# Modified two envelope paradox

This problem is a variation on a two envelope paradox.

This time Alice and Bob play the game. Envelopes X and Y, when opened contain money. One envelope has n dollars and the other has 2*n dollars. Alice opened envelope X, saw 20 dollars and decides if she wants to get this envelope (X)or the envelope Y. From her perspective, envelope Y has 40 or 10, thus mean is 30 > 20 and she should change the envelope.

Bob knows that Alice chose envelope X. Bob also knows something that Alice doesn't: when Alice chose X a machine first generated value for envelope Y, then produced value for envelope X based on a fair coin flip. From Bob's perspective envelope X has better average return.

So, Alice should choose Y and Bob should choose X for the same pair of envelopes. We can test this experimentally. Who's strategy is going to be correct?

My impression is that there are three cases

(1) In a single experiment this question is meaningless (one loses other wins with 50%/50% chances)

(2) In repeated experiments with narrow interval of valid values (more than a dollar, but less than 1000) Alice will lose whenever envelope X contains more than half of the cutoff sum. "Envelope X has 998, so there is a 50/50 chance of 1996 in other envelope!" - no. There is a 100% chance of 499 in Y due to the cutoff. Bob will never suffer from this cutoff effect.

(3) If we increase the cutoff value to decrease the devastating effect of cutoff on Alice then we go back to case (1). A lucky streak of 2 wins in 1000 rounds is negated by a single loss of in one billion vs two billion dollars round.

Is this analysis correct?

• Yea it's about right...Is that all? – Don Thousand Aug 19 '18 at 2:35
• In your experiment, how are you going to determine $n$? – Brian Borchers Aug 19 '18 at 2:40

As with the usual two-envelope paradox, it is impossible to answer without knowledge of the distribution of prizes (in this case, the distribution of $Y$). If Alice knows the distribution, she can make a simple calculation to decide which envelope to take. If she doesn't know the distribution, then she should probably base her decision on the state of her personal finances.
• @Stepan Certainly Bob should always choose $X$, since he knows that its expected value is $1.25Y$, regardless of how $Y$ was generated. The only reason to choose $Y$ would be "real life circumstances", e.g., if he desperately needs a fancy coffee, and $Y$ is $\$6$, then maybe he should take it rather than risking only getting$\$3$. – Théophile Aug 19 '18 at 20:16