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Whether to switch


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22 replies to this topic

#11 plasmid

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Posted 07 May 2013 - 02:55 PM

Spoiler for my take

 
Can you gain as much by switching back? Using the same reasoning?
 
There is a paradox: A gain for switching can be anticipated.
Yet, there is a preferred envelope, and if we initially chose it we should not switch.

 
Not sure what you mean by "switching back". The 50/50 comes from randomly picking one of 2 envelopes from which we know one has double the money of the other. I don't see a paradox.

The paradox comes from the fact that $1000 is arbitrary.
Suppose I were to point to an envelope before any were opened, and I said "That envelope contains some amount of money; call it $X. The other envelope therefore must contain $2X or $X/2." You could now argue that the expected earnings from picking the other envelope are $5X/4, so you should choose the other envelope. But in reality, have I actually given you any more information than you already had when you only knew that one envelope contains twice as much money as the other?
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#12 k-man

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Posted 07 May 2013 - 03:28 PM

Thanks, plasmid. It's clear now.

Spoiler for simply put

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#13 bushindo

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Posted 07 May 2013 - 05:32 PM

 

 

 

Spoiler for my take

 
Can you gain as much by switching back? Using the same reasoning?
 
There is a paradox: A gain for switching can be anticipated.
Yet, there is a preferred envelope, and if we initially chose it we should not switch.

 

 
Not sure what you mean by "switching back". The 50/50 comes from randomly picking one of 2 envelopes from which we know one has double the money of the other. I don't see a paradox.

 

The paradox comes from the fact that $1000 is arbitrary.
Suppose I were to point to an envelope before any were opened, and I said "That envelope contains some amount of money; call it $X. The other envelope therefore must contain $2X or $X/2." You could now argue that the expected earnings from picking the other envelope are $5X/4, so you should choose the other envelope. But in reality, have I actually given you any more information than you already had when you only knew that one envelope contains twice as much money as the other?

 

 

 

I think k-man is right. There is no infinite switching paradox in this formulation. That is because being able to see the envelope amount ($1000) nails down the value of one envelope. In other words, the amount $1000 is not arbitrary. The moment we see it, Reverend Bayes has already entered the room.

Spoiler for


Let me try to disentangle the paradox using Bayesian statistics
Spoiler for


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#14 Morningstar

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Posted 08 May 2013 - 02:41 PM

I wrote a quick Java program to generate the total amount gained from switching or not over 100,000 trials with an envelope value of 1,000. Although the random number generator isn't entirely random, I thought it might help nonetheless.

 

 

Spoiler for text of the program

 

Spoiler for the results


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The cake is a lie.

#15 plasmid

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Posted 08 May 2013 - 03:02 PM

Morningstar: what would happen if you changed your program so the amount of money in the first envelope was random? And would that prove that it's always better to switch from whichever envelope you're looking at?

Bushindo: It's certainly better to not switch if you know that there is a ceiling for how much money could be in an envelope and you see an envelope containing more than half of that. But the problem doesn't make any mention of a ceiling. And it may very well be that you don't know how much money is in the bank account, or even if you did then you know that the amount of money in the envelopes is small compared to the size of the bank account but not precisely how small.

That might lead one down the road of looking for a probability distribution with an interesting property. But a complete answer to the question should take into account that the probability distribution could be anything.
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#16 bushindo

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Posted 08 May 2013 - 03:51 PM

I wrote a quick Java program to generate the total amount gained from switching or not over 100,000 trials with an envelope value of 1,000. Although the random number generator isn't entirely random, I thought it might help nonetheless.

 

 

Spoiler for text of the program

 

Spoiler for the results

 

I think when you code this problem as plasmid suggested, you have to be careful to specify what kind of random distribution you are drawing from since that essentially will define your prior assumptions about A and B. Most random number generator have an explicit upper range, so it might be a problem to sample uniformly from a infinite real line.

 

Morningstar: what would happen if you changed your program so the amount of money in the first envelope was random? And would that prove that it's always better to switch from whichever envelope you're looking at?

