Monthly Archive for October, 2010

Weekend Roundup

pumpkinroundupThis week we celebrated the greatest logician, the greatest economist, and the greatest lyric poet of the 20th century, beginning on Monday with a nod to the 80th anniversary of Kurt Godel‘s Incompleteness Theorem, continuing Tuesday with an attempt to popularize Ken Arrow‘s Impossibility Theorem (which, incidentally, Arrow preferred to call the “General Possibility Theorem”) and on Wednesday with an homage to Dylan Thomas on his 96th birthday.

Tyler Cowen and Alex Tabarrok also took on Arrow’s Theorem this week, though it seems to me that Tyler got it wrong, for reasons I explained on Thursday. And on Friday I did my small part to publicize Bob Murphy‘s challenge to Paul Krugman, and how you can help.

More on Monday! In the meantime, happy Halloween!

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And in This Corner….

How much would you pay to see Paul Krugman debate the irrepressible Austrian economist Bob Murphy?

Murphy isn’t the first Austrian to challenge Krugman to a debate, but I bet he’s the cleverest. He’s calling for pledged donations to help the New York City Food Bank feed the hungry — with the pledges contingent on Krugman’s accepting the challenge. The pledge total is currently around $40,000 and Murphy is hoping to hit $100,000. Then Krugman can choose between facing off against Murphy or denying $100,000 worth of food assistance to the poorest of the poor — an option that in another context, Krugman himself might be quick to label as “callous”. Or worse yet, “Republican”. Here‘s where you go to pony up.

I would love to see this debate, all the moreso after watching Murphy’s two promotional videos, each so entertaining in its own way that they made me want to send him money independent of the Krugman thing. Watch, and enjoy:

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Arrow’s Theorem, Take Two

Tyler Cowen is of course one of the primary reasons to be grateful that you live in the age of the Internet. But none of us is infallible, and I believe Tyler has stumbled in his account of Arrow’s Theorem. His example:

Let’s say you had two people on a desert island, John and Tom, and John wants jazz music on the radio and Tom wants rap. Furthermore any decision procedure must be consistent, in the sense of applying the same algorithm to other decisions. In this set-up (with a further assumption), there is only dictatorship, namely the rule that either “Tom gets his way” or “John gets his way.”

Not true. A rule (or, in Arrow’s language, a social welfare function) has to prescribe a choice not just today, but every day, even as Tom’s and John’s preferences might change from one day to another. So there are in fact 16 possible rules. One is “Tom always gets his way.” Another is “John always gets his way.” Another is “Always turn the radio to jazz”, which seems pretty unreasonable since it prescribes jazz even on days when Tom and John both prefer rap. Yet another is:

  • If Tom and John agree, do whatever they agree on. If they disagree, turn the radio to jazz.

That last rule is particularly interesting because it satisfies every one of Arrow’s “reasonableness” criteria without anointing a dictator. What Arrow’s theorem says is that no non-dictatorial rule can meet all of those criteria.

Continue reading ‘Arrow’s Theorem, Take Two’

Blogpost in October

dylant

If Dylan Thomas hadn’t drunk himself to death in 1953, he might be celebrating his ninety-sixth birthday today, perhaps with a successor to the grand and glorious poem he wrote to celebrate his thirtieth.

He left us with a small number of poems so heart-wrenching that I cannot read them, even for the two hundredth time, without all of the symptoms of an emotional crisis. Take In Country Sleep, where a father reassures his daughter that she has nothing to fear from fairy tale villains—but only from the Thief who comes in multiple guises to take her faith and ultimately to leave her “naked and forsaken to grieve he will not come”. In Country Sleep was a standard bedtime poem in our house, and my daughter soon learned to anticipate “the part where Daddy cries”.

