Okay. As another question, to what extent should quantum effects be considered in the area?
1: If there are essentially no Quantum effects, then I don't have access to a source of true randomness, just pseudorandom numbers. This would influence my coding, because true randomness can be used to lengthen a program in ways that fake randomness cannot, so I would have to adjust my code to take that into account.
[My understanding may be off here, but I think that given a pseudorandom algorithm, there are events which can be defined as to never take place. Ergo, a bad pseudorandom algorithms might never generate "0.01" 4 times in a row. But given quantum randomness, any defined event will eventually happen with probabilities approaching 1 as runtime increases]
2: On the other hand, if there are quantum effects, I can attempt to make programs like the following:
X=0;
DountilHalt;
X=X+1;
Write "S" in Memory Register X;
If the Character in Memory Register X is "0" then halt.
Else goto DountilHalt;
Which would keep running until there is a Quantum bitflip of "S" into "0" at just the right moment (or some other bitflip in the program that also caused a halt.)
3: Alternatively, I could view it as "Your program can call Quantum Mechanical randomness, if you want it to, but neither your program, nor it's output, will be changed by Quantum Mechanical effects unless you program that in."
Which means that the Program in 2 would never halt because I did not call a Quantum function anywhere inside the program.
It seems sort of like the implicit scenario is 3, but I may be incorrect (or I may have cast 1,2 or 3 incorrectly.)
In an earlier post, I talked about how we could deal with variants of the Heaven and Hell problem - situations where you have an infinite number of options, and none of them is a maximum. The solution for a (deterministic) agent was to try and implement the strategy that would reach the highest possible number, without risking falling into an infinite loop.
Wei Dai pointed out that in the cases where the options are unbounded in utility (ie you can get arbitrarily high utility), then there are probabilistic strategies that give you infinite expected utility. I suggested you could still do better than this. This started a conversation about choosing between strategies with infinite expectation (would you prefer a strategy with infinite expectation, or the same plus an extra dollar?), which went off into some interesting directions as to what needed to be done when the strategies can't sensibly be compared with each other...
Interesting though that may be, it's also helpful to have simple cases where you don't need all these subtleties. So here is one:
Omega approaches you and Mrs X, asking you each to name an integer to him, privately. The person who names the highest integer gets 1 utility; the other gets nothing. In practical terms, Omega will reimburse you all utility lost during the decision process (so you can take as long as you want to decide). The first person to name a number gets 1 utility immediately; they may then lose that 1 depending on the eventual response of the other. Hence if one person responds and the other doesn't, they get the 1 utility and keep it. What should you do?
In this case, a strategy that gives you a number with infinite expectation isn't enough - you have to beat Mrs X, but you also have to eventually say something. Hence there is a duel of (likely probabilistic) strategies, implemented by bounded agents, with no maximum strategy, and each agent trying to compute the maximal strategy they can construct without falling into a loop.