I guess my position is thus:
While there are sets of probabilities which by themselves are not adequate to capture the information about a decision, there always is a set of probabilities which is adequate to capture the information about a decision.
In that sense I do not see your article as an argument against using probabilities to represent decision information, but rather a reminder to use the correct set of probabilities.
I don't think it's correct to equate probability with expected utility, as you seem to do here. The probability of a payout is the same in the two situations. The point of this example is that the probability of a particular event does not determine the optimal strategy. Because utility is dependent on your strategy, that also differs.
Hmmm. I was equating them as part of the standard technique of calculating the probability of outcomes from your actions, and then from there multiplying by the utilities of the outcomes and summing to find the expected utility of a given action.
I think it's just a question of what you think the error is in the original calculation. I find the error to be the conflation of "payout" (as in immediate reward from inserting the coin) with "payout" (as in the expected reward from your action including short term and long-term rewards). It seems to me that you are saying that you can't look at the immediate probability of payout
The point of this example is that the probability of a particular event does not determine the optimal strategy. Because utility is dependent on your strategy, that also differs.
which I agree with. But you seem to ignore the obvious solution of considering the probability of total payout, including considerations about your strategy. In that case, you really do have a single probability representing the likelihood of a single outcome, and you do get the correct answer. So I don't see where the issue with using a single probability comes from. It seems to me an issue with using the wrong single probability.
And especially troubling is that you seem to agree that using direct probabilities to calculate the single probability of each outcome and then weighing them by desirability will give you the correct answer, but then you say
probability by itself is not a fully adequate account of rationality.
which may be true, but I don't think is demonstrated at all by this example.
Thank you for further explaining your thinking.
The subtlety is about what numerical data can formally represent your full state of knowledge. The claim is that a mere probability of getting the $2 payout does not.
However, a single probability for each outcome given each strategy is all the information needed. The problem is not with using single probabilities to represent knowledge about the world, it's the straw math that was used to represent the technique. To me, this reasoning is equivalent to the following:
"You work at a store where management is highly disorganized. Although they precisely track the number of days you have worked since the last payday, they never remember when they last paid you, and thus every day of the work week has a 1/5 chance of being a payday. For simplicity's sake, let's assume you earn $100 a day.
You wake up on Monday and do the following calculation: If you go in to work, you have a 1/5 chance of being paid. Thus the expected payoff of working today is $20, which is too low for it to be worth it. So you skip work. On Tuesday, you make the same calculation, and decide that it's not worth it to work again, and so you continue forever.
I visit you and immediately point out that you're being irrational. After all, a salary of $100 a day clearly is worth it to you, yet you are not working. I look at your calculations, and immediately find the problem: You're using a single probability to represent your expected payoff from working! I tell you that using a meta-probability distribution fixes this problem, and so you excitedly scrap your previous calculations and set about using a meta-probability distribution instead. We decide that a Gaussian sharply peaked at 0.2 best represents our meta-probability distribution, and I send you on your way."
Of course, in this case, the meta-probability distribution doesn't change anything. You still continue skipping work, because I have devised the hypothetical situation to illustrate my point (evil laugh). The point is that in this problem the meta-probability distribution solves nothing, because the problem is not with a lack of meta-probability, but rather a lack of considering future consequences.
In both the OPs example and mine, the problem is that the math was done incorrectly, not that you need meta-probabilities. As you said, meta-probabilities are a method of screening off additional labels on your probability distributions for a particular class of problems where you are taking repeated samples that are entangled in a very particular sort of way. As I said above, I appreciate the exposition of meta-probabilities as a tool, and your comment as well has helped me better understand their instrumental nature, but I take issue with what sort of tool they are presented as.
If you do the calculations directly with the probabilities, your calculation will succeed if you do the math right, and fail if you do the math wrong. Meta-probabilities are a particular way of representing a certain calculation that succeed and fail on their own right. If you use them to represent the correct direct probabilities, you will get the right answer, but they are only an aid in the calculation, they never fix any problem with direct probability calculations. The fixing of the calculation and the use of probabilities are orthogonal issues.
To make a blunt analogy, this is like someone trying to plug an Ethernet cable into a phone jack, and then saying "when Ethernet fails, wifi works", conveniently plugging in the wifi adapter correctly.
The key of the dispute in my eyes is not whether wifi can work for certain situations, but whether there's anything actually wrong with Ethernet in the first place.
The exposition of meta-probability is well done, and shows an interesting way of examining and evaluating scenarios. However, I would take issue with the first section of this article in which you establish single probability (expected utility) calculations as insufficient for the problem, and present meta-probability as the solution.
