Regarding the first paradox, however you want to think about infintesimal credences, I think it's obvious that the probability that you would be in room 500 *given that you exist* doesn't depend on whether the coin landed heads or tails. After all, in either case, all of the rooms have the same number of people in them. In a finite case, the probability that you will be in some room given that you exist is just the proportion of people in that room. In the infinite case, that's not well-defined, but it seems clear that any analogue of the concept of proportion that does apply is going to say that there are the same proportion of people in Room 500 regardless of whether the coin landed heads or not, since the relative numbers of people in each room are the same.
So if there is an update at all, it will be an anthropic update coming from the fact that you exist at all, not from seeing your room number. It could be argued that there should be an anthropic update in favor of heads on the basis of SIA - after all, it seems like there's some sense in which you're a billion times more likely to be created if the coin lands heads. In that case, Premise 1 would be wrong, though given the messiness with infinity, it's hard to say for sure if this is the correct way to reason about it. Regardless of how to deal with the anthropics, Premise 3 will always be right.
Very interesting problems with infinity! I think I don't know enough about anthropics to provide much of a sensible comment on the first problem, although I guess your solution makes sense to me, approximately.
In the second problem, I think I also agree with your conclusion. However, I want to emphasise pretty strongly that the part of your conclusion I most agree with is:
// Certainly I think this holds in finite cases. But a lot holds in finite cases and not in infinite cases.//
I certainly do not think we should be making inductions from what moral principles we need in order to have some consistent ethical principles in the case of infinitely many people, and then generalise those to have to apply to finite people. It would be really bad if philosophers started doing these generalisations I think (for both my sanity and for the goodness of the world)!
However, something pretty interesting arises in your evaluation of the second claim. Most of the problems you're encountering are to do with the fact that you're dealing with aleph null, which we know, cannot be assigned a uniform measure (i.e. there is no way to define a value function on aleph null so that v(0) = v(1) = v(2) = ..., and that the sum of v(n) for all n is equal to 1). This is causing the issue with undefinedness.
If instead in problem 2, we were talking about continuum many people (a much weirder state of affairs, to be sure), so that we can pair up each person with a real number between 0 and 1, then we can totally deflate this issue! We can say that the state of affairs before rolling the dice is that we expect an average utility of 4/6, or 2/3. After rolling the die, we'll have partitioned the reals into two subsets, which we can call "happy" and "sad". These will both end up being some crazy unmeasurable subset of the reals.
I'm not going to prove this (hence I'm not absolutely certain about it), but I think if we allow ourselves to then apply principle 4, and rearrange them, it won't be too hard to show that we'll be able to create a measurable set with the same people, just reordered so that the people corresponding to [0, 5/6) are "happy" (i.e. +1) and the people corresponding to [5/6, 1] are "sad" (i.e. -1). Then we totally _can_ consistently define the happiness of the people. Just take the measure of the sets, and you get back 2/3, exactly what you'd want.
So I would be careful about the statement of premise 5, as it will only apply for certain types of unmeasurable infinites.
Your solution to the second problem doesn't quite work. You could also rearrange the groups so that the happy people were assigned to the Cantor set and have measure zero, and the sad people are assigned to the rest of the points and have measure one.
Of course, all of these rearrangements depend on the axiom of choice, so maybe it makes more sense to deny that instead.
//It’s not that clear what infinitesimal times 1 billion is, but it’s not obvious that it’s more than the original infinitesimal.//
Infinitesimals work pretty much like normal numbers, so 1 billion * an infinitesimal is 1 billion times bigger. A (relatively) easy way to think about infinitesimals and the hyperreals is that they correspond to polynomial functions, ordered by which are greater as x tends towards infinity. So for example, 1/x tends to 0, but never quite reaches it, and will tend towards 1/2 of 2/x as x tends to infinity.
I've been told that infinitesimals don't work for probabilities anyway though, so assuming that's true your paradox remains.
For your second paradox, you could do a fun variation where everyone's utilities form a conditionally convergent series, so that you can use the mathematical proof that the limit can be made to be literally any number you like, just be changing the ordering of the terms.
Personally, I think the issues are basically arising because we're taking infinity too seriously as an actual number and without properly defining it for the context.
Now hang on a moment! You can make infinitesimals that work like the real numbers, but that also requires admitting infinities that work like the real numbers, which isuch more fine grained than cardinality is.
(I think the main issue with infinitesimal probabilities is that they break countable additivity by no longer having unique limits of countable sums)
Also, if you admit hyperreals we can just define a version of the first problem with one room for every hypernatural number, and now there aren't infinitesimals small enough to assign uniform non-zero probabilities to every room
There's no uniform probability distribution on a countably infinite set. You should look that up if you don't know it. Your first puzzle depends on there being one.
