[This post assumes knowledge of decision theory, as discussed in Eliezer Yudkowsky’s Timeless Decision Theory and in Arbital’s Introduction to Logical Decision Theory.]

I recently discovered an interesting thought experiment, “Betting on the Past” by Cambridge philosopher Arif Ahmed. It can be found in his book Evidence, Decision and Causality, which is an elaborate defense of Evidential Decision Theory (EDT). I believe that Betting on the Past may be used to money-pump non-EDT agents, refuting Causal Decision Theories (CDT), and potentially even ones that use logical conditioning, such as Timeless Decision Theory (TDT) or Updateless Decision Theory (UDT). At the very least, non-EDT decision theories are unlikely to win this bet. Moreover, no conspicuous perfect predicting powers, genetic influences, or manipulations of decision algorithms are required to make Betting on the Past work, and anyone can replicate the game at home. For these reasons, it might make a more compelling case in favor of EDT than the Coin Flip Creation, a problem I recently proposed in an attempt to defend EDT’s answers in medical Newcomb problems. In Ahmed’s thought experiment, Alice faces the following decision problem:

Betting on the Past: In my pocket (says Bob) I have a slip of paper on which is written a proposition P. You must choose between two bets. Bet 1 is a bet on P at 10:1 for a stake of one dollar. Bet 2 is a bet on P at 1:10 for a stake of ten dollars. So your pay-offs are as in [Figure 1]. Before you choose whether to take Bet 1 or Bet 2 I should tell you what P is. It is the proposition that the past state of the world was such as to cause you now to take Bet 2. [Ahmed 2014, p. 120]

Ahmed goes on to specify that Alice could indicate which bet she’ll take by either raising or lowering her hand. One can find a detailed discussion of the thought experiment’s implications, as well as a formal analysis of CDT’s and EDT’s decisions in Ahmed’s book. In the following, I want to outline a few key points.

Would CDT win in this problem? Alice is betting on a past state of the world. She can’t causally influence the past, and she’s uncertain whether the proposition is true or not. In either case, Bet 1 strictly dominates Bet 2: no matter which state the past is in, Bet 1 always yields a higher utility. For these reasons, causal decision theories would take Bet 1. Nevertheless, as soon as Alice comes to a definite decision, she updates on whether the proposition is true or false. If she’s a causal agent, she then finds out that she has lost: the past state of the world was such as to cause her to take Bet 1, so the proposition is false. If she had taken Bet 2, she would have found out that the proposition was correct, and she would have won, albeit a smaller amount than if she had won with Bet 1.

Betting on the Past seems to qualify as a kind of Newcomb’s paradox; it seems to have an equivalent payoff matrix (Figure 1).

Figure 1: Betting on the past has a similar payoff matrix to Newcomb’s paradox

	P is true	P is false
Take Bet 1	10	-1
Take Bet 2	1	-10

Furthermore, its causal structure seems to resemble those of e.g. the Smoking Lesion or Solomon’s problem, indicating it as a kind of medical Newcomb problem. In medical Newcomb problems, a “Nature” node determines both the present state of the world (whether the agent is sick/will win the bet) and the agent’s decision (see Figure 2). In this regard, they differ from Newcomb’s original problem, where said node refers to the agent’s decision algorithm.

Figure 2: Betting on the past (left) has a similar causal structure to medical Newcomb problems (right).

One could object to Betting on the Past being a medical Newcomb problem, since the outcomes conditional on our actions here are certain, while e.g. in the Smoking Lesion, observing our actions only shifts our probabilities in degrees. I believe this shouldn’t make a crucial difference. On the one hand, we can conceive of absolutely certain medical Newcomb cases like the Coin Flip Creation. On the other hand, Newcomb’s original problem is often formalized with absolute certainties as well. I’d be surprised if probabilistic vs. certain reasoning would make a difference to decision theories. First, we can always approximate certainties to an arbitrarily high degree. We might ask ourselves why a negligible further increase in certainty would at some point suddenly completely change the recommended action, then. Secondly, we’re never really certain in the real world anyway, so if the two cases would be different, this would render all thought experiments useless that use absolute certainties.

