Two Envelopes Paradox -- 9/19/2019
I have a fondness for paradoxes, and among them, the Envelope Paradox stands out as one of my favorites. The concept can be articulated in several ways, but since I enjoy game shows, I prefer to present it within that context. Here’s how the paradox unfolds:
Imagine you are a contestant on a game show where the host introduces you to two sealed envelopes and prompts you to select one. After you've made your choice, but before opening the envelope, the host reveals that one envelope contains double the amount of money compared to the other. You're then given the chance to switch your selection to the other envelope.
As you ponder whether or not to switch, you deduce that the other envelope might contain either half or double the sum of what is in the envelope you’ve chosen. There’s a 50% probability that your selection is the lower or the higher amount. Let’s denote the amount in your chosen envelope as x. You then reason that the expected value of the contents in the other envelope can be calculated as the mean of half of x and double x. In a more formal mathematical expression, the expected value of the alternative envelope is (1/2)*2x + (1/2)*(x/2) = x + x/4 = 1.25 x.
At first glance, it appears that switching envelopes is a favorable choice. However, you could apply the same line of reasoning to switch back if given the chance. If unlimited exchanges were permitted, you could continuously alternate between envelopes without ever settling. Clearly, this suggests that you aren't actually gaining anything by switching. This raises the crucial question: where does the reasoning fall short in claiming that the expected value of the second envelope is 1.25 times that of the one you picked?
There's no straightforward resolution to this dilemma. Extensive articles published in sophisticated mathematical journals have explored it in depth. Personally, I've debated this issue with fellow mathematicians for hours. While everyone concurs that the 1.25x argument is flawed, there is much disagreement on articulating the reasoning behind its flaws in simple terms.
In my view, one of the clearest ways to illustrate the flaw in the expected value reasoning is that the 2 and 0.5 factors are mistakenly applied to the same x value from the first envelope. This implies that the second envelope must contain either 2x or 0.5x , leading to a ratio of 2x to 0.5x being 4 . However, the scenario specifies that the larger sum is merely twice that of the smaller amount, not four times it. Therefore, this reasoning can't be accurate.
Nevertheless, this clarification doesn’t completely satisfy me. While it may refute the expected value argument, it still leaves unanswered questions about the specifics of where that argument fails. The explanation I prefer is that the expected value formula is flawed because it presumes that x is a constant value, whereas it is actually subject to randomness. The multiplier is directly tied to the amount of x, leading to the collapse of the expected value argument.
A more permissible way to analyze this dilemma is to assess the potential gain or loss incurred when switching. This value represents the difference between what is contained in the two envelopes. For instance, if one envelope has y and the other contains 2y, then switching results in a net gain or loss of y. In simpler terms, the net outcome of switching is represented as *y + 0.5*-y = 0 .
That said, I still feel a bit uneasy with this explanation. While I can find solace in it, I'm not entirely convinced that someone unfamiliar with the topic would grasp my reasoning. They likely wouldn’t.
I apologize if this newsletter does not reach the heights of brilliance. If you find this subject intriguing, it occasionally resurfaces in discussions on my forum at Wizard of Vegas . Here are the two primary discussions about it: