Paradoxes of runoff voting

The New Zealand Labour party will soon have an election for leader of its Parliamentary caucus. The voting system is a weighted form of instant runoff using the single seat version of Hare’s method (instant runoff/IRV/alternative vote). IRV works as follows. Each voter submits a full preference order of the candidates (I am not sure what happens if a voter doesn’t rank all candidates but presumably the method can still work). In each round, the voter with smallest number of first preferences (the plurality loser) is eliminated, and the candidate removed from the preference orders, keeping the order of the other candidates the same. If there is a tie for the plurality loser in a round, this must be broken somehow.

The NZLP variant differs from the above only in that not all voters have the same weight. In fact, the caucus (34 members) has a total weight of 40%, the party members (tens of thousands, presumably) have total weight 40%, and the 6 affiliated trade unions have total weight 20%, the weight being proportional to their size. It is not completely clear to me how the unions vote, but it seems that most of them will give all their weight to a single preference order, decided by union leaders with some level of consultation with members. Thus in effect there are 34 voters each with weight 20/17, 6 with total weight 20, and the rest of the weight (total 40) is distributed equally among tens of thousands of voters. Note that the total weight of the unions is half the total weight of the caucus, which equals the total weight of the individual members.

IRV is known to be susceptible to several paradoxes. Of course essentially all voting rules are, but the particular ones for IRV include the participation paradoxes which have always seemed to me to be particularly bad. It is possible, for example, for a candidate to win when some of his supporters fail to vote, but lose when they come out to vote for him, without any change in other voters’ behaviour (Positive Participation Paradox). This can’t happen with three candidates, which is the situation we are interested in (we denote the candidate C, J, R). But the Negative Participation Paradox can occur: a losing candidate becomes a winner when new voters ranking him last turn out to vote.

The particular election is interesting because there is no clear front-runner and the three groups of voters apparently have quite different opinions. Recent polling suggests that the unions mostly will vote CJR. In the caucus, more than half have R as first choice, and many apparently have C as last. Less information is available about the party members but it seems likely that C has most first preferences, followed by J and R.

The following scenario on preference orders is consistent with this data: RCJ 25%, RJC 7%, CRJ 10%, CJR 30%, JRC 20%, JCR 8%. In this case, J is eliminated in the first round and R wins over C in the final round by 52% to 48%. Suppose now that instead of abstaining, enough previously unmotivated voters decide to vote JRC (perhaps because of positive media coverage for J and a deep dislike of C). Here “enough” means “more than 4% of the total turnout before they changed their minds, but not more than 30%”. Then R is eliminated in the first round, and C wins easily over J. So by trying to support J and signal displeasure with C, these extra voters help to achieve a worse outcome than if they had stayed at home.

The result of the election will be announced within a week, and I may perhaps say more then.

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