Caspar42

How to formalize predictors

As I mentioned elsewhere, I don't really understand...

>I think (1) is a poor formalization, because the game tree becomes unreasonably huge

What game tree? Why represent these decision problems as any kind of trees or game trees in particular? At least some problems of this type can be represented efficiently, using various methods to represent functions on the unit simplex (including decision trees)... Also: Is this decision-theoretically relevant? That is, are you saying, a good decision theory doesn't have to deal with 1 because it is cumbersome to write out (some) problems of this type? But *why* is this decision-theoretically relevant?

>some strategies of the predictor (like "fill the box unless the probability of two-boxing is exactly 1") leave no optimal strategy for the player.

Well, there are less radical ways of addressing this. E.g., expected utility-type theories just assign a preference order to the set of available actions. We could be content with that and accept that in some cases, there is no optimal action. As long as our decision theory ranks the available options in the right order... Or we could restrict attention to problems where an optimal strategy exists despite this dependence.

>And (3) seems like a poor formalization because it makes the predictor work too hard. Now it must predict all possible sources of randomness you might use, not just your internal decision-making.

For this reason, I always assume that predictors in my Newcomb-like problems are compensated appropriately and don't work on weekends! Seriously, though: what does "too hard" mean here? Is this just the point that it is in practice easy to construct agents that cannot be realistically predicted in this way when they don't want to be predicted? If so: I find that at least somewhat convincing, though I'd still be interested in developing theory that doesn't hinge on this ability.

Extracting Money from Causal Decision Theorists

On the more philosophical points. My position is perhaps similar to Daniel K's. But anyway...

Of course, I agree that problems that punish the agent for using a particular theory (or using float multiplication or feeling a little wistful or stuff like that) are "unfair"/"don't lead to interesting theory". (Perhaps more precisely, I don't think our theory needs to give algorithms that perform optimally in such problems in the way I want my theory to "perform optimally" Newcomb's problem. Maybe we should still expect our theory to say something about them, in the way that causal decision theorists feel like CDT has interesting/important/correct things to say about Newcomb's problem, despite Newcomb's problem being designed to (unfairly, as they allege) reward non-CDT agents.)

But I don't think these are particularly similar to problems with predictions of the agent's distribution over actions. The distribution over actions is behavioral, whereas performing floating point operations or whatever is not. When randomization is allowed, the subject of your choice is which distribution over actions you play. So to me, which distribution over actions you choose in a problem with randomization allowed, is just like the question of which action you take when randomization is not allowed. (Of course, if you randomize to determine which action's expected utility to calculate first, but this doesn't affect what you do in the end, then I'm fine with not allowing this to affect your utility, because it isn't behavioral.)

I also don't think this leads to uninteresting decision theory. But I don't know how to argue for this here, other than by saying that CDT, EDT, UDT, etc. don't really care whether they choose from/rank a set of distributions or a set of three discrete actions. I think ratificationism-type concepts are the only ones that break when allowing discontinuous dependence on the chosen distribution and I don't find these very plausible anyway.

To be honest, I don't understand the arguments against predicting distributions and predicting actions that you give in that post. I'll write a comment on this to that post.

Extracting Money from Causal Decision Theorists

Let's start with the technical question:

>Can your argument be extended to this case?

No, I don't think so. Take the class of problems. The agent can pick any distribution over actions. The final payoff is determined only as a function of the implemented action and some finite number of samples generated by Omega from that distribution. Note that the expectation is continuous in the distribution chosen. It can therefore be shown (using e.g. Kakutani's fixed-point theorem) that there is always at least one ratifiable distribution. See Theorem 3 at https://users.cs.duke.edu/~ocaspar/NDPRL.pdf .

(Note that the above is assuming the agent maximizes expected vNM utility. If, e.g., the agent maximizes some lexical utility function, then the predictor can just take, say, two samples and if they differ use a punishment that is of a higher lexicality than the other rewards in the problem.)

Extracting Money from Causal Decision Theorists

Excellent - we should ask THEM about it.

Yes, that's the plan.

Some papers that express support for CDT:

- https://philarchive.org/archive/ARMCDT-2
- https://link.springer.com/article/10.1007/s11229-011-0022-6 (In general, James Joyce is a well-known defender of CDT.)
- https://link.springer.com/article/10.1007/s11098-018-1206-4
- https://onlinelibrary.wiley.com/doi/full/10.1111/phpr.12466 (argues for two-boxing, but against CDT)
- In his book
*Causality*, Judea Pearl also argues in favor of CDT (though he doesn't explicitly discuss Newcomb's problem).

