A Defense of Functional Decision Theory

This post is an attempt to refute an article offering critique on Functional Decision Theory (FDT). If you’re new to FDT, I recommend reading this introductory paper by Eliezer Yudkowsky & Nate Soares (Y&S). The critique I attempt to refute can be found here: A Critique of Functional Decision Theory by wdmacaskill. I strongly recommend reading it before reading this response post.

The article starts with descriptions of Causal Decision Theory (CDT), Evidential Decision Theory (EDT) and FDT itself. I’ll get right to the critique of FDT in this post, which is the only part I’m discussing here.

“FDT sometimes makes bizarre recommendations”

The article claims “FDT sometimes makes bizarre recommendations”, and more specifically, that FDT violates guaranteed payoffs. The following example problem, called Bomb, is given to illustrate this remark:


You face two open boxes, Left and Right, and you must take one of them. In the Left box, there is a live bomb; taking this box will set off the bomb, setting you ablaze, and you certainly will burn slowly to death. The Right box is empty, but you have to pay $100 in order to be able to take it.

A long-dead predictor predicted whether you would choose Left or Right, by running a simulation of you and seeing what that simulation did. If the predictor predicted that you would choose Right, then she put a bomb in Left. If the predictor predicted that you would choose Left, then she did not put a bomb in Left, and the box is empty.

The predictor has a failure rate of only 1 in a trillion trillion. Helpfully, she left a note, explaining that she predicted that you would take Right, and therefore she put the bomb in Left.

You are the only person left in the universe. You have a happy life, but you know that you will never meet another agent again, nor face another situation where any of your actions will have been predicted by another agent. What box should you choose?”

The article answers and comments on the answer as follows:

“The right action, according to FDT, is to take Left, in the full knowledge that as a result you will slowly burn to death. Why? Because, using Y&S’s counterfactuals, if your algorithm were to output ‘Left’, then it would also have outputted ‘Left’ when the predictor made the simulation of you, and there would be no bomb in the box, and you could save yourself $100 by taking Left. In contrast, the right action on CDT or EDT is to take Right.

The recommendation is implausible enough. But if we stipulate that in this decision-situation the decision-maker is certain in the outcome that her actions would bring about, we see that FDT violates Guaranteed Payoffs.”

I agree FDT recommends taking the left box. I disagree that it violates some principle every decision theory should adhere to. Left-boxing really is the right decision in Bomb. Why? Let’s ask ourselves the core question of FDT:

“Which output of this decision procedure causes the best outcome?”

The answer can only be left-boxing. As wdmacaskill says:

“…if your algorithm were to output ‘Left’, then it would also have outputted ‘Left’ when the predictor made the simulation of you, and there would be no bomb in the box, and you could save yourself $100 by taking Left.”

But since you already know the bomb in Left, you could easily save your life by paying $100 in this specific situation, and that’s where our disagreement comes from. However, remember that if your decision theory makes you a left-boxer, you virtually never end up in the above situation! In 999,999,999,999,999,999,999,999 out of 1,000,000,000,000,000,000,000,000 situations, the predictor will have predicted you left-box, letting you keep your life for free. As Vaniver says in a comment:

“Note that the Bomb case is one in which we condition on the 1 in a trillion trillion failure case, and ignore the 999999999999999999999999 cases in which FDT saves $100. This is like pointing at people who got into a plane that crashed and saying ‘what morons, choosing to get on a plane that would crash!’ instead of judging their actions from the state of uncertainty that they were in when they decided to get on the plane.”

“FDT fails to get the answer Y&S want in most instances of the core example that’s supposed to motivate it”

Here wdmacaskill argues that in Newcomb’s problem, FDT recommends one-boxing if it assumes the predictor (Omega) is running a simulation of the agent’s decision process. But what if Omega isn’t running your algorithm? What if they use something else to predict your choice? To use wdmacaskill’s own example:

“Perhaps the Scots tend to one-box, whereas the English tend to two-box.”

Well, in that case Omega’s prediction and your decision (one-boxing or two-boxing) aren’t subjunctively dependent on the same function. And this kind of dependence is key in FDT’s decision to one-box! Without it, FDT recommends two-boxing, like CDT. In this particular version of Newcomb’s problem, your decision procedure has no influence on Omega’s prediction, and you should go for strategic dominance (two-boxing).

However, wdmacaskill argues that part of the original motivation to develop FDT was to have a decision theory that one-boxes on Newcomb’s problem. I don’t care what the original motivation for FDT was with respect to this discussion. What matters is whether FDT gets Newcomb’s problem right — and it does so in both cases: when Omega does run a simulation of your decision process and when Omega does not.

