Newcomb's Paradox

Why is it a Paradox?

This scenario is considered a paradox because two reputable decision making principles point to different strategies. Expected utility says that we should always choose box B because we will get $2 rather than $1. Strategic dominance says the boxes are already set before you make a decision, so why not pick both and always receive the extra $1 from box A.

Entanglement

I sought a way to implement a working version of Newcomb's Paradox in the hope that it might provide some insight into the cause of the paradox. After working through this task, my current view is that the paradox arises when we fail to treat our thoughts and decisions as part of the same physics that govern everything else. Our choices do not live in some separate realm but are influenced and influence the world in which they are embedded. When we acknowledge this, we see the possibility that our choice could be entangled with the boxes and the predictor. There are ways in which I could have allowed the boxes to influence both the choice and the predictors prediction, such as lighting box A on fire, but to make this function as a magic trick I really needed the causality to work the other way around, such that a spectators choice could influence the contents of the boxes. That is precisely what this device does. A decision to remove the contents of box A, mechanically causes box B to automatically become empty. This device is essentially a No Greedy Box. It appears that the all knowing predicting magician causes box B to be empty or full, but in fact it is the spectator, and the magician is not required at all. According to the formulation of the paradox, it is only the behavior of a perfect predictor that we really ever observe. The predictor need not even exist, as long as a two box choice always leads to box B appearing empty and one box choice always leads to it appearappearing full. I achieve this entanglement with a simple mechanism hidden within the box, but one can conceive of a more complex interaction such as an advanced technology that could scan our brain and vaporize the contents of box B when it detects a two box decision. Once one sees how the two boxes are mechanically entangled, it becomes intuitive that choosing both boxes is equivalent to choosing just box A. Hence it becomes clear that box B with the higher value is the correct choice.

My hope is that this mechanism demonstrates how our decisions are themselves a physical process. They do not live in some separate isolated realm and hence in theory there is nothing to stop them interacting with the environment in highly correlated ways.

Perhaps you will think that this is breaking the rules of the game, and that box B must really be set before the choice is made. In which case, below are two alternative methods of entanglement, though harder for me to implement in a working device.

Common Cause

My favourite solution is probably the existence of a common cause. The initial conditions that caused the predictor to have a particular prediction, also caused the chooser to make a particular choice. This will not be recognisable when the common cause is small or complex but easy to intuit if it had some large scale manifestation such as Box A is on fire. The predictor can clearly see that the chooser will not choose to open a burning box. Once again box B is the rational choice in this situation.

The Perfect Predictor is You

If the predictor indeed had all of the information and computational apparatus required to predict your decision, it would be performing an exact copy of your own decision making procedure. It would essentially be you. The prediction and choice are the same event. So the scenario now becomes a choice between taking both boxes but discarding the contents of box B, or choosing just the contents from box B. 

Free Will?

So if choices are a physical process, can we still have free will? I think yes, but my definition of free will may not be regarded as such by others. My free will is simply the deterministic processes that I identify with. The values and beliefs stored within my brain that shape incoming information into decision and action. It is still deterministic and could potentially be simulated on a powerful computer. But the closer that computer got to correctly predicting my decision, the closer it would be to becoming me.

Footnote

That is my conservative stance. Though in fact I have some deep seated reservations about the viability of causality in general. It seems to me that reductionist causal explanations will always lead to infinite regress. One will always be left looking for the next cause one layer down. Perhaps rather than cause, the laws of physics are more to do with constraint. We are constrained to only be conscious of those mathematical truths that are themselves a conscious experience.