I tested 'I, Mayor' a lot on different computers and have now experienced the same problems like users who wanted to play the game. On that computer the Java runtime environment was in the version of Java RE 6. I updated it manually to the newest version being Java runtime environment Standard Edition 7(JRE SE 7) , and it worked.
Here you can download the newest version of Java SE:
In the list on that site you can select the download for your system (Windows, Linux, etc.).
I hope this will work for you, if you have any further questions, or you are still experiencing problems, i will be happy to help you out.