## The two envelopes paradox : not really a paradox but interesting

Posted: January 10, 2016 in Economics.se relay
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There exists the Envelope Paradox, or Two Envelopes problem for which it appears that a lot of ink has been expended in order to explain away a false argument. Copying from wikipedia,

<<The problem typically is introduced by formulating a hypothetical challenge of the following type:

Of two indistinguishable envelopes, each containing money, one contains twice as much as the other.
The subject may pick one envelope and keep the money it contains.
Having chosen an envelope at will, but before inspecting it, the subject gets the chance to take the other envelope instead.
What is the optimal rational strategy for maximising the amount of money to be gained?

There is no point at all in switching envelopes as the situation is symmetric. However, the story now introduces the so-called switching argument that shows that it is more beneficial to switch. The problem is to show what is wrong with this argument.>>

I really don’t understand why is there a problem to show what is wrong with the “switching argument” (you can read it in the wikipedia article), since a basic formulation of the problem shows easily that there is no increase in expected gain if one switches the envelope. I do this in a recent post in economics.se.

My intention was mainly to showcase how beneficial, and in fact necessary, is to treat such cases formally, avoiding vague language and seemingly logical verbal arguments: expected utility theory, set-theoretic probability and game theory are all put to work (lightly) to provide definite answers.

But I also treat the more interesting variant of the paradox, where one gets to open the envelope, learn the amount in it, and then decide whether to switch or not: In this case, risk-takers and risk-neutral expected utility maximizers should always switch, as well as risk-averse people with risk-aversion lower  than that implied by a logarithmic utility function. At logarithmic utility, one becomes again indifferent with respect to switching, while for more risk-averse persons, it is optimal not to switch.

In both variants, selecting one of the envelopes is a given. So in both variants, what is in the chosen envelop is a given. What is interesting and fascinating here is that just the knowledge of what you got changes the optimal action you should take, even though it does not change the actual allocation of the two prizes.