Author Archive for Steve Landsburg

Another Probability Puzzle

Published
by
Steve Landsburg
on February 2, 2011
in Puzzles
. 74 Comments

medicineA man goes to the doctor, gets a prescription for a headache medicine, and dies the next day. It’s known that 60% of those who receive these prescriptions actually fill them and take the medicine. It’s also been established by investigators that if the man took the medicine, then there’s a 90% chance it killed him. What’s the probability that he took the medicine and it killed him?

The answer might depend on your auxiliary assumptions, but there is a particularly simple and natural set of auxiliary assumptions that leads to a nice clean answer. And no, that answer is not 54%. (Nor would 54% be an easy answer to defend under any reasonable assumptions I can think of.)

EDIT: I had written here For extra clarity, the phrase “the medicine killed him” should be interpreted to mean that if he hadn’t taken the medicine, he wouldn’t have died. . This seems to be confusing some readers, and I briefly posted an edit here saying to ignore it — but it actually is what I meant to say all along.

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Freedom, Prosperity, and the Future of Egypt

Published
by
Steve Landsburg
on February 1, 2011
in Current Events, Economics and Politics
. 57 Comments

With regime change perhaps imminent in Egypt and elsewhere in the Middle East, and amid all the calls for democracy and political freedom, it’s a good time to remind ourselves that desirable as political freedom may be, it’s no guarantee of prosperity. For that you need capitalism.

My colleague Alan Stockman and I looked into this question about 10 years ago; I have not updated the data since then but I expect it would still tell pretty much the same story. First, the following graph plots political rights (as defined and measured by Freedom House) against GDP per capita. Low scores indicate more political freedom (defined by criteria that include the existence of free and fair elections, the right to organize, the existence of opposition parties, the absence of domination by the military, religious heirarchies and economic oligarchies, open and transparent government operations, and full political rights for ethnic, religious and cultural minorities, ). There is a small postive relationship between political freedom and prosperity, but many of the freest countries are still poor. And there is very little difference in GDP per person between countries ranked between 2 and 7 on the political freedom scale.

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Nursery Tales — An Afterword

Published
by
Steve Landsburg
on January 31, 2011
in Economics and Paul Krugman
. 5 Comments

babyOne of Paul Krugman’s favorite stories is about the baby-sitting co-op that almost collapsed when members started hoarding scrip; similarly, he says, a lot of economic activity can dry up when people start hoarding money. Last Tuesday, in a post called Nursery Tales, I observed that money-hoarding can’t retard economic activity (at least in anything like Krugman’s sense) unless something prevents prices from adjusting. So absent an auxiliary story about what that “something” is, I don’t find the baby-sitting story terribly helpful.

Several commenters responded that in the real world, prices and/or wages are “known” to be sticky (that is, slow to adjust), and thought that this rescues Krugman’s metaphor. I don’t agree. Here’s why:

Continue reading ‘Nursery Tales — An Afterword’

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Foreign Policy

Published
by
Steve Landsburg
on January 27, 2011
in Policy, Politics and Rants
. 58 Comments

xenoThe LA Times reports that Republican lawmakers have called on the Obama administration to return to the Bush-era practice of sending jackbooted thugs into private workplaces to arrest illegal aliens — revealing (as if we didn’t already know) that virulent xenophobia is alive and well in the Republican party. (Note well the hypocrisy of complaining that foreigners sneak into our country to take advantage of the welfare system, and then addressing the problem by focusing your deportation efforts on foreigners who have obviously come here to work).

The same Times article observes that even without the workplace raids, deportations have reached new heights for two years running at the direction of President Barack Obama — revealing (as if we didn’t already know) that virulent xenophobia is alive and well in the Democratic party too. This is, after all, the same Barack Obama who said in his acceptance speech at the 2008 convention that nobody benefits when an employer undercuts American wages by hiring illegal workers. Well, sure. Nobody, that is, except the employer, his customers, and the illegal workers who, in Barack Obama’s universe, count as “nobody”.

