When Should We Push the Envelope?

A Version of this article is published in my book, ‘TWISTED LOGIC: PUZZLES, PARADOXES, AND BIG QUESTIONS’. CRC Press/Chapman & Hall, 2024. https://www.amazon.co.uk/Twisted-Logic-Puzzles-Paradoxes-Questions/dp/1032513349

A Dive into the Two Envelopes Paradox: Unravelling the Enigma

The ‘Two Envelopes Paradox’, also known as the ‘Exchange Paradox’, is a classic conundrum of choice and value. This deceptively simple dilemma presents us with two envelopes, each containing a certain amount of money. The rules of the dilemma are simple. One envelope contains exactly twice as much as the other. We choose an envelope, look inside, and then face the option of sticking with our original choice or switching to the other envelope.

Thanks for reading Twisted Logic! Subscribe for free to receive new posts and support my work.Subscribe

The Allure of the Switch

At first glance, the decision seems straightforward. We stand to double our money by switching. Let’s say the chosen envelope contains £100. In that case, the other envelope either contains £50 (half) or £200 (double). It’s tempting to switch, as we could either gain £100 or lose £50. It appears that on average we are better off switching, no matter what amount lies in the initial envelope, since we are equally likely to gain £100 as to lose £50.

The allure of the switch persists even when we do not know the envelope’s contents. We could argue that if the chosen envelope holds X pounds, then the amount in the other envelope would be either 2X or 1/2X, with equal likelihood. Mathematically, this can be shown to equate to an expected value of ½ (2½X), or 5/4X, which is greater than X. On this basis, it seems a good idea to switch envelopes.

An Infinite Dilemma: The Absurdity of the Switch

Following this line of logic might lead us to a bewildering conclusion. Why not switch back and forth between the envelopes endlessly? If each switch supposedly increases the expected value, would we not become ever wealthier by just continually swapping envelopes?

This conclusion defies our sense of reality. We know that there’s something fundamentally wrong with the idea of a perpetual money-making machine created just by swapping envelopes. Yet, where does our logic fail us?

Stepping Back: Viewing the Total Picture

A different approach to the problem is to consider the total sum of money present in both envelopes. Let’s represent this total as A. Since one envelope contains Y pounds and the other has twice as much, 2Y, we know that A equals 3Y.

If we initially picked the envelope with Y, switching to the 2Y envelope would give us an additional Y. However, if our first choice was the 2Y envelope, switching to the Y envelope would result in the loss of Y. So, there is an equal chance of gaining Y as it does of losing Y by making the switch. Balancing out these probabilities, we can conclude that the expected gain from switching is precisely zero.

Resolving the Paradox: Framing the Problem Correctly

A key reason why the paradox seems so puzzling lies in how we frame the situation. The argument for switching implies that there are three possible amounts of money in play: X, 2X, or 1/2X. However, we know that there are only two envelopes, hence only two possible amounts.

By accurately framing the problem with just two amounts of money, we realise there is no expected gain or loss from switching envelopes. This remains true whether we frame the amounts as X and 2X or as X and 1/2X. Regardless, the average gain from switching equals zero.

Conclusion: Embracing the Mystery of Probability

The Two Envelopes Paradox demonstrates the often counterintuitive nature of probability and expected value. Despite the tempting initial logic suggesting a continual switch might be profitable, careful consideration reveals that there is, in fact, no inherent benefit to switching—a twist that showcases the often-mystifying appeal of mathematical reasoning.