Regarding the second paradox, I think it makes sense to Reject 3 in the case you've described. The intuition for 3 comes from the fact that it seems like the average person is badly-off in the situation described, but that's only because we assume that the groups are formed in a "natural" way where any given person is 1000 times more likely to find themselves as one of the badly-off people in their group than they are to find themselves as the well-off person. But the dice example establishes that this isn't always the case. Because the groups are formed in a gerrymandered way after it's already known who is well-off, we end up with a strange situation where, despite each group having 1000 times as many badly-off people as well-off people, every given person is still 5 times as likely to find themselves as the well-off member of their group, rather than as one of the badly-off ones.
It also seems like there might be an issue here with treating events that are almost certain (probability 1, but not guaranteed) the same as events that are actually certain, even when the utility of the zero-probability outcomes is infinitely different than that of the probability-1 outcome. The paradox assumes that you will for sure be able to pair people up in groups with 1000 losers and 1 winner, but that's actually not certain - it's merely almost certain. There is a possibility, albeit an infinitely improbable one, that there will only be a finite number of losers.
The only way you can guarantee that you'll have groups with 1000 unlucky people and one lucky person in each is if you form the groups after the dice are rolled in a post-hoc fashion to make that happen - that's what I was referring to as gerrymandering. But any way of grouping people before knowing the outcome of the dice rolls will be extremely unlikely (probability 0) to end up with all the groups having 1000 unlucky people and 1 lucky person. It seems like you're assuming the metric is in some way related to the distribution of people in space after the dice rolls so that you can just move them around afterwards to make sure they are grouped into 1000-unlucky-1-lucky groups, but I'm suggesting that that just shouldn't be the metric you use.
The basic idea is this: Suppose you're one of the people in the world where everyone rolls a die, and you know that the people who roll a 1 through 5 will each be grouped with 1000 people who rolled a 6, with these groups exhausting everyone. The latter fact should not change the probabilities of your die roll, nor should the fact that infinitely many other people are also rolling dice, so you should conclude that you have a 5/6 chance of being the "lucky" person in your group, even though 1000/1001 people in your group are guaranteed to be "unlucky". Any grouping like this is unnatural: In a natural grouping, the probability of being lucky should match the expected proportion of people who are lucky. You can reject Premise 3 only in cases of unnatural groupings like this to get that the dice case is still good overall, and indeed anything that satisfies the ex-ante Pareto principle is good overall in expectation. There will be no way to rearrange people to make the groups natural by this definition that doesn't also remove the, "Each group is guaranteed to be mostly unlucky people," because by this definition, that aspect is precisely what makes a grouping unnatural.
No, there are the same number of people in the hotel either way, there are always a countably infinite number of people in the hotel.
In fact, even if when you flipped the coin, it landed on its side, and you added a countably infinite number of people to each room, there would still be the same number of people in the hotel.
Yeah, but in the hyperreal numbers, one infinite number can definitely be a billion times as big as the other!
However, after giving it more thought, it now seems to me BB was not talking about hyperreal numbers. (They may give a good way to resolve/avoid the paradox though.)
Hyperreal numbers may offer a solution though. If we define the number of guests to be (1, 2, 3, ...), you can't put a billion of them in each room of the hotel; but you can add 1 of them in each room.
Or you can have a total of (1, 2, 3, ...) people under tails and (10^9, 2 * 10^9, 3 * 10^9, ....) under heads. Then you do, in fact, have a billion times as many people under heads (which is what I originally meant).
Either way, there is no paradox :)
(But as I said, you weren't talking about hyperreal numbers. Your paradox seems to apply in your setup.)
They help because you would actually have a billion times as many people in the rooms (in total) under heads as under tails, not simply "infinitely many" under both.
Forget my last comment, I got confused. But more generally, if we use hyperreal numbers, we have one of the following situations (if I understand your problem correctly).
SITUATION 1
We have (1, 2, 3, ...) rooms and a total of (1, 2, 3, ...) people.
You flip the coin.
Tails: you put 1 person in every room.
Heads: you want to put a billion people in each room, but this is not possible: that requires a total of more than (1, 2, 3, ...) people.
This avoids the paradox completely.
SITUATION 2
We have (1, 2, 3) rooms and a total of (1 billion, 2 billion, 3 billion, ...) people.
You flip the coin.
Tails: you put 1 person in every room.
Heads: you put a billion people in every room. This is possible now.
However now we just have a billion times as many people in rooms as under tails, so I doubt there's any paradox left?
No, there is only one situation. There are not two. There is an infinite set of people. Then a coin is flipped. If heads you put a billion people in rooms with a label for each natural number. If tails you put one in a room for each natural number.
I'm saying that if we use hyperreal numbers, and have (1, 2, 3, ...) people, then you can't put a billion people in each room to begin with. Do you disagree with that?
