Scientists Crack Two-Envelope Problem

Here’s a fascinating story. Two Australian scientists have cracked the legendary two-envelope problem.
Let’s say that there are two envelopes with money in them, one has twice as much as the other. You can open one see how much is there. You now have the option of switching or holding, what do you do?
The CW says it doesn’t matter, you’re just playing the odds so why bother. Well, these two researchers have a way to beat the odds:

The formula relates a particular amount of money (y) found in the first envelope to the probability (P) that you should switch envelopes to gain in the long term.
For example, if you open the first envelope and see $10, the formula might tell you that the probability you should switch envelopes is 0.1 – that is once in every 10 games played.
The formula calculates a smaller probability of switching the larger the original amount (y) is.
Using the previous example, if y is $100, the probably might drop to 0.01, or 1 in every 100 games.
Random strategy
What’s key to this strategy is that the decision when exactly to switch – in which game – must be random, says McDonnell, who studies random processes in telecommunications and the brain.
McDonnell says their solution is different to those that have gone before because of this random element.
“Our solution is to switch randomly,” he says.
“Each time you are offered two envelopes, you observe the amount in one envelope, and then the larger the amount observed, the less likely it is that you should switch, but the choice is still random.”
“The key result is that this kind of random switching leads to a long-run financial gain in comparison with either (i) never switching or (ii) switching randomly in a manner that ignores the observed amount in the opened envelope.”

So is there a stock market connection? You betcha.

In real life, the actual gain will obviously depend on how much money is actually put in the envelopes, says McDonnell.
Abbott says the growth in money seen in the two envelope problem appears to have some similarities to a theory known as “volatility pumping”.
“Volatility pumping is a way of switching between poor investments and yet winning an exponentially increasing amount of money,” he says.
“It suggests the power of changing your portfolio of stocks periodically, buying low and selling high.”

Posted by on August 5th, 2009 at 10:33 am


The information in this blog post represents my own opinions and does not contain a recommendation for any particular security or investment. I or my affiliates may hold positions or other interests in securities mentioned in the Blog, please see my Disclaimer page for my full disclaimer.