Apologies for the rather mathematical nature of this post, but it seems to have some implications for topics relevant to LW. Prior to posting I looked for literature on this but was unable to find any; pointers would be appreciated.
In short, my question is: How can we prove that any simulation of a problem really simulates the problem?
I want to demonstrate that this is not as obvious as it may seem by using the example of Newcomb's Problem. The issue here is of course Omega's omniscience. If we construct a simulation with the rules (payoffs) of Newcomb, an Omega that is always right, and an interface for the agent to interact with the simulation, will that be enough?
Let's say we simulate Omega's prediction by a coin toss and repeat the simulation (without payoffs) until the coin toss matches the agent's decision. This seems to adhere to all specifications of Newcomb and is (if the coin toss is hidden) in fact indistinguishable from it from the agent's perspective. However, if the agent knows how the simulation works, a CDT agent will one-box, while it is assumed that the same agent would two-box in 'real' Newcomb. Not telling the agent how the simulation works is never a solution, so this simulation appears to not actually simulate Newcomb.
Pointing out differences is of course far easier than proving that none exist. Assuming there's a problem we have no idea which decisions agents would make, and we want to build a real-world simulation to find out exactly that. How can we prove that this simulation really simulates the problem?
(Edit: Apparently it wasn't apparent that this is about problems in terms of game theory and decision theory. Newcomb, Prisoner's Dilemma, Iterated Prisoner's Dilemma, Monty Hall, Sleeping Beauty, Two Envelopes, that sort of stuff. Should be clear now.)
The question is: How can I prove that all possible agents decide identically whether they're considering the simulation or the original problem?
To further illustrate the point of problem and simulation, suppose I have a tank and a bazooka and want to know whether the bazooka would make the tank blow up, but because tanks are somewhat expensive I build another, much cheaper tank lacking all parts I deem irrelevant such as tracks, crew, fire-control and so on. My model tank blows up. But how can I say with certainty that the original would blow up as well? After all, the tracks might have provided additional protection. Could I have used tracks of inferior quality for my model? Which cheaper material would have the same resistance to penetration?
Tank and bazooka are the problem, of which the tank is the impractical part that is replaced by the model tank in the simulation.
You... can't?