(a) I give you an envelope containing a certain amount of money, and you open it. I then put into a second envelope either twice this amount or half this amount, with a fifty-fifty chance of each. You are given the opportunity to trade envelopes. Should you?

(b) I put two sealed envelopes on a table. One contains twice as much money as the other. You pick an envelope and open it. You are then given the opportunity to trade envelopes. Should you?

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TopNewestThese are variations of the Two envelopes problem, which has received a surprising amount of analysis over the years. I posted a note on the subject several months ago, which can serve as an introduction to the supposed paradox.

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This reminds me of the Monty Hall problem which simply blew my mind the first time I learnt about it. For me, on both occasions, I would just choose randomly or based on the person's poker face because no new information came up to alter the probabilities. As far as I can see, there's equal probability for loss and gain at all times. So there's no advantage or disadvantage in switching or staying

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Yes, that's the most practical approach. If no useful information is provided, (and the information provided in version (a) is of no practical use), then there is no reason to switch. In the Monty Hall problem we are provided with useful information and thus have a logical reason to switch so as to improve our odds of winning. It is edifying though to look further into the two envelopes problem and see why certain seemingly convincing perspectives are flawed, (which I deal with (somewhat simplistically) in the note I've linked to in my previous comment). It's also interesting to consider the problem with more, (and even infinite), envelopes, with different multiples of money in each envelope.

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The subject of your note is whit in the scope of Game Theory. The choice depend of the criteria of the gambler then if he choose to keep the envelope he will be playing under a maximin policy since he is guaranteed the best of the worst scenario. I would recommend a textbook on Game Theory you may find resources in the web

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