Having the game_type_id field is correct and conforming to the general convention. I will be puzzled when the relationship and the table field having the same name like that admin relationship.
In this example, I suggest renaming this admin field to be admin_id. There's nothing wrong with having the game_type_id and admin_id fields.
If you don't want those fields, pls use resources, or simply do this:
$game->load(['admin', 'game_type']);
unset($game->admin_id);
unset($game->game_type_id);
return $game;