Riddle #2

OK, I posted this inside the thread comments to the previous riddle, but I wanted to call it out with it’s own post.

Riddle 2-A (taken from here):

Suppose we are presented with the opportunity to open our wallets. Whoever has more money has to give it to the other guy.

A simple analysis suggests that you have a 50/50 chance of winning, and if you do, you’ll gain more money than when you lose. So you should take the bet. But the same analysis suggests that I too should take the bet, and it’s a zero-sum game, so it can’t be advantageous to both of us!

Riddle 2-B (slightly rephrased)

I have two envelopes with checks (made out to ‘Cash’) in them. Andy and Bob randomly draw the envelopes, Andy ends up with envelope A and Bob with B. Andy is offered the opportunity, sight unseen, to make the same deal as in Riddle 2-A (i.e. if his envelope has a smaller check, he gets both envelopes, otherwise he loses his envelope). Should he do it?

Is there a fundamental difference between 2-A and 2-B above?

How about between these two and Riddle 1 from the previous post? (other than the concept of paying a premium to make the switch in the previous riddle…)

4 Responses to “Riddle #2”

  1. Esteban Says:

    Intuitively, I’d say the game has symmetrical opportunities, and being a zero-sum game it can’t be fundamentally good or bad for both players. So it must be neutral; either decision is equally rational. I would decide purely on risk lust or risk aversion.

    In this game, you have all reasons to want a rather empty wallet yourself: bigger chance to win, no less reward, and lower penalty if you lose. Therefore, if the other guy is proposing the game you can assume his wallet is starved and the game is not worth it for you. The other guy should think the same of you.

    I know this doesn’t debunk the expected value optimization argument. In this case, I feel this argument is more fallacious than in Riddle 1, but I can’t explain why..

  2. Phil Steinmeyer Says:

    Certainly, in real life, for 2-A, you’d have to be much more aware of the situation (i.e. if somebody else proposed it to you, he’s almost certainly got an empty or near-empty wallet).

    That’s why I also did the 2-B variant, which gets around that objection. But it makes the paradox more clear. Assume you swap, sight unseen. Then, before opening the envelopes, you’re offered the swap again (A for B this time instead of B for A). Theoretically, you’d have to take it again, and you’d be in an infinite loop (Rinse, repeat). 2-B is worded almost exactly as this wikipedia entry describes.

    Intuitively, it makes no sense to swap in 2-B, but it’s tough to explain away the expected value argument (and the wiki article doesn’t clear a lot up for me either).

  3. PhilSteinmeyer.com » Blog Archive » Riddle Answers Says:

    […] Phil Steinmeyer’s rumblings on the game biz, programming, and life « Riddle #2 […]

  4. Jeff Says:

    My wallet is always empty, or filled with a single dollar. I’ll take that bet :)

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