'Newcomb's Paradox': A thought experiment for the ages

This famous thought experiment has inspired a myriad of mathematical and philosophical discussion since its conception in the 1960s

Lets imagine there’s a supercomputer that can predict the future, and that in front of you there are 2 boxes. Box 1 always contains £1000, and box 2 contains either nothing or a £1,000,000 pounds, the value of which is decided based on whether the computer predicts you’ll choose. If you choose boxes 1 and 2, then box 1 will contain nothing, if you choose only box two then box 2 will contain £1,000,000.

Referred to as 'Newcomb's Paradox' or 'Newcomb's Problem', this particular concept was developed by theoretical physicist and professor, William Newcomb in 1960 during his tenure at the University of California. Now, this is a experiment which has prompted an array of philosophical debates regarding free will and simulation. While these have been discussed by hundreds if not thousands before myself, lets take a more probabilistic approach to the conundrum and attempt to calculate a set of possible outcomes for those who wish to participate; four to be exact.

Case One: Our computer is 100% accurate.

If the prediction computer is 100% accurate at predicting the future then you should always choose box 2, this will give an average return of, £1,000,000. If you were to choose both boxes then the return would be £1000 every single time.

Case Two: Our computer is 90% accurate.

Well if we take box 2 every time then we have an average expected return of £900,000, which means that in 1 out of every 10 occurrences the box will be empty. Now, if we were to take both boxes in this scenario then we will get an expected return of, £101000 which isn’t terrible however it is still clear that taking  box 2 is a better decision of our 2 options 

Case Three: The computer is 50% accurate.

Now let's say that our supercomputer salesman was slightly overselling the success rate. If we were to always take box 2 then our expected return would be £500,000, with a 50% chance of getting anything from the boxes. On the other hand in taking boxes 1 and 2 in this scenario our expected returns would be £501000, this is the first point where based on returns it is worth taking boxes 1 and 2

Case Four: The computer is 0% accurate

If we can guarantee the computer will always be incorrect, then expected returns of choosing only box 2 will be £0 and the expected returns of boxes 1 and 2 will be £1,000,1000.