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David_Chapman comments on Probability, knowledge, and meta-probability - Less Wrong
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Comments (71)
Jeremy, thank you for this. To be clear, I wasn't suggesting that meta-probability is the solution. It's a solution. I chose it because I plan to use this framework in later articles, where it will (I hope) be particularly illuminating.
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.
Yes, absolutely! I chose a particularly simple problem, in which the correct decision-theoretic analysis is obvious, in order to show that probability does not always determine optimal strategy. In this case, the optimal strategies are clear (except for the exact stopping condition), and clearly different, even though the probabilities are the same.
I'm using this as an introductory wedge example. I've opened a Pandora's Box: probability by itself is not a fully adequate account of rationality. Many odd things will leap and creep out of that box so long as we leave it open.
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
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
which may be true, but I don't think is demonstrated at all by this example.
Thank you for further explaining your thinking.
Yes, I see your point (although I don't altogether agree). But, again, what I'm doing here is setting up analytical apparatus that will be helpful for more difficult cases later.
In the mean time, the LW posts I pointed to here may motivate more strongly the claim that probability alone is an insufficient guide to action.