This is a post about hot hands in basketball. But first, some relevant history:

The single most controversial topic ever broached here on *The Big Questions* was not Obamacare, or tax policy, or the advantages of genocide, or the policy treatment of psychic harms. It was this:

The answer, of course, is that you can’t know for sure, because (for example) 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?

The “official” answer — the answer, for example, that Google was apparently looking for when they posed this as an interview question — is that no stopping rule can change the fact that each birth has a 50% chance of being either male or female. Therefore the expected fraction of girls in the population is 50%.

That turns out to be wrong. It’s true that no stopping rule can change the fact that each birth has a 50% chance of being either male or female. From this it **does** follow that the expected number of girls is equal to the expected number of boys. But it does **not** follow that the expected fraction of girls in the population is 50%. Instead, that expected fraction depends on the country size, but is always **less** than 50%.

If you don’t see why, I encourage you to browse the archive of relevant blog posts. If you still don’t get it, I encourage you to keep browsing. Whatever your objections might be, you’ll find them addressed somewhere in the archive. I’m not interested in relitigating this. I will, however, happily renew my offer to take $5000 bets on the matter, on the terms described here. Last time around, all takers changed their minds before putting any money on the table.

Now let’s get to the hot hands.

Thomas Gilovich, Robert Vallone and Amos Tversky (I’ll call them G, V and T) made quite a splash back in 1985, with a claimed debunking of the hot hand “myth”. According to the authors, a player who has just made his foul shot is thereby rendered neither more nor less likely to make the next one.

Now a new paper by Joshua Miller and Adam Sanjurjo (call them M and S) claims that G, V and T drew the wrong conclusions because they made **exactly the same mistake** that leads to the “official” but wrong answer to the boy/girl problem.

I’m particularly delighted by this, because back in 2010, when we were debating the boy/girl problem on this blog, a number of people defended the official answer vigorously, then realized they were wrong, and retreated to a position of “but it doesn’t matter for anything anyhow”. My experience in the classroom tells me that if you want to convince people that an idea **does** matter, the most effective strategy is to show them an application to sports. Where were M and S when I needed them five years ago?

Anyway—here’s why the two problems are the same problem. One of the several tests in the GVT paper consisted of placing ballplayers a distance from the basket where they could be expected to make just 50% of their shots. (This distance was, of course, different for every player). The players then made several attempts, and the researchers asked whether the observed sequence of hits and misses looked statistically identical to what you’d get from a sequence of fair coin flips. If so, then because we all know there’s no such thing as a hot hand in coin flips, we can conclude that there’s no such thing as a hot hand in basketball (or at least in this particular experimental setup).

So — what does a series of, say four coin flips (corresponding to four foul shots) look like? It looks like one of sixteen things: HHHH, HHHT, etc. If you’re trying to flip heads, a “hot hand” should mean that the sequence HH comes up more often than the sequence HT. If you **don’t** have a hot hand, then all sixteen sequences should be equally likely, and the frequencies of HH and HT are as follows:

Sure enough, across the sixteen sequences, HH and HT occur equally often — a total of 12 times each. So if you put the players at their foul lines, let each one take four shots, count the HHs and the HTs (where “H” now means “made it” and T now means “missed”), and if there are no hot hands, you should see about the same number of each.

But G,V and T did something a little different. For each player they computed the **ratio** HH/(HH+HT) — and, because they fell prey to **exactly the same mistake** that trips up so many would-be solvers of the boy-girl problem, they expected that for the average player, this should come out to about 50%. That’s not right. In fact, as the chart shows, it comes out to about 40.5%.

When G,V and T observed an average success ratio of about 50%, they said “Yup, that’s what we expected all along. No evidence of a hot hand here”. What they **should** have said was: “Wow! We expected 40.5%, not 50%. That’s a big difference. Only a hot hand could account for this.”

(Caveat: This was only one of the tests that G,V and T performed. I haven’t thought about which of their other tests were or were not affected by this slip up.)

Of course, if *The Big Questions* had been around in 1985, G, V and T would surely have been avid readers who had worked out the boy/girl puzzle and learned to avoid this mistake. Science marches on.

The frontispiece from Differentiable Germs and Catastrophes by Theodore Brocker:

There are about a million reasons why I hate my iPhone, but this one pretty much sums it all up.

On my phone, I’ve got quite a few files that were not downloaded from any of my other devices. These include pictures I’ve taken with the phone itself, pdfs I’ve downloaded through the phone’s browser, etc.

