a framework for multiverse decision theory

From: Wei Dai <weidai.domain.name.hidden>
Date: Sun, 11 Aug 2002 22:38:32 -0700

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.
Received on Sun Aug 11 2002 - 22:43:15 PDT