This raises the idle question: Which political party harbors more xenophobia? I have no careful documentation of this, but my impression in the 2008 election was that the Democrat John Edwards was the most despicable of the candidates in this dimension, with the Republican Mitt Romney running a somewhat distant but still unchallenged second. Going back to 2004, it was the Democrat John Kerry who called for federal contracts, whenever possible, to be performed by American workers, demanded tax incentives for firms that hired Americans instead of foreigners, and endorsed legislation encouraging consumers to “buy American”. (If that doesn’t strike you as virulent, ask yourself how you’d feel about a candidate who called for federal contracts, whenever possible, to be performed by white workers, demanded tax incentives for firms that hired whites instead of blacks, and endorsed legislation encouraging consumers to “buy White”.) But it was the Republican victor, George Bush, who followed in his Republican father’s footsteps by dispatching those jackbooted thugs who evoke such nostalgia in Republican leaders of today.

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Nursery Tales

Published
by
Steve Landsburg
on January 25, 2011
in Economics and Paul Krugman
. 62 Comments

babyPaul Krugman, not for the first time, invokes the Great Capitol Hill Baby Sitting Co-Op Crisis as a metaphor for the macroeconomy.

First things first: Krugman is absolutely right that we learn a lot from well-chosen simple examples. But this particularly example seems poorly chosen.

The Capitol Hill Baby-Sitting Co-Op consisted of about 150 couples who baby sat for each other. They paid each other in scrip — pieces of paper each worth a half hour of baby-sitting time. New members received 20 units of scrip, which they were expected to pay back upon retiring. Aside from that, you earned scrip by baby-sitting, and you purchased baby-sitting with scrip, so that in the long run you’d sit exactly as much as you were sat for.

The problem was that people started hoarding scrip, thinking they might need it someday. As a result, the demand for babysitting services dried up. This made it harder to earn scrip, which encouraged even more hoarding, and so on around the vicious circle. The solution was to issue more scrip — each member got 10 more units. This made the hoarders a little less frantic and a little more willing to go out, which meant more sitting jobs were available, which eased the hoarder’s minds still further, and soon the co-op entered a golden age.

That, says Krugman, is the story of most recessions. People hoard money, which makes it hard to earn money, which makes people hoard still more money, which makes it even harder to earn money. The solution is to issue more money.

But here’s the part of the baby-sitting story that never made sense to me: Continue reading ‘Nursery Tales’

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The State of the Union

Published
by
Steve Landsburg
on January 24, 2011
in Current Events, Economics and Policy
. 26 Comments

The New York Times reports that President Obama, in his State of the Union speech, will call, among other things, for encouraging exports.

Now, since exports must equal imports in the long run, encouraging exports is exactly the same thing as encouraging imports. And wouldn’t you expect that if you were out to encourage imports, your first step might be to stop discouraging imports, say by declaring an end to all tariffs and quotas on foreign-made goods?

In a sane world, that’s indeed what you might expect. But somehow I don’t expect it.

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Meager Means and Noble Ends

Published
by
Steve Landsburg
on January 20, 2011
in Birthdays, Economics and Education
. 8 Comments

impOn Monday we marked the hundredth birthday of the Nobel laureate and all-around intellectual curmudgeon George Stigler. I promised more Stigler quotes by the end of the week. Here, then, is Stigler on the consequences of competition in the market for higher education; the passage is from one of the two-dozen lively and provocative essays collected here. If he’d been born just a bit later, Stigler could have been a champion blogger.

For clarity: When Stigler refers to an academic “field”, he is referring to a sub-discipline. Economics is a discipline; industrial organization and public finance are fields. Physics is a discipline; particle physics and solid state physics are fields.

We cannot build universities that are uniformly excellent … I shall seek to establish this conclusion directly on the basis of two empirical propositions.