Of course, I’d like to have backups of all these files. And of course Apple makes this as difficult as possible by pushing me to use its abysmal iTunes software for creating the backup.

Now here is what iTunes does: I have photo files with names like IMG_0840.jpg — which, if not terribly descriptive, is at least immediately recognizable as a photo. I have pdfs with names like Dirac.QuantumMechanics.pdf, which is a nice, easily recognizable name. I download everything to my computer via iTunes, and here is a partial directory listing of what I get:

Yes, Dirac.QuantumMechanics.pdf has had its name changed to ffcc4678d5e0ab310f88d38926a4922e2b70ed0b. Or perhaps to ffff64402524baf7d1e3bc49780379e69fe69ed0. There is **no way to tell which file is which** (they are not even arranged in directories that mirror the directory structure on the iPhone) and **not even any way to tell which are jpgs, which are mp3s, which are pdfs, etc.**.

I can, of course, try randomly renaming these files things like “tryit.jpg”, “tryit.mp3″, “tryit.pdf”, etc., and see which ones open. I can do this with each of several hundred files separately. Or alternatively, I can have a life.

So in order to preserve **usable** backups of my files, I have to download them all over again, bypassing iTunes and hacking around a while to get my computer to believe that the iPhone is a hard drive (which Apple has gone out of its way to make as difficult as possible) and then copy the files over into another directory, which at least preserves their names, while creating exact duplicates of all the files you just saw listed above (and the hundreds of others like them) and thereby taking up twice as much space on my computer hard drive as should be necessary.

(And no, I can’t at this point erase the backups with the stupid names, because if I ever need to restore the state of my phone, it’s rigged so the phone won’t work properly unless you restore from the official Apple backup,)

You should get an iPhone only if either a) you never expect to move files back and forth between your phone and your computer, and in particular don’t care if all the files on your phone are in jeopardy of being lost because there is no decent way to back them up, or b) you are a very particular sort of masochist.

I’m sure this all has something to do with Apple’s paranoia that somebody might steal an mp3 file from the iTunes store, or something like that. I’m sure that if I fully understood the issues, I’d have more sympathy for them than I do. But the bottom line is that from the user’s viewpoint, the iPhone sucks, and if Apple is not evil, then it might as well be.

]]>In a bid for ongoing taxpayer support, Planned Parenthood president Cecile Richards will be appearing before Congress today. It’s reported that as part of her testimony, she will admit that only 1 percent of Planned Parenthood’s affiliates currently harvest fetal tissue, and that even those affiliates charge only modest fees of $60 per tissue specimen.

Which raises the question: Why should we give money to an organization that has access to a valuable resource but can’t be bothered to sell it to the highest bidder?

When your brother-in-law is out of work, you might be inclined to help him out. When your brother-in-law is out of work, deluged with job offers, and refusing even to consider them, you’ll probably be less inclined. Planned Parenthood is that brother-in-law.

This isn’t about your stance on abortion. Whether you’re pro-life, pro-choice, or pro-anything-in-between, surely we can all agree that valuable resources should not be lightly discarded, especially by those who are in the midst of pleading poverty, and doubly especially by those who are in the midst of pleading poverty as a rationale for conscripting other people’s money. This is so whether or not those valuable resources are byproducts of an event that you consider regrettable, or even appalling.

Of course those who recoil from abortion might want to prohibit the sale of fetal tissue in order to limit the incentive to abort. But that’s an argument about what the law should be going forward, not about what Planned Parenthood should be doing today.

Anyway, those who recoil from abortion don’t to need new reasons to oppose funding Planned Parenthood. Meanwhile, those who **don’t** recoil from abortion — or those who, at least, want abortion to be easily available — can still recoil from funding an organization that throws valuable tissue in the trash. Planned Parenthood has given every one of us a lot to recoil from.

(Related — and highly recommended — reading here.)

]]>For an upcoming Festschrift, I was recently asked to write an account of Dee (then Don) McCloskey‘s years as a brilliant teacher at the University of Chicago, her influence on a generation of economists, and my own enormous debts to her. This was a great pleasure to write. A draft is here.

Pope Francis is coming to New York, and Cardinal Timothy Dolan is disturbed about ticket-scalping:

“Tickets for events with Pope Francis are distributed free [via lottery] for a reason — to enable as many New Yorkers as possible, including those of modest means, to be able to participate in the Holy Father’s visit to New York,” Cardinal Dolan, the archbishop of New York, said in a statement. “To attempt to resell the tickets and profit from his time in New York goes against everything Pope Francis stands for.”

