My Understanding Of Three Puzzling Paradoxes
by Keep_Relentless on December 21st, 2020, 11:30 pm
Hi everyone :)
I had written this in correspondence with a friend recently and now I will paste it here for the interest of anybody who has seriously pondered these problems. If anybody is interested in a detailed discussion of a particular problem we might take it to another thread. Otherwise, I hope you find this post somewhat thought-provoking :)
"Newcomb's problem: You are presented with two boxes, box A and box B. Inside box A is $10,000 and inside box B is either $1,000,000 or nothing. You can choose to take either both boxes or just box B. A fortune-teller has predicted your choice, and if they predicted you would take only box B then they have placed $1,000,000 inside, and if they have predicted you would take both boxes, they have placed nothing in box B. The fortune-teller has a 100% accuracy rate and has done this many times. Yet the amount in box B is already fixed. So, do you take one box, or do you take both boxes?
Argh this is such a beautiful problem. And it divides people almost evenly, even professional philosophers.
The surprise hanging paradox: A judge tells a convicted prisoner that he will be hanged at noon next week (during the week), but that the hanging will be a surprise, he will not know the particular date until it happens. So the prisoner thinks to himself, well, it cannot be Friday then, because by Friday morning it would not then be a surprise. But if it's not going to be Friday, it also can't be Thursday, for the same reason, come Thursday morning, if it's not Friday, it won't be a surprise. But then it can't be Wednesday either. Or Tuesday. Or Monday. So the prisoner concludes it won't happen at all. But then it does happen, on Wednesday, and guess what? It's a surprise.
This is a lovely mess of a problem but I do believe it can be understood precisely.
The St. Petersburg paradox: This one is actually my favourite. So you are offered a game to play. Someone will flip a coin. If it comes up tails, you win $2, but if heads, your win is doubled to $4, and the coin is flipped again. If it comes up heads again, your win is doubled to $8, and the coin is flipped again. If it comes up heads again, your win is doubled to $16 and another flip is made. This happens again and again doubling as many times as it comes up heads, and the first time it hits tails, the game ends and you take home the win. So you have a 1/2 chance of getting $2, a 1/4 chance of getting $4, a 1/8 chance of getting $8, and so on.
To calculate how much this game is worth, you just add all possibilities together with their appropriate weight. So you get 1/2 times 2, plus a 1/4 times 4, plus a 1/8 times 8... So you get 1+1+1+1+1... forever, indicating this game is worth infinity.
The question is, why is this such an unrealistic result? You wouldn't sell your house to play this game. Why not? Where's the problem with the logic?"
"Firstly Newcomb's problem is so brilliant because either answer may be perfectly logical depending on your perspective. The people who say take one box (which is what I would do too, by the way) look at the fact that taking one box always nets a million dollars and taking two boxes always nets ten thousand. People who take both boxes, however, look at the fact that the money has already been placed in box B, so the choice is between the money in box B, and the money in box B plus an extra ten thousand.
I personally think there is something suspect in the second way of reasoning, in that if you do choose to take both boxes, that choice itself implies that there will be nothing in box B. Whereas if you choose to take one box, that choice implies there will be a million dollars inside.
I look at it by analogy of a golf swing. Just by analysing the motion of a golfer's swing after he hit the ball, you can infer whether it was a good hit or not - even though what happens with a golfer's swing after the hit has no causal effect on the path of the golf ball. So a good swing post-hit doesn't cause a good hit (which has already happened), but it does imply one. Similarly, a choice to take one box doesn't cause the million dollars to appear, but it does imply it.
I think I know what's going on with the surprise hanging, it's just quite confusing. Firstly, the judge says two things: 1. You will be hanged (on a particular day of the week), and 2. It will definitely come as a surprise to you. Note that after the prisoner followed his line of reasoning (which we'll get to), he decided to throw out claim #1, instead of claim #2. This was rather arbitrary of him. He could have concluded "Ok, I have a 20% chance of being hung on Friday, which means I have a 20% chance that it will not be a surprise, despite what the judge said". Anyway, he decided to cling onto claim #2, so he reasoned that it can't be Friday. Let's follow him and say hypothetically claim #2 must be correct. If it can't be Friday, and it must be a surprise, it also can't be Thursday. Do you see why? It's called (in game theory) iterative deletion: once you delete certain options you come up with a new situation, and then you may be able to delete more, and then more. The reason it couldn't be Thursday is because he would know 100% by Thursday morning that it was going to be Thursday (if it couldn't be Friday). Then it wouldn't be a surprise. So he eliminates Thursday, but now he has the same argument for Wednesday (come Wednesday morning, if he knew for sure it wouldn't be Thursday or Friday, he would know 100% that it must be Wednesday, and it would not be a surprise). And the same argument for Tuesday, and finally for Monday.
So where did he go wrong? In my estimation, by eliminating any of the days at all, and taking the judge's second claim too seriously. It's pretty neat that the fact that it was a surprise that it would happen at all was only an outcome of believing that it must be a surprise.
Shame you had no thoughts on the St. Petersburg paradox because it's quite neat. Basically this is a game where your chance of winning hundreds of thousands is one in hundreds of thousands, winning billions one in billions, and so on, and your average (median) win will be a few bucks, but the math says the game is worth infinity so you should be willing to bet literally any amount to play. And in terms of pure math, this is an unassailable conclusion, you should be willing to bet any amount to play if you are not risk-averse (which you probably should be).
But why in the real world wouldn't we judge this game to be worth all that much? Do you see why it doesn't seem worth that much? Double your money as many times as you get heads? You may get an astronomical amount of money, but your odds of that are astronomically small. And what if you do? Let's say you get mind-bogglingly lucky and you get 50 heads in a row for somewhere around a quadrillion dollars. What value does a quadrillion dollars have, over a billion? And who's going to pay you a quadrillion dolllars? So we see that for very high numbers the game completely loses its meaning. Would you rather win 50 nonillion dollars or 1,000 decillion dollars? With the mathematical assumption that money has a linear value, one should be WAY better than the other, but in reality that's not the case.
In fact money doesn't have a linear value to people, and that's the other assumption I want to attack. Studies show that money has a logarithmic value to people, that is, the more you have, the less valuable it is to have more. If we replace the calculation of 1/2 times 2 + 1/4 times 4 + 1/8 times 8 ... with 1/2 times log(2) + 1/4 times log(4) + 1/8 times log(8) etc... we get an infinite sum of 2log(2), taking the inverse log of that gives us an expected value in dollar terms of $4 for this game, which aligns much better with most people's intuition. Other variants of this game can be handled similarly, I calculated one to be realistically worth somewhere around $50,000."