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”.)
Continue reading ‘Miscellany’
Leonard 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’
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’
The 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’
The 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:
- A million dollars for certain
- 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:
- A lottery ticket that gives you an 11% chance at a million dollars (and an 89% chance of nothing)
- 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?’
If 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.
Click here to comment or read others’ comments.
You’re lost in a forest. What’s the best way to get out?
The great macroeconomist Bob Lucas once asked me this question, and I had no answer for him. I still don’t.
The assumption is that you know the size and shape of the forest, but you don’t know where you are or which way you’re facing. And the forest is so dense that you can never see any significant distance in front of you. What path should you follow?
Continue reading ‘Escaping the Forest’
Last week I posed this problem:
Several commenters did a wonderful job of explaining the answer. Let me just add a few words on the issue of “How can Tuesday be relevant?”
If the Tuesday part weren’t there, the problem would be easy. With two children, there are three equally likely ways to have (at least) one boy: The children in birth order might be Boy/Boy, Boy/Girl, or Girl/Boy. That gives a 1/3 chance of Boy/Boy.
So what does “Tuesday” have to do with it? Answer: Having (at least) one Tuesday boy is a lot more likely when you’ve got two boys than when you’ve got only one. So among those moms with a Tuesday boy, the Boy/Boy moms outnumber either of the other types. The three possibilities aren’t equally likely anymore.
Continue reading ‘A Big Answer’
With a hat tip to the mathematician John Baez, who in turn tips his hat to the science fiction author Greg Egan, who in turn credits the journalist Alex Bellos, who got this from the puzzle designer/collector Gary Foshee (who seems to have no website):
(For those who want more precision: We gather all those women in the world who have exactly two children, tell each of them to “go home unless you have a boy born on a Tuesday”, and select a woman randomly from those who remain. Assume that births are equally likely to occur on any day of the week, and that on any given day, boys and girls are equally likely.)
Click here to comment or read others’ comments.
Yesterday’s pop quiz posed this question:
Suppose that an acre of land in Iowa can yield either 50 bushels of wheat or 100 bushels of corn, while an acre of land in Oklahoma can yield either 20 bushels of wheat or 30 bushels of corn.
Which state has the comparative advantage in growing wheat? Which state has the comparative advantage in growing corn?
Suppose the residents of each state consume 200 bushels of wheat and 360 bushels of corn. If, instead of pursuing policies of self-sufficiency, each state specializes in its area of comparative advantage, how many acres of Iowa and Oklahoma farmland are freed up for other uses?
Quite a few people got this right in comments. In Iowa, the opportunity cost of a bushel of wheat is 2 bushels of corn. In Oklahoma, the opportunity cost of a bushel of wheat is 3/2 bushels of corn. Becauses 3/2 is less than 2, Oklahoma is the low-cost wheat producer, which is the same thing as saying that Oklahoma has the comparative advantage in wheat.
Continue reading ‘Pop Answers’
Commenting on this essay by former Intel chief Andy Grove, Tyler Cowen writes that “Only he who first shows he understands comparative advantage has license to partially reject it.”
Hear hear. When someone says “I understand comparative advantage, but in this case it doesn’t apply”, or “I understand comparative advantage but in this case it is overridden by other considerations”, my experience tells me that you can be nearly sure you’re talking to someone who does not in fact understand comparative advantage.
Continue reading ‘Pop Quiz’
A few years back, when Google acquired YouTube, I was heard to remark that the deal seemed kind of…imprudent. Given YouTube’s potential as a lawsuit generator, the best owners might not be the guys with some of the world’s deepest pockets.
A colleague points out that it seems equally odd for a company with pockets the depth of BP’s to be engaged in as risky an activity as deep water oil drilling. Why wasn’t this project sold off to someone with a lot less to lose?
Maybe BP expected to be protected by laws limiting its liability, but surely it was foreseeable that those laws might be circumvented, as it appears they’re about to be. So if that’s part of the answer, it’s only a small part.
Continue reading ‘Riddle Me This’
Here are some thoughts on last week’s absent-minded driver problem.
First a recap of the problem, with a bit more detail than last week:
Each day, Albert leaves his office (at the bottom of the map), gets on the Main Highway and attempts to drive home to his house on Second Street. If he turns too soon (onto First Street) or if he overshoots (going all the way to the north end of the Main Highway), he is mauled by dinosaurs.