In particular, you say
What’s interesting is that, when you have to decide whether or not to gamble your first coin, the probability is exactly the same in the two cases (p=0.45 of a $2 payout). However, the rational course of action is different. What’s up with that?
Here, a single probability value fails to capture everything you know about an uncertain event. And, it’s a case in which that failure matters.
I do not believe that this is a failure of applying a single probability to the situation, but merely calculating the probability wrongly, by ignoring future effects of your choice. I think this is most clearly illustrated by scaling the problem down to the case where you are handed a green box, and only two coins. In this simplified problem, we can clearly examine all possible strategies.
When put in these terms, it seems quite obvious that your choice to open the box would depend on more than the expected payoff from only the first box, because quite clearly your choice to open the first box pays off (or doesn't pay off) when opening (or not opening) the other boxes as well. This seems like an error in calculating the payoff matrix rather than a flaw with the technique of single probability values itself. It ignores the fact that opening the first box not only pays you off immediately, but also pays you off in the future by giving you information about the other boxes.
This problem easily succumbs to standard expected value calculations if all actions are considered. The steps remain the same as always:
In the case of two coins, we were able to trivially calculate the outcomes of all possible strategies, but in larger instances of the problem, it might be advisable to use shortcuts in the calculations. However, it still remains true that the best choice will still be the one you would have gotten if you had done out the full expected value calculation.
I think the confusion arises because a lot of the time problems are presented in a way that screens them off from the rest of the world. For example, you are given a box, and it either has $10.00 or $100.00. Once you open the box, the only effect it has on you is the amount of money you got. After you get the money, the box does not matter to the rest of the world. Problems are presented this way so that it is easy to factor out the decisions and calculations you have to make from every other decision you have to make. However, decision are not necessarily this way (in fact in real life, very few decisions are). In the choice of inserting the first coin or not, this is simply not the case, despite having superficial similarities to standard "box" problems.
Although you clearly understand that the payoffs from the boxes are entangled, you only apply this knowledge in your informal approach to the problem. The failure to consider the full effects of your actions in opening the first box may be psychologically encouraged by the technique of "single probability calculations", but it is certainly not a failure of the technique itself to capture such situations.
It's also irrelevant to the point I was making. You can point to different studies giving different percentages, but however you slice it a significant portion of the men she interacts with would have sex with her if she offered. So maybe 75% is only true for a certain demographic, but replace it with 10% for another demographic and it doesn't make a difference.
I was reading a lesswrong post and I found this paragraph which lines up with what I was trying to say
Some boxes you really can't think outside. If our universe really is Turing computable, we will never be able to concretely envision anything that isn't Turing-computable—no matter how many levels of halting oracle hierarchy our mathematicians can talk about, we won't be able to predict what a halting oracle would actually say, in such fashion as to experimentally discriminate it from merely computable reasoning.
Analysis of the survey results seems to indicate that I was correct: http://lesswrong.com/lw/fp5/2012_survey_results/
Yes, I agree. I can imagine some reasoning being concieving of things that are trans-turing complete, but I don't see how I could make an AI do so.
As mentioned below, we you'd need to make infinitely many queries to the Turing oracle. But even if you could, that wouldn't make a difference.
Again, even if there was a module to do infinitely many computations, the code I wrote still couldn't tell the difference between that being the case, and this module being a really good computable approximation of one. Again, it all comes back to the fact that I am programming my AI on a turing complete computer. Unless I somehow (personally) develop the skills to program trans-turing-complete computers, then whatever I program is only able to comprehend something that is turing complete. I am sitting down to write the AI right now, and so regardless of what I discover in the future, I can't program my turing complete AI to understand anything beyond that. I'd have to program a trans-turing complete computer now, if I ever hoped for it to understand anything beyond turing completeness in the future.
To me the part that stands out the most is the computation of P() by the AI.
From this description, it seems that P is described as essentially omniscient. It knows the locations and velocity of every particle in the universe, and it has unlimited computational power. Regardless of whether possessing and computing with such information is possible, the AI will model P as being literally omniscient. I see no reason that P could not hypothetically reverse the laws of physics and thus would always return 1 or 0 for any statement about reality.
Of course, you could add noise to the inputs to P, or put a strict limit on P's computational power, or model it as a hypothetical set of sensors which is very fine-grained but not omniscient, but this seems like another set of free variables in the model, in addition to lambda, which could completely undo the entire setup if any were set wrong, and there's no natural choice for any of them.