Then there’s the prose. Nobody is better at nostalgia and grief for time’s relentlessness:

Continue reading ‘Blogpost in October’

Straight Arrow

arrowTyler Cowen asks which economic ideas are hardest to popularize. Arnold Kling nominates the Arrow Impossibility Theorem. Tyler responds with an attempt to popularize it. Alex Tabarrok weighs in with another. Here’s my own attempt:

Every day, Alice, Bob and Charlie split a pizza with one topping — anchovies, mushrooms or pepperoni. Their preference orderings change from day to day — some days Alice is in the mood for mushrooms, other days the very thought of mushrooms makes her queasy. Every day, they have to call in their first, second and third choice pizza orders. (The pizza delivery place insists that you specify your second and third choices in case they run out of something.) So Alice, Bob and Charlie need a method for translating their individual preferences to a pizza order.

Continue reading ‘Straight Arrow’

Eighty Years of Incompleteness

godelThis weekend marked the 80th anniversary of the most significant event in the history of logic since the days of Aristotle. On October 23, 1930, the 24-year-old Kurt Godel presented his incompleteness theorems to the Vienna Academy of Sciences.

The first incompleteness theorem says this: No matter what axioms you start with, there will always be statements in arithmetic that you can neither prove nor disprove. (I am glossing over some technicalities here, but in this context they are not important.) Some of those statements take the form “Such-and-such an equation has (or does not have) any solutions”. You tell me your axioms, and I’ll produce an equation that you can neither prove solvable nor prove insolvable.

Continue reading ‘Eighty Years of Incompleteness’

Miscellany

1) I just had an extremely pleasant walk around the Beale Street area in Memphis, which strikes me, roughly, as Bourbon Street without the urine. (Also without the trash and the high general level of obnoxiousness — though also of course without the magnificent architecture, etc.) Yes, I realize it’s also a different musical genre (though in both cases it’s a sub-genre of “too loud”). But it’s astonishing to me how clean the streets are here, and how well-behaved the crowds, compared to what I’ve seen in Louisiana. If they can do that here, why can’t they do it there?

2) This weekend marks the anniversary of a world-changing event — an event that might be of particular interest to readers of The Big Questions, both the book and the blog. Who can tell me what event I have in mind? (Hint: It’s an anniversary ending in zero.) I’ll blog the answer on Monday.

3) The discussion of the Allais paradox rages on in comments on multiple posts. For the few of you who have not yet tuned this out, my latest comment is an attempt to cut through the fog and identify the locus of some commenters’ confusion, or disagreement, or both. I think it will very much help focus the discussion if the dissenters could tell us where they stand on these questions. (My answers are all “yes”.)

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The Noble Savage

savageLeonard Jimmie Savage was a pioneer in modern decision theory and a disciple of Frank Plumpton Ramsey, whose story occupies the final chapter of The Big Questions.

In 1954, Savage wrote a lovely and highly influential little book called The Foundations of Statistics, which starts with six simple axioms about human preferences — one of which says that if you prefer a dog to a cat, then you’ll prefer an 11% chance of a dog to an 11% chance of a cat (and likewise for any other percentage). From these axioms, he drew deep and surprising conclusions about human behavior. This work underlies much of modern game theory, decision theory and economics in general.

According to legend (and I have reason to suspect this legend is actually true), Professor Savage was giving a talk one day when he was interrupted by the French econometrician (and then-future Nobel Prize winner) Maurice Allais, who asked Savage if he’d be willing to answer two questions about his own preferences. Savage said sure. These were the questions:

Continue reading ‘The Noble Savage’

Wordplay

It has come to my attention that the word astasia refers to both

a. a colorless euglenoid that does not have plastids or a light-sensing spot

and

b. a lack of motor coordination which leaves the patient unable to stand or walk unassisted.

Where is the doggerel that plays off this double meaning? Where are the ironic little vignettes in which the hero (or heroine) is led charmingly astray through the confusion of one meaning with the other?

It’s not like the literary potential of other English double meanings has gone unexploited. (Think “pussy”.) Here is your opportunity to get in on the ground floor of a whole new subgenre. Give me your best astasia wordplay!

