The two envelopes paradox is a famous brain teaser of sorts, and not a true paradox. The problem is generally posed like this:

You are given a choice between two identical envelopes. One envelope contains some amount of money, and the other contains twice that amount of money. There is no way to distinguish between the two. However, when you choose one of the envelopes, before opening it, you are given the option of switching to the other envelope. Should you switch?

Why is this sometimes called a paradox? Well, if you choose to switch, you have a 50% chance of doubling your money, and a 50% chance of halving your money. If the amount of money in the envelope you initially chose is M, this reasoning suggests the expected amount in the other envelope is (2M + 0.5M) / 2 = 1.25M. This is more than M, so you should always switch.

But that would suggest once you’ve switched, you’re in the same position you were before you switched, so you should switch again. What is the problem with this reasoning?

Solution to Two Envelopes Problem

On the surface, the math seems to check out, it simply averages the two outcomes. However, the initial amounts in the two scenarios are not the same, which means averaging them is not a valid mathematical operation.

In other words, the scenario in which switching gives you 2M assumes you initially picked the smaller amount, and the scenario in which switching gives you 0.5M assumes you initially picked the larger amount—which means the “M” in the two scenarios are not the same, and therefore cannot be averaged.

Why did it appear to be correct at first glance? Labelling the initial amount “M” in both cases is a hand-waving trick, and it didn’t raise any alarm bells because you aren’t given actual amounts.

Instead, label the amounts M and 2M. Now, it’s clear that the two scenarios are:

  • Switch when you’re holding M: gain M money
  • Switch when you’re holding 2M: lose M money

Which averages out to zero, showing that switching is completely neutral.

Two Envelopes

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