Newcomb's Dialemma

Newcomb's paradox was thought up by a researcher named Newcomb; it was first explored and written up by Robert Nozick in the 1969 paper "Newcomb's Problem and Two principles of Choice".

The Situation

As narrated by an all knowing "predictor":

I am going to give you a choice. It is important to know that I really
pretty much know what you are going to do. I have been watching their whole
life and am additionally an immortal being; i've been doing this a long
time and always guess correctly. It's also important to know that I am
unbiased and don't care which decision you make, I have nothing to gain
either way.

Here are two boxes: a large and a small. The small has a 10 shekel coin
in it (show everybody). The large one may or may not have a thousand
shekels in it; you don't know. Your choice is to either take only the
large box or to take both the large and small boxes. The twist is that
I already knew which decision you will make and decided whether or not
to put the $1000 in the large box or not based on that knowledge.
If I knew you would "two box", then I left the large box empty. If I knew
you would "one box" then I filled it. 

Dominance Mindset

Regardless of what decision was made previously, and whether or not there is anything in the large box, the person is better off taking both boxes; either they will get just $10 (better than none) or $1010 (better than $1000). So two-box.

Trusting Mindset

The predictor is pretty much always right so we can just ignore the possibility that they are wrong. In this case, choosing to one-box implies that the Predictor knew you would and you get $1000; choosing to two-box implies that the predictor knew you would and you only get $10.

The predictor doesn't even have to be perfectly accurate; say they are 90%: If you one-box, your expected value is $900. If you two-box, your expected value is $110.

Discussion

It's disputed whether this is a paradox, and there are many deeper arguments that I don't have time to go into here. Ultimately, I am a one-boxer though this is something of a minority position.

Afterword

The person who taught me this paradox, Professor Augustin Rayo, a two-boxer, then had this to add. He was talking with his one-boxing friend and accused her of letting irrationality undermine her logic: she is so optimistic that if a statement S is unprovable, but it would be nicer if S was true than false, then she pretens that S is proven. So basically, even though there is no rationalization, she will accept a statement "just because it would be nice", and this isn't how logic works. To which she replied "but wouldn't it be nice if it was?".