Back in the day, the Georgia Tech Linux Users' Group had a very simple two-round system. The first round was approval voting; the second round was a run-off bet

Back in the day, the Georgia Tech Linux Users' Group had a very simple two-round system. The first round was approval voting; the second round was a run-off between the top two candidates. We voted like this every week to decide where to eat dinner. This extremely simple system satisfies participation, monotonicity and a weak form of Condorcet, where the winner always won pairwise against other candidates in terms of expressed preferences (but not all preferences are expressed). The participation criterion and Condorcet criterion are incompatible in the full sense, so this is as close as you can get. The run-off acts as a correction to AV's tendency to select underqualified moderates. However, it is a two-round election. A run-off is always necessary. But it can be made one-round via what I call rank-and-star: voters rank candidates by preference, and put a star next to the top N that they like. The two candidates with the most stars are then compared on rankings. It's pretty easy to prove this is mathematically equivalent to the two-round system, but it's not a ranked-choice method because the ballots (1-2-3) and (1-2-3) are different. Interestingly, it might still be possible to satisfy both participation and monotonicity if you expand the run-off to three candidates and use minimax Condorcet for the final round. For three candidates, minimax = Schulze = ranked-pairs and they satisfy participation; for four or more candidates, no Condorcet method satisfies participation. However, doing this costs you the attractive simplicity of the method (but minimax Condorcet is simple to understand, relative to IRV or ranked-pairs). Anyway, this puts me in the odd position of advocating a voting system nobody has ever heard of =p