Note: This post is in error, I've put up a corrected version of it here. I'm leaving the text in place, as historical record. The source of the error is that I set Pa(S)=Pe(D) and then differentiated with respect to Pa(S), while I should have differentiated first and then set the two values to be the same.
Nate Soares and Ben Levinstein have a new paper out on "Functional Decision theory", the most recent development of UDT and TDT.
It's good. Go read it.
This post is about further analysing the "Death in Damascus" problem, and to show that Joyce's "equilibrium" version of CDT (causal decision theory) is in a certain sense intermediate between CDT and FDT. If eCDT is this equilibrium theory, then it can deal with a certain class of predictors, which I'll call distribution predictors.
Death in Damascus
In the original Death in Damascus problem, Death is a perfect predictor. It finds you in Damascus, and says that it's already planned it's trip for tomorrow - and it'll be in the same place you will be.
You value surviving at $1000, and can flee to Aleppo for $1.
Classical CDT will put some prior P over Death being in Damascus (D) or Aleppo (A) tomorrow. And then, if P(A)>999/2000, you should stay (S) in Damascus, while if P(A)<999/2000, you should flee (F) to Aleppo.
FDT estimates that Death will be wherever you will, and thus there's no point in F, as that will just cost you $1 for no reason.
But it's interesting what eCDT produces. This decision theory requires that Pe (the equilibrium probability of A and D) be consistent with the action distribution that eCDT computes. Let Pa(S) be the action probability of S. Since Death knows what you will do, Pa(S)=Pe(D).
The expected utility is 1000.Pa(S)Pe(A)+1000.Pa(F)Pe(D)-Pa(F). At equilibrium, this is 2000.Pe(A)(1-Pe(A))-Pe(A). And that quantity is maximised when Pe(A)=1999/4000 (and thus the probability of you fleeing is also 1999/4000).
This is still the wrong decision, as paying the extra $1 is pointless, even if it's not a certainty to do so.
So far, nothing interesting: both CDT and eCDT fail. But consider the next example, on which eCDT does not fail.
Statistical Death in Damascus
Let's assume now that Death has an assistant, Statistical Death, that is not a prefect predictor, but is a perfect distribution predictor. It can predict the distribution of your actions, but not your actual decision. Essentially, you have access to a source of true randomness that it cannot predict.
It informs you that its probability over whether to be in Damascus or Aleppo will follow exactly the same distribution as yours.
Classical CDT follows the same reasoning as before. As does eCDT, since Pa(S)=Pe(D), as before, since Statistical Death follows the same distribution as you do.
But what about FDT? Well, note that FDT will reach the same conclusion as eCDT. This is because 1000.Pa(S)Pe(A)+1000.Pa(F)Pe(D)-Pa(F) is the correct expected utility, the Pa(S)=Pe(D) assumption is correct for Statistical Death, and (S,F) is independent of (A,D) once the action probabilities have been fixed.
So on the Statistical Death problem, eCDT and FDT say the same thing.
Factored joint distribution versus full joint distributions
What's happening is that there is a joint distribution over (S,F) (your actions) and (D,A) (Death's actions). FDT is capable of reasoning over all types of joint distributions, and fully assessing how its choice of Pa acausally affects Death's choice of Pe.
But eCDT is only capable of reasoning over ones where the joint distribution factors into a distribution over (S,F) times a distribution over (D,A). Within the confines of that limitation, it is capable of (acausally) changing Pe via its choice of Pa.
Death in Damascus does not factor into two distributions, so eCDT fails on it. Statistical Death in Damascus does so factor, so eCDT succeeds on it. Thus eCDT seems to be best conceived of as a version of FDT that is strangely limited in terms of which joint distributions its allowed to consider.