The Logician's Swindle

What makes a puzzle annoying ? When is solving a problem rewarding, and when is finding out the answer just frustrating ? If we could answer this, we might get a long way towards making the world a happier place. Getting people to actually enjoy solving problems, rather than being pissed off at their opponents for discovering a flaw in their arguments, would surely benefit political discourse enormously.

I don't propose to try and answer all of this today. Instead, what I can do is address one particular aspect of the problem. I say that at least one major cause of puzzles being annoying rather than enjoyable is when you've been outright cheated, and that this happens far more often than it should.

Specifically, consider Newcomb's Paradox as described on Veritasium. The video begins :

You walk into a room, and there's a supercomputer and two boxes on the table. One box is open, and it's got $1,000 in it. There's no trick. You know it's $1,000. The other box is a mystery box, you can't see inside.

Now, the supercomputer says you can either take both boxes, that is the mystery box and the $1,000, or you can just take the mystery box.

So, what's in that mystery box?

Well, the supercomputer tells you that before you walked into the room, it made a prediction about your choice. If the supercomputer predicted you would just take the mystery box and you'd leave the $1,000 on the table, well, then it put $1 million into the mystery box. But if the supercomputer predicted that you would take both boxes, then it put nothing in the mystery box.

The supercomputer made its prediction before you knew about the problem and it has already set up the boxes. It's not trying to trick you, it's not trying to deprive you of any money. Its only goal is to make the correct prediction.

So, what do you do? Do you take both boxes or do you just take the mystery box?

Don't worry about how the supercomputer is making its prediction. Instead of a computer, you could think of it as a super intelligent alien, a cunning demon, or even a team of the world's best psychologists. It really doesn't matter who or what is making the prediction. All you need to know is that they are extremely accurate and that they made that prediction before you walked into the room.

I highlight certain parts because they feel crucial. To me, this is saying very explicitly, "don't think about this aspect of the problem, it's not important at all". Were this not so, I would otherwise object to how such a thing could be possible, and the details would certainly matter : was the machine running over a diverse sample of people, or was there something particular about them that helped its accuracy ? But no, this apparently isn't important, so whatever misgivings I have about free will and suchlike, I willingly surrender for the purpose of the test. I put them aside, still fully expecting to be fooled (I suck at logical puzzles) but in some other way.

Having made that assumption, the answer is obvious. If the machine is essentially always accurate, I take one box. It knows, magically, that this box will contain a million dollars, and I walk out happy and rich and in search for a bank offering a good exchange rate to a proper currency. 

But later in the video we get :

Here's how I think about the problem in a way that actually makes sense. You know that the supercomputer has already set up the boxes, so whatever I decide to do now, it doesn't change whether there's zero or $1 million in that mystery box, and that gives us four possible options that I've written down here.

If there is $0 in a mystery box, then I could one-box and get $0 or I could two-box and get $1,000, but there could also be $1 million in a mystery box. And in that case, I would get $1 million if I one-box or I would get $1,001,000 if I two-box. So, I'm always better off by picking both boxes.

Rubbish. Complete twaddle. You just told us that the machine is accurate and we shouldn't factor this in to our calculations, but in this way of thinking you cannot possibly ignore how the machine works. This is not even self-consistent ! By saying that the machine is essentially perfectly accurate, you've eliminated the very possibility of $1,001,000. That can only happen if the machine actually is inaccurate in some cases, which to my mind you've all but told us directly to discount.

This, then is a swindle, and one common to various logical puzzles. "Don't think about this aspect of the problem", they say, only later to say, "Hah ! You should have thought about this aspect of the problem after all, you fool !". Right, so you expect me to think you're a liar ? How is that a fair test ?

The rest of the video is a perfectly decent discussion of free will etc. (Veritasium is one of my favourite YouTube channels), but the poor description from the outset makes the whole thing a mess. Having been told that accuracy was not an issue, I expect something else I've overlooked to come into play. Naturally I overlooked determinism and all that because you told me to overlook it. The pettiness of it all annoys me quite intensely.

Don't worry, I'm not going down the free will avenue with this post. Rather, I just want to briefly outline that this kind of swindle is common to logic problems, and is itself one particular expression of a more general reason they're so often very irritating.

The closest similarity is surely the Monty Hall problem (the one with the prize goats). That one always confused the heck out of me because people never properly explained that I should have been paying crucial attention to the host's knowledge, not how many goats there are or how many doors. But any logic puzzle can suffer if you're not properly informed about what the key aspect of the problem is, or worse, if you're actively told to ignore it.

Not that framing doesn't sometimes reveal something very interesting. Wason's selection is fascinating in showing how the same people can have much more difficulty solving the same task if it's described slightly differently – especially so when the alternative form is nothing they wouldn't also be familiar with. But there, the whole point is to study psychology. No deception is employed, no swindle pulls the solution out from beneath the solver's feet. The facts are laid bare and it presents a straightforward yet surprising challenge to many people who take it. No, framing is only annoying when it's done to deliberately thwart the participant. 

