And if at, say, the fifth turn, there is no alternative to seeing EEEEE, that means that the probability of observing it, conditioned on us still being in play, is
1, and entirely independent of n. No information can be derived to update our prior.
Well yes, conditioned on us being in play, we can’t get any more information from the sequence of events we observed. But the fact that we are still in play itself tells us that the gun is mostly empty. Now as it happens this isn’t very useful because we will never both update towards not playing, and be able to carry on playing, but I can still be practically certain after seeing 20 Es in a row that the gun is empty.
The way anthropics twists things is that if this were russian roulette I might not be able to update after 20 Es that the gun is empty, since in all the world’s where I died there’s noone to observe what happened, so of course I find myself in the one world where by pure chance I survived.
But the fact that we are still in play itself tells us that the gun is mostly empty.
No, it tells us that either the gun is mostly empty, or we were very lucky; but since either way that cancels out exactly with our probability of being still in play in the first place, no additional information can be deduced with which to decide our strategy. The bit about this being conditioned on being still in play is key! If we consider the usefulness of each bit of information acquired, then obviously they can only be useful if you’re still playing.
Why are we conditioning on still being in play though? Without the anthropic shadow there’s no reason to do so. There are plenty of world’s where I observe myself out of play, but with different information, why is the fact I’m not in one of them not telling me anything?
Let’s say each round there’s a 50 percent probability of adding an extra bullet to the gun.
If I didn’t update based on the fact I’m still playing then I would quickly stop after a few rounds, since the probability I would see a bullet constantly increases.
But if I do update, then it’s worth it carrying on, since the fact I’m still playing is evidence there still are plenty of empty chambers.
There are plenty of world’s where I observe myself out of play, but with different information, why is the fact I’m not in one of them not telling me anything?
There are! But here I am only interested in useful information, bits of information you can use to update your strategy. In those worlds you just acquired bits of information that have zero usefulness. While information would still be useful, though, you won’t acquire it. I should probably try to formalise this concept more, and I will maybe write something else about it.
See my comment from earlier below, which highlights how this information is in general useful, even if in this case it happens not to be:
To give a concrete counterexample.
Let’s say each round there’s a 50 percent probability of adding an extra bullet to the gun.
If I didn’t update based on the fact I’m still playing then I would quickly stop after a few rounds, since the probability I would see a bullet constantly increases.
But if I do update, then it’s worth it carrying on, since the fact I’m still playing is evidence there still are plenty of empty chambers
Not quite getting it—is the addition permanent or just for each round? Seems to me like all that does is make the odds even worse. If you’re already in a game in which you should quit, that only is all the more reason; if you’re not, it could tilt the scales. And in neither case does updating on the fact you’re still playing help in any way, because in fact you can’t meaningfully update on that at all.
The addition is permanent. Updating on the fact that you’re still playing provides evidence that the bullet was not in fact added in previous rounds, so it’s worth carrying on playing a little bit longer, whereas if you didn’t update, even if it was worth playing the first round, you would stop after 1 or 2.
OK, so if you get told that a bullet was added, then yes, that is information you can use, combined with the knowledge that the drum only holds maximum 6 bullets. But that’s a different game, closer to the second I described (well, it’s very different, but it has in common the fact that you do get extra information to ground your beliefs).
Even simpler, something similar would happen simply if you didn’t spin the chamber after each turn, which would mean the probability of finding a bullet isn’t uncorrelated any more. These details matter. I’d need to work it out to figure how it works, but I picked the game description very deliberately to show the effect I was talking about, so obviously changing it makes things different.
You don’t get told no, you just guess from the fact you’re still alive.
but I picked the game description very deliberately to show the effect I was talking about, so obviously changing it makes things different.
On the contrary, it doesn’t show any such effect at all. It’s carefully contrived so that you can update on the fact you’re still alive, but that happens not to change your strategy. That’s not very interesting at all. Often a change in probabilities won’t change your strategy.
I’m simply showing that with a slight change of setup, updating on the fact your still alive does indeed change your strategy.
If you don’t get told I don’t think it does then, no. It just changes the probabilities in a more confusing way, but you still have no particular way of getting information out of it.
As a player, who by definition must be still in play, you can’t deduce anything from it. You only know what whatever your odds of surviving an extra round were at the beginning, they will go down with time. This probably leads to an optimal strategy that requires you absolutely quit after a certain number of rounds (depending on the probability of the bullet being added). But that’s not affected by any actual in-game information because you don’t really get any information.
But that’s not affected by any actual in-game information because you don’t really get any information.
You keep on asserting this, but that’s not actually true—do the maths. A player who doesn’t update on them still being alive, will play fewer rounds on average, and will earn less in repeated play. (Where each play is independent).
