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My Current Take on Counterfactuals

Why is the price of the un-actualized bet constant? My argument in the OP was to suppose that PCH is the dominant hypothesis, so, mostly controls market prices.

Thinking about this in detail, it seems like what influence traders have on the market price depends on a lot more of their inner workings than just their beliefs. I was thinking in a way where each trader only had one price for the bet, below which they bought and above which they sold, no matter how many units they traded (this might contradict "continuous trading strategies" because of finite wealth), in which case there would be a range of prices that could be the "market" price, and it could stay constant even with one end of that range shifting. But there could also be an outcome like yours, if the agents demand better and better prices to trade one more unit of the bet.

My Current Take on Counterfactuals

So I don't see how we can be sure that PCH loses out overall. LCH has to exploit PCH -- but if LCH tries it, then we're seemingly in a situation where LCH has to sell for PCH's prices, in which case it suffers the loss I described in the OP.

So I've reread the logical induction paper for this, and I'm not sure I understand exploitation. Under 3.5, it says:

On each day, the reasoner receives 50¢ from T, but after day t, the reasoner must pay $1 every day thereafter.

So this sounds like before day t, T buys a share every day, and those shares never pay out - otherwise T would receive $t on day t in addition to everything mentioned here. Why?

In the version that I have in my head, theres a market with PCH and LCH in it that assigns constant price to the unactualised bet, so neither of them gain or lose anything with their trades on it, and LCH exploits PCH on the actualised ones.

But the special bundled contract doesn't go to zero like this, because the conditional contract only really pays out when the condition is satisfied or refuted.

So if I'm understanding this correctly: The conditional contract on (a|b) pays if a&b is proved, if a&~b is proved, and if ~a&~b is proved.

Now I have another question: how does logical induction arbitrage against contradiction? The bet on a pays $1 if a is proved. The bet on ~a pays $1 if not-a is proved. But the bet on ~a isn't "settled" when a is proved - why can't the market just go on believing its .7? (Likely this is related to my confusion with the paper).

My proposal is essentially similar to that, except I am trying to respect logic in most of the system, simply reducing its impact on action selection. But within my proposed system, I think the wrong 'prior' (ie distribution of wealth for traders) can make it susceptible again.

I'm not blocking Troll Bridge problems, I'm making the definition of rational agent broad enough that crossing is permissible. But if I think the Troll Bridge proof is actively irrational, I should be able to actually rule it out. IE, specify an X which is inconsistent with PA.

What makes you think that theres a "right" prior? You want a "good" learning mechanism for counterfactuals. To be good, such a mechanism would have to learn to make the inferences we consider good, at least with the "right" prior. But we can't pinpoint any wrong inference in Troll Bridge. It doesn't seem like whats stopping us from pinpointing the mistake in Troll Bridge is a lack of empirical data. So, a good mechanism would have to learn to be susceptible to Troll Bridge, especially with the "right" prior. I just don't see what would be a good reason for thinking theres a "right" prior that avoids Troll Bridge (other than "there just has to be some way of avoiding it"), that wouldn't also let us tell directly how to think about Troll Bridge, no learning needed.

My Current Take on Counterfactuals

The payoff for 2-boxing is dependent on beliefs after 1-boxing because all share prices update every market day and the "payout" for a share is essentially what you can sell it for.

If a sentence is undecidable, then you could have two traders who disagree on its value indefinitely: one would have a highest price to buy, thats below the others lowest price to sell. But then anything between those two prices could be the "market price", in the classical supply and demand sense. If you say that the "payout" of a share is what you can sell it for... well, the "physical causation" trader is also buying shares on the counterfactual option that won't happen. And if he had to sell those, he couldn't sell them at a price close to where he bought them - he could only sell them at how much the "logical causation" trader values them, and so both would be losing "payout" on their trades with the unrealized option. Thats one interpretation of "sell". If theres a "market maker" in addition to both traders, it depends on what prices he makes - and as outlined above, there is a wide range of prices that would be consistent for him to offer as a market maker, including ways which are very close to the logical traders valuations - in which case, the logical trader is gaining on the physical one.

Trying to communicate a vague intuition here: There is a set of methods which rely on there being a time when "everything is done", to then look back from there and do credit assignment for everything that happened before. They characteristically use backwards induction to prove things. I think markets fall into this: the argument for why ideal markets don't have bubbles is that eventually, the real value will be revealed, and so the bubble has to pop, and then someone holds the bag, and you don't want to be that someone, and people predicting this and trying to avoid it will make the bubble pop earlier, in the idealised case instantly. I also think these methods aren't going to work well with embedding. They essentially use "after the world" as a subsitute for "outside the world".

My claim is: eventually, if you observe enough cases of "crossing" in similar circumstances, your expectation for "cross" should be consistent with the empirical history

My question was more "how should this roughly work" rather than "what conditions should it fulfill", because I think thinking about this illuminates my next point.

