In this post, I'll argue that Joyce's equilibrium CDT (eCDT) can be made into FDT (functional decision theory) with the addition of an intermediate step - a step that should have no causal consequences. This would show that eCDT is unstable under causally irrelevant changes, and is in fact a partial version of FDT.
Joyce's principle is:
Full Information. You should act on your time-t utility assessments only if those assessments are based on beliefs that incorporate all the evidence that is both freely available to you at t and relevant to the question about what your acts are likely to cause.
When confronted by a problem with a predictor (such as Death in Damascus or the Newcomb problem), this allows eCDT to recursively update their probabilities of the behaviour of the predictor, based on their own estimates of their own actions, until this process reaches equilibrium. This allows it to behave like FDT/UDT/TDT on some (but not all) problems. I'll argue that you can modify the setup to make eCDT into a full FDT.
Death in Damascus
In this problem, Death has predicted whether the agent will stay in Damascus (S) tomorrow, or flee to Aleppo (F). And Death has promised to be in the same city as the agent (D or A), to kill them. Having made its prediction, Death then travels to that city to wait for the agent. Death is known to be a perfect predictor, and the agent values survival at $1000, while fleeing costs $1.
Then eCDT fleeing to Aleppo with probability 999/2000. To check this, let x be the probability of fleeing to Aleppo (F), and y the probability of Death being there (A). The expected utility is then
- 1000(x(1-y)+(1-x)y)-x (1)
Differentiating this with respect to x gives 999-2000y, which is zero for y=999/2000. Since Death is a perfect predictor, y=x and eCDT's expected utility is 499.5.
The true expected utility, however, is -999/2000, since Death will get the agent anyway, and the only cost is the trip to Aleppo.
The eCDT decision process seems rather peculiar. It seems to allow updating of the value of y dependent on the value of x - hence allow acausal factors to be considered - but only in a narrow way. Specifically, it requires that the probability of F and A be equal, but that those two events remain independent. And it then differentiates utility according to the probability of F only, leaving that of A fixed. So, in a sense, x correlates with y, but small changes in x don't correlate with small changes in y.
That's somewhat unsatisfactory, so consider the problem now with an extra step. The eCDT agent no longer considers whether to stay or flee; instead, it outputs X, a value between 0 and 1. There is a uniform random process Z, also valued between 0 and 1. If Z<X, then the agent flees to Aleppo; if not, it stays in Damascus.
This seems identical to the original setup, for the agent. Instead of outputting a decision as to whether to flee or stay, it outputs the probability of fleeing. This has moved the randomness in the agent's decision from inside the agent to outside it, but this shouldn't make any causal difference, because the agent knows the distribution of Z.
Death remains a perfect predictor, which means that it can predict X and Z, and will move to Aleppo if and only if Z<X.
Now let the eCDT agent consider outputting X=x for some x. In that case, it updates its opinion of Death's behaviour, expecting that Death will be in Aleppo if and only if Z<x. Then it can calculate the expected utility of setting X=x, which is simply 0 (Death will always find the agent) minus x (the expected cost of fleeing to Aleppo), hence -x. Among the "pure" strategies, X=0 is clearly the best.
Now let's consider mixed strategies, where the eCDT agent can consider a distribution PX over values of X (this is a sort of second order randomness, since X and Z already give randomness over the decision to move to Aleppo). If we wanted the agent to remain consistent with the previous version, the agent then models Death as sampling from PX, independently of the agent. The probability of fleeing is just the expectation of PX; but the higher the variance of PX, the harder it is for Death to predict where the agent will go. The best option is as before: PX will set X=0 with probability 1001/2000, and X=1 with probability 999/2000.
But is this a fair way of estimating mixed strategies?
Average Death in Aleppo
Consider a weaker form of Death, Average Death. Average Death cannot predict X, but can predict PX, and will use that to determine its location, sampling independently from it. Then, from eCDT's perspective, the mixed-strategy behaviour described above is the correct way of dealing with Average Death.
But that means that the agent above is incapable of distinguishing between Death and Average Death. Joyce argues strongly for considering all the relevant information, and the distinction between Death and Average Death is relevant. Thus it seems when considering mixed strategies, the eCDT agent must instead look at the pure strategies, compute their value (-x in this case) and then look at the distribution over them.
One might object that this is no longer causal, but the whole equilibrium approach undermines the strictly causal aspect anyway. It feels daft to be allowed to update on Average Death predicting PX, but not on Death predicting X. Especially since moving from PX to X is simply some random process Z' that samples from the distribution PX. So Death is allowed to predict PX (which depends on the agent's reasoning) but not Z'. It's worse than that, in fact: Death can predict PX and Z', and the agent can know this, but the agent isn't allowed to make use of this knowledge.
Given all that, it seems that in this situation, the eCDT agent must be able to compute the mixed strategies correctly and realise (like FDT) that staying in Damascus (X=0 with certainty) is the right decision.
Let's recurse again, like we did last summer
This deals with Death, but not with Average Death. Ironically, the "X=0 with probability 1001/2000..." solution is not the correct solution for Average Death. To get that, we need to take equation (1), set x=y first, and then differentiate with respect to x. This gives x=1999/4000, so setting "X=0 with probability 2001/4000 and X=1 with probability 1999/4000" is actually the FDT solution for Average Death.
And we can make the eCDT agent reach that. Simply recurse to the next level, and have the agent choose PX directly, via a distribution PPX over possible PX.
But these towers of recursion are clunky and unnecessary. It's simpler to state that eCDT is unstable under recursion, and that it's a partial version of FDT.