Wednesday, March 15, 2006

Infinite Spheres of Utility

Imagine a universe containing infinitely many immortal people, partitioned into two "spheres". In one sphere, all the inhabitants live a blissful existence, whereas the members of the other sphere suffer unbearable agony. Now compare the following two variations:

1) Everyone starts off in the blissful sphere. But each day, one more person gets permanently transferred across to the agony sphere, where they reside for the rest of eternity.

2) Everyone starts off in the agony sphere. But each day, one more person gets permanently transferred across to the blissful sphere, where they reside for the rest of eternity.

Which scenario is better? The answer, paradoxically, appears to be "both". At any moment in time, there will be infinitely many people in the original sphere, and only a finite number who have been transferred across. So option 1 is better.

However, each particular person will spend only a finite amount of time in the first sphere, whereas they will spend an eternity in their post-transfer home. So option 2 is better.

[A clarification is in order. As stated, it remains possible for some people to remain forever in their original sphere. (Suppose we assign each person a natural number. Each day we can transfer across the next even-numbered person. Then the infinitely many odd-numbered people never get transferred at all!) So let us stipulate that no-one is "skipped" in this way, and that every individual will indeed get transferred at some point.]

How are we to evaluate the options without falling into paradox?

(I owe this problem to recent discussion with ANU grad students. I think they in turn got it from Alan Hajek.)


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20 comments:

  1. I think 2 is better wholistically
    but partly because you seem to have "cheated" by making your infinite time bigger than your infinite population.

    Unless you mean that the last person is transfered at the last moment in the universe in which case the universe is a sort of mirror image of itself.

    Even in that case i guess I like improvement. all other things being Absolutly equal :)

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  2. I guess the relevant question is, in which scenario would you prefer to be born? I would certainly prefer the second one, because I would get infinite bliss after only finite misery, and surely any rational person would choose the same. If option 2 is the best for each person, it is the best for all of them, and the best simpliciter, don't you think? The appeal to an infinite number of blissful people in option 1 seems unreal by comparison.

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  3. Alejandro - dominance reasoning doesn't help here. I can make the analogous argument about times. For every moment, there are infinitely many people in the first sphere and only a finite number have been transferred. Thus option #1 contains more total happiness at every single moment in time. By your dominance principle, if it contains more happiness at every moment in time, then it contains more happiness overall. Thus #1 contains more total happiness, contrary to your argument.

    It isn't clear to me why we should privilege the individual perspective over the temporal one, as you do by asking "in which scenario would you prefer to be born?". To avoid question-begging, we must adopt a more objective perspective on the question. For example, imagine you're a benevolent God, faced with the choice of creating one or other of these worlds. Since you're not part of the world (either as an individual or a time), this will help us avoid bias. Also, I must insist that the infinitely many people are no less "real" than the infinitely long time each gets to live for. The infinities in either case are equally baffling.

    Genius - There is no last person, nor any last moment. We instead have an infinite sequence of people and times. But of course any particular person or moment we choose will necessarily be finite. So if we consider an arbitrary person, that makes the infinite time appear to outweigh the population, and we obtain your preference for #2. But we might just as well consider an arbitrary moment in time, in which case the infinite population seems comparatively larger, and we get the result that #1 is better. Neither sort of reasoning seems invalid, but they give contradictory results -- hence the paradox!

    I wonder if it would help to consider hyper-acceleration here. Suppose the first person is transferred after 1/2 an hour, the next after 1/4 more, then 1/8, etc. After one hour, every single one of the infinitely many people will have been transferred. To compensate, let us add that people's subjective experience of time slows down in corresponding increments, so that the first half hour seems no longer than the next 1/4 hour, and so forth. (We can thus fit an infinite amount of subjective time into the one objective hour.) Perhaps the universe explodes on the hour mark.

    That doesn't really change anything. (Note that there is still no last person or subjective moment, as we fit in more and more as we get asymptotically closer to the hour mark.) But it might help illuminate things slightly.

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  4. I realize that standard utility calculations can't solve the paradox, as we get a different result whether we look at persons or at times. When utilities are finite the answers are independent of the way the calculation is done, but when they are infinite as here they can be dependent on the calculation procedure. This seems to leave us only two possibilities: a) say the case is ill-defined in a similar way as an infinite non-convergent sum is ill-defined in mathematics, or b) declare some calculation procedure to be "privileged". I'm not advocating b) over a), just saying that if we choose b) then the most natural choice is to look at individuals. Given that the picture-yourself-as-God thought experiment gives ambiguous results, the next most reasonable thought experiment is picture-yourself-as-an-individual. After all, you cannot picture yourself as a time!

