Two Envelopes

So, this time I have a logic puzzle wrapped in a paradox. I'll introduce the paradox first, then the puzzle, then sometime over the next few weeks I'll resolve the paradox. Here is the setup:
A rich man has decided to give away some money to you. He selects a number, and writes a cheque for that amount of money and puts it in an envelope. Then he writes a cheque for twice as much money and puts that in another envelope. He shuffles up the envelopes and hands one to you. You look inside and see $100. Then he offers you the chance to switch envelopes. You know that you got the smaller or larger envelope with 50/50 chance, and the other envelope contains either $200 or $50, thus on average it has $125, therefore, you should switch. However, you can reach the conclusion that you should switch without even looking in the first envelope, as the other envelope contains 5/4 times as much, on average.

Clearly something is wrong here, one can find some time exploring to find out what the deal is. It basically comes down to the problem of saying "he selects a number", you cannot select a random real number without specifying a probability distribution. I'll go more into that later, but for now:
Assuming the initial amount of money is selected from a well defined (but unknown) probability distribution, there is a strategy that guarantees more money than one would get by just simply keeping the first envelope or by just switching blindly. What is the strategy?

Certainly the strategy must work for any choice of probability distribution, its sort of neat to know that it can exist.


edison said...

Hey, kory... if i remember right, there's a study that actually has some relevance to this one :)

It's been studied that the pain of losing money is 'felt' more than the joy of gaining more money.

Basically it means that the pain of losing 1 million dollar is more intense than the happiness of winning 1 million dollar.

You can see this effect in gameshows (Deal or No Deal, for example) where up to a certain point, people just won't risk what they have anymore...

I think, in real life, it would be quite different when you're gambling with 100 dollars (in effect risking 50 dollars) or 1 million dollars (in effect risking 500.000 dollars)

kstevens said...

Yeah, that is true, this relates to utility theory. Basically, utility is some quantifiable version of happiness. Usually this sort of problem implicitly assumes that a persons utility is proportional to the amount of money they gain, but in practice that will essentially always be false.

One also assumes that people are rational actors, in the sense that they consider "gaining" money and "keeping" money to give the same amount of utility, and that will also tend to be false in spite of the fact that they are mathematically and financially the same thing.