Obviously, Albert’s best strategy is to go straight at the first intersection and turn right at the second. Unfortunately, both intersections look identical. Doubly unfortunately, Albert can never remember whether he’s already passed the first intersection.
Continue reading ‘Absentminded Musings’
Until last week, I had never heard of the paradox of the absent-minded driver, but I was recently told that it has some relevance to my encyclopedia article on quantum game theory. That plus the fact that I am a notoriously absent-minded driver myself made me think I should check out the original source. Here’s what I extracted:
Each day, Albert leaves his office (at the bottom of the map), gets on the Main Highway and attempts to drive home to his house on Second Street. If he turns too soon (onto First Street) or if he overshoots (going all the way to the north end of the Main Highway), he is mauled by dinosaurs.
Obviously, Albert’s best strategy is to go straight at the first intersection and turn right at the second. Unfortunately, both intersections look identical. Doubly unfortunately, Albert can never remember whether he’s already passed the first intersection.
Continue reading ‘The Absent-Minded Driver’
In high school, we used to play four-dimensional tic-tac-toe. The board looks like this:
Here each four-by-four subsquare is an ordinary tic-tac-toe board (except that it’s four-by-four instead of the traditional three-by-three). You should think of the four subsquares in the first column (or any other column) as stacked above each other in the third dimension. The red x’s form a vertical line in that direction, so if you manage to place four x’s in those positions, you’re a winner.
You should also think of the four subsquares in the first row as stacked above each other in yet another dimension. The red o’s form a diagonal line passing from the bottom left to the top right (using “bottom” and “top” to refer to directions in this fourth dimension). And the black x’s form another kind of diagonal line, passing from one corner to another through all four dimensions. So there are a lot of ways to win this game.
Continue reading ‘Tic Tac Toe in Four Dimensions’
An unexpectedly full weekend leaves me caught short without a full fledged blog post for today. I’ll make up for it tomorrow. In the meantime, here are two tidbits to hold you over:
- A useful recipe for salted water. Do not fail to read the reviews.
- A puzzle I got from the mathematician Alexander Merkurjev. If I recall right, he told me that it had appeared on a college entrance exam in the old Soviet Union:
A regular 400-gon is tiled by parallelograms. Prove that at least 100 of those parallelograms must be rectangles.
(A regular 400-gon is a 400-sided figure with all sides equal and all angles equal. The parallelograms can all be of different sizes and shapes—or not. “Tiled” means that the interior of the 400-gon is entirely covered, with no overlaps.)
Last week, I posed some brain teasers and a riddle about special relativity.
The brain teasers were all solved by multiple commenters; I’ll summarize their answers at the end of this post. The special relativity problem proved trickier; here it is again:
A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?
Now to the answer.
Continue reading ‘The Big Answers’
There are a bazillion alleged “paradoxes” in special relativity, all based on exactly the same fallacy, but I might have just invented a brand-new one—-where “invented” is shorthand for “confused the hell out of myself for a while”. When I finally got up and drew a picture (as opposed to lying in bed with my eyes closed doing something that felt like thinking), it became clear that, sure enough, it was the same old fallacy again (how could it not have been?), but in a new enough guise that someone reading this might find it amusing.
Continue reading ‘Geek or Dork?’
My travel schedule for the next several days will probably keep me from posting anything too substantial. So let me leave you with three lovely brain teasers to keep you occupied in the meantime.
1) (Hat tip to Ben Tilly): I have thought of two numbers, which I call A and B. You know nothing about how I came up with these numbers. I plan to flip a fair coin and then tell you the value of A if the coin comes up heads or B if the coin comes up tails. Your job is to guess whether the number I quote is the larger or the smaller of A and B. Devise a strategy that guarantees you a better-than-even chance of winning, no matter what A and B are.
(To make this more precise: Your probability P of winning is a function of A, B and your strategy. Devise a strategy S such that P(A,B,S) is greater than 1/2 for all A and B.)
2) (Hat tip to Stan Wagon): Alice and Bob ran a marathon (assumed to be exactly 26.2 miles long) with Alice running at a perfectly uniform eight-minute-per-mile pace, and Bob running in fits and starts, but taking exactly 8 minutes and 1 second to complete each mile interval (so, for example, it takes him exactly 8 minutes and 1 second to get from the 3.78 mile mark to the 4.78 mile mark, exactly 8 minutes and 1 second to get from the 3.92 mile mark to the 4.92 mile mark, etc.). Is it possible that Bob finished ahead of Alice?