The first proposition is that there are at most fourteen really first-class men in any field, and more commonly there are about six. Where, you ask, did I get these numbers? I consider your question irrelevant, but I shall pause to notice the related question: Is the proposition true? And here I ask you to do your homework: gather with your colleagues and make up a numbered list of the twenty-five best men in one of your fields — and remember that these fields are specialized. Would your department be first-class if it began its staffing in each field with the twenty-fifth, or even the fifteenth, name? You have in fact done this work on appointment committees. I remember no cases of an embarrassment of riches, and I remember many where finding five names involved a shift to “promising young men”, not all of whom keep their promises. I leave it to the professors of moral philosophy and genetics to tell us whether the paucity of first-class men is a sort of scientific myopia, a love of invidious ranking, or a harsh outcome of imprudent marriages. But the proposition is true.

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The Intellectual and the Marketplace

Published
by
Steve Landsburg
on January 17, 2011
in Blogging, Economics and Memories
. 40 Comments

stiglerToday is the 100th birthday of the late George Stigler, who won a Nobel prize for his economics and would have won a second if they gave one for dry wit. This is not the best example of that wit, but it’s the one I remember most vividly: One day long ago I was walking across the quadrangle at the University of Chicago, when I felt a hand on my shoulder — a very large hand, because Stigler was a very large man (in the tall-and-lanky sense of large). He’d been away for a few months, so I was a little surprised to see him. Before I could say anything like “Welcome back”, Stigler asked me: “So, what’s become of that young lady you were squiring around before I left town?”. In a fit of circumspection, all I said was “Oh, she still exists”, and Stigler immediately replied, “Oh, how lovely. You know, I’ve never been a subscriber to this theory that says you should destroy them when you leave them.”

The Intellectual and the Market Place — Stigler’s classic defense of the marketplace against the discomfort felt by so many intellectuals — is well worth a quick read. Parts of it have been paraphrased so often by so many imitators that they’ve begun to seem almost trite, but none of the imitators has ever achieved Stigler’s panache. Besides, it’s been imitated so much precisely because there’s so much here worth saying. A few sample paragraphs to whet your appetite:

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Public Service Announcement

Published
by
Steve Landsburg
on January 13, 2011
in Blogging
. 1 Comment

I’m traveling, and blogging is likely to be pretty light for the rest of the week. I won’t be gone long.

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A Whole New Brain Teaser

Published
by
Steve Landsburg
on January 10, 2011
in Bad Reasoning and Puzzles
. 51 Comments

brainToday I have a new brain teaser for you. It’s similar in flavor to last week’s brain teaser, but it’s genuinely different, and it’s different in an instructive way. And it stands on its own; you don’t need to have followed last week’s discussion to tackle this.

Here goes: There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. In expectation, what is the ratio of boys to girls?

For those who were here last week, notice that this problem is genuinely different. Last week I asked about the fraction of the population that is female. If we exclude the parents, that’s the ratio G/G+B. Today’s problem asks about the ratio B/G.

Stop here if you don’t want spoilers.

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The Extra Half Boy

Published
by
Steve Landsburg
on January 7, 2011
in Puzzles
. 42 Comments

Last week’s series of posts on boys, girls and population ratios drew an astonishing 850 or more comments, and they’re still coming in. There’s a lot of stuff worth reading in those comments, but the conversation — which is spread out over four threads and often involves several overlapping subconversations — has become difficult to follow. Fortunately, one of our more insightful commenters has volunteered to write a guest post summarizing what he sees as some of the most important discoveries to come out of those discussions. He has signed his guest post “Tom M”, but in the comments, he is simply “Tom”.

Without further ado:

The Extra Half Boy

A Guest Post

by

Tom M.

1. Introduction

Steve Landsburg posed a problem in this post at his blog entitled Are You Smarter than Google?. Here was the problem statement:

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Well, of course, you can’t know for sure, because, by some extraordinary coincidence, the last 100,000 families in a row might have gotten boys on the first try. But in expectation, what fraction of the population is female? In other words, if there were many such countries, what fraction would you expect to observe on average?

As we’ve worked on that problem, one of the points that came up was something called the “extra half boy.”

In this post I’m going to try to summarize what we’ve said about that critter and some related issues.

A key reference, that everybody should read, is What is the expected proportion of girls?, by ‘Thomas Bayes’.

