Cannot authenticate Game Center

Game Center just will not authenticate. When I write the code below, nothing happens.

Social.localUser.Authenticate ((bool success) => {
                    if (success) {
                            // show leaderboard UI
                            Social.ShowLeaderboardUI ();

If I write Social.ShowLeaderboardUI (); on its own I get the error: Game Centre Unavailable - Player is not signed in. What am I doing wrong?

When you reject to sign in into Game Center three times it will never appear again. You should go to the Settings/Game Center and login manually into your account to make it work.