In case you just want to know why I believe support for CDT/two-boxing to be wide-spread among academic philosophers, see https://philpapers.org/archive/BOUWDP.pdf , which is a survey of academic philosophers, where more people preferred two-boxing than one-boxing in Newcomb's problem, especially among philosophers with relevant expertise. Some philosophers have filled out this survey publicly, so you can e.g. go to https://philpapers.org/surveys/public_respondents.html?set=Target+faculty , click on a name and then on "My Philosophical Views" to find individuals who endorse two-boxing. (I think there's also a way to download the raw data and thereby get a list of two-boxers.)

Extracting Money from Causal Decision Theorists

Note that while people on this forum mostly reject orthodox, two-boxing CDT, many academic philosophers favor CDT. I doubt that they would view this problem as out of CDT's scope, since it's pretty similar to Newcomb's problem.

How does this CDT agent reconcile a belief that the seller's prediction likelihood is different from the buyer's success likelihood?

Good question!

Extracting Money from Causal Decision Theorists

I agree with both of Daniel Kokotajlo's points (both of which we also make in the paper in Sections IV.1 and IV.2): Certainly for humans it's normal to not be able to randomize; and even if it was a primarily hypothetical situation without any obvious practical application, I'd still be interested in knowing how to deal with the absence of the ability to randomize.

Besides, as noted in my other comment insisting on the ability to randomize doesn't get you that far (cf. Sections IV.1 and IV.4 on Ratificationism): even if you always have access to some nuclear decay noise channel, your choice of whether to consult that channel (or of whether to factor the noise into your decision) is still deterministic. So you can set up scenarios where if you are punished for randomizing. In the particular case of the Adversarial Offer, the seller might remove all money from both boxes if she predicts the buyer to randomize.

The reason why our main scenario just assumes that randomization isn't possible is that our target of attack in this paper is primarily CDT, which is fine with not being allowed to randomize.

Extracting Money from Causal Decision Theorists

I think some people may have their pet theories which they call CDT and which require randomization. But CDT as it is usually/traditionally described doesn't ever insist on randomizing (unless randomizing has a positive *causal* effect). In this particular case, even if a randomization device were made available, CDT would either uniquely favor one of the boxes or be indifferent between all distributions over . Compare Section IV.1 of the paper.

What you're referring to are probably so-called ratificationist variants of CDT. These would indeed require randomizing 50-50 between the two boxes. But one can easily construct scenarios which trip these theories up. For example, the seller could put no money in any box if she predicts that the buyer will randomize. Then no distribution is ratifiable. See Section IV.4 for a discussion of Ratificationism.

Extracting Money from Causal Decision Theorists

Yeah, basically standard game theory doesn't really have anything to say about the scenarios of the paper, because they don't fit the usual game-theoretical models.

By the way, the paper has some discussion of what happens if you insist on having access to an unpredictable randomization device, see Sections IV.1 and the discussion of Ratificationism in Section IV.4. (The latter may be of particular interest because Ratificationism is somewhat similar to Nash equilibrium. Unfortunately, the section doesn't explain Ratificationism in detail.)

Extracting Money from Causal Decision Theorists

>I think information "seller's prediction is accurate with probability 0,75" is supposed to be common knowledge.

Yes, correct!

>Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutely anything if your premise contains contradiction.

Why would it be a logical contradiction? Do you think Newcomb's problem also requires a logical contradiction? Note that in neither of these cases does the predictor tell the agent the *result* of a prediction about the agent.

>What kinds of mistakes does seller make?

For the purpose of the paper it doesn't really matter what beliefs anyone has about how the errors are distributed. But you could imagine that the buyer is some piece of computer code and that the seller has an identical copy of that code. To make a prediction, the seller runs the code. Then she flips a coin twice. If the coin does *not* come up Tails twice, she just uses that prediction and fills the boxes accordingly. If the coin *does* come up Tails twice, she uses a third coin flip to determine whether to (falsely) predict one of the two other options that the agent can choose from. And then you get the 0.75, 0.125, 0.125 distribution you describe. And you could assume that this is common knowledge.

Of course, for the exact CDT expected utilities, it does matter how the errors are distributed. If the errors are primarily "None" predictions, then the boxes should be expected to contain more money and the CDT expected utilities of buying will be higher. But for the exploitation scheme, it's enough to show that the CDT expected utilities of buying are strictly positive.

