Functional Decision Theory (FDT) basically is the idea that, rather than considering your decisionmaking process to be a black box with free will (as in causal decision theory), you instead choose the algorithm that will have the best result. (hmm that sounds circular - but that's probably just my failure in attempting to summarize it).
This is relevant to, e.g. Newcomb's problem where the predictor can see your algorithm. Unfortunately, as I will show, the math presented as the formula for FDT doesn't seem to work as the verbal descriptions imply.
I was intrigued by the Functional Decision Theory response to the "Asteroids in Aleppo Scenario":
Asteroids Every night you have the option of staying either in Damascus for $1 or Aleppo for $2. Neither city is especially safe, unfortunately. Every night in Damascus, there’s a 1% probability that you will be killed by disease. Every night in Aleppo, there’s a 10% probability you will be killed by asteroids. These chances are, from your point of view, stochastically independent. However, they’re actually determined by a process that’s known to DEATH, who’s also a perfect predictor of your decisions. DEATH wants you to die as soon as possible. He predicts where you will end up staying if he warns you I AM COMING FOR YOU TOMORROW. If he predicts that you will end up in a city where you’ll die, he issues you this warning.
The response of Fiona the FDT-user is stated as follows:
When death gives Fiona a warning, she knows disease is coming, yet she doesn’t think this means she should flee. Rather, she imagines that if warnings could make her flee, she would not have received this warning but would have received other warnings and run headlong into asteroids. With this in mind, Fiona decides to act such that she’s told I AM COMING FOR YOU TOMORROW as seldom and as late as possible. When death finally does issue her a warning, she does not panic; for she imagines that if she acted differently, then she would have been given different, more frequent, and earlier warnings.
Fiona refuses to imagine a hypothetical world where she acts differently but receives the same warnings. This is the mechanism by which FDT outperforms CDT in so many situations. Both agents rank actions according to how they imagine the world would look if they took that action; but the FDT agent only imagines worlds where perfect predictors would have predicted differently had she acted differently.Now, this sounds to me very rational, from a "rationalists win" point of view. It also seems compatible with Gary Drescher's reasoning in "Good and Real", who supports things like one boxing on Newcomb's problem with both boxes transparent (from Drescher, but I use my own wording below)
Tranparent Newcomb v1 Boxes A and B are both transparent, box A contains $1000. Omega has put $1 million in box B iff he expects you to choose only box B given that you see $1 million in box B.
You can make it even more hardcore:
Tranparent Newcomb v2 Boxes A and B are both completely transparent, A has $1000. Omega has put $1 million in box B iff he expects you to choose only box B in both the following situations: given that you see $1 million in box B, and given that you see that box B is empty.It might be psychologically tough to take only an empty box B, but if you want to win, better be prepared to do it or else you won't get the $1 million.
In Asteroids, Fiona is choosing the decision-making process in part due to its effect on something that she already knows has happened (the I AM COMING FOR YOU TOMORROW warning). Since she is guaranteed to die after receiving the warning, if Fiona did not take the effect of her decision-making process on the occurrence of the warning into account, the only effect of her decision is the $1 or $2. So a version ignoring effects on already-observed events would still stay in Damascus, but only to save $1, and would switch if the dollar amounts were reversed.
Before reading this, I hadn't been imagining that FDT as proposed in the paper would try to influence already-observed things. I hadn't read the equations, but just had an impression. So I went back to read the equations, and read up on Pearl's do-calculus so I could try to understand the equations.
So: does FDT as presented mathematically in that paper actually work that way?
Here is their official equation formulating FDT:
The inner P is a probability distribution on the values assigned to nodes of the graph G which represents the problem under consideration, but is it the probability distribution that one would assume from the problem statement itself, without any further info obtained by the time the decision is made, or is it the probability distribution modified by conditioning on later obtained info? I'll leave this issue aside for the moment and go on considering the rest of the equation...
do(FDT(P,G) = a)
This seems to mean that you take the graph G and probability distribution P on its nodes - whatever that probability distribution P is since as I said I don't know whether it includes information obtained after the problem statement - and then assume that the output of FDT when applied to this problem description is the action a. This assumption is applied (the do notation) to modify the graph G so that a node of G representing the output of FDT is set to a, and the probability of nodes pointed to directly or indirectly by that node is adjusted accordingly.
P(s | do(FDT(P,G) = a) )
OK, so this means you take the graph outputted by the do and find the probability that the world is in the state s given that outputted graph. But is the probability obtained directly by reading it off the graph without plugging in the observed information? Or is it probability conditional on the observed information? I'll leave this issue aside for the moment too...
