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Topic: paradoxes (Read 10087 times) |
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towr
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Re: paradoxes
« Reply #75 on: Jul 19th, 2008, 12:07pm » |
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on Jul 19th, 2008, 11:49am, BenVitale wrote:| Things with different rate of growth can't be subtracted to give zero. |
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Sure they can.
Besides which, you never even wondered if that sum you wrote even converges. As a sum it is utterly meaningless.
Quote:| What's the limit of a2 - a as a approaches infinity? |
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But we're dealing with something completely different here.
Quote:And we're not, we're subtracting infinite sets. If you remove every element from the set of integers, what you're left with is an empty set. The cardinality of the set you start with and the set you subtract may both be infinite, and the set you end up with has a cardinality of 0, but at no point were we subtracting infinity from infinity to get 0.
So that simply isn't an objection.
{P[10 i] + .. + P[10i+9] | integer i >=0} - {P[i] | integer i >=0} = {}
SUM ({P[10 i] + .. + P[10i+9] | integer i >=0} - {P[i] | integer i >=0}) = SUM {} = 0
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Benny
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Re: paradoxes
« Reply #76 on: Jul 19th, 2008, 8:21pm » |
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Towr, you're right. Thanks. I see it clearly now. I got carried away with my summations and screwed up.
And, I looked at the "Paradox of Enchantress and Witch " two schemes, see link
http://www.suitcaseofdreams.net/Enchantress_Witch.htm
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| « Last Edit: Jul 19th, 2008, 8:22pm by Benny » |
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If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
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LeoYard
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Re: paradoxes
« Reply #77 on: Jul 22nd, 2008, 11:34am » |
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Sorry for interrupting. I'm not ready to celebrate your realization yet.
Is there a simple way to verify this?
As the time reaches 12, the number of cards increases without limits.
What about the sums ben posted? why don't they make sense?
What about infinite sums?
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ThudnBlunder
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Re: paradoxes
« Reply #78 on: Jul 22nd, 2008, 12:07pm » |
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on Jul 22nd, 2008, 11:34am, LeoYard wrote:
Is there a simple way to verify this? |
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Don't think so. It is not a real-world problem.
on Jul 22nd, 2008, 11:34am, LeoYard wrote:
As the time reaches 12, the number of cards increases without limits. |
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When a function is not continuous, as here,
LIM f(x) = b does not imply f(a) = b
x -> a
on Jul 22nd, 2008, 11:34am, LeoYard wrote:
What about the sums ben posted? why don't they make sense? |
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Ben's sums might make sense before 12 o'clock. But, as with Cinderalla, when the clock strikes 12 there will be no more ball.
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| « Last Edit: Jul 22nd, 2008, 12:42pm by ThudnBlunder » |
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towr
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Re: paradoxes
« Reply #79 on: Jul 22nd, 2008, 12:21pm » |
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on Jul 22nd, 2008, 11:34am, LeoYard wrote:| As the time reaches 12, the number of cards increases without limits. |
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But at the strike of twelve, each card that was added was also removed. There is a discontinuity there, which causes a result that may be counter-intuitive.
A card added in step i will be removed between step 10*i-9 and step 10*i; for any card we can precisely say when it was added and removed.
Quote:| What about the sums ben posted? why don't they make sense? |
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Well, maybe if we knew something about the values of Pi. But there's not much to say a priori about a sum from infinity to infinity (which is what the limit goes to).
If you take, e.g. Pi = 1/2i, the sum would nicely converge to 0
If you take Pi = 1/i, then we might get lim i-> inf ln(10i) - ln(i) = lim i-> inf ln(10) = ln(10); and unlike the previous one,  1/i goes to infinity.
In any case it's a bad model when we're dealing with sets of distinguishable objects.
Quote:| What about infinite sums? |
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What about them?
I prefer them unconditionally convergent.
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Benny
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Re: paradoxes
« Reply #80 on: Jul 23rd, 2008, 11:22am » |
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Allais paradox :
http://www.daviddarling.info/encyclopedia/A/Allais_paradox.html
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A paradox that stems from questions asked in 1951 by the French economist Maurice Allais (1911-).1 Which of these would you choose: (A) an 89% chance of receiving an unknown amount and 11% chance of $1 million; or (B) an 89% chance of an unknown amount (the same amount as in A), a 10% chance of $2.5 million, and a 1% chance of nothing? Would your choice be the same if the unknown amount were $1 million, or if it were nothing?
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Would your choice be the same if the unknown amount were $1 million, or if it were nothing?
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The answer in the linked document is not clear to me, especially when the "unknown" amount is nothing.
Would anyone like to clarify?
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towr
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Re: paradoxes
« Reply #81 on: Jul 23rd, 2008, 12:10pm » |
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Given a 100% probability of winning a million, and a 1% of not winning anything or winning slightly more on average, I'd hate to not win anything. You risk a sure million by going for B.
But if chances are stacked against me, with only an 11% chance of winning a million, I may as well push my luck and go for a 10% chance of winning 2.5 million. It's marginal increase in risk for a huge extra payoff.
It may not be rational according to game theory standards, but those are poor standards anyway. What is lacking is a proper mathematical expression for risk. So it's hard to say for what unknown value we should switch from A to B as a risk-averse agent. At what point is a chance of more worth the risk of getting nothing.
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Benny
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Re: paradoxes
« Reply #82 on: Jul 23rd, 2008, 1:27pm » |
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Thanks towr,
I'm gonna need time to digest this concept.
First, I need to read again about the expected utility theory and see how the Allais paradox is inconsistent with the expected utility theory.
And I'm going to read more about risk aversion.
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| « Last Edit: Jul 23rd, 2008, 1:28pm by Benny » |
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towr
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Re: paradoxes
« Reply #83 on: Jul 23rd, 2008, 3:17pm » |
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If we use a simple model of valuing a marginal increase with the square of the probability we get it, then option A is better than B if the unknown value is greater than about 725 thousand.
(for 0  u 1)
A gives a subjective value of 12uM+0.112*(1-u)M and
B gives a subjective value of 12*0M+0.992uM+0.102*(2.5-u)M
Might not necessarily work that way in reality (and of course some might be more risk averse than others), but at least it shows that for some models it's a perfectly rational choice to prefer A for high u and B for low u.
Instead of squaring you could use an arbitrary power as parameter for averseness, I suppose.
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| « Last Edit: Jul 23rd, 2008, 3:18pm by towr » |
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Benny
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Re: paradoxes
« Reply #84 on: Jul 23rd, 2008, 4:01pm » |
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Thanks for your insights, towr.
I find Game Theory the most exciting thing in math.
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If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
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