Marion, I am glad you put in the prisoner thing. I figured there was no advantage to switching, but some of the switch theories were a bit confusing. The prisoners thing makes it clear.

BTW the following is not a joke, or an ethnic slur. Just an easy way to for me to keep track.

[1] There are three of us in jail. Englishman, Scotsman, Irishman (me).
[2] We are told that one will hang and two go free. We are not told which is which.
[3] We all figure we have two chances in three of being freed.
[4] Next morning they free the Englishman.
[5]BAD news for me, 'cos I now have only 1 chance in 2 of being freed.
[5a] Also, bad news for the Scotsman, 'cos HE now has only 1 chance in 2 of being freed.

No problems with the logic so far.

[6] But the original question is, IF I SWITCH CUPS, am I better off. OR In this case, am I better off, If I pretend to be the Scotsman, and he pretends to be me?

[7] If I'M better off to switch, then clearly HE'S also better off (because from HIS point of view HE is the one who is switching.)

[8]How can BOTH of us improve our odds? If my odds improve, his must disimprove, since 1 of the 2 of us must hang?

Since [7] & [8] are mutually exclusive, switching cannot improve the odds for either of us. Therefore to switch or not to switch makes no difference!

Q. E. D.