Alternatively, wdmacaskill argues,

“Y&S could accept that the decision-maker should two-box in the cases given above. But then, it seems to me, that FDT has lost much of its initial motivation: the case for one-boxing in Newcomb’s problem didn’t seem to stem from whether the Predictor was running a simulation of me, or just using some other way to predict what I’d do.”

Again, I do not care where the case for one-boxing stemmed from, or what FDT’s original motivation was: I care about whether FDT gets Newcomb’s problem right.

“Implausible discontinuities”

“First, take some physical processes S (like the lesion from the Smoking Lesion) that causes a ‘mere statistical regularity’ (it’s not a Predictor). And suppose that the existence of S tends to cause both (i) one-boxing tendencies and (ii) whether there’s money in the opaque box or not when decision-makers face Newcomb problems. If it’s S alone that results in the Newcomb set-up, then FDT will recommending two-boxing.”

Agreed. The contents of the opaque box and the agent’s decision to one-box or two-box don’t subjunctively depend on the same function. FDT would indeed recommend two-boxing.

“But now suppose that the pathway by which S causes there to be money in the opaque box or not is that another agent looks at S and, if the agent sees that S will cause decision-maker X to be a one-boxer, then the agent puts money in X’s opaque box. Now, because there’s an agent making predictions, the FDT adherent will presumably want to say that the right action is one-boxing.”

No! The critical factor isn’t whether “there’s an agent making predictions”. The critical factor is subjunctive dependence between the agent and another relevant physical system (in Newcomb’s problem, that’s Omega’s prediction algorithm). Since in this last problem put forward by wdmacaskill the prediction depends on looking at S, there is no such subjunctive dependence going on and FDT would recommend two-boxing.

Wdmacaskill further asks the reader to imagine a spectrum of a more and more agent-like S, and imagines that at some-point there will be a “sharp jump” where FDT goes from recommending two-boxing to recommending one-boxing. Wdmacaskill then says:

“Second, consider that same physical process S, and consider a sequence of Newcomb cases, each of which gradually make S more and more complicated and agent-y, making it progressively more similar to a Predictor making predictions. At some point, on FDT, there will be a point at which there’s a sharp jump; prior to that point in the sequence, FDT would recommend that the decision-maker two-boxes; after that point, FDT would recommend that the decision-maker one-boxes. But it’s very implausible that there’s some S such that a tiny change in its physical makeup should affect whether one ought to one-box or two-box.”

But like I explained, the “agent-ness” of a physical system is totally irrelevant for FDT. Subjunctive dependence is key, not agent-ness. The sharp jump between one-boxing and two-boxing wdmacaskill imagines there to be really isn’t there: it stems from a misunderstanding of FDT.

“FDT is deeply indeterminate”

Wdmacaskill argues that

“there’s no objective fact of the matter about whether two physical processes A and B are running the same algorithm or not, and therefore no objective fact of the matter of which correlations represent implementations of the same algorithm or are ‘mere correlations’ of the form that FDT wants to ignore.”

… and gives an example:

“To see this, consider two calculators. The first calculator is like calculators we are used to. The second calculator is from a foreign land: it’s identical except that the numbers it outputs always come with a negative sign (‘–’) in front of them when you’d expect there to be none, and no negative sign when you expect there to be one. Are these calculators running the same algorithm or not? Well, perhaps on this foreign calculator the ‘–’ symbol means what we usually take it to mean — namely, that the ensuing number is negative — and therefore every time we hit the ‘=’ button on the second calculator we are asking it to run the algorithm ‘compute the sum entered, then output the negative of the answer’. If so, then the calculators are systematically running different algorithms.

But perhaps, in this foreign land, the ‘–’ symbol, in this context, means that the ensuing number is positive and the lack of a ‘–’ symbol means that the number is negative. If so, then the calculators are running exactly the same algorithms; their differences are merely notational.”

I’ll admit I’m no expert in this area, but it seems clear to me that these calculators are running different algorithms, but that both algorithms are subjunctively dependent on the same function! Both algorithms use the same “sub-algorithm”, which calculates the correct answer to the user’s input. The second calculator just does something extra: put a negative sign in front of the answer or remove an existing one. Whether inhabitants of the foreign land interpret the ‘-’ symbol different than we do is irrelevant to the properties of the calculators.

“Ultimately, in my view, all we have, in these two calculators, are just two physical processes. The further question of whether they are running the same algorithm or not depends on how we interpret the physical outputs of the calculator.”

It really doesn’t. The properties of both calculators do NOT depend on how we interpret their outputs. Wdmacaskill uses this supposed dependence on interpretation to undermine FDT: in Newcomb’s problem, it would also be a matter of choice of interpretation whether Omega is running the same algorithm as you are in order to predict your choice. However, as interpretation isn’t a property of any algorithm, this becomes a non-issue. I’ll be doing a longer post on algorithm dependence/​similarity in the future.