On the first problem, why _would_ statement 2 be correct? If the setup was "heads: a billion people in room1, one person in each other room; tails: one person in each room", I can see how SIA suggests that "you're in room1" encourages "heads" (at least, infinities aside). But if there are a billion people in _every_ room, does that really multiply each P(I'm in room N) by a billion? That's unintuitive to me, because if those probabilities summed to 1 before being multiplied by a billion, then now they sum to a billion!
(ultimately I don't think this can be well-defined though, as you can't uniformly sample from the natural numbers; there are probably lots of resulting paradoxes)
I think you're right about the uniform sampling from the natural numbers, but that's not really what's going on here, I think. To explain more clearly the maths of what's happening.
Before you know your room number, all you know is that there are countably infinite number of people in the hotel. Whether that's by adding 1 person to each room, or by adding 1 billion person to each room, there are a countably infinite number of people in the hotel.
Once you _do_ know your room number, you've moved to a different set of affairs. You are the person in room 500. There are either 1 billion people in room 500, or there is 1 person in room 500. Therefore, under SIA, you should think its more likely that the coin came up heads. We can actually start comparing probabilities reasonably in this case, because everything is finite.
It's kind of a "given" situation. I.e. the probability of A is different from the probability of A given B. The probability of there being a countably infinity of 1 billion people VS a countably infinite number of 1 person in your situation is (intuitively) the same, since either way, it's countably infinite (call that A, so P(A) is 50%).
But the probability of there being 1 person in your situation vs 1 billion people in your situation is 1/1billion (under SIA). Call that the probability of A given B, where B = "You are in room 500" So the numbers change without ever having to multiply up to an overall probability of 1 billion, which I totally agree with you about, would be absurd! :)
Maybe that's somewhat clarifying?
Edit: changed a bunch of small things for clarity.
Suppose it's true that P(heads | youAreInRoom500) > P(tails | youAreInRoom500). Applying Bayes' Rule and simplifying shows this is equivalent to P(youAreInRoom500 | heads) > P(youAreInRoom500 | tails). This is the bit I'm not sure of, even given SIA. If you spawned into the universe and saw room500, then I think it would make sense; but if you already knew you were going to be _somewhere_ in the hotel, then why does the billion-per-room arrangement increase _every_ P(youAreInRoomN)?
Here's another problem with this. Suppose we have an axiom where for any N we have P(youAreInRoomN | heads) > P(youAreInRoomN | tails). Notice for example that P(youAreInRoom1 | heads) = 1 - P(youAreInRoom2 | heads) - P(youAreInRoom3 | heads) - ...
By applying the axiom on all these right-hand-side terms, we see that P(youAreInRoom1 | heads) < 1 - P(youAreInRoom2 | tails) - P(youAreInRoom3 | tails) - ...
= P(youAreInRoom1 | tails)
But this inequality is a direct contradiction of our axiom!
In this case, P(youAreInRoom500) is actually zero, as it is for any specific room. Dividing by zeroes here is going to lead us to absurdities (like the one you mention where P(youAreInRoom500 | heads) > P(youAreInRoom500 | tails).
The same issue arises in your second objection, both of these probabilities are actually equal, and equal to zero, so yes, we can't accept it using the actual "given" from Bayes' rule, because the measure originally is undefined (as you pointed out in your original comment).
I'm talking about a more philosophical notion of "given," which I don't have the time to write out and make rigorous right now.
This Bayes' rule argument is super interesting however, and feels intuitively strong to me! I think it should definitely inform whatever formalisation of "given" we end up using (if indeed any such formalisation can be made rigorous), but we do have to be careful about 0-divisions.
Edit: "My comment was a pretty sloppy" -> "My comment was pretty sloppy"
I'm still trying to understand your view. Is your view that IF we use n = (1, 2, 3, ...) rooms and n people, we can still pair all rooms using your pairing function?
I don't understand what heads or tails has anything to do with room numbers? It is possible I didn't understand your setup, but can't we run the first experiment without infinity? Heads 1 person in 1 room and tails 1 billion people in 1 room. You now walk into a room. You don't know if anyone else will join you. So with very high probability it was heads. All this reasoning is independent of room numbers.
To be exact, and assuming people are brought to their rooms 1 by 1, the probability of P[Heads | alone] = 1 billion / (1 billion + 1).
Further note. People are talking about how you can't have a uniform probability measure on the natural numbers, and that is correct of course, but you can have a probability measure over the space of densities on the natural numbers. In essence we are describing a random (not probability) measure on the natural numbers, which, with probability 1/2 is all 1 and with probability 1/2 is all 1 billion. This is a perfectly fine object, and doesn't require you to consider a probability measure on the natural numbers.
What you're seeing here is that you can't have a uniform probability measure over an infinite set - it just doesn't work! But this also defeats the anthropic arguments for large universes, because having more people doesn't mean you have more existencestuff in total.
