Unable to update Remote Desktop using App Store
Is it in your purchase history? I'm guessing no. 

Apple changed the policy on ARD a while ago. It went from 'serial number' to App Store, &amp; any existing ARD on the machine should have been 'upgraded' to that at the time. Mine was showing as purchased, but not available as an update.  

The App Store app was showing it as 'Open' which it wouldn't because the client had already updated to 3.9, which sent me back to the App Store -  so I was locked in an infinite loop

My fix was to trash the app [or zip it &amp; trash] then go to it in the App Store https://itunes.apple.com/app/id409907375 which then showed it as 'Install' &amp; I was up &amp; running. 

Alternative approach I've heard to work was - remove ARD from Apps, launch App Store, check for the ARD app, which shows as to buy, quit App Store, put ARD back in Apps &amp; check again.

A third possibility - untested &amp; definitely worth having a backup - see if com.apple.pkg.RemoteDesktopAdmin_AppStore either .bom or .plist exist in /private/var/db/receipts. Trash &amp; try one of the above methods.
I'm not certain whether this is double-checked in /Library/Receipts/InstallHistory.plist hence the backup recommendation.