and interpretation. In math, a proof is a series of connected logical statements that draw a line between some things that you assume are true and something else that you are trying to demonstrate is true. The Pythagorean theorem is a good example: “If you have a right triangle then the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longer side.” You are assuming that you have a right triangle and, if you do, the sometimes helpful fact about the side lengths holds.
This week we look at one of the big achievements in math, figuring out the minimum number of colors needed to permit adjacent countries to be shown in contrasting colors on any map, and connect it with a type of conflict resolution. Applications include efficient timetables for meetings, relatively peaceful assignments of students to cars for a field trip, and even putting as many types of fish as you can into the display window of a store that sells tropical fish. All of these applications have two steps: make a graph of the situation and then color that graph properly. We will explain the terms in italics in the rest of the post.
The fascinating mathematical fact here is that all these applications use exactly the same math.
Long ago, in Iowa, Occupy Math taught a course entitled “Introduction to Mathematical Concepts”. It was the course for people who might not have mastered arithmetic yet, but had a major that required a math course. During this course, there were only two days (other than examinations) with nearly complete attendance. The first day, because people wanted to know what was going to happen, and the day Occupy Math used a fair division algorithm to figure out the property division of Donald Trump’s (first) divorce. Short version: Ivana did not do well. In this post, Occupy Math examines two kinds of fair division. The simple one is dividing cake and the hard one is dividing property. We will also discuss why, with fair and equitable methods of dividing things, why we still spend piles of money on things like divorce lawyers.
Mathematical formulas exist to divide almost anything as close to fairly as possible — why then all the squabbles?
A perennial problem that people have is when the organization they work for, or with, makes a bad decision. A common refrain when this happens is “why didn’t they listen to me?” In today’s post we examine how to get them to listen to you using a mathematical tool called an influence map. The key mathematics that permits you to build and use influence maps is the directed graph, an example of which appears at the top of the post. The idea is this: objects in the directed graph are actors — people or organizations with power or influence — and the arrows show which way influence flows and, possibly, what type of influence it is.
Influence maps can be applied to an incredible number of different situations. These include your workplace and community.
This week, Occupy Math looks at math tests — and some other tests — from the perspective of fairness. It turns out that questions that test the same skills can have extremely adjustable difficulty levels. There is also the issue of tests designed for failure. For that, we will look at some examples of cosmically unfair questions. On the issue of math tests, this post discusses the differences between easy and hard questions for the same topic. Occupy Math can probably dial the average grade on a test across a range of 20% by playing with the way questions are phrased. All this will give you some perspective on how to survive a test (it helps to be able to spot structurally hard questions) — but mostly the message is this.
Fairness is largely an illusion and enforcing it is close to impossible. Hope for competent teaching and mercy instead.
This post was inspired by a wonderful video about the effects on Yellowstone National Park of returning wolves to the ecosystem after an absence of seventy years. Wolves are keystone predators and kill and eat other animals. The effect of introducing them was to … sharply increase both the diversity and number of animals in the park and the number of green plants in the park; the wolves even moved where the rivers flowed.
As you will find if you watch the video, this is a wonderful example of a trophic cascade. The wolves showed up and the ecosystem’s health increased a whole bunch. This is also an example of a mathematical phenomenon called a nonlinear system. Systems we understand easily tend to be linear — the response of the system is proportional to the input. If you jump on the bed harder, you bounce higher, unless the ceiling introduces a painful non-linearity by impacting your head. Non-linear systems are far more common than linear ones, but harder to fathom.
The mathematics of ecology in highly non-linear. This is why ecology is more like whack-a-mole than science, sometimes.
This week Occupy Math pays off on a promise to look at Trump administration appointee Betsy DeVos. Now confirmed — by the lowest margin possible and the lowest margin ever for a cabinet level post — Ms. Devos is in charge of the federal government’s education policies. While Secretary of Education is not as big a deal as Secretary of State or Attorney General, this is an important government post with a good deal of influence on education in the United States. Today’s post looks at her potential impact on education, including math education.
This post examines what we know about possible impacts of Ms. DeVos proposed programs, based on their partial implementation in her home state of Michigan.