Regarding your first paradox. I don’t think infinity has anything to do with the problem. The basic issue is that the probability of you existing and the coin coming up heads is much greater from your own subjective viewpoint, compare to the probability of you existing and the coin coming up tails. It is essentially an anthropics problem. Basically, you are getting bizarre results because of the update from your own existence. Your own existence is from your own subjective viewpoint evidence that it came up heads. To get a better idea, imagine that there are two parallel universes one in which it came up heads and 1 in which it came up tales. Obviously, if you find yourself in one of these rooms, you should assume that it came up heads because out of the group of billion and one people, 1 billion exist in the universe where the result was heads. You could argue that the probability is not well defined because of weirdness with infinite numbers, but I think it’s reasonably obvious that for any actual number, my analysis is correct and we know with a certain probability of one that you have to be in a room with a definite number. And even if the rooms did not have numbers the existence of an identical room where the same procedure is being run, should not affect your probability or at least, that’s my intuition.
For the second paradox, the problem seems to be that our method for determining the utility of infinite states. I don't know which way of determining the utility of infinite states is best, so I don't know which of 3-5 to reject--but any coherent method of determining the utilities will tell you one to reject.
So the paradox doesn't seem to add any weirdness on top of the weirdness that we already know there is with determining values of infinite states. As soon as you give a solution to that problem, you have a solution to the paradox.
I really think that 3 is mathematically incoherent, and jumps out of the page as a weird assertion, exactly because you can produce any ratio you want depending on in which order you count them: the rearrangement in 4 is a mathematically ordinary operation. It's normal to count things in a different order!
Traditionally , when dealing with infinities, you can't use ratios like 1000:1 for comparisons: you can only use mappings (one to one functions and onto functions). Rejecting this is going to have deep consequences for your maths, since eg this means that infinite sets are not a thing (sets have no predefined order or grouping and can have their elements considered in any arbitrary order).
I’m still reading through this, but I had a comment about the hotel premises. For 1, do we assume that each individual is not able to see whether there are more people in the room or not? If we do then I agree with 50% odds. However, if we assume that one would be able to see some person in the room then the evidence makes the coin coming up heads about 100% since if the coin came up tails, we would be the only person in the room. Just a small inquiry. I don’t really see anything controversial so far, though! Mainly because I think 2 appears false on its face. I believe it assumes SIA is true. I don’t think that more people having some evidence state increases the odds that I have that evidence state. My evidence states are (as far as I can tell) independent of others evidence states or how many people have those evidence states.
1st update: “you’re less likely to be in room one than in rooms 1 billion-2 billion” seems true, but I don’t understand how this is supposed to be less plausible by dropping out p2 or accepting p1 and p3. These are separate propositions. There is equal chance I am in any of the infinite rooms. By adding together the chance of being in one room times a billion (for being in any room between 1 billion-2 billion) you have a higher chance of being in those rooms versus being in room one specifically. This doesn’t conflict with there being more people not affecting your probability of being in any specific room.
2nd update: regarding the next paradox, 3 and 4 don’t seem particularly obvious. I sense that a world with beings that have equal goodness and equal badness is a better world than a world with no beings. As far as 4, it isn’t obvious that there can’t be at least one world where the arrangement of groups doesn’t change the overall value in that world. Of course there could be a world where it is obvious that rearranging the groups doesn’t affect the overall value of the world. I just don’t see that as the only possibility.
3rd update: overall I agree that infinity is weird, but I don’t think infinity being weird means we should drop plausible propositions entirely. Maybe they should be revised slightly, though. For p1 in the second paradox, it can possibly be revised to: For any reasonably affected state of affairs, if that state of affairs is better for everyone, in expectation, than non-existence, then it’s good that it exists. For example, if rolling six sided die doesn’t change the value in the world with enough rolls, then we can rule out the infinitely long term considerations with rolling that die (it’s the same either way). However, if we consider the first 10,000, 100,000, or 1 billion die rolls then we can consider them because those outcomes are reasonably affected by the die rolls. In which case we should find it more likely that these would result in better worlds overall.
Regarding the first paradox, however you want to think about infintesimal credences, I think it's obvious that the probability that you would be in room 500 *given that you exist* doesn't depend on whether the coin landed heads or tails. After all, in either case, all of the rooms have the same number of people in them. In a finite case, the probability that you will be in some room given that you exist is just the proportion of people in that room. In the infinite case, that's not well-defined, but it seems clear that any analogue of the concept of proportion that does apply is going to say that there are the same proportion of people in Room 500 regardless of whether the coin landed heads or not, since the relative numbers of people in each room are the same.
So if there is an update at all, it will be an anthropic update coming from the fact that you exist at all, not from seeing your room number. It could be argued that there should be an anthropic update in favor of heads on the basis of SIA - after all, it seems like there's some sense in which you're a billion times more likely to be created if the coin lands heads. In that case, Premise 1 would be wrong, though given the messiness with infinity, it's hard to say for sure if this is the correct way to reason about it. Regardless of how to deal with the anthropics, Premise 3 will always be right.