The hope is that we can block the troll argument completely if proving B->A does not imply cf(A|B)=1

This doesn't help against what I'm imagining, I'm not touching indicative B->A. So, standard Troll Bridge:

  • Reasoning within PA (ie, the logic of the agent):
    • Suppose the agent crosses.
      • Further suppose that the agent proves that crossing implies U=-10.
        • Examining the source code of the agent, because we're assuming the agent crosses, either PA proved that crossing implies U=+10, or it proved that crossing implies U=0.
        • So, either way, PA is inconsistent -- by way of 0=-10 or +10=-10.
        • So the troll actually blows up the bridge, and really, U=-10.
      • Therefore (popping out of the second assumption), if the agent proves that crossing implies U=-10, then in fact crossing implies U=-10.
      • By Löb's theorem, crossing really implies U=-10.
      • So (since we're still under the assumption that the agent crosses), U=-10.
    • So (popping out of the assumption that the agent crosses), the agent crossing implies U=-10.
  • Since we proved all of this in PA, the agent proves it, and proves no better utility in addition (unless PA is truly inconsistent). On the other hand, it will prove that not crossing gives it a safe U=0. So it will in fact not cross.

But now, say the agents counterfactual reasoning comes not from PA, but from system X. Then the argument fails because "suppose the agent proves crossing->U=-10 in PA" doesn't go any further because examining the sourcecode of the agent doesn't say anything about PA anymore, and "suppose the agent proves crossing->U=-10 in X" doesn't show that PA is inconsistent, so the bridge isn't blown up. But lets have a troll that blows up the bridge if X is inconsistent. Then we can make an argument like this:

  • Reasoning within X (ie, the logic of counterfactuals):
    • Suppose the agent crosses.
      • Further suppose that the agent proves in X that crossing implies U=-10.
        • Examining the source code of the agent, because we're assuming the agent crosses, either X proved that crossing implies U=+10, or it proved that crossing implies U=0.
        • So, either way, X is inconsistent -- by way of 0=-10 or +10=-10.
        • So the troll actually blows up the bridge, and really, U=-10.
      • Therefore (popping out of the second assumption), if the agent proves that crossing implies U=-10, then in fact crossing implies U=-10.
      • By Löb's theorem, crossing really implies U=-10.
      • So (since we're still under the assumption that the agent crosses), U=-10.
    • So (popping out of the assumption that the agent crosses), the agent crossing implies U=-10.
  • Since we proved all of this in X, the agent proves it, and proves no better utility in addition (unless X is truly inconsistent). On the other hand, it will prove that not crossing gives it a safe U=0. So it will in fact not cross.

Now, this argument relies on X and counterfactual reasoning having a lot of the properties of PA and normal reasoning. But even a system that doesn't run on proofs per se could still end up implementing something a lot like logic, and then it would have a property thats a lot like inconsistency, and then the troll could blow up the bridge conditionally on that. Basically, it still seems reasonable to me that counterfactual worlds should be closed under inference, up to our ability to infer. And I don't see which of the rules for manipulating logical implications wouldn't be valid for counterfactual implications in their own closed system, if you formally separate them. If you want your X to avoid this argument, then it needs to not-do something PA does. "Formal separation" between the systems isn't enough, because the results of counterfactual reasoning still really do effect your actions, and if the counterfactual reasoning system can understand this, Troll Bridge returns. And if there was such a something, we could just use a logic that doesn't do this in the first place, no need for the two-layer approach.

a convincing optimality result could

I'm also sceptical of optimality results. When you're doing subjective probability, any method you come up with will be proven optimal relative to its own prior - the difference between different subjective methods is only in their ontology, and the optimality results don't protect you against mistakes there. Also, when you're doing subjectivism, and it turns out the methods required to reach some optimality condition aren't subjectively optimal, you say "Don't be a stupid frequentist and do the subjectively optimal thing instead". So, your bottom line is written. If the optimality condition does come out in your favour, you can't be more sure because of it - that holds even under the radical version of expected evidence conservation. I also suspect that as subjectivism gets more "radical", there will be fewer optimality results besides the one relative to prior.

My Current Take on Counterfactuals

Because we have a “basic counterfactual” proposition for what would happen if we 1-box and what would happen if we 2-box, and both of those propositions stick around, LCH’s bets about what happens in either case both matter. This is unlike conditional bets, where if we 1-box, then bets conditional on 2-boxing disappear, refunded, as if they were never made in the first place.

I don't understand this part. Your explanation of PCDT at least didn't prepare me for it, it doesn't mention betting. And why is the payoff for the counterfactual-2-boxing determined by the beliefs of the agent after 1-boxing?

And what I think is mostly independent of that confusion: I don't think things are as settled. 

I'm more worried about the embedding problems with the trader in dutch book arguments, so the one against CDT isn't as decisive for me.

In the Troll Bridge hypothetical, we prove that [cross]->[U=-10]. This will make the conditional expectations poor. But this doesn’t have to change the counterfactuals.