(Extra credit: Which chapter of The Big Questions inspired me to look up the word astasia?)

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Ignorance, Bliss, and Rationality Re-Redux

twothinkers

Can ignorance be bliss?

There is allegedly a tradition of issuing a blank cartridge to one (randomly chosen) member of each firing squad, so that no shooter knows for certain that he contributed to a death. Let’s assume that tradition really exists and let’s assume that it exists because the shooters want it. Does that prove that shooters (at least in some instances) value ignorance?

Not necessarily. It might just mean that each shooter prefers a 5/6 chance of firing a real bullet over a 100% chance of firing a real bullet. That’s not the same thing as preferring to be ignorant.

So here’s the key experiment. Offer the shooters a choice:

Continue reading ‘Ignorance, Bliss, and Rationality Re-Redux’

Weekend Roundup

This week we had two posts on the foundations of rationality and two on whether there’s been a recent surge in government spending.

The government spending posts are here and here. It seems to me that the graphs in the second post are dispositive.

The rationality posts are here and here. These led to a lot of convoluted discussion, so let me give you the executive summary.

Continue reading ‘Weekend Roundup’

Bah

Here’s where you’d ordinarily see our weekly roundup post. Unfortunately, I haven’t been able to get a web connection to my provider for the past several hours (though they’re definitely up and running); all attempts seem to stall while trying to pass through a downed machine in Chicago. Isn’t the whole point of the Internet supposed to be that there are multiple paths from everywhere to everywhere so this kind of thing can’t happen?

Be that as it may, I am logged into a shell account and posting this via lynx (if you don’t know what that means, you’re probably not as old as me), and the interface is far too painful to type anything substantive. So I’ll plan to post the usual roundup sometime tomorrow, after they’ve cleared the gunk out of the Intertubes.

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Rationality Redux

thinkerThe rationality quiz that I posted on Tuesday has drawn a lot of comments from folks who think they can reconcile inconsistent answers by appealing to risk aversion. That’s surely incorrect. To see why, let’s start with another quiz.

Question 0: Which do you like better, dogs or cats?

Economists would not presume to declare either choice an irrational one. There’s no accounting for tastes.

Now I have two more questions for you:

Continue reading ‘Rationality Redux’

Feeling the Surge

Paul Krugman says there’s been no surge in government spending. I say there has been. Paul offers a graph in evidence. I look at that same graph and see a four-year surge.

Several commenters have insisted that I am ignoring Paul’s point — the point being that if we take the Bush years as a baseline, then there’s been no surge relative to that baseline. Really? Here’s federal government expenditure from the beginning of the Bush years to today. The blue line projects a continuation of the average annual spending increases under Bush. The vertical bar marks the advent of the Obama age.

(I constructed this graph from the invaluable FRED database at the St. Louis Fed.)

Now personally, I look at this graph and the main thing I see is a ten-year surge in federal government expenditures. But if you insist on taking the Bush years as a baseline — well, then what you see, starting in 2009, is an Obama surge. You might or might not want to argue that the surge was justified (or compelled) by economic conditions, but I don’t see how you can deny it’s there.

Continue reading ‘Feeling the Surge’

Surgin’ USA

Paul Krugman offers the following graph as evidence that “spending hasn’t surged”:

Now, what I’m seeing here is something like a 25% increase in spending under the Bush/Obama policies of the past four years. Which makes me wonder exactly what it would take to count as a surge in Krugman-land.

********

On separate notes:

1) Yesterday’s post on rationality generated several comments that deserve responses. For the most part, I am reserving those responses for a separate blog post a few days down the line.

2) I just now hit a wrong button and deleted about 100 comments that my software had classified as spam, without first skimming through them. There is therefore a small but non-zero chance that I deleted a legitimate comment or two. If so, I very much apologize and hope you’ll try again.