“But FDT gets the most utility!”

Here, wdmacaskill talks about how Yudkowsky and Soares compare FDT to EDT and CDT to determine FDT’s superiority to the other two.

“As we can see, the most common formulation of this criterion is that they are looking for the decision theory that, if run by an agent, will produce the most utility over their lifetime. That is, they’re asking what the best decision procedure is, rather than what the best criterion of rightness is, and are providing an indirect account of the rightness of acts, assessing acts in terms of how well they conform with the best decision procedure.

But, if that’s what’s going on, there are a whole bunch of issues to dissect. First, it means that FDT is not playing the same game as CDT or EDT, which are proposed as criteria of rightness, directly assessing acts. So it’s odd to have a whole paper comparing them side-by-side as if they are rivals.”

I agree the whole point of FDT is to have a decision theory that produces the most utility over the lifetime of an agent — even if that, in very specific cases like Bomb, results in “weird” (but correct!) recommendations for specific acts. Looking at it from a perspective of AI Alignment — which is the goal of MIRI, the organization Yudkowsky and Soares work for — it seems clear to me that that’s what you want out of a decision theory. CDT and EDT may have been invented to play a different game — but that’s irrelevant for the purpose of FDT. CDT and EDT — the big contenders in the field of Decision theory — fail this purpose, and FDT does better.

“Second, what decision theory does best, if run by an agent, depends crucially on what the world is like. To see this, let’s go back to question that Y&S ask of what decision theory I’d want my child to have. This depends on a whole bunch of empirical facts: if she might have a gene that causes cancer, I’d hope that she adopts EDT; though if, for some reason, I knew whether or not she did have that gene and she didn’t, I’d hope that she adopts CDT. Similarly, if there were long-dead predictors who can no longer influence the way the world is today, then, if I didn’t know what was in the opaque boxes, I’d hope that she adopts EDT (or FDT); if I did know what was in the opaque boxes (and she didn’t) I’d hope that she adopts CDT. Or, if I’m in a world where FDT-ers are burned at the stake, I’d hope that she adopts anything other than FDT.”

Well, no, not really — that’s the point. What decision theory does best shouldn’t depend on what the world is like. The whole idea is to have a decision theory that does well under all (fair) circumstances. Circumstances that directly punish an agent for its decision theory can be made for any decision theory and don’t refute this point.

“Third, the best decision theory to run is not going to look like any of the standard decision theories. I don’t run CDT, or EDT, or FDT, and I’m very glad of it; it would be impossible for my brain to handle the calculations of any of these decision theories every moment. Instead I almost always follow a whole bunch of rough-and-ready and much more computationally tractable heuristics; and even on the rare occasions where I do try to work out the expected value of something explicitly, I don’t consider the space of all possible actions and all states of nature that I have some credence in — doing so would take years.

So the main formulation of Y&S’s most important principle doesn’t support FDT. And I don’t think that the other formulations help much, either. Criteria of how well ‘a decision theory does on average and over time’, or ‘when a dilemma is issued repeatedly’ run into similar problems as the primary formulation of the criterion. Assessing by how well the decision-maker does in possible worlds that she isn’t in fact in doesn’t seem a compelling criterion (and EDT and CDT could both do well by that criterion, too, depending on which possible worlds one is allowed to pick).”

Okay, so we’d need an approximation of such a decision theory — I fail to see how this undermines FDT.

“Fourth, arguing that FDT does best in a class of ‘fair’ problems, without being able to define what that class is or why it’s interesting, is a pretty weak argument. And, even if we could define such a class of cases, claiming that FDT ‘appears to be superior’ to EDT and CDT in the classic cases in the literature is simply begging the question: CDT adherents claims that two-boxing is the right action (which gets you more expected utility!) in Newcomb’s problem; EDT adherents claims that smoking is the right action (which gets you more expected utility!) in the smoking lesion. The question is which of these accounts is the right way to understand ‘expected utility’; they’ll therefore all differ on which of them do better in terms of getting expected utility in these classic cases.”

Yes, fairness would need to be defined exactly, although I do believe Yudkowsky and Soares have done a good job at it. And no: “claiming that FDT ‘appears to be superior’ to EDT and CDT in the classic cases in the literature” isn’t begging the question. The goal is to have a decision theory that consistently gives the most expected utility. Being a one-boxer does give you the most expected utility in Newcomb’s problem. Deciding to two-box after Omega made his prediction that you one-box (if this is possible) would give you the most utility — but you can’t have your decision theory recommending two-boxing, because that results in the opaque box being empty.

In conclusion, it seems FDT survives the critique offered by wdmacaskill. I am quite new to the field of Decision theory, and will be learning more and more about this amazing field in the coming weeks. This post might be updated as I learn more.