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Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value x) and after you have seen it, the dealer offers you an opportunity to swap without anyone having looked at the other card.

If you choose to swap, your expected value should be the same, as you still have a 50% chance of getting the higher card and 50% of getting the lower card.

However, the other card has a 50% chance of being 0.5x and a 50% chance of being 2x. If we keep the card, our expected value is x, while if we swap it, then our expected value if 05.*0.5x+0.5*2x=1.25x, so it seems like it is better to swap. Can anyone explain this apparent contradiction.

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To whoever downvoted this: why? I think it's a perfectly good question and it's pretty obvious that this is math so it's in the scope of the site. – Ben Alpert Jul 21 '10 at 2:17
Your question is very hard to understand. However, if I'm understanding you correctly, I fail to see any paradox. If I know the value of the other card is either double or half the value of my card (and I'm trying to maximize the value of the card in my hand) of course its beneficial for me to swap cards. – Ami Jul 21 '10 at 3:07
You never define what you mean by "value." After reading your question a few times I assumed that each player is trying to maximize "value." The math that you do is not well explained or justified. I don't know what you mean by: "so using expected value." Lastly, I don't see any paradox here. From a probabilistic standpoint, if the player switches cards he may lose or he may gain, but he stands to gain more than he stands to lose, thats what your math shows. Whats the problem? – Ami Jul 21 '10 at 3:23
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This is exactly the two envelopes problem, if anyone wants to write up an explanation. – Larry Wang Jul 21 '10 at 3:49
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An even more interesting problem (similar, but not the same) has been discussed to death on mathoverflow – BlueRaja - Danny Pflughoeft Jul 21 '10 at 4:05
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2 Answers

up vote 3 down vote accepted

This puzzle is known as the two envelope paradox. This paper contains a nice explanation of the two envelope paradox, and some references to further literature regarding the puzzle.

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in the interests of full disclosure, I know one of the authors of the paper, but know very little about the subject. it might well be that there is a much better review paper on the subject – Seamus Aug 4 '10 at 17:20
The article in the link seems to propose an axiomatization to explain the a priori indifference between the envelopes. It does not attempt to resolve the paradox itself. The paradox is discussed quite well in en.wikipedia.org/wiki/Two_envelopes_problem. – Ittay Weiss Sep 23 '12 at 7:12
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This paradox has always interested me. Something to think about is that there does not exist a uniform probability distribution over the positive real numbers (since they are infinite). In arriving at your paradox, it seems you are assuming that any real number is equally likely, but this cannot be the case.

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There is a uniform distribution on the real numbers. I think you meant to say that the paradox is resolved when one realizes there is no uniform distribution on the natural numbers. However, that realization only solves one variant of the paradox. The paradox is recovered by using certain other distributions which still produce the paradox (see 'a new variant' in en.wikipedia.org/wiki/Two_envelopes_problem) – Ittay Weiss Sep 23 '12 at 7:10

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