There are two envelopes in which I, the host of the game, put two different natural numbers, chosen by any distribution I like, that you don’t have access.
Now, before everything I said happens, you must devise a strategy that guarantees that you have a greater than 1⁄2 chance of winning.
Well natural numbers and simple greater than satisfying makes it easy. “If one THEN swap ELSE keep.”
Maybe I didn’t express myself well, but this strategy should work regardless of the distribution I choose. For example, if I choose a distribution in which 1 has probability 0, than your strategy yield 1⁄2 chance.
Maybe I didn’t express myself well, but this strategy should work regardless of the distribution I choose. For example, if I choose a distribution in which 1 has probability 0, than your strategy yield 1⁄2 chance.
If that kind of selection of distributions is possible then there is no free lunch to be found.
For any strategy of envelope switching a hostile distribution selector who knows your strategy in advance can trivially select distributions to thwart it.
Have you looked at the “solution”? There really isn’t a counter-strategy that reduces it to 1⁄2 chance, although there are strategies moving it arbitrarily close to 1⁄2.
Well natural numbers and simple greater than satisfying makes it easy. “If one THEN swap ELSE keep.”
Maybe I didn’t express myself well, but this strategy should work regardless of the distribution I choose. For example, if I choose a distribution in which 1 has probability 0, than your strategy yield 1⁄2 chance.
If that kind of selection of distributions is possible then there is no free lunch to be found.
For any strategy of envelope switching a hostile distribution selector who knows your strategy in advance can trivially select distributions to thwart it.
Have you looked at the “solution”? There really isn’t a counter-strategy that reduces it to 1⁄2 chance, although there are strategies moving it arbitrarily close to 1⁄2.