Continue reading ‘The Extra Half Boy’

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Another Rationality Test

Published
by
Steve Landsburg
on January 5, 2011
in Puzzles and Rationality
. 135 Comments

spockHow rational are you? I once posted a test, based on ideas of the economist Maurice Allais, that most people seem to fail. Today’s test, based on ideas of the economist Dick Zeckhauser and the philosopher Richard Jeffrey, is one that almost everyone fails.

Suppose you’ve somehow found yourself in a game of Russian Roulette. Russian roulette is not, perhaps, the most rational of games to be playing in the first place, so let’s suppose you’ve been forced to play.

Question 1: At the moment, there are two bullets in the six-shooter pointed at your head. How much would you pay to remove both bullets and play with an empty chamber?

Question 2: At the moment, there are four bullets in the six-shooter. How much would you pay to remove one of them and play with a half-full chamber?

In case it’s hard for you to come up with specific numbers, let’s ask a simpler question:

The Big Question: Which would you pay more for — the right to remove two bullets out of two, or the right to remove one bullet out of four?

The question is to be answered on the assumption that you have no heirs you care about, so money has no value to you after you’re dead.

Almost everybody says they’d pay more in the first case than the second. Arguably, that means that in this scenario almost nobody is rational — because a rational person would give the same answer to both questions.

The reason for that is not immediately obvious. To understand it, you’ve got to think about four questions:

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Status of the Bets

Published
by
Steve Landsburg
on January 4, 2011
in Puzzles
. 21 Comments

rouletteAs promised, I am providing an update on the status of the various bets that got placed here last week.

The problem is this: In a certain country, each family continues having children until it has a boy, then stops. In expectation, what fraction of the population is female?

The answer is: not 1/2. A more precise answer depends on your additional assumptions — the number of families, the mortality rate, whether the parents count, whether there are grandchildren, etc. It might be possible to engineer those additional assumptions in some way that makes the answer come out to 1/2, but it would surely be impossible to argue that those ad hoc assumptions, whatever they are, are implicit in the statement of the problem.

Pretty much everyone seems to understand this now. (Here‘s one more learning aid, if you’re still struggling.) What, then, are we betting about? Here is a list of all the open and closed bets:

Continue reading ‘Status of the Bets’

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Happy New Year

Published
by
Steve Landsburg
on December 31, 2010
in Uncategorized
. 1 Comment

Happy New Year, one and all. I’ll see you in 2011.

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Slippery Lube

Published
by
Steve Landsburg
on December 30, 2010
in Bad Reasoning, Math, Puzzles and Rants
. 170 Comments

xkcdIt has been said of Lubos Motl that he’s hard to ignore, but it’s always worth the effort. I will, soon enough, take this advice to heart. But not quite yet. Lubos’s penchant for twisting other people’s words, just so he can have something to argue about, is well known and widely remarked. As his most recent victim (though “victim” is of course too strong a word, no actual harm having been done), I thought it would be both fun and instructive to challenge him to a bet. True to form, he continued to bluster but of course refused to back up his misrepresentations with actual cash.

Now of course Lubos will say that it is I who am twisting words, and in particular that I either “changed the question” or “changed the answer” (or both) between the original post and the offer to bet. That, however, won’t wash, since I’ve agreed, as part of the terms of the bet, to let an impartial panel of statistics professors determine the answer to the question as it was originally posed. So even if I had changed the question (which I haven’t), this would prevent me from getting away with it. (And no, I haven’t changed the answer either. If Lubos claims I have, we can put that to the stats profs also.)

I’m feeling annoyed enough to say a little more along these lines, but first I’d like to make it crystal clear that my annoyance does not extend to readers who are still puzzling this out. The problem with Lubos isn’t that he’s got it wrong; it’s that he’s not the least bit interested in getting it right. A few particulars:

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Centenary

Published
by
Steve Landsburg
on December 29, 2010
in Birthdays and Economics
. 18 Comments
The great Ronald Coase is 100 years old today.

One year ago today, he celebrated with a 100th birthday party, though he was only 99. I’m not sure what festivities are planned for today but I hope it’s a very good day for him.

