Expected Value and the Two Envelope Problem

September 4th, 2011
expected_value
Someone puts $X in one envelope and $2X in another. They hand you one at random. Before you open it and keep the contents, they offer to let you switch your envelope for the other one. You can't detect any differences between the two envelopes. Should you accept their offer to switch?

Common sense says no: the two are identical, you have even odds of getting $X and $2X. Expected value, however, says yes. Whatever envelope you have has some amount of money $Y in it. The other envelope has, with even odds, $2Y and $0.5Y. The expected value of switching, then, is the average of $2Y and $0.5Y, or $1.25Y, minus the $Y in the envelope you have. That would be $0.25Y as the expected value of switching, so you should switch. Unfortunately, now that you have switched, you would repeat all this logic, determine that the other envelope had more money, and switch again. Forever. Does expected value trap us in eternal loops of envelope cycling?

This is often considered to be a demonstration of how expected value can give unreasonable answers. But perhaps a solution to this is to frame things differently. We know one of the envelopes has $X and the other has $2X. In switching you are either going from $X to $2X or the reverse; either gaining $X or losing $X. So the expected value of switching is the average of $X and -$X, or zero.

The second framing gives a more reasonable answer, but is there any other reason to prefer it to the first?

Referenced in: Internal Statistics

Comment via: google plus, facebook, substack

Recent posts on blogs I like:

Tuberculosis Considered As Dating Strategy

Against some evopsych

via Thing of Things July 8, 2025

Retrospective on life tracking and effectiveness systems

I’ve been doing life tracking for around 10 years, and this post is looking back at some things I learned from the data (since my previous retrospective in 2017). Highlights include what I get out of the Oura ring, correlations between sleep and deep work…

via Victoria Krakovna July 4, 2025

Elixir's Last Dance

On May 18th, the contra dance band Elixir had their last gig ever. The dance was packed: there were three hundred people. It was the only dance BIDA has ever done where they sold tickets. People flew from across the country just to hear Elixir play one la…

via Lily Wise's Blog Posts June 5, 2025

more     (via openring)