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  1. #1
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    Exponential growth question

    Suppose you have N factories. This constitutes the 0th generation. Each factory can produce either a widget or another factory.

    So, for example, if N = 2, then you have two factories F1 and F2. If you decide that each produces a widget, then in generation 1 you'd have 2 factories, F1 and F2, and 2 widgets. If you decide that each produces a factory, then in generation 1 you'd have 4 factories, F1 through F4, and 0 widgets.

    1.) Is there an ideal strategy that maximizes widgets at every generation?
    2.) What is the ideal strategy to maximize the number of widgets by generation k?
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  2. #2
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    Re: Exponential growth question

    Quote Originally Posted by CliveStaples View Post
    1.) Is there an ideal strategy that maximizes widgets at every generation?

    No. For every factory that produces a widget rather than another factory, the potential number of widgets produced in future generations is reduced exponentially.
    The more you maximize early widget production, the less you maximize later widget production (including cumulative widgets produced).


    2.) What is the ideal strategy to maximize the number of widgets by generation k?

    Generate only factories thru generations k-2, (in generation k-1, whatever is produced this generation makes no difference to the cumulative total of widgets produced by generation k) then generate only widgets in generation k.


  3. #3
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    Re: Exponential growth question

    Quote Originally Posted by Galendir View Post
    Generate only factories thru generations k-2, (in generation k-1, whatever is produced this generation makes no difference to the cumulative total of widgets produced by generation k) then generate only widgets in generation k.

    It would make a difference to the cumulative total. If at k-1 you decided to start making only widgets, then the cumulative total would include all widgets produced at k-1 + k-2 + k-3. . . + 2nd Generation + 1st Generation + 0th Generation. Once it was decided to only do it at k-1, then it would look like 2^k-1 widgets produced +0+0. This would still be the cumulative total. For something to be cumulative the value of any other value added onto it does not have to be greater than zero.
    But even if it did, you could have just one factor make a widget in generation k-2, then in generation k-1 you could make all widgets.



    1.) Is there an ideal strategy that maximizes widgets at every generation?
    I am not sure what you mean by this. Because the generations would theoretically never stop, then it would approach infinity no matter what method you chose, right?

    2.) What is the ideal strategy to maximize the number of widgets by generation k?
    I think this could be done by some type of linear programming method, but anyways - you would always want to make only factories until generation k-1 then make only widgets... Unless k < or = 2. That's because for generation 1 you would have to make only widgets to maximize widgets, and for generation 2 it is the same (=4) either way because 2*2=4 -> 4 widgets is the same as 2-> 2 widgets + 2-> 2 widgets = 4 widgets.

    The strategy is to basically compare the value at the functions:
    f(k)=2k and f(k)=2^k and see for which function f(k) is greater. The reason it is f(k)=2^k instead of 2^k-1 is because you defined generation 0 as having 2 and in essence "shifted" the exponential graph to the left 1.
    Last edited by Soren; January 2nd, 2012 at 09:21 PM.

  4. #4
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    Re: Exponential growth question

    Quote Originally Posted by Soren View Post
    Quote Originally Posted by Galendir View Post

    Generate only factories thru generations k-2, (in generation k-1, whatever is produced this generation makes no difference to the cumulative total of widgets produced by generation k) then generate only widgets in generation k.


    It would make a difference to the cumulative total. If at k-1 you decided to start making only widgets, then the cumulative total would include all widgets produced at k-1 + k-2 + k-3. . . + 2nd Generation + 1st Generation + 0th Generation. Once it was decided to only do it at k-1, then it would look like 2^k-1 widgets produced +0+0. This would still be the cumulative total. For something to be cumulative the value of any other value added onto it does not have to be greater than zero.

    "Once it was decided to only do it at k-1, then it would look like 2^k-1 widgets produced +0+0."
    I don't understand this sentence.


    As stipulated, each factory produces only factories thru generation k-2; thus, at generation k-2 no widgets have yet been produced.
    Also as stipulated, generation k produces only widgets -- one widget for every factory produced as of k-1.
    If any given factory (f1) produces a widget in both generations k-1 and k, then that factory (f1) will have produced two widgets total (by generation k which is all we care about). If, instead, the factory (f1) produces another factory (f2) in generation k-1 and both factories then produce a widget each in generation k, there will still be a net production of two widgets from the original factory (f1) (one directly and the other from f2). So, whatever is produced in generation k-1 makes no difference to the cumulative total of widgets produced by generation k.



    Note: I altered the typeface from my original post from italics to bold where I used it for emphasis since this distinction is lost by the quote function.

  5. #5
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    Re: Exponential growth question

    Quote Originally Posted by Soren
    I am not sure what you mean by this. Because the generations would theoretically never stop, then it would approach infinity no matter what method you chose, right?
    Sure, they might all tend toward infinity, but I'm talking about an ideal strategy S that maximizes widgets at every generation. Let f(S,k) denote the number of widgets, under strategy S, at generation k. If S is an ideal strategy that maximizes widgets at every generation, then it must satisfy the following statement:

    "For any strategy T, and for all k, f(S,k) >= f(T,k)" That is, any other strategy produces no more widgets at ANY generation than S.


    Say that strategy T produces widgets in the following sequence, starting at generation 0: {0,1,3,5,7,9,11,...} Thus, f(T,k) tends toward infinity. Suppose some other strategy U produces widgets in the following sequence: {0,2,4,6,8,10,...} f(U,k) also tends toward infinity, but U produces more units than T in every generation--even though both tend toward infinity, U converges "to" infinity faster.
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
    **** you, I won't do what you tell me

    HOLY CRAP MY BLOG IS AWESOME

  6. #6
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    Re: Exponential growth question

    Quote Originally Posted by Galendir View Post

    No. For every factory that produces a widget rather than another factory, the potential number of widgets produced in future generations is reduced exponentially.
    The more you maximize early widget production, the less you maximize later widget production (including cumulative widgets produced).

    True, although a nice slick proof would be interesting to see.

    Generate only factories thru generations k-2, (in generation k-1, whatever is produced this generation makes no difference to the cumulative total of widgets produced by generation k) then generate only widgets in generation k.


    What about where generation k-2 is undefined? I.e., k=1.

    Your solution for k > 1 is correct.
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
    **** you, I won't do what you tell me

    HOLY CRAP MY BLOG IS AWESOME

 

 

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