The tiny blue notification button isn’t exactly obvious, and honestly it’s a hassle considering it’s main objective is to be filled up with notifications of games started which you already know about.
So this is a 3-parter. First, don’t let a “This game has started” notification linger if I’ve already played a move in it. Starting play = I already know, x it automatically.
Secondly, notify with more than just a number. Show the text of the most recent alert, so people see the private challenges and friend request promptly, rather than the next time they decide to clean out that list of boxes. An optional non-intrusive sound alert would be great too.
Thirdly, make it possible to offer a rematch at the end right away. Once the game’s over, a final blue button of “offer rematch” to make it easy to start a new match with the same settings, rather than both people challenging, or both waiting for the rematch challeng.