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How Rational Are You?

rationalThe death this week of Nobel laureate (and relativity denier!) Maurice Allais reminds me that I’ve been meaning to blog about Allais’s famous challenge to the way economists think about rational decision making.

I’m going to ask you two questions about your preferences. In neither case is there a right or a wrong answer. A perfectly rational person could answer either question either way. But I do want you to think about your answers, and to write them down before you read any further.

Question 1: Which would you rather have:

  1. A million dollars for certain
  2. A lottery ticket that gives you an 89% chance to win a million dollars, a 10% chance to win five million dollars, and a 1% chance to win nothing.

Try taking this seriously. What would you actually do if you faced this choice? Don’t bother trying to figure out the “right” answer, because there is no right answer. Some perfectly rational people choose A, and other perfectly rational people choose B.

Okay, ready for the next question?

Question 2: Which would you rather have:

  1. A lottery ticket that gives you an 11% chance at a million dollars (and an 89% chance of nothing)
  2. A lottery ticket that gives you a 10% chance at five million dollars (and a 90% chance of nothing)

Once again, this is a matter of preference. There is no right or wrong answer. But decide what your answer is and write it down before you continue.

Continue reading ‘How Rational Are You?’

Weekend Roundup

The Internet seems to have bred a peculiar subspecies of troll that cheerfully devotes enormous effort to refuting arguments nobody ever made. While they seem to have infinite time to construct these pointless rebuttals, these troll-types seem to have no time at all in which to actually digest the arguments they think they’re rebutting. They start with a guess as to what someone else might have said, and seem all but incapable of entertaining the notion that they might have guessed wrong. Is there a name for these people? “Crank” and “troll” are too general. If it were up to me, we’d reserve the word “Bozo” for this purpose, but it too is already in more general use. We need a new word! Give me your suggestions!

A title like More Sex is Safer Sex is like red meat to these folks and on Monday we took a moment to defend that argument against the latest Bozo Barrage (I’ll stick with this name till you give me a better one). Then on Tuesday we confronted a different subspecies of troll — the statistical obfuscator. Our reader Windypundit did some detective work and discovered that the offending graph was produced by an agency of the United States government. That, of course, is no excuse for perpetrating the deception.

Our graphical escapade led to a discussion of the gender gap in wages and whether it can be plausibly explained by employer discrimination. It’s often argued that it can’t, because that would require employers to sacrifice a profit opportunity. On Wednesday, I rejected that argument on the grounds that employers sacrifice profit opportunities all the time — but offered a (slightly) more sophisticated version that rejects the employer-discrimination hypothesis because it would require employers to sacrifice a very large profit opportunity. Of course, as our reader Patrick observes, this still doesn’t rule out the hypotheses of customer-discrimination or employee-discrimination. (In the latter case, male workers refuse to accept female colleagues. And again, the argument I gave can’t reject this. On the other hand, a different argument probably can — if the wage gap were driven by employee-discrimination, firms could profit not by hiring a few more females, but by hiring only females. In equilibrium, you’d expect half of all firms to be all female, half all male, and wages to be equalized across firms.

Thursday I reran a year-old post on how to add all the positive integers and get -1/12. This post generated just one comment! I’m not sure whether this was because none of you like this kind of thing, or whether you found it too awesome to remark on.

Continue reading ‘Weekend Roundup’

Gathering for Gardner

gardnerIf you like this blog, then you either are or should be a fan of the late Martin Gardner, the long-time “Mathematical Games” columnist for Scientific American. On the 21st of October (what would have been Gardner’s 96th birthday), “Gatherings for Gardner” will take place around the world, where fans can share their favorite puzzles, ideas, magic tricks and reminiscences in what’s being billed as a global “celebration of mind”. You’re welcome to attend one of these events — or to host one.

(Potential attendees would surely benefit from a list of locations in lieu of having to navigate that idiotic map, but that’s what’s there.)

