How to download a game twice with an apple ID?
Anything you buy on the App Store is able to be downloaded again as many times as you like, it's not like it used to be on iTunes where you downloaded it once, and then couldn't do it again if you lost the file.

To do this, just go back to the app store, and search for the app in question.  Instead of listing a price, or saying FREE etc, you should have a little icon showing a cloud and an arrow which you can tap to redownload.  Note, you can only download the latest version of the App, so it it has been updated since you last used it, then you cannot go back to an older version (unless you have a local backup of the file on your iTunes machine).  Also, you must be logged into the store using the same Apple ID that was used to make the original purchase.

The only way I know of that you cannot use this method, is if the app is no longer available on the App Store, in which case you can still get it, but you need to go through your purchase history to get to the download link instead of getting it from searching the store.