The madman murders only almost always. It is possible but vanishingly unlikely that he just never rolls snake eyes (or he runs outside of the total population with the growth so he can't get a full patch). Option 1 doesn't care whether the doom ultimately happens while option 2 assumes that the doom will happen.
The proper enlish version of option two would be "Given that the dice came up snake eyes and that you were kidnapped at some point what is the probabilty that it did so while you were kidnapped?". Notice also that this is independent off what dice readings result in doom. That is if the world is only saved on snake eyes the chance is still "only" 9/10.
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To my view, the 1/36 is "obviously" the right answer, what's interesting is exactly how it all went wrong in the other case. I'm honestly not all that enlightened by the argument given here nor in the links. The important question is, how would I recognise this mistake easily in the future? The best I have for the moment is "don't blindly apply a proportion argument" and "be careful when dealing with infinite scenarios even when they're disguised as otherwise". I think the combination of the two was required here, the proportion argument failed because the maths which normally supports it couldn't be used without at some point colliding with the partly-hidden infinity in the problem setup.
I'd be interested in more development of how this relates to anthropic arguments. It does feel like it highlights some of the weaknesses in anthropic arguments. It seems to strongly undermine the doomsday argument in particular. My take on it is that it highlights the folly of the idea that population is endlessly exponentially growing. At some point that has to stop regardless of whether it has yet already, and as soon as you take that into account I suspect the maths behind the argument collapses.
Edit: Just another thought. I tried harder to understand your argument and I'm not convinced it's enough. Have you heard of ignorance priors? They're the prior you use, in fact the prior you need to use, to represent a state of no knowledge about a measurement other than an invariance property which identifies the type of measurement it is. So an ignorance prior for a position is constant, and for a scale is 1/x, and for a probability has been at least argued to be 1/x(1-x). These all have the property that their integral is infinite, but they work because as soon as you add some knowledge and apply Bayes rule the result becomes integrable. These are part of the foundations of Bayesian probability theory. So while I agree with the conclusion, I don't think the argument that the prior is unnormalisable is sufficient proof.
Actually, no, improper priors such as you suggest are not part of the foundations of Bayesian probability theory. It's only legitimate to use an improper prior if the result you get is the limit of the results you get from a sequence of progressively more diffuse priors that tend to the improper prior in the limit. The Marginalization Paradox is an example where just plugging in an improper prior without considering the limiting process leads to an apparent contradiction. My analysis (http://ksvanhorn.com/bayes/Papers/mp.pdf) is that the problem there ultimately stems from non-uniform convergence.
I've had some email discussions with Scott Aaronson, and my conclusion is that the Dice Room scenario really isn't an appropriate metaphor for the question of human extinction. There are no anthropic considerations in the Dice Room, and the existence of a larger population from which the kidnap victims are taken introduces complications that have no counterpart when discussing the human extinction scenario.
You could formalize the human extinction scenario with unrealistic parameters for growth and generational risk as follows:
Let n be the number of generations for which humanity survives.
The population in each generation is 10 times as large as the previous generation.
There is a risk 1/36 of extinction in each generation. Hence, P(n=N+1 | n >= n) = 1/36.
You are a randomly chosen individual from the entirety of all humans who will ever exist. Specifically, P(you belong to generation g) = 10^g / N, where N is the sum of 10^t for 1 <= t <= n.
Analyzing this problem, I get
P(extinction occurs in generation t | extinction no earlier than generation t) = 1/36
P(extinction occurs in generation t | you are in generation t) = about 9/10
That's a vast difference depending on whether or not we take into account anthropic considerations.
The Dice Room analogy would be if the madman first rolled the dice until he got snake-eyes, then went out and kidnapped a bunch of people, randomly divided them into n batches, each 10 times larger than the previous, and murdered the last batch. This is a different process than what is described in the book, and results in different answers.