Very interesting problems with infinity! I think I don't know enough about anthropics to provide much of a sensible comment on the first problem, although I guess your solution makes sense to me, approximately.
In the second problem, I think I also agree with your conclusion. However, I want to emphasise pretty strongly that the part of your conclusion I most agree with is:
// Certainly I think this holds in finite cases. But a lot holds in finite cases and not in infinite cases.//
I certainly do not think we should be making inductions from what moral principles we need in order to have some consistent ethical principles in the case of infinitely many people, and then generalise those to have to apply to finite people. It would be really bad if philosophers started doing these generalisations I think (for both my sanity and for the goodness of the world)!
However, something pretty interesting arises in your evaluation of the second claim. Most of the problems you're encountering are to do with the fact that you're dealing with aleph null, which we know, cannot be assigned a uniform measure (i.e. there is no way to define a value function on aleph null so that v(0) = v(1) = v(2) = ..., and that the sum of v(n) for all n is equal to 1). This is causing the issue with undefinedness.
If instead in problem 2, we were talking about continuum many people (a much weirder state of affairs, to be sure), so that we can pair up each person with a real number between 0 and 1, then we can totally deflate this issue! We can say that the state of affairs before rolling the dice is that we expect an average utility of 4/6, or 2/3. After rolling the die, we'll have partitioned the reals into two subsets, which we can call "happy" and "sad". These will both end up being some crazy unmeasurable subset of the reals.
I'm not going to prove this (hence I'm not absolutely certain about it), but I think if we allow ourselves to then apply principle 4, and rearrange them, it won't be too hard to show that we'll be able to create a measurable set with the same people, just reordered so that the people corresponding to [0, 5/6) are "happy" (i.e. +1) and the people corresponding to [5/6, 1] are "sad" (i.e. -1). Then we totally _can_ consistently define the happiness of the people. Just take the measure of the sets, and you get back 2/3, exactly what you'd want.
So I would be careful about the statement of premise 5, as it will only apply for certain types of unmeasurable infinites.
Your solution to the second problem doesn't quite work. You could also rearrange the groups so that the happy people were assigned to the Cantor set and have measure zero, and the sad people are assigned to the rest of the points and have measure one.
Of course, all of these rearrangements depend on the axiom of choice, so maybe it makes more sense to deny that instead.
The first problem seems like extreme version of sleeping beauty dilemma
This is great stuff.
//It’s not that clear what infinitesimal times 1 billion is, but it’s not obvious that it’s more than the original infinitesimal.//
Infinitesimals work pretty much like normal numbers, so 1 billion * an infinitesimal is 1 billion times bigger. A (relatively) easy way to think about infinitesimals and the hyperreals is that they correspond to polynomial functions, ordered by which are greater as x tends towards infinity. So for example, 1/x tends to 0, but never quite reaches it, and will tend towards 1/2 of 2/x as x tends to infinity.
I've been told that infinitesimals don't work for probabilities anyway though, so assuming that's true your paradox remains.
For your second paradox, you could do a fun variation where everyone's utilities form a conditionally convergent series, so that you can use the mathematical proof that the limit can be made to be literally any number you like, just be changing the ordering of the terms.
Personally, I think the issues are basically arising because we're taking infinity too seriously as an actual number and without properly defining it for the context.
Now hang on a moment! You can make infinitesimals that work like the real numbers, but that also requires admitting infinities that work like the real numbers, which isuch more fine grained than cardinality is.
(I think the main issue with infinitesimal probabilities is that they break countable additivity by no longer having unique limits of countable sums)
Also, if you admit hyperreals we can just define a version of the first problem with one room for every hypernatural number, and now there aren't infinitesimals small enough to assign uniform non-zero probabilities to every room
There's no uniform probability distribution on a countably infinite set. You should look that up if you don't know it. Your first puzzle depends on there being one.
Regarding the second paradox, I think it makes sense to Reject 3 in the case you've described. The intuition for 3 comes from the fact that it seems like the average person is badly-off in the situation described, but that's only because we assume that the groups are formed in a "natural" way where any given person is 1000 times more likely to find themselves as one of the badly-off people in their group than they are to find themselves as the well-off person. But the dice example establishes that this isn't always the case. Because the groups are formed in a gerrymandered way after it's already known who is well-off, we end up with a strange situation where, despite each group having 1000 times as many badly-off people as well-off people, every given person is still 5 times as likely to find themselves as the well-off member of their group, rather than as one of the badly-off ones.
It also seems like there might be an issue here with treating events that are almost certain (probability 1, but not guaranteed) the same as events that are actually certain, even when the utility of the zero-probability outcomes is infinitely different than that of the probability-1 outcome. The paradox assumes that you will for sure be able to pair people up in groups with 1000 losers and 1 winner, but that's actually not certain - it's merely almost certain. There is a possibility, albeit an infinitely improbable one, that there will only be a finite number of losers.