There's also a common tendency for the puzzle-setter to declare the rational solution from authority, saying "this is obviously the correct solution because the alternative doesn't make sense to me". A classic example concerns people refusing small amounts of compensation when they would normally expect a much bigger payout. Time and time again we hear people declaring that accepting the small offer would be rational since they come out with a net cash gain. But to any sensible person there are a multitude of reasons why this would be an extremely foolish thing to do : accepting the initial offer may deny them any chance at the larger amount; they may simply feel insulted and disrespected, and responding to such behaviour is essentially letting the bully get away with it. It is only rational in an incredibly narrow and naive economic sense, and more broadly simply isn't rational at all*.

* Veritasium does this with a unique peculiarity, openly acknowledging that the "irrational" decision of choosing one box is the more profitable. I find this is going deep into "what's wrong with you ?" territory.

Again, this is a sort of swindle, denying the opposing argument by forbidding debate rather than engaging with it on an equal footing. You thought things were going to be fair and above-board only to find out that they were anything but, that the answer had already been decided without you.

Another similarity is the pettiness. Veritasium didn't have to pull the rug out from under the viewer's feet any more than anyone has to accept that getting a smaller payout is somehow rational. 

Very occasionally, I've run public surveys to help me with my own research. I've tried to ensure the wording was extremely careful, including omitting details when this would bias the result. For example I once ran a public poll on how many groups of points – galaxies – people could see in a plot, deliberately not telling them what they were looking at. Some people objected that there wasn't enough information (e.g. what sort of scales they should be considering), and I sympathise that they might find this annoying. But for me this was the whole point, to gauge what people's natural reactions were : I wanted to know if they would instinctively identify the same groupings that appeared natural and obvious to me (most of them did, as it turned out). I needed to know if my additional knowledge was biasing me, or if the groupings I identified would be readily visible without this extra information. 

The point here is that there's absolutely no reason for misdirection. It's perfectly possible to account for this in a way that will give you a meaningful result to the question you're asking. Sometimes, this can only become apparent after the fact, but in those cases the participant should feel relieved, not annoyed. Annoyance only happens when the misdirection was unnecessary. 

A second personal example : group meetings back in my PhD days. These served the valuable purpose of getting the students used to dealing with tough questions. But they also turned the experience into a weekly grilling that made the whole thing quite intensely annoying... instead of having an enjoyable, low-stakes discussion about science, we had to deal with supervisors being deliberately over-critical. That we all knew full well what was going on didn't help in the least. It would have been fine if such sessions had been clearly demarcated and set aside as such, with regular meetings more about science for its own sake. Trying to pretend this was how scientific discussion should happen, though, was just unfair.

Again, there was no reason for the misdirection. This too was a sort of swindle. Oh, you think you're here to discuss your work ? You thought I was being harsh because I wanted to be ? Hah hah, fooled you ! The idea that maybe they could have just not done that was never raised.

On an grander scale, problems with the alternatives to dark matter. This too feels like something of a swindle : proponents often raise objections to dark matter which are based entirely on the properties of the ordinary matter we can see. They make highly dubious inferences about the necessary connection to the dark matter they're trying to demonstrate doesn't exist, saying that the lack of a naively-expected correlation proves it can't exist. Some of these problems can become obvious, but sometimes it's worth spelling this out at the high level because it's all too easy to lose sight of the forest for the trees. Once you start questioning the underlying assumption and realising that maybe the connection isn't so direct after all, often the whole thing falls apart.

And in other arenas too we find possible swindles. As I've covered before, thought experiments become extremely annoying when changing a small detail would profoundly alter the result but the instigator refuses to consider any variation : no, you must focus on this aspect of the problem because I said so, even if my scenario is actually bunk. Just like insisting someone should accept a miniscule payout, it's disrespectful not to think the other person's opinion might have some value. 

Likewise with analogies. An indirect analogy can be extremely powerful when the relevant aspect is sufficiently similar to its comparison subject, becoming thought-provoking in both its similarities and in its minor, extraneous differences. When an analogy is intended to be direct, though, the seemingly-extraneous details can become crucial, so expecting people to shut up and ignore them is not realistic. It's extremely difficult to focus on the "relevant" bit (usually declared by authority) when there are obvious deficiencies in the whole thing. Conversely, it does no good to pretend similarities don't exist when they do, or to overlook them on grounds which are actually minor details or only quantitative differences.

All this sets out some conditions for when puzzles becoming annoying, and gives us a rough working definition : The Logician's Swindle is the use of unnecessary misdirection from a position of unjustified authority.

This is similar to but not quite the same as the Magician's Choice. In the latter, we know we're being denied crucial information, misdirected, and otherwise deceived. We go in with eyes open knowing we'll almost certainly be tricked and often paying for the privilege of suspension of disbelief. We know we won't be able to solve the problem and we enjoy our failed attempts to work out what's going on.

The Logician's Swindle is altogether nastier. Here, we're supposed to have all the information we need to reach the "correct" conclusion, but we find only afterwards that actually we don't – with the swindler often denying this for the sake of making us look foolish. And the conclusion itself may be open to dispute but the proponent argues from a completely artificial authority that it isn't. Worst of all is that "mistakes" can (though do not always actually) carry real-world consequences. In short, it's a scam : a discussion that should be in good faith which actually isn't.

And that's why I hate logic puzzles.