Change the Apple ID for a game and not lose the progress made?

Question

Short version of my question: can I change the apple id associated with a game to another one and not lose the progress made in the game thus far?
Long version: when there were only a few iGadgets in the house, all games and music were downloaded through Alpha's apple id and the credit card tied to that account. Now, we two folks have multiple iGadgets between the both of us and Beta has her own apple ID. Because we only used one apple id  for so long, Beta has some games installed through Alpha's account and her own Beta account. Beta would now like to change the account the game PocketTimeSuck is associated with from Alpha's account to Beta's account without losing any of the progress made in this game which she has been playing for over a year.
Is there any way of re-installing PocketTimeSuck onto Beta's device under Beta's apple ID and maintain the progress in the game she has made after installing it under Alpha's apple id? Alpha would like to start playing PocketTimeSuck but cannot install it under her own apple id because it comes with all of Beta's progress.
(Also, if this is not the right forum, please let me know where I should post this question. Don't flame me, just tell me.)

Answer 1

I'm getting confused by the Greek alphabet here so I will explain it this way and hope I understand what you are asking.
Beta has been playing a game that she downloaded using Alpha's ID but now Beta would like to transfer the game and all of its progress to her own ID. That is not possible to do. The game was bought with Alpha's ID and is now associated with that ID and that progress is part of Alpha's backup.
Alpha now wants to play the game under her own ID but the game is associated with her ID already. If she wanted to, she could just pick up where beta left off and leave Beta out in the cold. Somehow I don't think that is a good solution for either Alpha or Beta.
Alpha should probably create a new Game Center account and start all over again with her own Game Center account. Beta will obviously have to do the same thing if she wants to play this game using her own Apple ID on her own iGadget.
So if I do understand correctly, Alpha and Beta will both have to start over again. Howver, I could be confused because this is all "Greek to me".