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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

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

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