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                                      Michael Lee's Mathematics and Physics

Newcomb's Problem


Suppose you enter a room that contains two containers (i.e. boxes).  One of them is transparent and the other is opaque.  Here's the deal; I have a machine called the "predictor" that will tell me what you shall do before you do it and it is absolutely reliable.   You have two options;  you may choose the contents of the opaque box only and leave the transparent one alone  OR  take the contents of both boxes.


The machine shall inform me which option of the two you shall make to me long before you enter the room and so I placed a good cheque for $100,000 in the transparent box and the contents of the opaque box depends upon the choice you shall make.  If you choose only the opaque box and leave the transparent box alone, I shall place a cheque for $1,000,000 in the opaque box.  On the other hand, if you choose both boxes, I shall leave you nothing in the opaque box.  The rules of the game are made aware to you.


To make it as simple as I can, there is always $100,000 in the transparent box and there will either be nothing or $1,000,000 in the opaque box depending upon whether you select "just opaque"  or "both".  Remember, the "predictor" machine is absolutely reliable.


So upon entering the room, which are you going to choose,  "just opaque" or "both"?  Suppose you select "both".  You already know there is $100,000 in the transparent glass and you can also see the cheque in it.  So, you open the opaque one and find nothing because the predictor told me you would take "both."  So you take the contents of the transparent box for a paltry $100,000.  Obviously then, the "best choice" is to take the contents of only the opaque box and walk away as fast as you can to the bank with the $1,000,000 cheque.


But here is the problem, before leaving the room with a cheque for one million in your hand.  Look back at the transparent box and tell me what you see in it.  It contains a cheque for 100,000 you say.  Well, what is preventing you from going back there and taking it too?  It can't just disappear into thin air; the transparent box always contains a cheque for $100,000 regardless of your choice.  So instead of getting $1,000,000 you'll obtain $1,100,000 which is greater.



The above problem is a disagreement between two well accepted principles of choice theory:  maximizing expected utility (i.e. money in our example) verses the principle of dominance.  For the former, the probability of receiving 1,000,000 by only taking the opaque box is very high whereas the probability of getting an amount at least as high as that is very low if you choose both (i.e. $100,000).  So you should choose the opaque box only.  But for the latter, the principal of dominance says picking "both" boxes is always better than picking just the opaque box.  In this case, the contents of the boxes have already been determined and set up before you make your ultimate selection and so your decision has no bearing on the contents of the boxes at the time you make your choice.  If there is $1,000,000 in the opaque box, you should also take the transparent box for $1,100,000.  If there is nothing in the opaque box, you should also take the contents of the transparent box for $100,000 which is better than nothing at all.  On the other hand, if you find nothing in the opaque box, and suppose you already have plenty of money in the bank, you can prove the predictor wrong (which denies our original assumption) by simply leaving the transparent box and its $100,000 contents alone.