P(s | do(FDT(P,G) = a) )u(o[a,s])
This means you multiply that probability that the world is in the state s determined from that outputted graph by the utility u of the outcome o that results from taking action a in state s.
Finally, you sum this over all possible states s of the world to get the utility U assigned by FDT to action a.
So that's what (I think) I understand about that equation. What I don't understand is how this equation makes use of information that wasn't in the problem statement, but obtained later. Clearly a sensible decision maker has to make use of some information not in the problem statement or it will only get the money half the time in the following problem:
Should be free money for any decision theory A random process deposits some money in either box A or box B. You can only choose 1 box. Omega helpfully tells you which box it is in, no catch.
It seems the information must come in via either the inner P, the outer P or both, but I don't know which, as I noted above when discussing those Ps.
First, I'll assume that the outer P is conditional on the known information, regardless of what the inner P is.
In that case FDT doesn't take into account the effects on already observed info. Because that's all undone by the conditionalization on it. Consider the application to the Asteroids in Aleppo problem:
P((disease in Damascus) | do(FDT(P,G) = Damascus), observation of warning) = 1
and
P((asteroids in Aleppo) | do(FDT(P,G) = Aleppo), observation of warning) = 1.
So the only u's attached to nonzero probabilities are the ones that involve dying, resulting in FDT choosing just according to price, not to avoid death.
What if, on the other hand, it's the inner P that is conditional on the known information? (In this case the outer P could be interpreted as simply "reading off" values from the graph as modified by do).
The above argument will still apply, unless the do overrides this known information so that the final probabilities need not be compatible with it. Can do do that?
Since I didn't know how do worked, except a vague impression, I had to read up on it to determine if it could or not. After some review I think I've more or less figured out how it works. The arrows in the graph represent what nodes are dependent from others, so that the probability distribution for each node can be represented as conditional (only, directly) on the nodes with arrows pointing to it. These are in turn conditional on the nodes that point to them and so on. So, the probability distribution of each node is a product of a bunch of conditional probabilities. do sets one node (or a set of nodes) to a particular value or values, deletes all arrows pointing inward to those nodes, and projects forwards from the nodes set along the arrows pointing outward. This thing I call "projecting forward" is just keeping all the conditional probability distributions the same, except for the one(s) set to particular values by do.
Now, all probabilities are implicitly conditional on a body of knowledge. So, regardless of what other methods there may be to represent gaining information, one valid way ought to be adding the observation to the body of knowledge that all the probability distributions represented in the graph are conditional on. This means, changing the probability distributions to account for the new information (which may also change the graph, depending on the circumstances).
Now, if due to an observation one of the conditional probability distributions now say that the node "warning received" is set to "yes" no matter what the other nodes are, then do cannot change that, because do doesn't change the conditional probabilities, it only adjusts the values of nodes keeping the conditional probabilities the same.
(Unless of course that node is itself the one set by do (but in fact, I would regard the observation as restricting the allowed set of values so that do conflicting with it would be invalid).)
So, the FDT formula doesn't work as the verbal descriptions imply, regardless of what the inner and outer P's are conditional on, as either it doesn't take information into account at all, or it views information already received fatalistically.
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As an aside, the paper provides an analogous formula for CDT, but I think the analogy is somewhat misleading. The CDT formula looks superficially similar:
But since DEATH is assumed to have supernatural prediction capability, it can't make a difference if do is applied to the act or the algorithm in the Asteroids scenario. So, clearly some other difference must be intended, but none is apparent from the formulas provided.
I might speculate that CDT is supposed to have the outer P be conditional on observations, and maybe FDT is supposed to have the inner P conditional on observations with some super-do that does override observations. If so, these differences ought to be made clear.
Note: originally, before I figured out how do works, my argument was based on the analogy to the formula for CDT - i.e. that if the analogy was not misleading, then the differences that were apparent could not account for the differences in their treatment of Asteroids. But I considered that a weak argument since sloppy notation can easily result in misleading analogies even if an underlying correct meaning of the math was actually intended.
P.S. this was supposed to be one of those "simpler stuff" but it got a bit more complicated...I just persevered enough to think I may have got something worth posting this time.
This post has been heavily edited since I made it, as I learned more (initially I was not confident due to insufficient knowledge on how do worked). I may have done more with the edits in the end to cover my tracks (how I bumblingly came across the knowledge I present) than to actually clarify things...anyway I haven't posted a link to this blog at the time of this edit, so I'm a bit more liberal with my edits than I would have been if it had been likely that anyone else had read it.
