This is my response to all those who mentioned me in this thread, so @Suny, @Banno and just now, @Joshs.
You’re right I made a massive mistake here. Of course it’s true that P(\text{survive}) = 1, but the absolutely probability is not the right measure, it should be the difference in the survival probability:
\begin{align} \Delta P(\text{survive}) &= -(-P(\text{you die} \mid \text{you pick blue})) \\[2ex] &= P(\text{you die} \mid \text{you pick blue}) \end{align}
And that changes everything…
I’m quite busy these days, but I had a window of time where I could respond to this thread. As such, I rushed my answer too much and wound up with some fatal flaws in my mathematics.
So, I wasn’t trying to assume the Principle of Indifference. My integrals did assume the Principle of Indifference, but my actual assumption was this:
Any reasonable non-uniform distribution of likely P_r-values will not be enough to sufficiently bias them for the bluey interval’s contribution to matter…
That assumption was probably correct given my incorrect mathematics…
But after fixing my EV mistake above, I now see that the distribution of possible P_r-values is very important. The more density to the right, the better redding becomes. The more density at 0.5, the better blueing becomes. The more density to the left, assuming an okay density at 0.5, the better blueing becomes.
If it’s a multimodal distribution, high density to the left or at 0.5 might not be enough… For example, if it has a symmetric, somewhat polarized multimodal distribution, redding wins, because the hump on the right matters far more than the hump on the left…
A multimodal distribution would only arise given that we’d combine the evidence from different, competing models… which might be relevant here. Although we could just combine all the questionnaires on specifically the blue/red dilemma, see the total vote, and just model the likelihood of different P_r-values unimodally, I think that people will find this insufficient.
I introduce the idea of C_{br} and C_{rb}, which are the conversion rates of “professed bluers/redders” to “actual redders/bluers”, respectively.
Since we cannot know the latter, we cannot know what these conversion rates are, but we do know they exist… Because of course, some people judge themselves incorrectly… Especially on matters of life and death.
People could construct models taking data from comparable setups, where similar, hypothetical dilemmas became real dilemmas. I don’t think it’s a controversial claim to suggest that C_{br} > C_{rb}, because people usually underestimate their selfishness more than they understimate their altruism.
Which means, even if all our polls give us P_r, a model that takes into account some pair of conversion rates it has calculated, we’d be looking at:
\begin{align} \text{model}(P_r) &= P_r + P_bC_{br} -P_rC_{rb} \end{align}
But of course, there’d be a lot of debate regarding what the best model here is, and if we’d apply the Principle of Indifference to all plausible models, we could wind up with a multimodal distribution of what values of \text{model}(P_r) are likely. This complicates matters.
I want to give this topic the care that it deserves, which I don’t really have the time for nowadays… (my chance was yesterday, but I over-simplified and made some errors).
But now, I cannot say I agree with the crux of my previous posts anymore.
My position has shifted from being a redder to mostly withholding my assent. I need to figure out two things before I make up my mind.
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I need to figure out the mathematics properly, as the above stemmed from a quick investigation after I realized the P(\text{survive}) mistake, so I’m not fully sure about it.
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I need to see the polling numbers as they change and become statistically significant*. Preferably not just for TPF, since TPF is not even remotely a representative slice of humanity. I imagine someone will eventually set up a bigger poll somewhere on the internet, but even that could have some representativity issues since roughly 20 to 30 percent of the world does not have internet and would thus not participate…
*In the Insulated Game however, I would not have access to 2., which means once I’ve figured out 1., I could give my mathematically-reflected, but not statistically-informed view, a view that I could plausibly have/attain in the Insulated Game.
But ending with the above would be a cop-out… Because what if I got hauled away to the red/blue game right now? I would not be able to work out the mathematics in time (unless they’d give me unlimited time in there).
To withhold one’s assent is merely a privilege of circumstance… So, if I was forced to make a decision right now, what would I do? Well, given that my current tentative mathematical understanding is such that even in a case where our estimate for what P_r is a multimodal distribution concentrated on the left, even just a small hump to the right could be enough to trigger a very aggressive term in the EV formulae, thus making redding best.
I don’t think the density on 0.5 is great enough, and/or the right-side density low enough, that blueing has the highest EV. I also have a very strong survival instinct and you can call me selfish, and maybe I am, but given my current partial and tentative understanding, I believe the EV of redding is still best.
But now it’s actually a matter of how the data updates, for me. Earlier, I thought no plausible data set could possibly make blueing the highest EV choice, but that was based on my mathematical errors. Now, I’m genuinely unsure. Thank you @Suny for pointing out the errors! I hope I’ll have time to return to this thread so I can set up the proper EV formulae, maybe set up a Python script to figure out some stuff, and then also look at what the statistics say at that point. Maybe make a good model for what we can expect P_r to be.