How a three-way race in democracy can be “fair” in electing any of them.

If you knew that three candidates, let’s call them Clinton, Sanders and O’Malley, had achieved 49%, 48% and 3% respectively in a first-past-the-post election (used extensively in the US and UK) you’d correctly proclaim Clinton as the winner. However, suppose two systems that took into account second and third rankings not only went against Clinton, but disagreed on which of the other two candidates should win. You’d probably be puzzled. Yet this is perfectly possible and will be demonstrated using voting intentions that are rather close to a real life example from 2016.

The Iowa Democrat Party caucus in 2016 saw two “major candidates”, Hillary Clinton and Bernie Sanders, square off. However, there was a third, minor candidate, Martin O’Malley who polled in the single digits and realistically was never going to be proclaimed the winner under existing rules (and he withdrew mid-contest). Whilst ranked choice voting has been adopted in Australia and in State elections in several parts of the USA, it remains contentious and was rejected in a referendum a few years ago in the UK. Ranked choice voting effectively would likely, if Iowa had used it in a primary vote, have allowed O’Malley’s supporters to act as “King-makers”: whoever their 2nd ranks went to would get “over the line” and be the winner. Yet there is another way to treat the second and third rankings that can be argued to be fairer, but give victory to O’Malley. This method is Most-Least Voting (MLV).

To illustrate how O’Malley could win, the first preference percentages will be retained. For purposes of exposition, it will be assumed that:
(1) Supporters of Clinton and Sanders strongly dislike the other candidate and would always place them in last place (third rank). This assumption is not unreasonable, given the polarised nature of debate at the time.
(2) O’Malley’s supporters regard Sanders as the “lesser of two evils” and all put Clinton as rank three. It will be shown that this assumption only matters for ranked choice voting – if they hate Sanders the result under MLV is unchanged.

Under ranked choice voting O’Malley is eliminated in the first round, since no candidate got 50%+1 vote. The ballot papers of the 3% of voters who picked O’Malley are re-examined. It is found that they all put Sanders as rank two. Their 3% is added to Sanders’s 48% making 51% and he wins. Had one third (1% of total) of O’Malley’s supporters put Clinton as 2nd rank then she would have got over the line and would have won.* MLV tallies things differently as it uses a profoundly different philosophical approach.

Under MLV, a ballot is only valid if the voter has indicated their “most preferred” candidate AND they have indicated their “least preferred” candidate AND they have not chosen the same candidate for both. A corollary of this is that election tampering/miscounting is very easy to spot: this will be explained later.

Given the above two assumptions, this is how MLV would play out. MLV simply subtracts the total number of “least preferred” votes from the total number of “most preferred”. The “net voting total” is very similar in nature to “net approval scores” often quoted in the media for high profile politicians.

Clinton’s net score would be 49% (her positive vote) minus (48+3=51%) – all Sanders and O’Malley voters put her as least preferred. Her net score is -2%. For Sanders it is 48-49=-1% (Sanders support is more than cancelled by the Clinton supporters but O’Malley supporters have no effect on his net vote). For O’Malley, although his most preferred score is only 3%, since nobody put him as least preferred, he obtains the highest net count and is declared winner. It should be noted that the net scores (-2, -1, +3) sum to zero. They must by definition and provide the aforementioned test for election miscounting or fraud. A major issue here is “how do we translate these three net scores into meaningful delegate counts?” Perhaps, as in most states at the level of the electoral college, “winner takes all” is applied, meaning that all of Iowa’s 44 (non-Super Delegate) votes would go to O’Malley. A subject for another day!

If O’Malley winning provokes disquiet I will not criticise you. MLV is just as vulnerable to Arrow’s Impossibility theorem as all other voting systems**. I will refrain from attempting to explain how the likelihood function of MLV differs from ranked choice and so why with three candidates, identical rankings can give different winners. Instead, I will simply detail the philosophical basis of MLV.

MLV, at its simplest, does two things:
(1) It gives EVERY voter identical weight in deciding the winner; ranked choice essentially gives the eliminated candidate’s supporters “another bite at the cherry” in playing kingmaker;
(2) It penalises candidates who might have a strong primary vote but are highly polarising. Essentially Iowa would have been “a plague on both your houses”.

One final thought on electing someone who is probably largely unknown. After (if going all the way) O’Malley had been president for 4 years, you can bet everyone will have formed a firm opinion on him and would have had a chance to turf him out. MLV was used for some elections in certain Baltic states in the immediate aftermath of their secession from the USSR. It is an open question as to whether it may be appropriate in elections in the 2020s. However, it certainly generates interesting debate over how the electoral landscape might be shaken up if it were to be adopted.

* Ranked choice voting can be argued to be destructive because the two major candidates simply try to get enough supporters of the minor candidate to help get them over the line. Under MLV, the second preferences of the minor candidate are irrelevant to the outcome: If O’Malley’s supporters all regarded Clinton as the lesser evil she still wouldn’t win. One of the two major candidates can only guarantee a win by making themself acceptable to some of the supporters of the OTHER major candidate. Getting some of “your” supporters to put the “minor candidate” as “least preferred” can offer a path to victory but is a very risky strategy – it can easily backfire and allow the other major candidate to come through.

** I and one of my co-authors of the definitive textbook on Best-Worst Scaling (of which MLV is a special case) believe this. My other co-author was unsure as to whether BWS variants were vulnerable. I stick with the math psych co-author!

PS: The MLV article is lost somewhere on a drive I currently can’t access. Authors from Belgium/Netherlands (IIRC) around 2014 detailed it. They failed to note that MLV is a special case of BWS but they probably had a limit to the number of references allowed in their bibliography. But it is curious that they knew of Tony Marley’s 1960s work but didn’t reference his post 2010 work.