But the thing is, updating on being still alive doesn’t change anything—it can never drive your estimate of n up and thus save you from losing out. It could convince you to play if you aren’t playing—but that’s an absurdity, if you’re not playing you won’t get any updates! All updating gives you is a belief that since you’re still alive, you must be in a low-bullets, high-probability world. This belief may be correct (and then it’s fine, but you would have played even without it) or wrong (in which case you can never realise until it’s too late). Either way, it doesn’t swing your payoff.
In your added bullets scenario thinking about it there’s a bit of a difference because now a strategy of playing for a certain amount of turns can make sense. So the game isn’t time-symmetric, and this has an effect. I’m still not sure how you would use your updating though. Basically I think the only situation in which that sort of updating might give a genuine benefit is one in which the survival curve is U-shaped: there’s a bump of mortality at the beginning, but if you get through it, you’re good to go for a while. In that case, observing that you survived long enough to overcome the bump suggests that you’re probably better off going on playing all the way to the end.
But the thing is, updating on being still alive doesn’t change anything—it can never drive your estimate of n up and thus save you from losing out. It could convince you to play if you aren’t playing—but that’s an absurdity, if you’re not playing you won’t get any updates! All updating gives you is a belief that since you’re still alive, you must be in a low-bullets, high-probability world. This belief may be correct (and then it’s fine, but you would have played even without it) or wrong (in which case you can never realise until it’s too late). Either way, it doesn’t swing your payoff.
In your added bullets scenario thinking about it there’s a bit of a difference because now a strategy of playing for a certain amount of turns can make sense. So the game isn’t time-symmetric, and this has an effect. I’m still not sure how you would use your updating though. Basically I think the only situation in which that sort of updating might give a genuine benefit is one in which the survival curve is U-shaped: there’s a bump of mortality at the beginning, but if you get through it, you’re good to go for a while. In that case, observing that you survived long enough to overcome the bump suggests that you’re probably better off going on playing all the way to the end.