Instant runoff voting (IRV) is known to display a number of perverse problems when there are more than two strong candidates. As a thought experiment, and mental exercise, Professor Warren Smith--mathematician and founder of the Center for Range Voting--try to determine the simplest IRV election that can showcase the greatest number of these issues.
9 voters: A > B > C
12 voters: B > C > A
8 voters: C > A > B
The example has only three candidates, and needs just 29 voters, but still illustrates:
- Reversal Paradox: if all the ballots have their order reversed, the "winner" stays the same, so the "best" choice is also the worst choice. (It's A in both cases.)
- Less-is-More: one of two forms of non-monotonicity, where by lowering the rankings for a losing candidate on certain ballots, they become the winning candidate. (Change two B-first voters to C-first, and B wins instead of A.)
- More-is-Less: the other form of non-monotonicity, where raising the winning candidate's rank on certain ballots makes them lose. (Raise A from bottom to top on 5 of the B-first ballots, and C wins instead of A.)
- Participation Paradox: where, if certain voters had just stayed home instead of voting, the result of the election would have been better for them. (Remove 5 B-first voters (who rank A last!) and C wins instead of A.)
- Anti-Participation Paradox: where, if more of a certain group of voters had shown up and voted, the results of the election would have been worse for them. (Add 2 C-first voters (who rank B last!) and B wins instead of A.)
- Precinct Paradox: the ballots can be divided into precincts, and the winner in each precinct is the same, but is not the same as the winner of the overall election. (Break into 3 precincts such that A-first/B-first/C-first in each is 3/4/4, 3/4/4, 3/4/0; B wins in each precinct, but A wins the aggregate.)
- Tactical Opportunity: some voters could prevent their least-favorite choice from winning by ranking a different candidate above their honest favorite. (If at least 2 B-first voters instead rank C above B, then A doesn't win; either B or C wins.)
I want to emphasize the last point, tactical opportunity; because this is the number-one problem that I see IRV-proponents falsely claiming that IRV doesn't have. It is simply not true that IRV removes the incentive to vote for the "lesser of two evils". There are less-contrived examples that can show this too, but it's worth pointing out every time it comes up.
How would approval voting handle this election? That, of course, depends on exactly how many voters in each faction approve of their second choice, but the Nash equilibrium is a win for B. I believe this is also true for IRV, but it involves the majority of voters ranking their second-favorite above their true favorite. Meanwhile, under approval it never helps you to lower your vote for your true favorite. I'll explore equilibria more in a future post.