I’m happy to report that Apple has evidently fixed whatever the problem was with the Mac App Store versions of Photoshop Elements. (See this post if you missed the problem.)
The solution is quite simple. Go to your Applications folder and send the old copy of Photoshop Elements to the trash, then launch the App Store, go to your Purchases and reinstall PSE. Contrary to what I said in the first version of my previous post, you will find out of date versions available for download now.
Note that this only applies to the Mac App Store version of PSE. If you got Elements from anywhere else, you cannot uninstall it by trashing it, and nothing in this post applies to any problems you may be having with your version of PSE.
EDIT 11/14/15: A few people have found that simply downloading and reinstalling isn’t enough. In that case, empty the trash with the old version in it, sign out of the App Store and back in, then try downloading and reinstalling. You will need a fresh download, not a copy from a backup, in this instance.
Second EDIT 11/16/15: If you have PSE 13 from the App Store, you may need to follow the direcions in this Adobe kbase document to fix things:
https://helpx.adobe.com/photoshop-elements/kb/editor-does-not-start-error-mac-app-store.html