So according to Cardinal Dolan, “everything Pope Francis stands for” consists of the proposition that for New Yorkers of modest means, nothing should take precedence over turning out to see Pope Francis — not groceries, not medicine, not car repairs, not any of the other things that people can buy with the proceeds from selling their tickets.

I doubt that Pope Francis is quite as egomaniacal as the Cardinal paints him. But apparently the Cardinal himself would rather see poor people cheering for the Pope than improving their lives.

]]>My sister snapped this picture of my Dad and me sitting on a couch:

No, this wasn’t posed. It’s just how we happened to be sitting.

To review the bidding:

Two days ago I posed a puzzle about 10 pirates dividing 100 coins.

Yesterday, I presented what appears to be an airtight argument that the coins must be divided 96-0-1-0-1-0-1-0-1-0.

But yesterday I also told you that the “airtight argument” is in fact not airtight, and that other outcomes are possible. I challenged you to find another possible outcome, and to pinpoint the gap in the “airtight argument”.

Our commenter Xan rose to the occasion. (Incidentally, his website looks pretty interesting.) Here’s his solution:

The most ferocious pirate proposes to take all the coins, leaving everyone else with nothing. The other nine all want this proposal to fail. Nevertheless, they all vote yes and the proposal is accepted. |

Note that this is perfectly consistent with rational behavior. No matter how much you want the proposal to fail, as long as everyone else is voting “yes”, your vote can’t affect on the outcome, so voting “yes” can’t hurt you. (Neither, of course, can voting “no”.)

Another possible outcome is that one of the pirates (perhaps even the most ferocious pirate!) votes no, while the rest vote yes. Or three vote no while seven vote yes.

In each of these cases, no single voter can change the outcome so there’s no particular reason for anyone to change his vote. In other words, everyone is being perfectly rational, taking as given the choices made by everyone else. In the jargon of economics, Xan’s solution is a perfectly good Nash equilibrium.

Likewise, if the 2016 election comes down to Trump versus Sanders, and if every single voter prefers Sanders to Trump, it’s perfectly consistent with rational behavior for Trump to get 100% of the votes. If everyone else is voting for Trump anyway, you might as well vote for him too. (You also, of course, might as well vote for Sanders.)

That shows that yesterday’s conclusion was wrong (or at least incomplete). The next question is: What is the flaw in the logic that led to that conclusion?

Answer: The argument (as you’ll see if you read it carefully) makes the implicit (and quite unwarranted) assumption that people always vote for their preferred outcomes. **Given** that assumption, the argument really **is** airtight (despite some of the skepticism that was expressed in comments.) If pirates always vote for their preferred outcomes, then the only possible Nash equilibrium is indeed 96-0-1-0-1-0-1-0-1-0.

But if all we assume is that pirates behave rationally, then we can’t predict much about how they’ll behave in cases where their votes don’t matter. And that’s a lot of cases.

A historical note: I assigned this problem to classes for years, expecting (and usually getting) the 96-0-1-0-1-0-1-0-1-0 answer. One day in class, while I was reviewing that “correct” answer, my student Matt Wampler-Doty raised his hand to point out the plethora of alternative solutions and the unjustified assumption in the “official” analysis. I was sure he was wrong, and made some inane attempt to dismiss him, but to his credit, he persisted until I got the point. He made my day.

]]>Today I’ll offer the “official” solution to yesterday’s puzzle — that is, the solution that Google has apparently expected from its job candidates. This is also the solution I gave when I first saw the puzzle, and the solution I usually get from my best students, and the solution given yesterday by some astute commenters.

But this solution has a gaping hole in it. Can you find it?

We have, as you recall, 10 ferocious pirates in possession of 100 gold coins. The most ferocious pirate proposes a division, which is put up for a vote. If half or more of the pirates approve, the plan is implemented and the game is over. Otherwise, the most ferocious pirate is thrown overboard and the game begins again, with the second most ferocious pirate proposing a divison. A pirate’s first priority is to stay alive, his second is to amass as many gold coins as possible, and his third is to see others thrown overboard.

Let’s start with some simpler problems:

**Problem 1.** Suppose there’s just **one** ferocious pirate. Call him James. Obviously, James allocates 100 coins to himself, votes for his own plan, and wins.

