If you just want a representation, you can use Tableau's new Density feature with a scatter plot.
Your 10 X/Y values can be pivoted (from 10 columns into 2), plotted, and then mapped.
You'll want to IMPORT the picture and use that as your background, though you'll need to figure out the correct coordinates for it (I'm thinking 1000/750?).
Here's a view using *just* the first answers, superimposed on your map. Another grey-scale map might offer a more polished view for your users; I think you could generate the Lat/Long with two relatively simple calcs.
What this lacks:
Easy visibility into the cities beneath the "hotspots". A worksheet action could remove the dots on a click, though.
Feedback about what/where is being selected. While you can see it on the map, no variable holds the value "Austin"... so you can't perform tasks like "shows me the 10 most preferred cities"
Note that it appears not everyone followed the directions: someone clicked "Canada" as well as the state NAME of "MD"...
Please let me know if you'd like more info about this viz.