Question Regarding Proposed Solution to the (Closed Envelope Version of) Two Envelope Paradox

Question

Su, Francis, et. al. have a short description of the paradox here: https://www.math.hmc.edu/funfacts/ffiles/20001.6-8.shtml

I used that link because it concisely sets forth the paradox both in the basic setting but also given the version where the two envelopes contain $( \, \$2^k, \$2^{k+1}) \,$ with probability $\frac{( \,\frac{2}{3}) \,^k}{3}$ for each integer $k \geq 0$.

Where the paradox is formulated by considering one person’s odds when choosing to swap an envelope, my question is whether the paradox might be resolved by considering the paradox from both swapper’s perspective instead of just one (i.e. for one person to swap, there must be another person for the original to swap with).

From a single person’s perspective, the paradoxical odds are traditionally given by the equation: $$0.5( \,0.5x) \, + 0.5( \,2x) \, = 1.25x$$ To incorporate a two-person perspective, the equation would be [what one person stands to gain] less [what they stand to lose = what their opponent stands to gain]: $$[ \,0.5( \,0.5x) \, + 0.5( \,2x) \,] \, - [ \,0.5( \,0.5x) \, + 0.5( \,2x) \,] \, = 0$$ The result is that neither person improves their odds by swapping. Paradox resolved.

Comments, suggestions, agree, disagree…? I’m just fishing here. Thank you!

Answer 1

The result is that neither person improves their odds by swapping.

This isn't what the equation you've written says. To see this, consider the following game: there are two players, player 1 does nothing, and player 2 can choose whether or not to flip a coin. If 2 says no flip, then neither player gets anything; if 2 says flip, then player 1 gets 1 point if the coin comes up heads and player 2 gets one point if the coin comes up tails. Clearly 2 has an incentive to flip, but $$\mbox{(expected gain for 1 from flipping)-(expected gain for 2 from flipping)=0}.$$

All that your equation indicates is that the action is appropriately symmetric; but that doesn't say anything about either player having an incentive to switch.

Answer 2

We can also break it down into the four possible scenarios from a two-person perspective when swapping where, when ambiguously starting with $x$, we have 25% of the time you’ll lose 0.5x to your opponent, 25% of the time you’ll gain 0.5x from your opponent, 25% of the time you’ll lose x to your opponent, and 25% of the time you’ll gain x from your opponent. The same is true for your opponent.

$$x - 0.25( \,0.5x) \, + 0.25( \,0.5x) \, - 0.25( \,x) \, + 0.25( \,x) \, = x - 0.25( \,0.5x) \, + 0.25( \,0.5x) \, - 0.25( \,x) \, + 0.25( \,x) \,$$ $$x = x$$

In addition to what is noticed in the original question, the above shows that there is no incentive to swap.

The same general logic works for all consistent versions of the paradox that prey on the ambiguity of the variable $x$ (see the link in the original question for more insight into that statement).

I assert that the paradox is resolved.