3) (Hat tip to my old friend Steve Maguire): The border between Delaware and its neighbors includes a section with a circular arc: on the circle ten miles from a church in Dover Delaware. Can you name another state border that is partially defined by a circular arc?
I’ll continue to check comments while I’m on the road, but perhaps just a tad less diligently than usual.
Yesterday I posted a portrait gallery honoring 60 of my personal heroes; readers were quick to identify 47, with remarkably few mistakes, all of which were quickly corrected. As of this writing, thirteen remain. Among these thirteen are the greatest mathematician of the 17th century (assuming we classify Newton as a physicist) and the three greatest mathematicians of the 20th; one of these is quite probably the greatest mathematician of all time. (All in my educated-but-not-fully-educated opinion, of course.) Musical, literary and cinematic greatness are also well represented here.
Over the next couple of weeks, I will try to tell you a little bit more about some of these 60 people. Meanwhile, here are the thirteen mystery men/women. I’ve retained the numbering from yesterday’s post. Who can you identify?
Continue reading ‘Unidentified Persons’
Merry Christmas. As my gift to you, I present the long overdue answers to the remaining problems from my Oberlin honors exam. The original questions are here and here; the first round of answers is here.
Continue reading ‘The Big Answers, Part II’
This quiz amused the hell out of me. I hope it does the same for you.
Edited to add: In comments, Mike H points me (and you) to this even better quiz, which seems to have been the model for the one I linked to. Enjoy your day.
A little while back, I posted the first half and then the second half of the honors exam in economics that I administered at Oberlin College. Since then, I’ve slowly doled out a few answers, but I’m getting more and more requests for the complete set. Here, then, are the questions and answers for the first half; I warn you that some of these are pretty technical. I’ll post the second half soon.
Continue reading ‘The Big Answers, Part I’
Here are solutions to the two game theory problems from my honors exam:
Continue reading ‘Playing Games’
To the many people who have recently requested answers to my Honors Exam, Part I and Part II:
I’ve already posted answers to the Snidely Whiplash and “Rank the Taxes” problems. I’ll post solutions to the “Jack and Jill” and “Dukes of Earl” problems in the next day or two, and the remainder soon thereafter. Thanks for your patience.
Two weeks ago, I posted the first half of the honors exam that I administered last spring at Oberlin college. I am following up today with the second half. Once again, I’ve translated some of the questions from economese to English, but am fairly confident that nothing significant has been lost in the translation. This starts with Question 6:
Continue reading ‘The Honors Class, Part II’
Today I’ll give the solution to another of the problems from my honors exam:
Question 5. Rank these taxes in order of how much you’d dislike paying them:
- A tax on consumption
- A tax on wages
- A tax on income (including wages, interest and dividends)
Assume that the tax rates are adjusted so that your total tax bill is the same in each case.
Continue reading ‘The Best of Taxes and the Worst of Taxes’
If you’re at work on this post-Thanksgiving morning, it’s probably a slow day around the office (unless you’re in retail, in which case you’re probably not reading this). So to help you while away the hours, here are a few of my favorite logic puzzles from around the net:
Warning: These are majorly addictive. Enjoy, but resolve not to let them take over your life. You have a blog to get back to.
On his blog A Blank Slate, Vishal Patel posts a cute little brain teaser (with a hat tip to the Cosmic Variance blog):
Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?
(a) Yes
(b) No
(c) Can not be determined
This reminded me of one of my favorite little “zinger” math proofs. (If you think about the brain teaser long enough, you’ll see the connection.)
Continue reading ‘Rational Irrationality’
I’m going to dole out the answers to the first half of my honors exam slowly over the next several days. After that I’ll post the second half of the exam.
Let’s start with this one:
Question 3. Snidely Whiplash owns all the grocery stores and all the houses in the Yukon Territory. He charges a competitive price for groceries, and rents the houses at the highest price residents (who are all identical) are willing to pay. (If he charged any more, they’d all leave town). True or False: If Snidely raises the price of groceries, he’ll have to lower the price of housing, so he’ll be no better off than before.
Continue reading ‘Snidely Whiplash’
