I think one useful thing is to try and find out why some explanations are more plausible than others (which seems standard, the fact of which explanation is actually true then, won't affect the guess that much).
When asked a question by an experimenter, I imagine myself trying to give a somewhat quick answer (rather than ask the experimenter to repeat the experiment isolating some variables so that I can answer accurately). I imagine my mind going through reasons until it hits a reason that sounds like ok, i.e. would convince me if I heard it from someone else, and pick that up.
Many of those researches don't seem to give "the time to answer" as a variable. What if the subjects were asked to think it over for 30 minutes before answering? I am not suggesting they will get the right answer, but perhaps a different answer, since different brain parts may be included in the decision then.
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When calculting the odds of the winning/losing/machine defect, shouldn't you add the odds of the Many Worlds hypothesis being true? Perhaps wouldn't affect the relative odds, but might change odds in relation to "moderately rich/poor and won't try Quantum Immortality"