Here's a simple case of interest between these methods. This case makes UIRV's top bias obvious. Let A and B be two strong parties, and C is a "weaker" centrist. A and B supporters are scared and some compromise to C, while C's core supporters all bullet vote in the middle. Approval Ballots: A=30%, B=28%, C=22%, AC=10%, BC=10% Approval: A=40%, B=38%, C=42% C wins UIRV: Round 1: A=35%, B=33%, C=32% Eliminate C Round 2: A=40, B=38, NOTA=22 A wins Now pretend that 2% of voters supporting B decided that B can't win and instead of compromising, bullet voted for C. Approval Ballots: A=30%, B=28%, C=24%, AC=10%, BC=8% Approval: A=40%, B=35%, C=42% UIRV: Round 1: A=35%, B=32%, C=33% Eliminate B Round 2: A=35, C=37, NOTA=28 C wins This shows that supporters of the weakest candidate C may be better off ignoring their favorite in Unranked-IRV and voting fully for their compromise in order to not have their least favorite candidate win. Approval empowers voters to vote always vote for one's favorite. UIRV makes supporting a favorite and a compromise a risky proposition if elimination order is uncertain. One other thing's clear now. Approval makes compromise votes more likely to help a favorite lose. UIRV is more likely to eliminate one or the other to compete against a polar opposite candidate. I expect educated Voters would tend to offer compromises less often in Approval. Pretend here that A supporters were more brave and only half as many compromised: Approval Ballots: A=35%, B=28%, C=24%, AC=5% BC=8% Approval: A=40%, B=38%, C=37% A wins for standing tall in Approval In Approval, if you vote for your favorite and compromise, your compromise has a good chance to beat your favorite. In Unranked-IRV, if you vote for your favorite and compromise, both are more likely to lose! This is the fundamental risk difference in offering compromises. Both are safe when order is safe and both are dangerous when candidates are close. Approval is always safe if you really don't care which wins. In approval, the top two candidates will tend to be similar (rather than diverse) while in runoffs the top two candidates will tend to be polar opposites. Another way to look at it, runoffs generate progressively smaller subsets of candidates which collectively best represent the most voters, but if you're only electing one, it may not really matter! Well, that's my best thoughts about the differences for now! Tom Ruen