Also one year ago today, I published a 99th birthday tribute here on this blog. I’m re-running it today.

Happy Birthday, Ronald Coase

In the theory of externalities—that is, costs imposed involuntarily on others—there have been exactly two great ideas. The first, forever associated with the name of Arthur Cecil Pigou (writing about 1920) is that things tend to go badly when people can escape the costs of their own behavior. Factories pollute too much because someone other than the factory owner has to breathe the polluted air. Nineteenth century trains threw off sparks that tended to ignite the crops on neighboring farms, and the railroads ran too many of those trains because the crops belonged to someone else. Farmers keep too many unfenced rabbits when they don’t care about the lettuce farmer next door.

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Win Landsburg’s Money!!!

Published
by
Steve Landsburg
on December 27, 2010
in Bad Reasoning, Puzzles and Statistics
. 278 Comments

winLast week I posted a little brain teaser that shows up frequently in recreational puzzle books — and reportedly in Google job interviews. The interesting thing about that puzzle is that the “official” answer is wrong. Not only that, but it’s wrong for an interesting reason.

I explained the official answer, I explained exactly where it goes wrong, and I explained how to get the right answer, citing Douglas Zare’s post here as inspiration.

The physicist Lubos Motl, however, still defends the official “50%” answer on his own blog. I am therefore offering to bet him $15,000 that I’m right (with detailed terms described below). If you agree with Lubos, this is your chance to get in on the action. I will take additional bets up to $5000 per person from all comers until such time as I decide to cut this off. You can place your bet by commenting on this post with the amount you’d care to stake. Be sure to include your email address (which does not show up in the post) so I can email you and verify that you’re for real.

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A Big Answer

Published
by
Steve Landsburg
on December 22, 2010
in Puzzles
. 280 Comments

I was going to wait a few days before posting the answer to yesterday’s puzzler, but we’re up well over 100 comments already, the holidays are almost upon us, and I think it’s time to settle this so you can all give your full attention to whatever festivities you’ve got coming up.

Here’s the puzzle again:

More precisely: What fraction of the population should we expect to be female? That is, in a large number of similar countries, what would be the average proportion of females?

Stop reading here if you don’t want spoilers:

Continue reading ‘A Big Answer’

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Are You Smarter Than Google?

Published
by
Steve Landsburg
on December 21, 2010
in Puzzles
. 134 Comments

chestnutThere’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Well, of course, you can’t know for sure, because, by some extraordinary coincidence, the last 100,000 families in a row might have gotten boys on the first try. But in expectation, what fraction of the population is female? In other words, if there were many such countries, what fraction would you expect to observe on average?

I first heard this problem decades ago, and so, perhaps, did you. It comes up in job interviews at places like Google. The answer they expect is simple, definitive and wrong.

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Lord Russell’s Nightmare

Published
by
Steve Landsburg
on December 20, 2010
in Anniversaries, Heroes, Logic, Math and Philosophy
. 31 Comments

bertieBertrand Russell, that most rational of men, was nonetheless plagued by intermittent depression and the occasional nightmare. Including this one, as reported by Russell’s confidante, the mathematician G.H. Hardy:

[Russell] was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated….

Principia Mathematica, to which Russell had devoted ten years of his life, was his (and co-author Alfred North Whitehead‘s) audacious and ultimately futile attempt to reduce all of mathematics to pure logic. It is a failure that enabled some of the great successes of 20th century mathematics. And — the first volume having been published in December, 1910 — this is its 100th birthday.

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Why Yes. The Law Is An Ass.

Published
by
Steve Landsburg
on December 15, 2010
in Bad Reasoning, Law, Policy and Rants
. 72 Comments

According to U.S. District Judge Henry E. Hudson, the same government that requires you to buy retirement insurance (via Social Security) is constitutionally barred from requiring you to buy health insurance.

