How do I sign out of Game Center iOS7 iPhone 5S

Question

I can not sign out of Game Center on my iPhone 5S, iOS7. I have searched for a solution, and chatted with AppleCare. No results at all.
In Game Center: Tapping on anything on the home page will not result in an option to sign out.
In Settings: I can "sign out", but that only works for 2 seconds. As soon as I close Settings, I'm signed in again. I hoped it was a setting that cause automatic sign in. AppleCare Chat could not say that is was.
Live Chat Jan. 28, 2014: Solution 1: (Doesn't work)
"When the Game Center screen appears, you should be able to tap on your account name and choose the option to Sign Out. If you are signed out, your friends will not be able to see what you are doing. Also, you can go to Settings > Notification Center and change everything for Game Center to Off."
Solution 2: (Doesn't work, I am NEVER asked to sign in, I am ALWAYS signed in)
"Sign out the Game Center in Settings. Later, when asked to sign in, choose Cancel. Exit the game, then start the game again. Do this two more times and the final attempt will ask if you want to disable Game Center. Choose Yes. Be aware that you will get automatically signed in if you make a purchase in the iTunes Store or the App Store."
     My reply: I am NEVER asked to sign in. It's just automatically signed in as soon as I close Settings.
AppleCare: "If that is the case, something is not being done correctly, because signing out is completely possible. You can also open the Game Center application from the Home Screen and choose to Sign Out from there."
     My Reply: No, I can't.
AppleCare: "Why is that?"
     Me: Nothing I tap on in Game Center brings up a Sign Out option.
AppleCare: "The Sign Out option is one of the only things to be done in the Game Center application. Does your Apple ID appear there?"
     Me: No. My Apple ID is my email and it's not showing in Game Center, just in settings, then Game Center.
AppleCare: "These are the way to sign out of the Game Center. If your phone doesn’t give you these options, there’s not much left to do."
     Me: These are NOT the way to sign out of the Game Center!
AppleCare: "I am happy to listen to how to sign out of the Game Center, if you have found another method.  http://help.apple.com/iphone/7/#/iph6c493cac Using the Settings > Game Center method is the official Apple method to sign out."
     Me: I appreciate, but it still won't sign out for me.
AppleCare: "The one last thing I have here for you is that you can turn on Airplane Mode" (to be invisible, while signed in to Game Center)
     Me: What is wrong? Is my phone defective?
AppleCare: "It is a software issue. You can restore the software through iTunes, but that is an awfully drastic thing to do to your phone for this issue."
     Me: Will it be fixed?
AppleCare: If Apple deems it to be an issue happening to enough people to be addressed.
     Me: So, are you telling me, Odds are, Apple won't ever satisfy my request to be able to log out of Game Center on my new phone?
AppleCare: "You can logout by going to Settings > Game Center. Because it isn’t working right now does not mean the ability is not in place for you to sign out".
     Me: Not getting this, I can but I can't?
AppleCare: "The ability to sign out is there. Maybe it will work tomorrow or in five minutes."
     Me: .... do you mean it might be fixed tomorrow??...(I don't intend to use airplane mode)
AppleCare: "If that is the case, then I’m sorry none of my suggestions are to your satisfaction. The Airplane Mode trick will satisfy your purposes and it is up to you if you want to have friends see you or not, when the Airplane Mode trick will fulfill your needs."
     Me: Where to I submit an issue?
AppleCare: "www.apple.com/feedback" "Yes, the support communities are not quite as formal as the feedback URL I gave you....That is truly where to submit your opinions, complaints and suggestions. Everything does get read and made a note of. Same goes for when people chat in with issues, the issue gets seen by more people..."

Answer 1

I used this method yesterday and everything was done:
On your iOS device, open Settings menu then choose the Game Center menu.
Tap on the Apple ID (yellow line in the above image) menu then choose Sign Out.
You have done.
Source: http://cocland.com/tutorials/how-to-log-out-of-game-center