>When you write "$1−P (money in Bi | buyer chooses Bi ) · $3 = $1 − 0.25 · $3 = $0.25.", you assume that P(money in Bi | buyer chooses Bi )=0.75.

I assume you mean that I assume P(money in Bi | buyer chooses Bi )=0.25? Yes, I assume this, although really I assume that the seller's prediction is accurate with probability 0.75 and that she fills the boxes according to the specified procedure. From this, it then follows that P(money in Bi | buyer chooses Bi )=0.25.

>That is, if buyer chooses the first box, seller can't possibly think that buyer will choose none of the boxes.

I don't assume this / I don't see how this would follow from anything I assume. Remember that if the seller predicts the buyer to choose no box, both boxes will be filled. So even if *all* false predictions would be "None" predictions (when the buyer buys a box), then it would still be P(money in Bi | buyer chooses Bi )=0.25.

Sorry for taking some time to reply!

>You might wonder why am I spouting a bunch of wrong things in an unsuccessful attempt to attack your paper.

Nah, I'm a frequent spouter of wrong things myself, so I'm not too surprised when other people make errors, especially when the stakes are low, etc.

Re 1,2: I guess a lot of this comes down to convention. People have found that one can productively discuss these things without always giving the formal models (in part because people in the field know how to translate everything into formal models). That said, if you want mathematical models of CDT and Newcomb-like decision problems, you can check the Savage or Jeffrey-Bolker formalizations. See, for example, the first few chapters of Arif Ahmed's book, "Evidence, Decision and Causality". Similarly, people in decision theory (and game theory) usually don't specify what is common knowledge, because usually it is assumed (implicitly) that the entire problem description is common knowledge / known to the agent (Buyer). (Since this is decision and not game theory, it's not quite clear what "common knowledge" means. But presumably to achieve 75% accuracy on the prediction, the seller needs to know that the buyer understands the problem...)

3: Yeah, *there exist* agent models under which everything becomes inconsistent, though IMO this just shows these agent models to be unimplementable. For example, take the problem description from my previous reply (where Seller just runs an exact copy of Buyer's source code). Now assume that Buyer knows his source code and is logically omniscient. Then Buyer knows what his source code chooses and therefore knows the option that Seller is 75% likely to predict. So he will take the other option. But of course, this is a contradiction. As you'll know, this is a pretty typical logical paradox of self-reference. But to me it just says that this logical omniscience assumption about the buyer is implausible and that we should consider agents who aren't logically omniscient. Fortunately, CDT doesn't assume knowledge of its own source code and such.

Perhaps one thing to help sell the plausibility of this working: For the purpose of the paper, the assumption that Buyer uses CDT in this scenario is pretty weak, formally simple and doesn't have much to do with logic. It just says that the Buyer assigns some probability distribution over box states (i.e., some distribution over the mutually exclusive and collectively exhaustive s1="money only in box 1", s2= "money only in box 2", s3="money in both boxes"); and that given such distribution, Buyer takes an action that maximizes (causal) expected utility. So you could forget agents for a second and just prove the formal claim that for all probability distributions over three states s1, s2, s3, it is for i=1 or i=2 (or both) the case that

(P(si)+P(s3))*$3 - $1 > 0.

I assume you don't find this strange/risky in terms of contradictions, but mathematically speaking, nothing more is really going on in the basic scenario.

The idea is that everyone agrees (hopefully) that orthodox CDT satisfies the assumption. (I.e., assigns some unconditional distribution, etc.) Of course, many CDTers would claim that CDT satisfies some *additional* assumptions, such as the probabilities being calibrated or "correct" in some other sense. But of course, if "A=>B", then "A and C => B". So adding assumptions cannot help the CDTer avoid the loss of money conclusion if they also accept the more basic assumptions. Of course, *some* added assumptions lead to contradictions. But that just means that they cannot be satisfied in the circumstances of this scenario if the more basic assumption is satisfied and if the premises of the Adversarial Offer help. So they would have to either adopt some non-orthodox CDT that doesn't satisfy the basic assumption or require that their agents cannot be copied/predicted. (Both of which I also discuss in the paper.)

>you assume that Buyer knows the probabilities that Seller assigned to Buyer's actions.

No, if this were the case, then I think you would indeed get contradictions, as you outline. So Buyer does *not* know what Seller's prediction is. (He only knows her prediction is 75% accurate.) If Buyer uses CDT, then of course he assigns some (unconditional) probabilities to what the predictions are, but of course the problem description implies that these predictions aren't particularly good. (Like: if he assigns 90% to the money in box 1, then it immediately follows that *no* money is in box 1.)