Bushindo: It's certainly better to not switch if you know that there is a ceiling for how much money could be in an envelope and you see an envelope containing more than half of that. But the problem doesn't make any mention of a ceiling. And it may very well be that you don't know how much money is in the bank account, or even if you did then you know that the amount of money in the envelopes is small compared to the size of the bank account but not precisely how small.

That might lead one down the road of looking for a probability distribution with an interesting property. But a complete answer to the question should take into account that the probability distribution could be anything.

 

I think the discussion should be about what prior distribution is more representative of the puzzle conditions

Spoiler for


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#17 plasmid

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Posted 09 May 2013 - 02:42 PM

I believe there is a good reason why a probability distribution was not stated in the problem.

I can think of an "experiment" that I believe would be considered satisfactory by most people, that doesn't depend on the probability distribution of how much money is in the smaller or larger envelope. The answer would be appropriate if our only information were that $1000 is small compared to the available bank account (ruling out the trivial possibility that the bank account is less than $3000 which would make it impossible for the other envelope to hold $2000 and obvious that you should not switch).
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#18 bushindo

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Posted 09 May 2013 - 04:46 PM

I can think of an "experiment" that I believe would be considered satisfactory by most people, that doesn't depend on the probability distribution of how much money is in the smaller or larger envelope.

 

I'd love to hear about this experiment that does not depend on the probability distribution of how much money is in A and B. My feeling is that Reverends Bayes is hiding somewhere, possibly heavily disguised, in the set-up. But I may be wrong, I often am.


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#19 plasmid

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Posted 09 May 2013 - 11:23 PM

The experiment is fairly simple. Randomly generate howevermany "smaller" envelopes you want {s1, s2, s3 ... sn} and for each of them generate a matching "greater" envelope {g1, g2, g3 ... gn} where the value in gx = 2sx. Let the value of all sx and gx be small relative to the bank account. The participant is given a random envelope and given the choice of whether or not to switch -- if he was originally given an envelope sx then he will switch to envelope gx, and if he was originally given an envelope gy then he will switch to envelope sy. Now compare what happens if the participant takes a strategy of switching vs a strategy of staying with the initial envelope.

 

The probability distribution is random, and results should generalize to any probability distribution that you face.

 

One could argue that it doesn't fit the OP because the player isn't presented a value of $1000. I would counter that the precise value that the player finds when he opens the envelope is arbitrary (you could multiply all of the s and g terms by any value you like) and would not affect the conclusions of the experiment.


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#20 bushindo

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Posted 10 May 2013 - 12:09 AM



The experiment is fairly simple. Randomly generate however many "smaller" envelopes you want {s1, s2, s3 ... sn} and for each of them generate a matching "greater" envelope {g1, g2, g3 ... gn} where the value in gx = 2sx. Let the value of all sx and gx be small relative to the bank account. The participant is given a random envelope and given the choice of whether or not to switch -- if he was originally given an envelope sx then he will switch to envelope gx, and if he was originally given an envelope gy then he will switch to envelope sy. Now compare what happens if the participant takes a strategy of switching vs a strategy of staying with the initial envelope.

 

The probability distribution is random, and results should generalize to any probability distribution that you face.

 

One could argue that it doesn't fit the OP because the player isn't presented a value of $1000. I would counter that the precise value that the player finds when he opens the envelope is arbitrary (you could multiply all of the s and g terms by any value you like) and would not affect the conclusions of the experiment.

 

Can you clarify the part highlighted in red? Do you mean specifically to generate N random numbers from the uniform distribution between, say, 0 and L?

 

If I'm writing code for this experiment, I can't generate a random number without telling the computer precisely which probability distribution to use (and the corresponding distribution parameters). Most computer programs, for instance, will allow one to generate a random number uniformly between [0, L], but then you will need to supply the value for the upper limit L. (Reverend Bayes, is that you?)

 

Randomness comes in many forms (e.g., normal, uniform, exponential, etc. ) and I don't think it is possible to generate random numbers without specifying which probability distribution we are working with.


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