I'm a long time app-designer and I always strive for ease of use and efficiency in my interfaces.
I think the "jump to your next playable game" link would be greatly improved by changing the link-text to something different if you have no more playable games to 'jump to'. (If it isnt your turn in any of your games) The link could become "return to My Games" or something... doesnt really matter what it says as long as its different.
This way if i just log in to make my moves (which is frequently all i do) I dont have to bother clicking the link and can instead just close my browser after i finish my final move.