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Qiaochu_Yuan comments on Simulating Problems - Less Wrong Discussion
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It was not obvious to me that you were talking about game-theoretic problems. "Problem" is not a word owned solely by game theorists.
It's unclear to me what you mean by this. If a problem contains elements which are impossible to construct in real life, in what sense can a practical version be said to be identical in terms of rules, interactions, results, and so on?
I have edited my top-level post to clarify what kind of problems I mean.
For a trivial example, Omega predicting an otherwise irrelevant random factor such as a fair coin toss can be reduced to the random factor itself, thereby getting rid of Omega. Equivalence can easily be proven because regardless of whether we allow for backwards causality and whatnot, a fair coin is always fair and even if we assume that Omega may be wrong, the probability of error must still be the same for either side of the coin, so in the end Omega is exactly as random as the coin itself no matter Omega's actual accuracy. Of course this wouldn't apply if the result of the coin toss was also relevant in some other way.
Okay, so right now I don't understand what your question is. It sounds to me like "how can we prove that simulations are simulations?" given what I understand to be your definition of a simulation.
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?
This is obviously not about bazookas and tanks. If you want to know whether real tanks really blow up, you need real evidence. If you want to know whether CDT defects in PD, you don't. You can do maths just with logic and reason, und fortunately this is 100% about maths.
You have not given me anything like a precise statement of a mathematical problem.
Here you go:
Given a problem A which is impossible or impractical in real life, find a practical problem B (called simulation) with the same payoff matrix for which it can be proven that any possible agent will make analogous decisions in analogous states.
Solve for Newcomb or other problems at will. Bonus points for finding generalized approach.
That is not a precise statement of a mathematical problem. What do "impractical" and "practical" mean? What does "analogous" mean?
"Impractical" means that you don't want to or can't realize the problem in its original form, for example because it would be too expensive or because you don't have a prison handy and can't find any prisoner rental service.
"Practical" pretty much means the opposite, for example because it's inexpensive or because you happen to be a prison director and are not particularly bent on interpreting the law orthodoxly.
"Analogous" basically means that if you can find isomorphisms between the set of the states of problem A and the set of the states of problem B as well as between the set of decisions of problem A and the set of decisions of problem B, then each thus mapped pair of decisions or states is called analogous if analagous decisions lead to analogous states and analogous states imply analogous decisions.