I’ve run into this issue twice now, so into the blog it goes.
The issue manifested once with a recovered Ubuntu instance and once with a completely fresh install of Ubuntu on a new laptop. Both instances used the default Unity graphical shell.
Here’s the error: when you do something in the shell that requires elevated permissions, you should be prompted for your password:

Instead you get something like this. Here I’m attempting to install the “GNOME Tweaks” application via the “Ubuntu Software” application:

Here’s the fix:
sudo apt-get install policykit-1-gnome
Then:
sudo reboot
Something apparently zapped the policy kit for the gnome desktop environment and this restored it.
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