It was at a previous Gathering for Gardner that puzzle designer Gary Foshee posed his notoriously tricky probability puzzle about the mom with a son born on a Tuesday. (Spoilers here.) If you host or attend a gathering, do come back here and share your favorite finds.

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A Little Arithmetic

About a year ago, when I was a novice blogger, I posted a piece called “A Little Arithmetic”. The arithmetic all looked fine in my browser, but I failed to realize that it might not look fine in everyone’s. As a result, some of you found it unreadable. But I like to think it was worth reading, so now that I’ve figured out how to make it look nice, I’m posting it again:

The mathematician John Baez has been dazzling science lovers on the web for over 15 years with his weekly Finds in Mathematical Physics. (He was a blogger long before there were blogs). Baez recently gave a lovely series of talks on his favorite numbers (they are 5, 8 and 24) in which he mentions Euler’s observation that if you sum up all the positive integers (1 + 2 + 3 + 4 + …) you get -1/12. (I promise, this is not a joke.)

Baez’s “proof” uses a little calculus, but I’ve reworked it into a form you can share with your middle schoolers—and better yet, have them share with their teachers.

Continue reading ‘A Little Arithmetic’

Women’s Wages and the Back of My Envelope

Yesterday’s breathtakingly dishonest graph from the AFL-CIO touched off some discussion in comments about whether the male/female wage differential could plausibly be driven by employer discrimination.

The usual argument to the contrary runs like this: If the differential is driven by employer discrimination (as opposed to, say, the abilities and/or preferences of the workers), then non-discriminating employers (i.e. those who care only about making a buck, regardless of who they have to hire to do it) would draw only from the relatively cheap female labor pool. It wouldn’t take many of these non-discriminating employers to drive women’s wages up to the same level as men’s. We don’t see that happening, ergo the hypothesis of employer discrimination is refuted.

The problem with that argument is that it assumes employers won’t ignore a profit opportunity, whereas in fact employers ignore profit opportunities all the time — by keeping on their incompetent nephews, taking Wednesday afternoons off to play golf, or, yes, hiring people they like having around instead of people who could do a better job.

Continue reading ‘Women’s Wages and the Back of My Envelope’

Topsy Turvy

The AFL-CIO is calling for passage of the Paycheck Fairness Act to close the wage gap between men and women, a problem they say is increasingly urgent, with the above graph as Exhibit A. Get a load of that plummetting dotted gray line!

Now have a look at the right hand axis, which the perpetrators have conveniently drawn upside down for no apparent reason other than the obvious dishonest one.

Continue reading ‘Topsy Turvy’

The Python Misinterpreter

moresexI once wrote a book called More Sex is Safer Sex”. If you’re wondering what that means, you can read the essence of the argument in Chapter 12 of The Big Questions and/or watch me explain it on video.

Python programmer Jack Trainor has posted a simulation that he believes is somehow relevant to this argument. (Comments on his post are here.) I’d thought this was too nonsensical to respond to, but more than one reader has asked for a response, so here goes: Except for the fact that his code runs, Trainor’s managed to get this argument wrong in every possible way. He’s misstated the assumptions, he’s misstated the logic, and he’s misstated the conclusions.

Continue reading ‘The Python Misinterpreter’

Weekend Roundup

Reminder: The Big Questions is out in paperback, and Amazon has it for 33% off. Do your Christmas shopping early!

This week we tackled the biggest of all questions: Why is there something rather than nothing? Then we tackled it again. If you like this stuff, you’ll like the opening chapters of The Big Questions .

In between, we paused for a few words about religion — another topic you’ll find a lot more about in The Big Questions.

I also told you where to find me in Memphis. I’ll see you there, or at least back here on Monday.

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Between Soft Covers

The paperback edition of The Big Questions goes on sale today — and Amazon has it for 33% off. This is a good time to stock up on gift copies for all your friends who are curious about math, physics, economics, philosophy or rational inquiry. Thanks to all of you for making this such a rewarding enterprise, and for the many blog comments that will make the next book even better.

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