11 ms·

Your A/B explanation misses the point that the selection doesn't reveal the relationship, only the absolute value. Once you reveal both options, the selection r

Your A/B explanation misses the point that the selection doesn't reveal the relationship, only the absolute value. Once you reveal both options, the selection reveals the absolute value and the position in the set. Using your example, ignoring selection and envelopes and simplifying the goal of the answer: "Here's the letter B from a set of two consecutive letters. What's the other letter?"

dools 13y ago
Your A/B explanation misses the point that the selection doesn't reveal the relationship, only the absolute value. The selection doesn't actually reveal anything. You're given the opportunity to switch prior to opening the envelope. Using your example, ignoring selection and envelopes and simplifying the goal of the answer: "Here's the letter B from a set of two consecutive letters. What's the other letter?" A and B are just labels for the 2 options present at the start of the exercise. You can't assume you've selected B and then say that the remaining option is either A or C because only A and B existed to begin with (or B and C, whatever the labels are is irrelevant). To put it another way: if you select an envelope and attribute a dollar value to it (in the Wikipedia article this is $20) and then suppose it's either the higher or lower value, then compare the expected value of switching in each case, the things you're comparing have absolutely no relationship to each other. If you assume the value you chose was $20, and it's the lower value, then your initial values were $20 and $40, and you should switch. If you assume the value you chose was $20, and it's the higher value, then your initial values are $20 and $10 and you shouldn't switch. In either case the only relevant piece of information is whether or not you chose the higher or lower value and you can't include all 3 values ($10, $20 and $40) in an "expected value" calculation because only 2 values actually exist. Actually it becomes even more starkly obvious if you change the dollar value between each statement, eg: assume you chose $20 and it's the higher value. If you switch, you lose $10. But what if you chose $5,000 and it's the lower value? If you switch you'd gain $5,000 so you should always switch because you're only risking a 50/50 chance of losing $10 for a 50/50 chance of gaining $5,000. This is clearly ridiculous!
- pencilcheck 13y ago
  I will try my best to explain this paradox. This is a paradox only if you act like a computer. So what do I mean by that? This problem simply means, that if you selected an envelope, and you know that there is a chance that the other envelope will have higher value than then one you selected, then by probability, you could conclude that you are better off swapping to the other envelope. However, the paradox starts after you switch envelope because now you can start this all over again and think, hey, even though I just swapped, but it is still possible that the other envelope has value higher than the one I just selected! To explain it in mathematical terms, when the value of the currently selected envelope is Z, the expected value of the other envelope will always be Z/2 * 0.5 + 2Z * 0.5 = 5/4Z higher than the one you selected. The trick is to think procedurally, instead of trying to be realistic. In other words, it is always statistically possible that the other envelope has higher value than the one you selected, therefore you should always swap.
  - dools 13y ago
    Right ... except that's not true :) I actually found a great article today on lesswrong.com which puts things far more elegantly than I ever could, but fundamentally you need to know something about how the values get into the envelope: http://lesswrong.com/lw/dy9/solving_the_two_envelopes_problem/ http://lesswrong.com/lw/dy9/solving_the_two_envelopes_proble... I've submitted this as a fresh link on HN now. Hopefully we see some lively discussion there :)

dools 13y ago

Your A/B explanation misses the point that the selection doesn't reveal the relationship, only the absolute value. The selection doesn't actually reveal anything. You're given the opportunity to switch prior to opening the envelope. Using your example, ignoring selection and envelopes and simplifying the goal of the answer: "Here's the letter B from a set of two consecutive letters. What's the other letter?" A and B are just labels for the 2 options present at the start of the exercise. You can't assume you've selected B and then say that the remaining option is either A or C because only A and B existed to begin with (or B and C, whatever the labels are is irrelevant). To put it another way: if you select an envelope and attribute a dollar value to it (in the Wikipedia article this is $20) and then suppose it's either the higher or lower value, then compare the expected value of switching in each case, the things you're comparing have absolutely no relationship to each other. If you assume the value you chose was $20, and it's the lower value, then your initial values were $20 and $40, and you should switch. If you assume the value you chose was $20, and it's the higher value, then your initial values are $20 and $10 and you shouldn't switch. In either case the only relevant piece of information is whether or not you chose the higher or lower value and you can't include all 3 values ($10, $20 and $40) in an "expected value" calculation because only 2 values actually exist. Actually it becomes even more starkly obvious if you change the dollar value between each statement, eg: assume you chose $20 and it's the higher value. If you switch, you lose $10. But what if you chose $5,000 and it's the lower value? If you switch you'd gain $5,000 so you should always switch because you're only risking a 50/50 chance of losing $10 for a 50/50 chance of gaining $5,000. This is clearly ridiculous!

pencilcheck 13y ago
I will try my best to explain this paradox. This is a paradox only if you act like a computer. So what do I mean by that? This problem simply means, that if you selected an envelope, and you know that there is a chance that the other envelope will have higher value than then one you selected, then by probability, you could conclude that you are better off swapping to the other envelope. However, the paradox starts after you switch envelope because now you can start this all over again and think, hey, even though I just swapped, but it is still possible that the other envelope has value higher than the one I just selected! To explain it in mathematical terms, when the value of the currently selected envelope is Z, the expected value of the other envelope will always be Z/2 * 0.5 + 2Z * 0.5 = 5/4Z higher than the one you selected. The trick is to think procedurally, instead of trying to be realistic. In other words, it is always statistically possible that the other envelope has higher value than the one you selected, therefore you should always swap.
- dools 13y ago
  Right ... except that's not true :) I actually found a great article today on lesswrong.com which puts things far more elegantly than I ever could, but fundamentally you need to know something about how the values get into the envelope: http://lesswrong.com/lw/dy9/solving_the_two_envelopes_problem/ http://lesswrong.com/lw/dy9/solving_the_two_envelopes_proble... I've submitted this as a fresh link on HN now. Hopefully we see some lively discussion there :)