Newcomb’s Bitcoins: Schrödinger’s Other Cat
There are simple concepts out there that you think you understand… until you are asked to define them. Time is one of them. And when mankind does not fully grasp a concept, there will always be a genial scientist that comes up with a paradox to point to its trickiest part.
Schrödinger brought the concept of quantum entanglement to the macroscopic level: has anybody ever seen a cat that is both alive and dead simultaneously? Newcomb brought the concept of predictability to the macroscopic level. And if you liked the former, you will love the latter.
Determinism vs. randomness
The second law of thermodynamics states that there is something known as entropy (which can be roughly described as disorder) that, in a closed system, can only increase over time. Parents telling their kids to tidy up their bedrooms? Not going to stop.
This general principle has dramatic consequences: it means that there is something asymmetric about the elapse of time: there is a past and a future. This is in direct contradiction with Newtonian physics, where the differential equations are reversible. Even in quantum computing, reversibility is a very desirable property and quantum gates always have as many output bits as input bits.
There is a conflict between, on the one hand, the idea that the future can be predicted (that’s what scientists spend their life trying to do) and the existence of true randomness, by essence unpredictable. That’s exactly what Newcomb’s problem brings into light.
Newcomb’s Problem (slightly adapted to 2022)
Two boxes lie in front of you. The first box is transparent and contains 1,000 bitcoins, thereafter “the transparent box.” The second box is opaque, yet you know that it contains either nothing at all, or 1,000,000 bitcoins, premium quality, mined with ASICs, 120+ confirmations. Thereafter “the opaque box.”
You are given the choice between:
- Taking the opaque box, and its contents belong to you.
- Taking both boxes, and their contents belong to you.
Yes, both. And yes, there is a catch.
Somebody, in the past, back in 2009, made a prediction about your choice, 1 or 2 boxes. The details of how they did it are left out. Maybe it’s AlphaGo. Maybe it’s a very good friend of yours who discussed this problem with you and already knows that you feel very strong towards the one or the other side. But what you know is: this predictor already played this game 1,000 times with others, maybe also with you. And 1,000 times their prediction was right.
Depending on their prediction, this person filled the opaque box. If they predicted you will take one box only, they will have put 1,000,000 BTC into the opaque box. If they predicted that you will take both boxes, they will have left the box empty.
The predictor is not venal and has no agenda besides getting their prediction right.
So… do you take one box, or both of them?
…
…
Scroll down once you have made up your mind.
…
…
…
Two opposite reasonings
If you ask people randomly about this problem, you will notice three things: first, many people take one box (thereafter “one-boxers”), while many other people take both boxes (thereafter “two-boxers”).
Second, their reasonings sound totally flawless.
And third, this is very polarising and people usually feel very strong about their choice.
One boxers
One-boxers typically reason as follows.
“If I pick one box and the prediction is right, I’ll have 1,000,000 BTC. If I pick both boxes and it was predicted correctly, I’ll have 0 + 1,000 = 1,000 BTC. So, I’m better off picking one.”
And they pick one. And find 1,000,000 BTC inside. And are happy to have 1,000,000 BTC rather than 1,000 BTC.
Two boxers
Two-boxers (among them many theoretical physicists and game theorists) typically reason as follows.
Let us call x the number of bitcoins in the opaque box. If I pick one box, I’ll have x BTC. If I pick both, I’ll have x + 1000 BTC. With a dominant strategy argument, I’ll be better off with two boxes irregardless of the value of x.
And they pick two. And find 1,000 BTC inside: the prediction was indeed correct and x=0. And are happy to have 1,000 BTC rather than 0 BTC.
The resolution
The resolution of the apparent paradox lies in the fact that the problem, on purpose, leaves an assumption open in its formulation: whether the prediction is counterfactually dependent on the choice or not. One-boxers assume it is. Two-boxers assume it is not. So, both are right with their respective assumptions.
This demonstrates that there are two competing visions for the account of time. They also correspond to both visions depicted in a former post. The lesson here is that not all dependencies must be causal. While causal dependencies can only go forward in time, counterfactual dependencies ignore the arrow of time.
Just a short disclaimer: I am not sure yet whether the scientific community has reached a unanimous consensus on the fact that Newcomb’s paradox is solved, with the solution stated above, but several publications [2] [3] converging to it have now been out for a while.
References
[1] (First publication of Newcomb’s Problem) Robert Nozick (1969). “Newcomb’s Problem and Two Principles of Choice”. In Rescher, Nicholas. Essays in Honor of Carl G Hempel (PDF). Springer.
[2] Jean-Pierre Dupuy, Two temporalities, two rationalities: a new look at Newcomb’s Paradox, In Economics and Cognitive Science, edited by Paul Bourgine and Bernard Walliser, Pergamon, Amsterdam, 1992, Pages 191–220, ISBN 9780080410500, http://dx.doi.org/10.1016/B978-0-08-041050-0.50023-6.
(http://www.sciencedirect.com/science/article/pii/B9780080410500500236)
[3] Wolpert D, Benford G (2013) The lesson of newcomb’s paradox. Synthese 190(9):1637-
1646, DOI 10.1007/s11229–011–9899–3, URL http://dx.doi.org/10.1007/
s11229–011–9899–3
Picture copyright: serezniy, cammep / 123RF Stock Photo
Originally published at https://www.linkedin.com.
