I call this a framework because there are lots of details left unspecified, problems unsolved, etc. However I expected any multiverse decision theory will probably look something like this. My goal in writing this down is to have a framework for formalizing problems and proposed solutions.

This decision theory is more complex than the one in [Joyce] for two reasons. First is to take into account the lack of logical omniscience (following [Lipman]). This is necessary because if the multiverse consist of all logically consistent universes, then logical omniscience implies no decision is possible, so we have to assume lack of logical omniscience in that case. Second is to take into account the fact that you don't know which universe you're in or which observer moment you are at in each universe, and the different observer moments that you may be at can have different utility functions. This implies we have to use the game theoretic concept of equilibrium. Start with the set A of possible strategies to choose from. Consider an element a from this set. We want to determine if a is an equilibrium strategy. Consider another set S, where each element is a full description of the entire multiverse (including the history of all of the universes inside it), which may be false or even inconsistent or unsatisfiable. You know that one and only one of the descriptions is true, but you're not sure which one. You have a function P such that P(s) is your subjective probability that s is true given the assumption that all observer moments that you may be at do choose strategy a. For each s, you have: - a set M_s of observer moments, which are the possible observer moments that you may be at if s is true. Let M be the union of all M_s. - a set C_s of possible consequence functions c: M_s x A -> S x M. If <s',m'> = c(m,a'), then s' is what s would be if m chose a' instead of a (while all other observer moments in M_s still choose a), and m' is the observer moment corresponding to m in M_s'. - functions Pm_s and Pc_s with the interpretation that Pm_s(m) is the probability that you are at m if s is true and Pc_s(c) is the probability that c is the true consequence function if s is true. - utility function u_s: M_s -> R, where u_s(m) gives the utility of s being true if you are m. To check whether it is an equilibrium to choose a, we compute the expected utility of all strategies under the assumption that other observer moments choose a, and see if choosing a gives the maximum expected utility. The expected utility of a' is: \sum_{s \in S} \sum_{m \in M_s} \sum_{c \in C_s} P(s) * Pm_s(m) * Pc_s(c) * u_s'(m'), where <s',m'> = c(m,a') Some problems and discussion: What does each element of S look like exactly? It would seem to take an infinite string to fully describe the entire multiverse, so how can we think about them and assign probabilities to them? Perhaps it's better to make them finite descriptions of the parts of the multiverse that we care about or are relevant to the decision. What should P be? Suppose the multiverse consists of all mathematical structures (as proposed by Tegmark). In that case each element in S would be a conjunction of mathematical statements and P would assign probabilities to these mathematical statements (most of which we have no hope of proving or disproving). How should we do that? Of course we already do that (e.g. computer scientists bet with each other on whether polytime = non-deterministic polytime, and we make use of mathematical facts that we don't understand the proofs for), but there does not seem to be a known method for doing it optimally. This is also where anthropic reasoning would come in. What about Pm? That brings up the discussions we had about whether running an experience again doubles its measure, whether a bigger computer generates more measure, etc. Following up on an earlier point I made, I'll note the interchangeability between P_m(s) and the utility u_s'(m'). You can increase one, decrease the other, and keep expected utilities the same. (Joyce also talks about this in his book.) So you can't really talk about what Pm should be without also talking about what u should be. Pc is where all the causality stuff goes of course. Beyond that it's an open question how it should be derived. Is this framework general enough so that any rational decision maker can be modeled by choosing the appropriate A, S, M, C, P, Pm, Pc, and u? Can it be simplified? Barton L. Lipman. "Decision Theory without Logical Omniscience: Toward an Axiomatic Framework for Bounded Rationality," Review of Economic Studies, April 1999. James M. Joyce. The Foundations of Causal Decision Theory. Cambridge: Cambridge University Press, April 1999.