**Problem 2:** Suppose there are **two** ferocious pirates — Igor and James, with Igor the most ferocious. Now Igor has half the votes and can therefore impose any plan he wants to. He therefore allocates 100 coins to himself, votes for his own plan, and wins. James gets nothing.

**Problem 3** adds a third pirate — Howard, who is more ferocious even than Igor. If Howard wants to avoid being thrown overboard, he needs at least two votes for his plan. He’s got his own vote, so he’s got to buy a vote from either Igor or James — both of whom realize that if Harold goes overboard, they’ll be playing the game from Problem 2. Igor likes that game so much that he can’t be bought. James, on the other hand, knows he gets **zero** in that game, and can therefore be bought for a single coin. So Howard’s proposal is: 99 for Howard, 0 for Igor, 1 for James. Howard and James vote yes and the plan is approved.

**Problem 4:** Now let’s add a fourth pirate, the even more ferocious George. To pass a plan, George needs two votes — his own and someone else’s. If his plan fails, we’ll be playing Problem 3, where Howard gets 99 coins, Igor gets 0, and James gets 1. Therefore Howard won’t sell his vote for less than 100 coins, while Igor will sell for just 1, and James will sell for 2. Igor’s vote is the cheapest, so that’s the one George buys. So George’s proposal is: 99 for George, 0 for Howard, 1 for Igor, 0 for James. George and Igor vote yes and the plan is approved.

**Problem 5:** If there’s a fifth and even more ferocious pirate — call him Freddy — then Freddy needs three out of five votes to pass his plan. In addition to his own vote, he’lll buy the two cheapest votes available. In view of what happens in Problem 4, George’s price is 100, Howard’s is 1, Igor’s is 2, and James’s is 1. Therefore Freddy buys Howard’s and James’s votes for 1 coin each. Freddy gets 98, George gets 0, Howard gets 1, Igor gets 0, James gets 1.

And so on (you can fill in Problems 6, 7, 8 and 9 yourself!) until we reach:

**Problem 10:** With 10 pirates, Arlo (the most ferocious) gets 96, Bob gets 0, Charlie gets 1, Dave gets 0, Eeyore gets 1, Freddy gets 0, George gets 1, Howard gets 0, Igor gets 1 and James gets 0. Arlo, Charlie, Eeyore, George and Igor all vote yes, because if they vote no, Arlo will be thrown overboard and Bob will propose a division in which Arlo is thrown overboard while Charlie, Eeyore, George and Igor all get nothing.

Conclusion: The only possible outcome is 96-0-1-0-1-0-1-0-1-0. You’ll find the same solution, with pretty much the same argument, in yesterday’s comments. But the conclusion is wrong.

It’s **partly** right, in the sense that that 96-0-1-0-1-0-1-0-1-0 is **one** possible outcome. But the argument above seems to prove it’s the **only** possible outcome. That’s wrong. Many other outcomes are also possible.

So here are today’s puzzles:

1) What other outcomes are possible?

2) What’s wrong with the above argument, which appears to rule those outcomes out?

I’ll follow up later in the week.

]]>I’m not sure where this problem originated. I heard it first from John Conley, and have often assigned it to my classes. Google has used it to weed out job candidates. The answer that Google expects is the same answer I gave John Conley, and the answer I usually get from my best students. That answer is wrong. (Long time readers might feel a sense of deja vu.)

Can you get it right?

Here’s the problem:

Ten ferocious pirates have come into possession of 100 gold coins, which they must divide among themselves. The procedure for dividing them is as follows:

- The most ferocious pirate proposes a divison (such as: “I’ll take 50, the second most ferocious pirate gets 10, and everyone else gets 5 apiece”.)
- The pirates vote on whether to accept the division. As long as half or more of the pirates vote yes, the division is accepted, the coins are divided, and the game is over.
- If, instead, fewer than half the pirates vote yes, the most ferocious pirate is thrown overboard. The second most ferocious pirate now becomes the most ferocious pirate and we return to Step One.
You should also know that each pirate enjoys seeing other pirates thrown overboard, but not as much as he enjoys receiving one additional coin. So, for example, any pirate, contemplating the following three outcomes, prefers A to B and B to C:

- I get six coins and another pirate is thrown overboard.
- I get six coins and no other pirate is thrown overboard.
- I get five coins and another pirate is thrown overboard.
Finally, you should know that each pirate’s

lastchoice is being thrown overboard.Each pirate devises a strategy that is designed to make himself as happy as possible, taking the other pirates’ strategies as given.

What happens?

I’ll reveal the correct answer a little later in the week.

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