Wednesday, July 31, 2013

Newcomb’s Paradox and Bell Theorem

Today I watched a lovely presentation of Newcomb’s paradox by Norm Wildberger

and I could not help but notice the deep relationship with Bell theorem. In particular Bell result shows clearly the solution for Newcomb’s paradox and I am interrupting the series to present the solution. Since I did not follow Newcomb’s paradox developments, I don’t know if my take on it is new or not.

So what is Newcomb’s paradox ? This paradox was popularized by Martin Gardner and is described in his book: “The Colossal Book of Mathematics” on page 580. Let me quote its description from this book to achieve maximum clarity:

Two closed boxes, B1 and B2, are on a table. B1 contains $1,000. B2 contains either nothing or $1 million. You do not know which. You have an irrevocable choice between to actions:

  1. Take what is in both boxes.
  2. Take only what is in B2.

At some time before the test a superior Being has made a prediction about what you will decide. It is not necessary to assume determinism. You only need be persuaded that the Being’s predictions are “almost certainly” correct. If you like, you can think of the Being as God, but the paradox is just as strong if you regard the Being as a superior intelligence from another planet or a supercomputer capable of probing your brain and making highly accurate predictions about your decisions. If the Being expects you to choose both boxes, he has left B2 empty. If he expects you to take only B2, he has put $1 million in it. (If he expects you to randomize your choice by say, flipping a coin, he has left B2 empty.) In all cases B1 contains $1,000. You understand the situation fully, the Being knows you understand, you know that he knows, and so on.

What should you do? Clearly it is not to your advantage to flip a coin, so that you must decide on your own. The paradox lies in the disturbing fact that a strong argument can be made for either decision. Both arguments cannot be right. The problem is to explain why one is wrong.

Let us look first at the argument for taking only B2. You believe the Being is an excellent predictor. If you take both boxes, the Being almost certainly will have anticipated your action and have left B2 empty. You will get only the $1,000 in B1. On the other hand, if you take only B2, the Being, expecting that, almost certainly will have placed $1 million in it. Clearly it is to your advantage to take only B2.

Convincing? Yes, but the Being made his prediction, say a week ago, and then left. Either he put the $1 million in B2, or he did not. If the money is already there, it will stay there whatever you choose. It is not going to disappear. If it is not already there, it is not going to suddenly appear if you choose only what is in the second box. It is assumed that no “backwards causality” is operating; that is, your present actions cannot influence what the Being did last week. So why not take both boxes and get everything that is there? If B2 is filled, you get $1,001,000. If it is empty, you get at least $1,000. If you are so foolish as to take only B2, you know you cannot get more than $1 million, and there is even a slight possibility of getting nothing. Clearly it is to your advantage to take both boxes!

So here is the solution.  Let us call the first solution (pick only B2): trust in correlations, and the second solution (pick both B1 and B2): trust in local realism.

If you trust correlations, you obviously should pick only B2 and the video above has a nice mathematical formulation of why and when you should choose this. If you trust local realism clearly you should pick both boxes. Now here is where Bell’s Theorem comes into play. As originally written, the paradox is based on a mathematical contradiction: local realism imposes a limit on the maximum possible correlation and Bell inequalities captures precisely this limit. As stated, Newcomb’s Paradox violates Bell’s theorem.

If the correlations are higher than Bell inequality limit, this cannot be achieved with classical objects like money. Therefore the physically correct solution is to pick both boxes.

On the other hand we can introduce a variation to the problem by populate the two boxes with quantum objects instead of money. Suppose we have may identically prepared copies of the quantum objects and the “Being” perform a weak ago half measurement in an EPR-B type experiment. Then the choice for the player is to either perform the other half (and get $1 million if the correlation is above Bell limit) or to perform a random intermediate measurement (to be discarded) and then complete the other half measurement and this reduces the correlation into Bell’s limit. What happens in this case is the violation of this: “If the money is already there, it will stay there whatever you choose. It is not going to disappear. If it is not already there, it is not going to suddenly appear if you choose only what is in the second box.” because unperformed measurements have no pre-defined values in quantum mechanics.

Therefore both solutions are valid in their range of validity and the hidden contradiction in the original paradox was the incompatibility between local realism and correlations which Bell theorem explains. The original setup of the paradox is a physical and a mathematical impossibility as shown by Bell theorem.


I did some reading to see other people's take on this paradox, and I found a paper by Eric Cavalcanti who had the same idea as mine, but presented it in a physics context with emphasis on the Bayesian approach. 

After the paradox was introduced, for the two solutions of the paradox, two schools of thought developed: Causal Decision Theory and Evidential Decision Theory (see‎)

The way to think about the paradox is to rename "Being" as "a potential Con Artist" and then assert the degree of trust into his ability to predict your mind. If the laws of physics prevents him to achieve the correlation, then pick both boxes, if not, go with the solution dictated by the Evidential Decision Theory and pick one box.

Bell theorem sets a limit on the maximum achievable correlations and Cavalcanti showed explicitly that the standard Bell analysis for local hidden variable theories (LHV) corresponding to local realism is same as the analysis advocated by the Causal Decision Theory. This being the case, Bell inequalities set up the maximum achievable correlations for local realism. 

The domain of validity of the Evidential Decision Theory is limited by the Bell limit for classical resources, or by the Tsirelson limit for quantum resources. 

In the case of this paradox, Bell limit demands p_a = p_b = 50% where p_a and p_b are the probability of "a potential Con Artist" to correctly predict your choices (see the above video for definition) 

By Bell theorem, p_a,b > 50% OR p_a,b < 50% demands that "a potential Con Artist" is definitely a "Con Artist" and if you follow the course of action suggested by Evidential Decision Theory I have a bridge to sell you. 

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