The reason is simple—they’re not going to be able to play for very long when there are lots of bullets added, so the times when they find themselves still playing are disproportionately those where bullets weren’t added, so they should play for longer.
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.
The way anthropics twists things is that if this were russian roulette I might not be able to update after 20 Es that the gun is empty, since in all the world’s where I died there’s noone to observe what happened, so of course I find myself in the one world where by pure chance I survived.
This is incorrect due to the anthropic undeath argument. The vast majority of surviving worlds will be ones where the gun is empty, unless it is impossible to be so. This is exactly the same as a Bayesian update.
This post seems incorrect to me. Here’s the crux:
Well yes, conditioned on us being in play, we can’t get any more information from the sequence of events we observed. But the fact that we are still in play itself tells us that the gun is mostly empty. Now as it happens this isn’t very useful because we will never both update towards not playing, and be able to carry on playing, but I can still be practically certain after seeing 20 Es in a row that the gun is empty.
The way anthropics twists things is that if this were russian roulette I might not be able to update after 20 Es that the gun is empty, since in all the world’s where I died there’s noone to observe what happened, so of course I find myself in the one world where by pure chance I survived.
No, it tells us that either the gun is mostly empty, or we were very lucky; but since either way that cancels out exactly with our probability of being still in play in the first place, no additional information can be deduced with which to decide our strategy. The bit about this being conditioned on being still in play is key! If we consider the usefulness of each bit of information acquired, then obviously they can only be useful if you’re still playing.
Why are we conditioning on still being in play though? Without the anthropic shadow there’s no reason to do so. There are plenty of world’s where I observe myself out of play, but with different information, why is the fact I’m not in one of them not telling me anything?
To give a concrete counterexample.
Let’s say each round there’s a 50 percent probability of adding an extra bullet to the gun.
If I didn’t update based on the fact I’m still playing then I would quickly stop after a few rounds, since the probability I would see a bullet constantly increases.
But if I do update, then it’s worth it carrying on, since the fact I’m still playing is evidence there still are plenty of empty chambers.
There are! But here I am only interested in useful information, bits of information you can use to update your strategy. In those worlds you just acquired bits of information that have zero usefulness. While information would still be useful, though, you won’t acquire it. I should probably try to formalise this concept more, and I will maybe write something else about it.
See my comment from earlier below, which highlights how this information is in general useful, even if in this case it happens not to be:
Not quite getting it—is the addition permanent or just for each round? Seems to me like all that does is make the odds even worse. If you’re already in a game in which you should quit, that only is all the more reason; if you’re not, it could tilt the scales. And in neither case does updating on the fact you’re still playing help in any way, because in fact you can’t meaningfully update on that at all.
The addition is permanent. Updating on the fact that you’re still playing provides evidence that the bullet was not in fact added in previous rounds, so it’s worth carrying on playing a little bit longer, whereas if you didn’t update, even if it was worth playing the first round, you would stop after 1 or 2.
OK, so if you get told that a bullet was added, then yes, that is information you can use, combined with the knowledge that the drum only holds maximum 6 bullets. But that’s a different game, closer to the second I described (well, it’s very different, but it has in common the fact that you do get extra information to ground your beliefs).
Even simpler, something similar would happen simply if you didn’t spin the chamber after each turn, which would mean the probability of finding a bullet isn’t uncorrelated any more. These details matter. I’d need to work it out to figure how it works, but I picked the game description very deliberately to show the effect I was talking about, so obviously changing it makes things different.
You don’t get told no, you just guess from the fact you’re still alive.
On the contrary, it doesn’t show any such effect at all. It’s carefully contrived so that you can update on the fact you’re still alive, but that happens not to change your strategy. That’s not very interesting at all. Often a change in probabilities won’t change your strategy.
I’m simply showing that with a slight change of setup, updating on the fact your still alive does indeed change your strategy.
If you don’t get told I don’t think it does then, no. It just changes the probabilities in a more confusing way, but you still have no particular way of getting information out of it.
As a player, who by definition must be still in play, you can’t deduce anything from it. You only know what whatever your odds of surviving an extra round were at the beginning, they will go down with time. This probably leads to an optimal strategy that requires you absolutely quit after a certain number of rounds (depending on the probability of the bullet being added). But that’s not affected by any actual in-game information because you don’t really get any information.
You keep on asserting this, but that’s not actually true—do the maths. A player who doesn’t update on them still being alive, will play fewer rounds on average, and will earn less in repeated play. (Where each play is independent).
The reason is simple—they’re not going to be able to play for very long when there are lots of bullets added, so the times when they find themselves still playing are disproportionately those where bullets weren’t added, so they should play for longer.
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.
This is incorrect due to the anthropic undeath argument. The vast majority of surviving worlds will be ones where the gun is empty, unless it is impossible to be so. This is exactly the same as a Bayesian update.