How Not To Be Wrong

I read a good mathematics book by Jordan Ellenberg called “How Not to Be Wrong: The Power of Mathematical Thinking”.  The book covered just four topics: linearity, expectation, regression to the mean, and inference.

The book opened with a discussion of a World War II mathematician who was asked to analyze the bullet holes in the fuselage of bomber planes to determine where to put extra armor.  He looked at that bullet holes in the returning planes and told the air force they should put the armor where the bullet holes weren’t because planes hit in those places didn’t return.  This was an example of the power of mathematical thinking.

In the linearity section, Ellenberg described the Laffer curve in political economy.  The statement was made by a right-wing journalist in United States: Why would the United States want to be more like Sweden when Sweden wants to be more like United States?

The fallacy in this argument is the assumption of linearity.  The economist, Arthur Laffer, explained it on the back of a napkin to Dick Cheney and Donald Rumsfeld at an afternoon meeting during the Ford administration.  The Laffer curve explains the value of taxation policy from left leaning countries (high taxes and many government services) to right leaning countries (low taxes and few government services).  The curve has a hump in the middle.  He explained that the United States may be too low on the curve and should move towards the middle in the direction of Sweden while Sweden might be too high on the curve and should move towards the middle in the direction the United States.  Since the curve has a hump in the middle, the optimal amount of taxes and government services is between the amounts in the United States and Sweden.

To explain expectation, Ellenberg describes the expected winnings in lotteries.  He tells about a mathematician at Stanford who has won four major lotteries by knowing the distribution of winning "scratch and win" ticket sales.  He also describes a syndicate who could obtain a positive expected winnings by buying all the tickets in the Virginia lottery when the pot got high enough because of roll-over.  He also talked about the law of very large numbers where if you have enough observations almost anything can happen.  With all of the lotteries in the world every week, it is not unusual for somewhere at sometime to have the same set of numbers come up twice in a row.  When this actually happened in Bulgaria, the government conducted an investigation into whether the lottery was fixed.

In the inference section, Ellenberg talked about how medical treatments are studied for their effectiveness.  He described hypothesis testing and how there are many unreplicable studies because of false positive results.  One of the items he discussed was mutual fund performance and survivor bias.  All mutual fund results look good because poor performing funds are subsumed into more successful funds. Thus only data on survivors is available.

Finally, Ellenberg talked about regression to the mean.  He described a study at the turn of the century in which a business analyst studied many successful firms and then 30 years later found that these firms were no longer successful while firms that were not very successful in the earlier time might now be successful.  He tried to publish his finding as a remarkable law of business.  The study was reviewed by a famous mathematician who tried to explain that the finding was nothing more than regression to the mean and the success of the firms in both periods was probably luck not business acumen.

Later after reading “How Not to Be Wrong: The Power of Mathematical Thinking”, I heard an interview with the author.  He said he had written a book proposal with thirteen sections.  But after four sections, he had written 300 pages.  So he asked for permission to stop there and save the rest of the topics for future books.  I look forward to reading volume II and III, etc.