Achei outra maneira de resolver e que provavelmente foi o que aconteceu comigo.
Segue abaixo o que achei no fórum da Apple.
"Other way to solve this problem, is to download Xcode from the Mac App Store, it's free.
So, when it's installed, you can enter Xcode>Preferences>Downloads and install the iOS 5.0 Simulator, it will ask the developers apple id, for me worked as well the Apple ID.
When Xcode's installed, go to the Application folder, find Xcode, right click on it and select "show content" then navigate to Contents>Applications and double click on iPhone Simulator...
With that you can run an iphone on the mac ! Start on it game center, use the Apple ID... and follow the steps. Then Game Center works on the Mac ...(It worked for me)."
fonte: https://discussions....054527#19054527