You never know the shenanigans of a machine, and one million is largely enough for me until I die, or if science gives us the option to live forever I bet machines will do the work for us :-)
Edit: as I believe the machine can be wrong, I’d probably take A + B
I think the major unanswered question is how reliable do we think the machine is? 50%? 100%? I think the most interesting scenario is one where we are convinced that the machine actually predicts the future and always predicts correctly, so I’ll continue with that assumption in mind.
From one point of view, we have no reason not to take both boxes, since we can’t alter the machine’s prediction now, it’s already happened. I think however that this undermines my premise. Choosing both boxes only makes sense if we don’t actually believe the machine predicts the future.
One would be tempted to say "alright, then I will choose only box B, as the machine will have predicted that and I will get lots of money. If I were to choose both boxes, the machine would have predicted that too, and I would get much less money.
My argument is that both answers are wrong in a sneaky way: assuming an actual perfect predictor, my answer is box B only. However, the important part here is that this will not be, in fact, a choice. The result was already determined ahead of time, so I really only had that one option.
I meant, let’s imagine the machine predicted B and is wrong (because I take A+B). I would call that scenario „I have free will - no determinism.“ Then I will have 1.000.000.000 „only“. That’s a good result.
Except that's not the worst case. If the machine predicted you would pick A&B, then B contains nothing, so if you then only picked B (i.e. the machine's prediction was wrong), then you get zero. THAT'S the worst case. The question doesn't assume the machine's predictions are correct.
ka = the potential prize in box B; i.e. "k times larger than a"
p = the odds of a false positive. That is, the odds that you pick box B only and it got nothing, because dumb machine assumed that you’d pick A too.
n = the odds of a false negative. That is, the odds that you pick A+B and you get the prize in B, because the machine thought that you wouldn’t pick A.
So the output table for all your choices would be:
pick nothing: 0
pick A: a
pick B: (1-p)ka
pick A+B: a + nka
Alternative 4 supersedes 1 and 2, so the only real choice is between 3 (pick B) or 4 (pick A+B).
You should pick A+B if a + nka > (1-p)ka. This is a bit messy, so let’s say that the odds of a false positive are the same as the odds of a false negative; that is, n=p. So we can simplify the inequation into
a + nka > (1-n)ka // subbing “p” with "n"
1 + nk > (1-n)k // divided everything by a
1 + nk - (1-n)k > 0 // changed sides of a term
1 + 2nk -k > 0 // some cleaning
n > (k-1)/2k // isolating the junk constant
In OP’s example, k=1000, so n > (1000-1)/(2*1000) → n > 999/2000 → n > 49.95%.
So you should always pick B. And additionally, pick A if the odds that the machine is wrong are higher than 49.95%; otherwise just B.
Note that 49.95% is really close to 50% (a coin toss), so we’re actually dealing with a machine that can actually predict the future somewhat reliably, n should be way lower, so you’re probably better off picking B and ignoring A.
If I wanted to use logic, I'd say taking both A and B is the only way to have a guaranteed $1,000,000 outcome, because B only could get you money but also nothing.
But, if I choose B only, I'm sort of "forcing" the machine into that kind of prediction, right? I don't know about this experiment, but since your post says it's a paradox, I think that's how it works.
So my choice is B, the machine has predicted it and I get a nice $1,000,000,000.
I’d much rather take a sure million with a (slight?) chance of a bonus billion, versus an unknown chance at 0 or a billion. I could do plenty with a million that would significantly change my life for the better.
But I would probably do the opposite if A contained $1000 and B contained a potential million as in the original example. $1000 is a tolerable amount to risk missing out on.
Hehe thats why I think the original question of Box A being $1000 and Box B being a million was kinda boring, since $1000 is barely anything in today's world. 3 more zeroes does making things more interesting
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