I feel like math and english are often seen by people on two opposite ends on a spectrum of subjects. On one side you have the “strict” and “rigorous” math, where there is only “one right answer.” On the other hand, you have “free” English Literature, where any answer can be correct “as long as you can justify it.”
But this had always seemed just plain incorrect to me. Literature is NOT a wishy-washy dream subject where every answer can be right. If you think J.B Priestley’s An Inspector Calls is a pro-capitalism play, you’re wrong. If you think Steinbeck’s Of Mice and Men is a celebration of opportunity and freedom in America, you’re just plain wrong. There’s no way about it, it’s just wholly incorrect.
And math questions don’t always have only one right answer. Quadratics have 2 solutions, and trigonometric identities can have infinite solutions within an unbounded domain. And in some, strange, strange cases, we can have 2 solutions that are contradictory. The existence of either one should disprove the other, but both seem to be true. This is the crux at the heart of Newcomb’s Paradox.
Now, we consider a scenario. Two closed boxes lay in front of you, labelled A and B. Now, you can either take only Box B or both boxes A and B. You can’t choose only Box A because of… reasons. Anyways, overseeing your choice is a master psychologist and human reader. Let’s call this guy Kevin because… you know… reasons. He tries to predict your decision, and allocates money into the boxes accordingly before you actually make the choice.
Now, before anything begins Box A contains a $1,000 cheque. Sweet, a brand new PC, I hear you think to yourself. But here’s the thing. If Kevin predicts you’ll take both Box A and B and guarantee yourself that sweet grand, he’ll leave Box B empty. However, if he predicts you’ll take only Box B, he’ll slip in a cool $1,000,000 cheque. Yeah, forget the brand new PC, you can go buy a brand new MANSION with that kind of money. And Kevin is pretty smart, so his predictions are right 90% of the time.
To recap: if you choose both boxes, you’re guaranteed $1,000 but only have a 10% chance of getting that extra $1,000,000 in Box B. If you choose only Box B you’ll have a 90% chance of getting a quick one million, but you’re missing out on that $1,000 in Box A. The table below nicely summarises the possibilities:
What do you do to maximise your winnings?
The correct answer is…non existent.
Ok, well looking at the table the answer may seem obvious. No matter what the Kevin predicts, your winnings are ALWAYS higher if you choose both boxes than if you don’t. After all, he makes his prediction and puts the money in the boxes before your choice. So whatever is in Box B, your earnings are greater if you choose Box A as well. No-brainer. Both boxes is CLEARLY the right choice. And you’re right!
Except… the thing is Kevin is 90% accurate. You taking both boxes means there’s only a 10% chance of Box B having a million bucks. But Box B would have a 90% chance of being filled with money if you don’t take Box A. Let’s use Expected Value to calculate our expected earnings. Now EV = k=0nakP(ak)
What that mess of symbols means is that you multiply the value of each outcome by the probability of that outcome occurring, and then add all of them up.
So if we choose both boxes: $1000 x 1 + $1,000,000 x 0.1 = $101,000
This is because the probability of getting $1000 in Box A is guaranteed, but we’re relying on the 10% chance that Kevin is wrong and predicted we’d choose only one box.
But if we choose one box: $1,000,000 x 0.9 = $900,000
Because there’s a 90% chance that Kevin predicted we’d choose just one box. And since the expected earnings for choosing just Box B are much higher than choosing both boxes, we choose Box B. No brainer. And you’re right!
Except…the dominant strategy clearly dictates that both boxes have more value than just one. And herein lies the crux of the Newcomb Paradox. Two different, mathematically valid lines of reasoning that result in contradictory conclusions. And down comes crumbling the facade of math’s indestructible logic and rigor.
While it may seem to nullify the beauty that comes from the universal truths of math, I prefer to think it somewhat adds to it. To know that math isn’t just this black or white, yes or no, right or wrong. It can be mysterious, nuanced, paradoxical even. That, to me, is the true beauty of Newcomb’s Paradox.