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Topic: A Selfless Act? (Read 4391 times) |
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alien2
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A Selfless Act?
« on: Jan 4th, 2013, 9:32am » |
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The wretch, concentred all in self,
Living, shall forfeit fair renown,
And, doubly dying, shall go down
To the vile dust, from whence he sprung,
Unwept, unhonour'd, and unsung.
Sir Walter Scott
100 imprisoned men, living in a parallel world to ours, have been found guilty of the crime of excessive selfishness. According to the State, their punishment is as follows. The warden tells them in the courtyard that in one hour, each of them will be teleported to a random cell. Each cell is windowless and soundproof. Only 50 cells have a button, which is emitting a dim light in the dark. All those who push the button will serve a two-month stretch in the hole, and others will be released after 24 h in solitary confinement. If nobody pushes the button within above 24 h, all prisoners will spend a year in the cells. The warden then leaves them so that they can discuss a plan.
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| « Last Edit: Jan 6th, 2013, 1:54pm by alien2 » |
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Grimbal
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Re: A Selfless Act?
« Reply #1 on: Jan 4th, 2013, 9:59am » |
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If some people do press the button, what happens to the others? Are they released?
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alien2
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Re: A Selfless Act?
« Reply #2 on: Jan 4th, 2013, 11:08am » |
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on Jan 4th, 2013, 9:59am, Grimbal wrote:| If some people do press the button, what happens to the others? Are they released? |
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I modified the riddle. I hope everything is clear now.
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Grimbal
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Re: A Selfless Act?
« Reply #3 on: Jan 4th, 2013, 12:07pm » |
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I will assume we want to minimize the average total prison time.
I considered 2 methods:
1. Choose n prisoners randomly. These must press on the button if they can.
I find an optimum at n=9 with an average 10.44 months total prison time.
2. Choose a probability p. Every prisoner who sees the button presses with probability p.
I find an optimum at p=0.121 with an average 13.88 months total prison time.
Method 1 is better.
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towr
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Re: A Selfless Act?
« Reply #4 on: Jan 5th, 2013, 2:48am » |
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on Jan 4th, 2013, 12:07pm, Grimbal wrote:
1. Choose n prisoners randomly. These must press on the button if they can.
I find an optimum at n=9 with an average 10.44 months total prison time. |
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So, one of these prisoner is in his cell, sees the light of the button and considers: do I want a two month prison sentence or take my chances that one of the other 8 prisoners pushes the button and only spend a day in prison? The chance is pretty good at least one of them is in a cell with a button, so if they then press it, I'm much better off not pressing my button.
Don't forget these are selfish bastards.
So at the very least, the selected prisoners will have to decide for themselves what probability of pushing the button maximizes their own individual prospects.
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| « Last Edit: Jan 5th, 2013, 3:44am by towr » |
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Grimbal
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Re: A Selfless Act?
« Reply #5 on: Jan 5th, 2013, 5:57am » |
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on Jan 5th, 2013, 2:48am, towr wrote:| Don't forget these are selfish bastards. |
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Ah, well, yes, that changes everything.
So we need a Nash equilibrium. I would propose to choose 3 prisoners. These prisoners are asked to press the button if they see it.
For a chosen prisoner, the probability that nobody else presses the button is ~1/4, if he doesn't press, he serves ~3 months on average. So he will press the button.
For the remaining prisoners, the probability that no chosen prisoner presses the button is ~1/8, that is an average prison time of ~1.5. So they don't press.
A more accurate calculation gives 144 months of total prison time. That's more than the 10.44 months they could serve, but that is the price for falling for the prisoner's dilemma.
You could make the selection secret. A jar has 100 marbles. 97 white and 3 black. Each prisoner takes a marble without showing it. Everybody is requested to press the button if and only if he got a black marble.
With this setting, you could improve the odds by actually putting in 9 black marbles, but without telling the prisoners.
But that would require an external person. If it is a prisoner who prepares the jar and he is so smart to think of putting more black marbles, being selfish he would put in at least 52 black marbles.
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| « Last Edit: Apr 21st, 2013, 2:30am by Grimbal » |
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cartoonle
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Re: A Selfless Act?
« Reply #6 on: Jan 6th, 2013, 3:36am » |
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What happens if more than one prisoner presses the button?
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alien2
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Re: A Selfless Act?
« Reply #7 on: Jan 6th, 2013, 7:11am » |
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on Jan 6th, 2013, 3:36am, cartoonle wrote:| What happens if more than one prisoner presses the button? |
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I modified the riddle again.  Everything should be clear now.
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| « Last Edit: Jan 6th, 2013, 11:09pm by alien2 » |
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marlonmark
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Re: A Selfless Act?
« Reply #8 on: Jan 6th, 2013, 10:24pm » |
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my friend dont keep on modifying the riddle again n again as it becomes difficult to understand this one.
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Annettagiles
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Re: A Selfless Act?
« Reply #9 on: Oct 14th, 2014, 4:27am » |
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In that you haven't mentioned about minimum how many members need to press the button
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rmsgrey
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Re: A Selfless Act?
« Reply #10 on: Oct 14th, 2014, 6:15am » |
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on Oct 14th, 2014, 4:27am, Annettagiles wrote:| In that you haven't mentioned about minimum how many members need to press the button |
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At the end of 24 hours, if at least one button has been pressed, all prisoners who didn't press a button are released, while those who did press a button serve 2 months. If no-one has pressed by that time, then everyone serves 1 year instead.
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