I had the same problem, also with Mint. The "Game" (desktop) file does not work so you tried to run "Game" (the binary file) instead? Wrong (I made the same mistake first because it seems obvious). But if you take a look into the "Game" desktop file, it does NOT link to "Game" (binary) but to a binary called "nw" instead. (No idea why they did this, it seems confusing on purpose). So to run the game make "nw" executeable, open a terminal and run "./nw". Now it works.