Yet another Probability Paradox
Brian Weatherson from Thoughts, Arguments and Rants has just reminded us of the wonderful Monty Hall Problem. Here is a even stranger problem, taken from Raymon Smullyians wonderful Satan, Cantor and infinity: The Two Envelope Paradox. It goes like this: There are two envelopes, one contains twice as much as the other envelope. You see no difference between them from the outside. You choose one envelope and open it, it contains X>>0 dollars. Now you can either keep the money in that envelope or take the money from the other envelope. You are assumed to be risk averse, only caring for the expected amount of money you can get from a decision.
According to one train of thought, you should choose the other envelope since it contains 2X$ with a probability of 0.5 and 0.5X$ with the same probability. That gives you an expected net gain of 3/4X$, a strictly positive sum. Maybe you are convinced now, but there is another train of thought. Suppose you would have chosen the other envelope. You still would want to choose another envelope according to the argument above. But that means the new information you got doesn't influence your decision in any way. So why is that different from the initial position, where you could have chosen any envelope? So it doesn't matter whether you choose the other envelope or not.
The Baysian solution is very simple: You must first have a (subjective) probability distribution and calculate the expected gain of changing afterwards, which will probably lead to other probablities of the higher amount being in the other envelope than 0.5. A analysis for people with a background in basic probability heory can be found here.
According to one train of thought, you should choose the other envelope since it contains 2X$ with a probability of 0.5 and 0.5X$ with the same probability. That gives you an expected net gain of 3/4X$, a strictly positive sum. Maybe you are convinced now, but there is another train of thought. Suppose you would have chosen the other envelope. You still would want to choose another envelope according to the argument above. But that means the new information you got doesn't influence your decision in any way. So why is that different from the initial position, where you could have chosen any envelope? So it doesn't matter whether you choose the other envelope or not.
The Baysian solution is very simple: You must first have a (subjective) probability distribution and calculate the expected gain of changing afterwards, which will probably lead to other probablities of the higher amount being in the other envelope than 0.5. A analysis for people with a background in basic probability heory can be found here.
