Quote:
Originally Posted by AaronBrown
The only distribution you know in the two envelopes problem is that your envelope was chosen at random so it has 50% probability of having the higher amount and 50% probability of having the lower amount.
In the version of the problem treated in the book, you are told that one envelope contains twice the amount of the other envelope.
If you represent the amount in your envelope as X, you know the other envelope has a 50% chance of holding X/2 and a 50% chance of holding 2X. So the expected holding is 1.25*X. You are not assuming anything about the distribution of X, only the facts given. In other variants of the problem the same principle applies although the exposition is more complicated.
I suppose it depends on exactly what you mean by "the expected holding". To make things clear, let the "Small Envelope" amount be chosen from some random distribution like the one BruceZ gave in his post (uniform on {1,2,...100}) and let the "Large Envelope" be 2*(Small Envelope). Shuffle the two envelopes to get a "First Envelope" and "Second Envelope". Then,
P( [Second Envelope] = 2*[First Envelope]) = 1/2
P( [Second Envelope] = .5*[First Envelope]) = 1/2
Then it is NOT the case that,
E[Second Envelope] = 1.25*[First Envelope]
That would make no sense because E[Second Envelope] is constant and in BruceZ's example equals 75, while "First Envelope" remains a random varibiable.
What is true is that the expected value of the ratio is 1.25. ie.
E[(Second Env)/(First Env)] = E[(First Env)/(Second Env)] = 1.25
Since the expected value of the two ratios above are equal it's hard to see how a case can be made for an advantage in switching based on the expected value of the ratio being 1.25. The expected value of the respective ratio when not switching is also 1.25.
Quote:
Originally Posted by AaronBrown
Therefore, if X is your numeraire, you increase your expected value by switching. This is rational in some cases. For example, suppose the value in the envelopes is points to be used in an auction for valuable real goods. One envelope is drawn at random to determine the point total everyone bidding will be assigned. You are offered the chance to instead take the point total from the other envelope. You should do it, because half the time you'll approximately double your goods from the auction and the other half you'll cut it approximately in half, on average you'll wind up with more goods.
I don't understand this example.
PairTheBoard