But how is the counterfactual supposed to actually think? I don't think just having the agent unrevisably believe that crossing is counterfactually +10 is a reasonable answer, even if it doesn't have any instrumental problems in this case. I think it ought to be possible to get something like "whether to cross in troll bridge depends only on what you otherwise think about PAs consistency" with some logical method. But even short of that, there needs to be some method to adjust your counterfactuals if they fail to really match you conditionals. And if we had an actual concrete model of counterfactual reasoning instead of a list of desiderata, it might be possible to make a troll based on the consistency of whatever is inside this model, as opposed to PA.

I also think there is a good chance the answer to the cartesian boundary problem won't be "heres how to calculate where your boundary is", but something else of which boundaries are an approximation, and then something similar would go for counterfactuals, and then there won't be a counterfactual theory which respects embedding.

These later two considerations suggest the leftover work isn't just formalisation.

Identifiability Problem for Superrational Decision Theories

are the two players physically precisely the same (including environment), at least insofar as the players can tell?

In the examples I gave yes. Because thats the case where we have a guarantee of equal policy, from which people try to generalize. If we say players can see their number, then the twins in the prisoners dilemma needn't play the same way either.

But this is one reason why correlated equilibria are, usually, a better abstraction than Nash equilibria.

The "signals" players receive for correlated equilibria are already semantic. So I'm suspicious that they are better by calling on our intuition more to be used, with the implied risks. For example I remember reading about a result to the effect that correlated equilibria are easier to learn. This is not something we would expect from your explanation of the differences: If we explicitly added something (like the signals) into the game, it would generally get more complicated.

Identifiability Problem for Superrational Decision Theories

Well, if I understand the post correctly, you're saying that these two problems are fundamentally the same problem

No. I think:

...the reasoning presented is correct in both cases, and the lesson here is for our expectations of rationality...

As outlined in the last paragraph of the post. I want to convince people that TDT-like decision theories won't give a "neat" game theory, by giving an example where they're even less neat than classical game theory.

Actually it could. 

I think you're thinking about a realistic case (same algorithm, similar environment) rather than the perfect symmetry used in the argument. A communication channel is of no use there because you could just ask yourself what you would send, if you had one, and then you know you would have just gotten that message from the copy as well.

I can use my knowledge of distributed computing to look at the sort of decision problems where you play with copies

I'd be interested. I think even just more solved examples of the reasoning we want are useful currently.

Identifiability Problem for Superrational Decision Theories

"The same" in what sense? Are you saying that what I described in the context of game theory is not surprising, or outlining a way to explain it in retrospect? 

Communication won't make a difference if you're playing with a copy.

Phylactery Decision Theory

Another problem with this is that it isn't clear how to form the hypothesis "I have control over X".

You don't. I'm using talk about control sometimes to describe what the agent is doing from the outside, but the hypothesis it believes all have a form like "The variables such and such will be as if they were set by BDT given such and such inputs".

One problem with this is that it doesn't actually rank hypotheses by which is best (in expected utility terms), just how much control is implied.

For the first setup, where its trying to learn what it has control over, thats true. But you can use any ordering of hypothesis for the descent, so we can just take "how good that world is" as our ordering. This is very fragile of course. If theres uncountably many great but unachievable worlds, we fail, and in any case we are paying for all this with performance on "ordinary learning". If this were running in a non-episodic environment, we would have to find a balance between having the probability of hypothesis decline according to goodness, and avoiding the "optimistic humean troll" hypothesis by considering complexity as well. It really seems like I ought to take "the active ingredient" of this method out, if I knew how.

Reflective Bayesianism

From my perspective, Radical Probabilism is a gateway drug.

This post seemed to be praising the virtue of returning to the lower-assumption state. So I argued that in the example given, it took more than knocking out assumptions to get the benefit.

So, while I agree, I really don't think it's cruxy. 

It wasn't meant to be. I agree that logical inductors seem to de facto implement a Virtuous Epistemic Process, with attendent properties, whether or not they understand that. I just tend to bring up any interesting-seeming thoughts that are triggered during conversation and could perhaps do better at indicating that. Whether its fine to set it aside provisionally depends on where you want to go from here.

Reflective Bayesianism

Either way, we've made assumptions which tell us which Dutch Books are valid. We can then check what follows.

Ok. I suppose my point could then be made as "#2 type approaches aren't very useful, because they assume something thats no easier than what they provide".

I think this understates the importance of the Dutch-book idea to the actual construction of the logical induction algorithm. 

Well, you certainly know more about that than me. Where did the criterion come from in your view?

This part seems entirely addressed by logical induction, to me.

Quite possibly. I wanted to separate what work is done by radicalizing probabilism in general, vs logical induction specifically. That said, I'm not sure logical inductors properly have beliefs about their own (in the de dicto sense) future beliefs. It doesn't know "its" source code (though it knows that such code is a possible program) or even that it is being run with the full intuitive meaning of that, so it has no way of doing that. Rather, it would at some point think about the source code that we know is its, and come to believe that that program gives reliable results - but only in the same way in which it comes to trust other logical inductors. It seems like a version of this in the logical setting.

By "knowing where they are", I mean strategies that avoid getting dutch-booked without doing anything that looks like "looking for dutch books against me". One example of that would be The Process That Believes Everything Is Independent And Therefore Never Updates, but thats a trivial stupidity.

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