If Betting on the Past is indeed a kind of medical Newcomb problem, this would be an interesting conclusion. It would follow that if one prefers Bet 2, one should also one-box in medical Newcomb problems. And taking Bet 2 seems so obviously correct! I point this out because one-boxing in medical Newcomb problems is what EDT would do, and it is often put forward as both a counterexample to EDT and as the decision problem that separates EDT from Logical Decision Theories (LDT), such as TDT or UDT. (See e.g. Yudkowsky 2010, p.67)

Before we examine the case for EDT further, let’s take a closer look at what LDTs would do in Betting on the Past. As far as I understand, LDTs would take correlations with other decision algorithms into account, but they would ignore “retrocausality” (i.e. smoke in the smoker’s lesion, chew gum in the chewing gum problem, etc.). If there is a purely physical cause, then this causal node isn’t altered in the logical counterfactuals that an LDT agent reasons over. Perhaps if the bet was about the state of the world yesterday, LDT would still take Bet 2. Clearly, LDT’s algorithm already existed yesterday, and it can influence this algorithm’s output; so if it chooses Bet 2, it can change yesterday’s world and make the proposition true. But at some point, this reasoning has to break down. If we choose a more distant point in the past as a reference for Alice’s bet – maybe as far back as the birth of our universe – she’ll eventually be unable to exert any possible influence via logical counterfactuals. At some point, the correlation becomes a purely physical one. All she can do at that point is what opponents of evidential reasoning would call “managing the news” (Lewis, 1981) – she can merely try to go for the action that gives her the best Bayesian update.

So, do Logical Decision Theories get it wrong? I’m not sure about that; they come in different versions, and some haven’t yet been properly formalized, so it’s hard for me to judge. I can very well imagine that e.g. Proof-Based Decision Theory would take Bet 2, since it could prove P to be either true or false, contingent on the action it would take. I would argue, though, that if a decision theory takes Bet 2 – and if I’m right about Betting on the Past being a medical Newcomb problem – then it appears it would also have to “one-box”, i.e. take the option recommended by EDT, in other medical Newcomb problems.

If all of this is true, it might imply that we don’t really need LDT’s logical conditioning and that EDT’s simple Bayesian conditioning on actions could suffice. The only remaining difference between LDT and EDT would then be EDT’s lack of updatelessness. What would an updateless version of EDT look like? Some progress on this front has already been made by Everitt, Leike, and Hutter 2015. Caspar Oesterheld and I hope to be able to say more about it soon ourselves.

Acknowledgement

I wrote this post while working for the Foundational Research Institute, which is now the Center on Long-Term Risk.

While I disagree with James M. Joyce on the correct solution to Newcomb’s problem, I agree with him that the standard framing of Newcomb’s problem (from Nozick 1969) can be improved upon. Indeed, I very much prefer the framing he gives in chapter 5.1 of The Foundations of Causal Decision Theory, which (according to Joyce) is originally due to JH Sobel:

Suppose there is a brilliant (and very rich) psychologist who knows you so well that he can predict your choices with a high degree of accuracy. One Monday as you are on the way to the bank he stops you, holds out a thousand dollar bill, and says: “You may take this if you like, but I must warn you that there is a catch. This past Friday I made a prediction about what your decision would be. I deposited $1,000,000 into your bank account on that day if I thought you would refuse my offer, but I deposited nothing if I thought you would accept. The money is already either in the bank or not, and nothing you now do can change the fact. Do you want the extra $1,000?” You have seen the psychologist carry out this experiment on two hundred people, one hundred of whom took the cash and one hundred of whom did not, and he correctly forecast all but one choice. There is no magic in this. He does not, for instance, have a crystal ball that allows him to “foresee” what you choose. All his predictions were made solely on the basis of knowledge of facts about the history of the world up to Friday. He may know that you have a gene that predetermines your choice, or he may base his conclusions on a detailed study of your childhood, your responses to Rorschach tests, or whatever. The main point is that you now have no causal influence over what he did on Friday; his prediction is a fixed part of the fabric of the past. Do you want the money?

I prefer this over the standard framing because people can remember the offer and the balance of their bank account better than box 1 and box 2. For some reason, I also find it easier to explain this thought experiments without referring to the thought experiment itself in the middle of the explanation. So, now whenever I describe Newcomb’s problem, I start with Sobel’s rather than Nozick’s version.

Of course, someone who wants to explore decision theory more deeply also needs to learn about the standard version, if only because people sometimes use “one-boxing” and “two-boxing” (the options in Newcomb’s original problem) to denote the analogous choices in other thought experiments. (Even if there are no boxes in these other thought experiments!) But luckily it does not take more than a few sentences to describe the original Newcomb problem based on Sobel’s version. You only need to explain that Newcomb’s problem replaces your bank account with an opaque box whose content you always keep; and puts the offer into a second, transparent box. And then the question is whether you stick with one box or go home with both.