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  5. I'll advocate something like Alejandro's option a). In variation 1, there are infinitely many person-days (or subjective person-days) of bliss and infinitely many person-days of misery. In variation 2, there are also infinitely many person-days of bliss and infinitely many person-days of misery. Neither option is better.

    Your paradox comes from arguing that, for all n, person n is better off under option 2 than under option 1, and that, for all n, day n is better under option 1 than under option 2. But these kinds of "for all n" statements don't necessarily hold once you let things go infinite. To take the two examples from your clarification (people switch spheres in the order of the natural numbers or the even numbers) we could say that, for all n, the two situations are identical after n days in terms of the number of people in each sphere (they're equivalent up to a re-numbering, we could say). This equivalence breaks once we go to infinity, and I think that both of your opposing arguments break as well.

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  6. A) what is infinity
    1) infinity has funny tricks in maths because you dont know what the number is or how you will compare it. that is not realy the case here.
    2) Infinities are generally NOT equal. You might argue that depends on how they are compared theoretically but even if I accept that - we KNOW how these will be compared - we will use whatever your standard utilitarian time/individual comparison is.

    B) which one is bigger?
    if every person gets transfered to the good sphere then at some point the time infinity has begun to over power the popultion infinity because at some point there were more people in the good sector than the bad one as a result of them moving there. Infinitly in the future there must be a time when one sphere has more than the other. you can use hyper accelration to close off that universe to you but we are measuring from inside the spheres so all that does is close you of from the universelike dropping you into a black hole and preventing you from being able to solve the problem.

    C) the contradiction
    Basically I dont see why you seem to wnat to limit your analysis in time to only those events less than infinitely far away when you are willing to count infinitly many people. The way you change what you wont look at between the two scenarios (because of your view of infinity) creates the contradiction.

    D) the Clearest perspective
    ANYWAY - let us put you in a sphere..... If you were to find yourself born in one of these spheres there would certainly be an infinite number of people in both spheres - because the odds of you being born at any other time are ridiculous (1/infinity) so rational analysis will be that at any particular instance net utility is 0 (and you could equaly be in either sphere). Also I think conceptually your "there is no last person" is valid only from a specific frame of reference. To a certain individual there IS a last person and it is him, there is also a first person (which in a sense is equally ridiculous from other perspectives - ie a infinitly long sequence with a first person?). But Why would you intentionally chose the frame of reference which is not relevant and where you cant solve the problem?

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  7. Ok maybe not "born" maybe "swaped with a member of the population for 80 years" would fit the example better

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  8. I will add that infinity is not a quantity. It is a general direction, an indication of extremity. Their difference in magnitutde is more relational than quantitative.

    --

    I don't like this hypothetical. It is (1) far-fetched (2)I am not clear on what it is suppose to be demonstrating. Further, what are we suppose to be maximizing in this scenario.

    My sense is that at some point, t(n), everyone willl be in the opposite sphere that they began in, presumably spending a comparable amount of time in each, quantified by the amount of time they spent in their original sphere.

    So what options are to be evaluated? This seems a contrived and stilted example. For option 2: why would one want to live in eternal bliss?

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  9. I have to agree with erik; infinity is not a number. The idea that the question is a valid one itself requires some kind of justification, because infinity does not allow itself to be subjected to the normal rules of any number.

    Each day, one person is transferred from one sphere to another. What are the parameters that decide _who_ gets selected? This is a necessary input for a person to decide which sphere to be born in. Without this information, no useful choice can be made, because -
    (1) If you are born in the first sphere, you might be transferred the very next day and suffer forever afterward
    (2) If you are born in the second sphere, it might take an infinitely long time for you to get transferred, and again you suffer for an infinitely long time

    Let's put ourselves in the position of God with a choice before Him: which universe should He create?
    Again, like erik hinted, there's no "objective function" to be maximized. It is a reasonable assumption that God would want to maximize the total happiness in the universe. If so, I think the first scenario is the better option, because then the sphere of bliss (with an infinite number of people) will _always_ have more individuals than the sphere of agony (with a finite number of people). Sum it up over an infinite amount of time and you'll get more happiness than agony.

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  10. Ram/Erik,
    Depending on how you efine infinity (and we seem intent on defining it 'as a direction" as erik sugggested). The question does indeed become nonsense.
    But ram I think you are still missing my point.

    Lets say we draw the time line
    --> 0 infinity 1/4 infinity 1/2 infinity 3/4 infinity 1 infinity -->
    now pick a randon point how many people are in each sphere?
    infinity?
    on basically every point in the timeline? if that is the case how can you say
    " the sphere of bliss (with an infinite number of people) will _always_ have more individuals than the sphere of agony"
    What created the perspective that there will always be more in the first sphere relates to the "we are at the beginning" perspective of infinity where infinity obscures the future. (of course from god's perspective that would be wrong because he would see it al at the same "time").