Apparently some idiot lawyers have gotten it into their heads that the Social Security mandate is okay because it’s called a “tax”, whereas the Obamacare mandate is not okay because it’s enforced by what’s called a system of “fines”. From which I infer that if the government taxes you $1000 and uses it to buy you some health insurance, that’s constitutional. Or, if the government gives you a tax credit for buying insurance (after raising taxes to cover the cost of everyone’s credits, of course), then that’s constitutional — just as tax credits for home insulation are constitutional. Whereas if they just require you to buy $1000 worth of health insurance directly, that’s not constitutional even though it has exactly the same consequences as other policies that are constitutional. From which I infer that the law is an ass.

Continue reading ‘Why Yes. The Law Is An Ass.’

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Bars Versus Churches

Published
by
Steve Landsburg
on December 14, 2010
in Empirical Puzzles and Sex
. 17 Comments

Tyler Cowen asks: What variable best predicts your number of sex partners? My answer, which I stand by, was that it’s the same variable that best predicts your number of tennis partners, your number of checkers partners, and your number of visits to Disneyland — namely preferences. But Jason Malloy of Gene Expression steps in with something a little more empirically verifiable. Drawing on data from the respected General Social Survey, he’s correlated number of partners with everything from patriotism to empathy to “ever been punched” (all self-reported), and has invited me to share his results with you. Number of sex partners (also self-reported) is defined as number of partners since your 18th birthday. Jason’s calculations are restricted to heterosexual couplings. A negative number means that that a high response to the variable predicts fewer sex partners and a positive number means that a high response predicts more.

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The Great Compromise

Published
by
Steve Landsburg
on December 10, 2010
in Current Events, Economics, Policy and Politics
. 24 Comments

A few scattered thoughts on the great compromise (numbered for the convenience of commenters, so you can easily say which part you’re responding to):

  1. There were never any such thing as a “Bush tax cut”. There were only tax deferrals. In the absence of spending cuts, lower taxes today mean higher taxes tomorrow. So all this talk about how, in the absence of an extension, the average family will pay so-and-so many more thousands in taxes — it’s sheer balderdash. We will collectively pay exactly the same amount in taxes, present and future combined, whether or not this extension goes through.
  2. Although the average long-run tax burden is unaffected, changes in the tax code can of course shift the burden from one class of taxpayers to another. The Bush “tax cuts”, for example, probably made the tax code somewhat more progressive, shifting the burden from the poor to the rich. (You might have heard the opposite, but I suggest paying more attention to numbers than to rhetoric.)

    3. Continue reading ‘The Great Compromise’

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The Most Counterintuitive Theorem EVER

Published
by
Steve Landsburg
on December 9, 2010
in Cool Stuff and Math
. 41 Comments

comicbookguyToday I want to tell you about the single most counterintuitive-but-true thing I’ve ever heard.

Suppose you’re observing something that changes over time — say the Dow Jones average, or the temperature in Barrow, Alaska, or the number of people who have been shot by terrorists so far this year. Suppose you have absolutely no prior information about how this thing behaves — in particular, you might have no way of knowing whether it changes continuously (like the temperature) or whether it’s subject to sudden changes (like the number of terrorist victims). You have no formula for it; you don’t even know whether there is a formula. It could be absolutely anything.

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The Return of Depression Economics

Published
by
Steve Landsburg
on December 7, 2010
in Bad Reasoning, Economics, Paul Krugman and Policy
. 23 Comments

Paul Krugman writes that trade does not equal jobs and concludes that trade restrictions cannot even in principle trigger a depression. After all, restricting trade means restricting exports (less jobs!) but it also means restricting imports (more jobs!) so everything washes out.

Well, let’s try an extreme example. Suppose I prevent everyone in America from trading with anyone outside their own households. We’d eat only what we could raise in our own gardens, burn only the fuel we could gather from our own backyards, and wear only the clothes we could make for ourselves. In other words, we’d all be living pretty much at the subsistence level. Would you be willing to call that a Depression? I would. Krugman, apparently, would not.

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That Decision Theory Problem

Published
by
Steve Landsburg
on December 6, 2010
in Puzzles
. 17 Comments

Last week I posed a (perhaps imperfectly remembered) problem from Nick Kiefer’s course on Decision Theory. I’m very sorry that I haven’t found time to work out a complete solution (or even to read carefully through all the comments). Today I’ll post some hints from my notes toward a solution. Warning One: Only the math nerds will care about this post. Warning Two: This has all been double checked, but none of it’s been triple checked. It could be wrong.

