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#1
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Before you there are two boxes, one white and one black. Each box contains money; one has twice as much as the other. You may choose either box and keep whatever money is inside.
You choose the white box and find $100 in it. You are now given the option of switching. You may either keep the contents of the white box, or you may instead opt for the contents of the black box. What's the EV of switching? |
#2
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<font color="white">The black box will have $50 in it 50% of the time, and $200 in it 50% of the time. If you choose to switch, you will have $125 on average. If you stay, you will have $100 on average. So the EV of switching is $25. </font>
GoT |
#3
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If you'd originally picked the black box, you'd have come to the exact same conclusion -- that switching is +EV. So no matter which box you choose first, you should switch.
But that can't be right. |
#4
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This is a classic paradox, (not really a paradox, but the way most people naturally think about it) almost as famous as the monty hall problem.
aloiz |
#5
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This is known as the "two-envelope paradox." I used boxes instead of envelopes to make it harder for people to Google the answer.
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#6
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But that can't be right.
It can, and it is. Problems like this one is why people who know a decent amount of game theory get frustrated watching game shows. GoT |
#7
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[ QUOTE ]
It can, and it is. [/ QUOTE ] Actually, it's not. Let me know if you'd like the answer. |
#8
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A decent, but somewhat geeky explanation for this is the fact that the dollar amounts are a geometric progression, with a fast rising curve ahead of you, and a slow falling one behind. With the big upside, and small downside you always want to take any chance to advance up the steep slope.
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#9
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[ QUOTE ]
Problems like this one is why people who know a decent amount of game theory get frustrated watching game shows. [/ QUOTE ] Problems like this one are why some people should step back and think about problems in a common-sense way before jumping straight into EV calcs. |
#10
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Relatively simple explanation as to why GoT's approach is invalid here: Two-envelope paradox.
More detailed explanation Just thinking logically about this problem before jumping straight into EV calculations should make it clear that one can't just apply GoT's basic EV calculations here when dealing w/ finite numbers. See Brocktoon's posts for more on that logic. |
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