Newcomb_s_Paradox.html Newcomb's Paradox The Newcomb's paradox is a decision problem described as following:
The player of the game is presented with two boxes, one transparent (labeled A) and the other opaque (labeled B). The player is permitted to take the contents of both boxes, or just the opaque box B. Box A contains a visible $1,000. The contents of box B, however, are determined as follows: At some point before the start of the game, the Predictor makes a prediction as to whether the player of the game will take just box B, or both boxes. If the Predictor predicts that both boxes will be taken, then box B will contain nothing. If the Predictor predicts that only box B will be taken, then box B will contain $1,000,000.

A mathematical representation of the predictor for an ideal agent

Let U(a) be our environment, which takes action a and returns the amount of money that the agent receives after choosing the action a.
Let A=argmax U(a) , the action of an ideal agent that obtains maximum amount of money from U .
Let a=1 be the action of taking one box, and a=2 be the action of taking both boxes.
First attempt at representation of the environment with the predictor in it:
Let U(a)=
argmax U(a) = 1 and a = 1 : 1 000 000
argmax U(a) = 1 and a = 2 : 1 001 000
argmax U(a) = 2 and a = 1 : 0
argmax U(a) = 2 and a = 2 : 1000
where argmax U(a) is the action that is ultimately chosen, and the action that is predicted.
Note that the function U is recursive. The predictor can not work by simple simulation of the agent, as such simulation would keep running itself forever. It seems to be a common misunderstanding of the problem to picture the predictor as a machine which runs a "copy" of the agent, which is only possible for agents that are simpler than their environment.
In this particular case, if argmax U(a) = 1 , then U(a) would have a new maximum at a=2 . By exaustion, the only consistent U is:
U(a)= a = 1 : 0 a = 2 : 1000 This is commonly described as to mean that the rational choice in Newcomb's problem is to take both boxes, and walk away with 1 thousand.
I find this verbal representation of a solution to the above system of equations to be rather deceptive, which becomes particularly clear in the following example:
There are two transparent boxes, P and Q. The predictor puts money into the box Q if it predicts that you'll choose the box P, but into the box P if it predicts that you'll choose the box Q. The solution to the corresponding equations is to take the box that contains no money; this is not because this is the best choice, but because if you simply took the box where you see the money the predictor would be wrong and thus the equations would not be satisfied.
Mathematical solutions to such decision problems represent also the pre-requisities for existence of the postulated objects in its environment. In particular, existence of predictor requires that player can not see with it's eyes or deduce content of the box A and then take 1 box if the box A is empty, and both boxes if box A is not empty. Either the knowledge has to be restricted, or the choice.

Relation to Godel's first and second incompleteness theorems

Similarly to Godel's incompleteness theorem, for any consistent formal system employed by the player, there will exist an environment E in which the best choice of the action can not be proved by the formal system in question, unless this formal system is inconsistent. Sketch of a proof: consider an environment consisting of a copy of the agent, and 2 boxes; if the copy decides to take one box, the money are put into the other box.

Bounded agent in an environment larger than the agent

The dominance principle does not hold (the choice is not guaranteed to be the best) and there is no paradox of any note. Application of dominance principle is not guaranteed to produce optimum results for bounded agents; the paradox essentially dissolves into a matter of representation of environments.