But however you're grouping to decide gerrymandering you can rearrange the people to be better by that metric.
As for your second point, we can just stipulate that you are informed that some of the dice came up 1-5 and others came up 6.
The only way you can guarantee that you'll have groups with 1000 unlucky people and one lucky person in each is if you form the groups after the dice are rolled in a post-hoc fashion to make that happen - that's what I was referring to as gerrymandering. But any way of grouping people before knowing the outcome of the dice rolls will be extremely unlikely (probability 0) to end up with all the groups having 1000 unlucky people and 1 lucky person. It seems like you're assuming the metric is in some way related to the distribution of people in space after the dice rolls so that you can just move them around afterwards to make sure they are grouped into 1000-unlucky-1-lucky groups, but I'm suggesting that that just shouldn't be the metric you use.
The basic idea is this: Suppose you're one of the people in the world where everyone rolls a die, and you know that the people who roll a 1 through 5 will each be grouped with 1000 people who rolled a 6, with these groups exhausting everyone. The latter fact should not change the probabilities of your die roll, nor should the fact that infinitely many other people are also rolling dice, so you should conclude that you have a 5/6 chance of being the "lucky" person in your group, even though 1000/1001 people in your group are guaranteed to be "unlucky". Any grouping like this is unnatural: In a natural grouping, the probability of being lucky should match the expected proportion of people who are lucky. You can reject Premise 3 only in cases of unnatural groupings like this to get that the dice case is still good overall, and indeed anything that satisfies the ex-ante Pareto principle is good overall in expectation. There will be no way to rearrange people to make the groups natural by this definition that doesn't also remove the, "Each group is guaranteed to be mostly unlucky people," because by this definition, that aspect is precisely what makes a grouping unnatural.
"Before seeing your room number, you should think at 50% odds that the coin came up heads (after all, you have no evidence either way)."
This seems false to me. Regardless of your room number, under heads, there are a billion as many people in the hotel as under tails!
No, there are the same number of people in the hotel either way, there are always a countably infinite number of people in the hotel.
In fact, even if when you flipped the coin, it landed on its side, and you added a countably infinite number of people to each room, there would still be the same number of people in the hotel.
Useful link:
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel#Infinitely_many_coaches_with_infinitely_many_guests_each
Edited: Whoops misread the article, but the point stands. I thought we flipped the coin independently for each room!
Yeah, but in the hyperreal numbers, one infinite number can definitely be a billion times as big as the other!
However, after giving it more thought, it now seems to me BB was not talking about hyperreal numbers. (They may give a good way to resolve/avoid the paradox though.)
It's exactly the same set of people either way.
Yes, I realized this later.
Hyperreal numbers may offer a solution though. If we define the number of guests to be (1, 2, 3, ...), you can't put a billion of them in each room of the hotel; but you can add 1 of them in each room.
Or you can have a total of (1, 2, 3, ...) people under tails and (10^9, 2 * 10^9, 3 * 10^9, ....) under heads. Then you do, in fact, have a billion times as many people under heads (which is what I originally meant).
Either way, there is no paradox :)
(But as I said, you weren't talking about hyperreal numbers. Your paradox seems to apply in your setup.)
Hyperreals don't help because there still is a bijection between rooms and people.
They help because you would actually have a billion times as many people in the rooms (in total) under heads as under tails, not simply "infinitely many" under both.
Forget my last comment, I got confused. But more generally, if we use hyperreal numbers, we have one of the following situations (if I understand your problem correctly).
SITUATION 1
We have (1, 2, 3, ...) rooms and a total of (1, 2, 3, ...) people.
You flip the coin.
Tails: you put 1 person in every room.
Heads: you want to put a billion people in each room, but this is not possible: that requires a total of more than (1, 2, 3, ...) people.
This avoids the paradox completely.
SITUATION 2
We have (1, 2, 3) rooms and a total of (1 billion, 2 billion, 3 billion, ...) people.
You flip the coin.
Tails: you put 1 person in every room.
Heads: you put a billion people in every room. This is possible now.
However now we just have a billion times as many people in rooms as under tails, so I doubt there's any paradox left?
No, there is only one situation. There are not two. There is an infinite set of people. Then a coin is flipped. If heads you put a billion people in rooms with a label for each natural number. If tails you put one in a room for each natural number.
Again, which of the three premises do you reject?
I'm saying that if we use hyperreal numbers, and have (1, 2, 3, ...) people, then you can't put a billion people in each room to begin with. Do you disagree with that?
Yes! Because there's a bijection between them.
This is easy to illustrate. Let's give the rooms names.
R1, R2, R3, R4, R5, etc.
Let's give the people names.
