Consider an urn filled with balls, each labelled with one of several possible collective decisions. Now, draw two balls from the urn, let a random voter pick her more preferred as the collective decision, relabel the losing ball with the collective decision, put both balls back into the urn, and repeat. In order to prevent the permanent disappearance of some types of balls, a randomly drawn ball is labelled with a random collective decision once in a while. We prove that the empirical distribution of collective decisions converges towards the outcome of a celebrated probabilistic voting rule proposed by Peter C. Fishburn (Rev. Econ. Stud., 51(4), 1984). The proposed procedure has analogues in nature recently studied in biology, physics, and chemistry. It is more flexible than traditional voting rules because it does not require a central authority, elicits very little information, and allows agents to arrive, leave, and change their preferences over time.