A Majority is a Majority is a Majority (warning, contains algebra!)

I've seen a few people object that AV has a weakness because the people who vote for the least popular candidates first, transfer their votes before everyone else, meaning that minority candidates get to have their say on the outcome between the most popular candidates before everyone else. It looks like a reasonable objection at first glance..

However I've argued that a majority is a majority, and by definition, all the other votes put together couldn't beat a majority. Yet some people are understandably not convinced, so this blog post aims to demonstrate algebraically why having everyone else transfer their preferences would not alter the outcome.

Consder, after the first round, there are 2 leading candidates and the 3rd and 4th placed candidates also receiving a reasonable share of the vote each. The remaining minority of votes is divided between fringe candidates. The fringe candidates are eliminated first and those who voted for them have their votes transferred while the voters for the top four candidates have their votes remain with their first preference, and the first placed candidate eventually wins.

Let's say P = No. of voters who put the 1st placed Candidate 1st.
Q = No. of voters who put the 2nd placed candidate 1st.
R = No. of voters who put 3rd/4th placed candidates 1st
S is everyone who put the eliminated candidates 1st.

No-one gets a majority immediately, so we can say 50% > P > Q > R > S


So suppose the 1st placed candidate wins with a majority from a combination of 1st prefs and a minority's second prefs.

Let's consider the extreme case where all of S have transferred their votes to the 1st placed candidate and all of R would transfer to the 2nd place if 3rd&4th are eliminated.



P+S > 50% > Q+R

The question is if R transferred their votes first before S could it have given Q a majority to beat P?

Could Q+R > P? Yes, of course it could. P is less than 50% so there's no reason why that cannot be true.

However, could Q+R win with a majority? Since Q+R is less than 50% they can't get a majority for the 2nd place candidate on their own, so no, it wouldn't have made a difference if R got to transfer their votes first.

There would be another round, those who voted for the least popular candidates would transfer their votes, and P+S would have a majority for the first placed candidate who would be declared the winner.

I hope this serves to satisfy any doubters that AV is fair and doesn't lead to a minority defeating a majority, something that only FPTP can be said to do.