P1, P2, P3, P4, P5.
I'll just do 2:1 rather than a billion to one to make the math easier.
A coin is flipped. If heads P1 and P2 go to R1. P3 and P4 go to R2. And so on.
If tails, P1 goes to R1, P2 to R2, and so on.
This is a perfectly consistent function that pairs people and it results in everyone being paired with a room.
On the first problem, why _would_ statement 2 be correct? If the setup was "heads: a billion people in room1, one person in each other room; tails: one person in each room", I can see how SIA suggests that "you're in room1" encourages "heads" (at least, infinities aside). But if there are a billion people in _every_ room, does that really multiply each P(I'm in room N) by a billion? That's unintuitive to me, because if those probabilities summed to 1 before being multiplied by a billion, then now they sum to a billion!
(ultimately I don't think this can be well-defined though, as you can't uniformly sample from the natural numbers; there are probably lots of resulting paradoxes)
I think you're right about the uniform sampling from the natural numbers, but that's not really what's going on here, I think. To explain more clearly the maths of what's happening.
Before you know your room number, all you know is that there are countably infinite number of people in the hotel. Whether that's by adding 1 person to each room, or by adding 1 billion person to each room, there are a countably infinite number of people in the hotel.
Once you _do_ know your room number, you've moved to a different set of affairs. You are the person in room 500. There are either 1 billion people in room 500, or there is 1 person in room 500. Therefore, under SIA, you should think its more likely that the coin came up heads. We can actually start comparing probabilities reasonably in this case, because everything is finite.
It's kind of a "given" situation. I.e. the probability of A is different from the probability of A given B. The probability of there being a countably infinity of 1 billion people VS a countably infinite number of 1 person in your situation is (intuitively) the same, since either way, it's countably infinite (call that A, so P(A) is 50%).
But the probability of there being 1 person in your situation vs 1 billion people in your situation is 1/1billion (under SIA). Call that the probability of A given B, where B = "You are in room 500" So the numbers change without ever having to multiply up to an overall probability of 1 billion, which I totally agree with you about, would be absurd! :)
Maybe that's somewhat clarifying?
Edit: changed a bunch of small things for clarity.
Suppose it's true that P(heads | youAreInRoom500) > P(tails | youAreInRoom500). Applying Bayes' Rule and simplifying shows this is equivalent to P(youAreInRoom500 | heads) > P(youAreInRoom500 | tails). This is the bit I'm not sure of, even given SIA. If you spawned into the universe and saw room500, then I think it would make sense; but if you already knew you were going to be _somewhere_ in the hotel, then why does the billion-per-room arrangement increase _every_ P(youAreInRoomN)?
Here's another problem with this. Suppose we have an axiom where for any N we have P(youAreInRoomN | heads) > P(youAreInRoomN | tails). Notice for example that P(youAreInRoom1 | heads) = 1 - P(youAreInRoom2 | heads) - P(youAreInRoom3 | heads) - ...
By applying the axiom on all these right-hand-side terms, we see that P(youAreInRoom1 | heads) < 1 - P(youAreInRoom2 | tails) - P(youAreInRoom3 | tails) - ...
= P(youAreInRoom1 | tails)
But this inequality is a direct contradiction of our axiom!
Ah very interesting! I think there is a way around this issue though.
My comment was pretty sloppy about the given thing. I think that we can't do actual "given" calculations in this case, for the following reason.
P(heads | youAreInRoom500) = P(heads ∩ youAreInRoom500)/P(youAreInRoom500).
In this case, P(youAreInRoom500) is actually zero, as it is for any specific room. Dividing by zeroes here is going to lead us to absurdities (like the one you mention where P(youAreInRoom500 | heads) > P(youAreInRoom500 | tails).
The same issue arises in your second objection, both of these probabilities are actually equal, and equal to zero, so yes, we can't accept it using the actual "given" from Bayes' rule, because the measure originally is undefined (as you pointed out in your original comment).
I'm talking about a more philosophical notion of "given," which I don't have the time to write out and make rigorous right now.
This Bayes' rule argument is super interesting however, and feels intuitively strong to me! I think it should definitely inform whatever formalisation of "given" we end up using (if indeed any such formalisation can be made rigorous), but we do have to be careful about 0-divisions.
Edit: "My comment was a pretty sloppy" -> "My comment was pretty sloppy"
I'm still trying to understand your view. Is your view that IF we use n = (1, 2, 3, ...) rooms and n people, we can still pair all rooms using your pairing function?
I don't understand what heads or tails has anything to do with room numbers? It is possible I didn't understand your setup, but can't we run the first experiment without infinity? Heads 1 person in 1 room and tails 1 billion people in 1 room. You now walk into a room. You don't know if anyone else will join you. So with very high probability it was heads. All this reasoning is independent of room numbers.
