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As specified here, the Google Play Games API uses the signature provided by the game developer to authenticate the login of each person that attempt to sign in using a Google account.
As I've said here other times, whenever you make any kind of modification in an apk file using Lucky Patcher, the original signature provided by the developer is changed. Because Google Play Games uses a unique signature provided by the developer to perform logins authentication, it will no longer work after this, as the original signature has changed.
Read more about signing apps on Android:
Read more about the Google Play Games API on Android:
Maybe this will help solve your problem:How to make unsigned APK and Google+ / Facebook login | Android Republic