    Anyway Some might say that heaven then hell is better because you get "burnt/harmed" by the bad environment and helped by the good one (presumably) if this happens at all then good then heaven to hell could be better.

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  11. Ramnath, I am not treating infinity as a number. [By contrast, it is nonsense to speak of "1/2 infinity" like Genius does in the above comment. Note that merely the even natural numbers have the same cardinality as the evens + odds, for they can be put into one-one correspondence via the function f(n) = 2n. Thus both sets are equinumerous, by the mathematical definition.] Anyway, my point is, this paradox I point to arises from standard mathematical treatment of the infinite.

    I guess one could restate the question as concerning which scenario contains greater total happiness.

    Note that I have stipulated that nobody waits for an infinite amount of time. Each person is assigned a (finite) natural number n, and they get transferred on the n-th day.

    Finally, your endorsement of the temporal dominance reasoning ignores my earlier comments which explain how equivalent reasoning over the infinitely many individuals yields the opposite result.

    Blar - I think I might have to agree with you.

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  12. Richard,

    I guess you're right. I think the original problem would have been much clearer if it had been stated mathematically. (This is a long comment, bear with me.)

    (1) Think of a n-by-n matrix in which n tends to infinity.

    (2) Each row identifies a single time-instant, and each column identifies one individual.

    (3) The value of an element (i, j) may be 1, indicating that individual j is happy at instant i, or -1, indicating that he/she is unhappy.

    (4) The problem requires that along any column (one individual), there are initially a finite number of ones (assuming scenario #1), followed by an infinite number of -1's.

    (5) The problem also requires that along any row (at an instant), there are only a finite number of -1's and an infinite number of ones.

    Since there are an infinite number of -1's in each column, the total along any column is negative. Add up the totals of all the columns and the result is still negative.
    Since there are an infinite number of 1's in each row, the total along any row is positive. Add up the totals of all the rows and the result is still positive.
    [This is the paradox.]
    At every instant, one individual moves from happy-land to agony-land. I'll restate point (5) above, and say that the individual represented by column j (j = 1, 2, 3 etc.) stays in happy-land for j instants. This is simply a matter of rearrangement of how individuals are numbered and therefore a valid step. (Check)

    The resultant matrix is actually a triangular matrix, if I have my terminology right, and all non-diagonal elements are equal and opposite, adding up to zero. The diagonal elements are all 1, which leads us to a happy universe.
    [1, -1, -1, -1...; 1, 1, -1, -1...; 1, 1, 1, -1...;...]

    Instead, if I redefine the terms slightly, and say that at time t = 0 itself, I transfer one person from the happy-sphere to the agony-sphere, and therefore the individual represented by column j (j = 1, 2, 3 etc.) stays in happy-land FOR j - 1 INSTANTS then the diagonal elements become all -1's instead of ones and we have a bad universe.
    [-1, -1, -1, -1...; 1, -1, -1, -1...; 1, 1, -1, -1...;...]

    Does this make any sense? Can we conclude something from this?

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  13. Ramnath, thanks, the matrix does seem illuminating, and may support Blar's contention that it makes no difference which sphere we start in. Though I too am suspicious of the summing / "cancelling out" procedure, given the results obtained last time we tried that! ;-) Is there a principled reason why it's more legitimate to sum anti-diagonals (to get zero) than rows or columns (where we get +/- infinity)?

    The left-over diagonal elements seem especially arbitrary (and prevent a full vindication of Blar's position here, since whether they're positive or negative will depend on which sphere people begin in). We might get rid of the diagonal elements by stipulating that individuals spend the day before they transfer in a neutral "limbo" (with utility = 0). Intuitively, this shouldn't make any difference to the problem. But if I understand the mathematical model correctly, this would yield a matrix with diagonal elements all of zero.

    I think perhaps we should think of this 'anti-diagonal summation' as a third form of illegitimate sum, on a par with the rows and the columns. It just seems paradoxical to say the universe might be an infinitely bad one (say if the diagonals are all -1 because we start in the sphere of agony), when every single individual in it has net utility that's infinitely positive! (And similarly, mutatis mutandis, for calling it 'good' when each moment is uniformly bad!) Perhaps the only way to avoid paradox here is to throw up our hands and say that there is no determinate answer how to evaluate these bizarre worlds.

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  14. I'm in favor of hand throwing-up. Certainly far superior to hand-waving.

    There is nothing special about the method of summing that cancels out everything except the diagonal elements. One way to see this is to realize that our way of defining the rows was completely arbitrary. We have one person switching spheres each day, and each row representing 1 day, but we could just as easily have each row representing 1 hour (so that there would be 24 identical rows between sign switches) or each row representing 1 week (if you wanted, you could have the matrix contain numbers between -7 and 7 to indicate each person's utility during that week, so that no information is lost). Then, looking at the upper left n x n matrix, we would either see mostly positive numbers or mostly negative numbers, and the diagonal (except for possibly the first entry) would all have the same sign, but this wouldn't mean anything at all.