The goal is to guess Nick’s secret number, which is drawn randomly from a uniform distribution on the numbers from 0 to 100. Each day, he draws a new “Number of the Day” and tells you whether it’s larger or smaller than the secret number.

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First Things

Published
by
Steve Landsburg
on December 2, 2010
in Knowledge and Math
. 29 Comments
God created the integers.

All else is the work of man.

—Leopold Kronecker

The problem with knowledge is that you have to start somewhere. Once you know something, you can start deducing other things. But how do you know the first thing?

Descartes’s famous solution was “I think, therefore I am”. I have direct knowledge of my own thoughts, and this in turn tells me that I exist. Now we can go on.

Mathematics, like all other knowledge, needs a starting point. Most of our mathematical knowledge is deduced from prior mathematical knowledge. I know that every positive integer is the sum of four squares because I know how to deduce this fact from other things I know. But where do I start?

There seems to be a widespread misconception (widespread, that is, among non-mathematicians — mathematicians know better) that all we know is what we can derive from axioms. This is wrong for several reasons. Two of these I’ve blogged about repeatedly in the past (e.g. here, here, and here and here), but the third is even more fundamental:

Continue reading ‘First Things’

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Public Service Announcement

Published
by
Steve Landsburg
on December 1, 2010
in Blogging
. 1 Comment

Things have gotten a little busy around here (in entirely a good way), so blog posts are likely to appear a little less frequently for the next couple of weeks — maybe two or three times a week instead of the usual five. I hope yesterday’s puzzle will keep you entertained for another day or two. See you soon!

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How to Decide

Published
by
Steve Landsburg
on November 30, 2010
in Puzzles
. 45 Comments

kieferMany years ago, when I was teaching at Cornell, my then-colleague Nick Kiefer kept me endlessly amused with the creative assignments he dreamed up for the students in his Decision Theory class. I’m reporting this from memory, so I don’t guarantee that I have it exactly right, but I think this was one of those problems:

  • On Day Zero (before classes start), Nick uses his pocket calculator to generate, at random, a secret number between 0 and 100. This number is drawn from a uniform distribution, which means (roughly) that the drawing is unbiased, so any number is as likely to come up as any other. The students’ job is to guess this number.
  • On Day One — the first day of class — Nick uses his pocket calculator to generate a Number of the Day. He doesn’t tell the students what the Number of the Day is, but he does tell them whether it’s larger or smaller than the secret number.
  • On Days Two, Three, and so forth, he does the same thing. He generates a Number of the Day and announces whether it’s bigger or smaller than the secret number.
  • You, the student, can submit your guess on the day of your choice.
  • You are then assessed two penalties. The first penalty is equal to the square of the error in your guess. So if the correct number is 3 and your guess is 5, your penalty is 22, or 4. The second penalty is equal to the natural logarithm of the day when you submit your guess. So if you submit your guess on Day 7, your penalty is the log of 7, which is 1.94591.
  • These penalties are subtracted from some some fixed number of points that you’re given to start out with (say 1000). The result, after the subtraction, is your score for the assignment. This score counts for a significant fraction of your course grade.

Continue reading ‘How to Decide’

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Thanksgiving

Published
by
Steve Landsburg
on November 29, 2010
in History and Musings
. 12 Comments

When I review the blessings of my extraordinarily blessed life, this one always appears near the top of my list: I am an adult male who has never been to war. I have always assumed — without thinking about it too hard — that in the historical scheme of things, this is a great privilege, and a great rarity.

Am I right about that? Over the course of human history, what is your estimate of the fraction of males who have reached adulthood without participating in a military conflict?

(Obviously, there’s some fuzziness about what counts as military conflict. I’m thinking here not about the occasional street fighter, but about the guy living in mud and getting shot at for weeks at a time — or things equally dangerous/traumatic/uncomfortable.)

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