Simulate a process of state change with transition probability dependent on proportion in state in previous time
The principle of your voting system makes me think of Instant Run off, which forces the participants to vote X number of time when there's X+1 candidates on the poll. For Instant Runoff it is the choice that gather the less votes which is out of the competition, but i think you can easily adapt this system to your situation. (Inverting the max and min condition to get the right state out of the game)

I think you can figure it out in R, but you can also try to do it in Python. Take a look at this post : List sorting/modify problem if you want to see how IR can be implemented in Python. You might have to try and post what you've done (even not working) or to ask more specific questions if you want further help from the community.

You can also try other voting systems if it matches with your expectations, i would strongly recommand you the Condorcet voting method if you don't want to bother with multiple round processus : http://short.open-agora.com/m0ZQMPjMQV