    FYI, Ramnath, a triangular matrix is one with all zeros either below the main diagonal (then it is "upper triangular") or above the main diagonal (then it is "lower triangular"). I don't know if there is a name for a matrix where the i,j entry is the negative of the j,i entry (excluding the main diagonal).

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  15. There is one good reason for summing up diagonally as opposed to rows-and-then-columns (or vice versa). In the former case, by moving diagonally across a well-defined matrix, we have effectively converted our double-infinity into a single one.

    @blar Certainly this is not a rigorous proof that I'm proposing; I'm just trying to get the idea across. The point of representing the matrix in this particular way is to convert the problem into something that can be solved, a very standard practice. Sure, you can change days into hours or anything else, but unless you can solve it and get a _different_ answer, I don't see why you should object to this method, especially when you can't point out something _wrong_ with it. You can't reject a possible solution simply because you manage to represent it in a form that you don't know how to solve. The numbers 1 and -1 only imply that one day/hour/minute in the happy-sphere is balanced by one day/hour/minute in the other sphere. Any number will do.

    The obvious limitation of this method is that it assumes that the period between transfers is constant.

    One more thing...the diagonal elements are arbitary even if we stick to one scenario, because it depends on how we handle the first instant, whether we start transfering people at t = 0 or t = 1.

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  16. In the finite case, if there are more days than people option 2 is better. If there are more people than days option 1 is better.
    In the infinite case, the same thing is true. I don't see any reason why we can't compare the respective sizes of two infinities.
    If we can't then the problem is undefined.
    I guess I'm just saying the same as genius.

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  17. On one hand, everyone only stays in the initial sphere for a finite amount of time before transferring.

    On the other hand, the average time each person stays in the initial sphere is infinite.

    So if you're a random person, I wouldn't anticipate transferring any time soon, because no matter what time bound you select, you have probability 1 of transferring after that.

    It's on days like this that I'm glad I'm an infinite set atheist and don't believe in "particular moments" in time.

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  18. Just found this post through a hypertext on a recent post.

    It's an interesting problem that I think has a solution. First of all, as to the question which is "better," the term is ambiguous. "Better" can be used to mean which world has more total utility. In that case the two worlds are identical. But it could also mean that which world is more rational for one to choose to live in. In that case I think it's the second world. Here's my reasoning.

    No matter who you turn out to be in the second world, you will eventually be transferred into the sphere where you will experience an eternity of positive utility. Your experiences of negative utility, no matter how large up to that point just before transfer, will be finite whereas your experiences in the world with positive utility will be infinite so it's rational to choose the second world.

    If you had chosen the first world, this would be reversed. No matter who you are, you will eventually be sent to a sphere with infinite negative utility for you. So no matter how much positive utility you experienced prior to being sent there i while still in the first sphere, you will experience an infinite amount of total negative utility.

    So choose option two. The paradox as I see it comes from the fact that the two ambiguous uses of "better" can come apart in such scenarios. But that's not surprising for me. There is a similar situation I talked about on my own blog before.

    Consider this scenario:

    An immortal who has alternating good and bad days (sum of his utility on those days are either positive or negative). His bad days all come on odd numbered days. So days 1, 3, 5, 7,... are all bad and the rest are good days for him where he has +1 utility.

    But let's also say that on his bad days, they are a diminishing sequence, say on his first bad day, his summed negative utility is -1 and on his second bad day, -1/2, 3rd bad day, -1/3, 4th day, -1/4 and so on. So we will have a sequence (-1 + -1/2 + -1/3, + -1/4...). As the negative days approach infinity, his negative utility approaches infinity as that series is a non divergent series increasing without bound. Thus there will be the same amount of positive vs negative utility in the end for him (both infinite) but only a fool cannot see that this is a very good world to live in. For *any* continuous finite span of time beginning with the first day and is longer than two days, the sum of his utilities for those days will be positive. The longer than span, the more positive it will be. The further he is along his life span, the more positive benefit he will have accrued.

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  19. Might I ask about the point of this thought-experiment? What's the point of cashing it out in ethical terms, as opposed to, say, purely mathematical terms? It is just a (putative) mathematical paradox, is it not?

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  20. Nick,

    I think what I tried to show is that the utilitarian assumption that total sum utility is the only criterion for judging ultimate value is wrong or at least it should be distinguished from the alternate sense I described. I don't really think it is mathematically paradoxical at all. It is paradoxical because it conflates two different senses of the term "better".

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