To be exact, and assuming people are brought to their rooms 1 by 1, the probability of P[Heads | alone] = 1 billion / (1 billion + 1).
Further note. People are talking about how you can't have a uniform probability measure on the natural numbers, and that is correct of course, but you can have a probability measure over the space of densities on the natural numbers. In essence we are describing a random (not probability) measure on the natural numbers, which, with probability 1/2 is all 1 and with probability 1/2 is all 1 billion. This is a perfectly fine object, and doesn't require you to consider a probability measure on the natural numbers.
What you're seeing here is that you can't have a uniform probability measure over an infinite set - it just doesn't work! But this also defeats the anthropic arguments for large universes, because having more people doesn't mean you have more existencestuff in total.
Regarding your first paradox. I don’t think infinity has anything to do with the problem. The basic issue is that the probability of you existing and the coin coming up heads is much greater from your own subjective viewpoint, compare to the probability of you existing and the coin coming up tails. It is essentially an anthropics problem. Basically, you are getting bizarre results because of the update from your own existence. Your own existence is from your own subjective viewpoint evidence that it came up heads. To get a better idea, imagine that there are two parallel universes one in which it came up heads and 1 in which it came up tales. Obviously, if you find yourself in one of these rooms, you should assume that it came up heads because out of the group of billion and one people, 1 billion exist in the universe where the result was heads. You could argue that the probability is not well defined because of weirdness with infinite numbers, but I think it’s reasonably obvious that for any actual number, my analysis is correct and we know with a certain probability of one that you have to be in a room with a definite number. And even if the rooms did not have numbers the existence of an identical room where the same procedure is being run, should not affect your probability or at least, that’s my intuition.
For the second paradox, the problem seems to be that our method for determining the utility of infinite states. I don't know which way of determining the utility of infinite states is best, so I don't know which of 3-5 to reject--but any coherent method of determining the utilities will tell you one to reject.
So the paradox doesn't seem to add any weirdness on top of the weirdness that we already know there is with determining values of infinite states. As soon as you give a solution to that problem, you have a solution to the paradox.
I'm pretty convinced of the ubiquitous incomparability method--and the paradox illustrates that that requires we abandon ex ante pareto.
I really think that 3 is mathematically incoherent, and jumps out of the page as a weird assertion, exactly because you can produce any ratio you want depending on in which order you count them: the rearrangement in 4 is a mathematically ordinary operation. It's normal to count things in a different order!
Traditionally , when dealing with infinities, you can't use ratios like 1000:1 for comparisons: you can only use mappings (one to one functions and onto functions). Rejecting this is going to have deep consequences for your maths, since eg this means that infinite sets are not a thing (sets have no predefined order or grouping and can have their elements considered in any arbitrary order).
I’m still reading through this, but I had a comment about the hotel premises. For 1, do we assume that each individual is not able to see whether there are more people in the room or not? If we do then I agree with 50% odds. However, if we assume that one would be able to see some person in the room then the evidence makes the coin coming up heads about 100% since if the coin came up tails, we would be the only person in the room. Just a small inquiry. I don’t really see anything controversial so far, though! Mainly because I think 2 appears false on its face. I believe it assumes SIA is true. I don’t think that more people having some evidence state increases the odds that I have that evidence state. My evidence states are (as far as I can tell) independent of others evidence states or how many people have those evidence states.
1st update: “you’re less likely to be in room one than in rooms 1 billion-2 billion” seems true, but I don’t understand how this is supposed to be less plausible by dropping out p2 or accepting p1 and p3. These are separate propositions. There is equal chance I am in any of the infinite rooms. By adding together the chance of being in one room times a billion (for being in any room between 1 billion-2 billion) you have a higher chance of being in those rooms versus being in room one specifically. This doesn’t conflict with there being more people not affecting your probability of being in any specific room.
2nd update: regarding the next paradox, 3 and 4 don’t seem particularly obvious. I sense that a world with beings that have equal goodness and equal badness is a better world than a world with no beings. As far as 4, it isn’t obvious that there can’t be at least one world where the arrangement of groups doesn’t change the overall value in that world. Of course there could be a world where it is obvious that rearranging the groups doesn’t affect the overall value of the world. I just don’t see that as the only possibility.
3rd update: overall I agree that infinity is weird, but I don’t think infinity being weird means we should drop plausible propositions entirely. Maybe they should be revised slightly, though. For p1 in the second paradox, it can possibly be revised to: For any reasonably affected state of affairs, if that state of affairs is better for everyone, in expectation, than non-existence, then it’s good that it exists. For example, if rolling six sided die doesn’t change the value in the world with enough rolls, then we can rule out the infinitely long term considerations with rolling that die (it’s the same either way). However, if we consider the first 10,000, 100,000, or 1 billion die rolls then we can consider them because those outcomes are reasonably affected by the die rolls. In which case we should find it more likely that these would result in better worlds overall.