Wizard Of Odds Weekly Update September 19, 2019
I love a good paradox and the Envelope Paradox is one of my favorites. There are various ways of wording it. I like game shows, so prefer to word it in that format. That said, here is the paradox:
You are on a game show where the host presents you with two sealed envelopes and asks you to choose one, which you do. Without opening it, the host then explains that one envelope has twice as much money as the other. He then gives you the option to switch to the other envelope.
In considering the decision to switch, you reason that the other envelope has either half or twice as much money as the one you chose. There is a 50% chance you chose the lower or higher envelope. Let x be the amount in your chosen envelope. You calculate that the expected value of the other envelope is the average of half of x and double x. In more mathematical words, the expected value of the other envelope equals (1/2)*2x + (1/2)*(x/2) = x + x/4 = 1.25 x.
This seems to make switching look like a good bet. However, you could apply that same reasoning to switch back, if given the opportunity. If unlimited switching were allowed, you would keep switching back and forth infinitely. Clearly, you aren't gaining anything in the process. So, the question is, where is the flaw in the argument that the expected value of the other envelope is 1.25 times the one your chose?
There is no simple answer to the question. Long articles in advanced mathematical journals have been written about it. I personally have argued with other mathematicians on a peer level for hours about it. Everybody agrees the 1.25x arguments is flawed, but not in how to explain why it is flawed, especially in plain simple English.
In my opinion, the easiest way to explain the flaw in the expected value argument is the 2 and 0.5 multipliers are applied to the same value of x in the first envelope. For one thing, this suggests that the amount in the other envelope is either 2x or 0.5x. The ratio of 2x and 0.5x is 4. The problem itself states the larger amount is twice the smaller amount, not four times the amount. So, that can't be right.
Still, that argument leaves me unsatisfied. It may disprove the expected value argument, but where does the expected value argument go wrong? The way I prefer to explain it is the expected value formula doesn't work because it assumes that x is a fixed amount. It's not, it is random. The multiplier is 100% correlated to the amount of x. That causes the expected value argument to fall apart.
A more kosher way to think about the problem is to consider the amount gained or lost in switching. That amount is the difference between the two envelopes. For example, if the two envelopes contain y and 2y, then the switching will cause a gain or loss of y. In other words, the gain by switching is *y + 0.5*-y = 0.
Still, I am not entirely satisfied with this explanation. I can sleep at night with it, but don't know that a layman would understand my argument. He probably wouldn't.
Sorry if this newsletter is less than brilliant. If the topic interests you, it comes up from time to time in my forum at Wizard of Vegas. Here are the two main threads about it: