Let the candidates be A,B,C,D,E,F and suppose rank-order voting (Hare/Droop STV-PR system) is used to elect the top 5. Assume there are 100 voters in all.
#voters | their rank-order vote |
---|---|
17 | B>A>C>D>E>F |
17 | C>A>D>E>F>B |
17 | D>A>E>F>B>C |
17 | E>A>F>B>C>D |
17 | F>A>B>C>D>E |
15 | A>B>C>D>E>F |
In this example, A is preferred versus every rival (such as B) by a huge 83-to-17 majority, far larger than the greatest landslide in US presidential election history. So if it were a head-to-head A versus B race (or A versus anybody else), A would win huge.
However, the Hare/Droop PR-STV system used in, e.g. Ireland, Australia, and for nominations for most OSCAR cinema awards, will eliminate A in round 1. In other words the 5 worst candidates B,C,D,E,F are the 5 winners but not A who would seem fairly clearly to be the best. This is even though we are trying to make it maximally easy to win by having 5 winners out of 6 candidates – still the hugely-favored A loses!
Why does this happen? Two ways to look at it:
This election also serves as a good example where RRV (reweighted range voting) could behave significantly better than PR-STV. RRV would almost-certainly elect A as the first winner, in this scenario. I find it hard to believe that the virtues of "balancing the badness" by fully-satisfying PR-STV's notions of "proportionality," here are worth the cost to society of eliminating the clearly-best candidate.
In the example at top, many of the details do not matter. For example
#voters | their rank-order vote |
---|---|
17 | B>A>xxx |
17 | C>A>xxx |
17 | D>A>xxx |
17 | E>A>xxx |
17 | F>A>xxx |
15 | A>xxx |
where xxx means "it does not matter how the rest are ordered" also works to demonstrate the same pathology.
We should also note that this pathology is not merely hypothetical. It appears to have occurred in various OSCAR votes in which superb movies, flagged by the judgment of history as among the best of all time, failed even to get nominated for "best picture" that year!