Then it was on to the justice system. On Tuesday I offered a little quiz on recognizing reasonable doubt (or its absence) in an exceptionally simple environment. Some readers thought that environment was too simple to be interesting. On the contrary, it’s simplicity is what makes it so interesting. If we can’t recognize reasonable doubt in such a simple environment, how can we ever recognize it in the courtroom?

On Wednesday, we talked about the appropriate numerical cutoff for reasonable doubt, and on Thursday we took a step back and asked what principles we should apply in choosing that cutoff. On Friday, I decried the dereliction of duty by judges and legislators who refuse to tell us what cutoff they have in mind when they use the word “reasonable”.

Some commenters thought that giving jurors a precise numerical standard was asking them to think more “mathematically” (whatever that means) than we can reasonably expect. But there’s no mathematics involved in telling a juror that he should convict if he believes that in 100 similar cases, at least 93 of the defendants will be guilty. No mathematics, that is, beyond the ability to count to 100, which is, I think, something we already expect of our jurors.

If I don’t tell you what “reasonable” means, then “beyond a reasonable doubt” makes as much sense as “beyond a gribzle doubt”. Judges could, if they wanted to, tell juries to convict if the evidence convinces them beyond a gribzle doubt, and then refuse to reveal what “gribzle” means. I don’t see how that system would differ substantially from the one we have now.

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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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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.

I claim these things:

Claim One: I have here a mystery pet that might be a dog and might be a cat. If you prefer dogs to cats, you’ll prefer a dog to the mystery prize — and vice versa.

Claim Two: If you prefer dogs to cats, then you’ll prefer an 11% chance at a dog to an 11% chance at a cat — and vice versa.

Claim Three: The same things are true if you replace “dog” and “cat” with any other prizes you care to think of.

Claim Four: The desirability of a lottery depends only on the prizes and the probabilities of winning.

Claim Five: If you accept Claims One through Four, then you must answer either A to both of the following questions or B to each of the following questions:

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.

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)

Claim Five, as I’ve explained in the earlier posts, is a logical necessity. So if you want to defend inconsistent answers to Questions 1 and 2, then you are forced to deny at least one of Claims One through Four. It would be very helpful if participants in the ongoing discussion would specify exactly which claim they dispute.

I’ll have more to say about rationality — and a variety of other topics — over the next week.

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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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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.

And Friday I pointed you to the upcoming Gatherings for Gardner and encouraged you to attend one in your area — or to consider hosting one.

(Yes, I was in fact put up to this by a publicist for the Gathering-for-Gardner organization. No, I don’t get any compensation for this beyond the satisifaction of doing a good turn for my readers. For the record, I get several emails a day from publicists wanting me to hawk their clients’ events and books, and only the tiny handful that I believe my readers will care about actually make it to the blog.)

I have a full weekend coming up and am likely to be AWOL on Monday. I’ll see you Tuesday at the latest.

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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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First, why we should care about efficiency.

Second, why the efficiency criterion is sometimes incoherent, why those episodes of incoherence are fortunately rare, and why efficiency therefore remains, in most cases, a good guide to policy.

Third, why our instinctive recoil from cold-blooded efficiency is often misplaced.

We also revisited last week’s probability puzzle and reposted some past videos in a new improved format.

I am off in the woods and largely away from the Internet for the rest of the weekend, so I might be a little slow to see your comments, but rest assured that I’ll all get read—and that I’ll be back on Monday.

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The former brought us the honor of an extended visit from Uwe Reinhardt, who, as far as I can tell, objects not to the concept of efficiency or to its usefulness, but to its name. But any crusade to change a well-established technical term is, I think, doomed to failure.

Efficiency, of course, is only one of the normative criteria in the economist’s arsenal. I pointed, for example, to an earlier post where I’d outlined a toy framework for evaluating some of the normative claims made by one of Professor Reinhardt’s Princeton colleagues. That toy framework employs a utilitarian criterion that goes beyond efficiency. It evaluates policies on the basis of “what an amnesiac would prefer”, which is very different than a pure efficiency criterion. This kind of analysis is perfectly standard in economics, so any allegation that we fixate exclusively on efficiency is a bum rap.

On the other hand, some fixation on efficiency can be an extremely valuable exercise, for reasons that I hope this week’s post made clear.

Re the foundations of arithmetic, I posted to dismiss the view that the natural numbers are fictitious. As one commenter pointed out, this was largely an attack on a straw man, because almost nobody believes otherwise. Indeed it was. This was intended as an educational post, not a contentious one, and attacking straw men can be a very effective form of education. When I teach students about continuous functions, I ask them to imagine a hostile party who insists that the function f(x) = x is not continuous, and we talk about how you could most effectively convince him otherwise. The hostile party is imaginary, but there’s a lot to be learned from thinking about how you’d refute him.

We also speculated on the defining idea of the next decade and the ideal reading list for a course on how economists view the world.

And then there was the probability problem: A woman has two children, one of whom is a boy born on a Tuesday. What is the probability they’re both boys? Several commenters explained the answer very clearly. In case you haven’t read the comments and don’t want me to give away the answer, I’ll just say that it’s greater than 45% but less than 49%. See the comments on the original post for the reason why.

We’re coming up on a long weekend, and I’m taking Labor Day off. I’ll see you Tuesday.

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We moved on to the perils of interpreting data, in this case with regard to the ingredients of a happy marriage. Then a look back to what the world of 1985 thought would constitue a marvelous future; we seem to have met expectations pretty well. And finally, we came in a sense full circle — from lamenting those focus single-mindedly on energy costs to the exclusion of all else to lamenting those who fault others for failing to focus single-mindedly on one political issue to the exclusion of all others.

I’ll be back next week with some thoughts on why we should care about economic efficiency, a little more on the foundations of arithmetic, and some surprises.

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