By Stephen H. Unger (Professor Emeritus, Computer Science and Electrical Engineering, Columbia University, NY)
In order to check my understanding of how STV works in a proportional representation (PR) election, I made up a fairly simple example. This was essentially a random example just to see what the computation is like. I did not try to rig it to prove or demonstrate anything. But it turned out to be quite interesting in that it makes STV look bad (the way IRV looks bad) in several ways. The example follows:
Assume 99 voters, 6 candidates, with 3 seats to be filled. The names of the candidates are A,B,C,D,E, and F. Using the Droop formula for the quota, Q=1+⌊v/(s+1)⌋, where s=#seats to be filled, v=#voters, we get Q=25=1+⌊99/4⌋. Assume the votes are:
| #Voters | Their Vote |
|---|---|
| 20 | A>B>C>D |
| 20 | B>D>E>F |
| 20 | F>A>C |
| 20 | D>C>E>F |
| 19 | C>D>B |
E is immediately eliminated, and then C, with all 19 votes going to D. So D is elected with 39 votes, 14 over the quota. These surplus votes are assigned to F and B (7 to each). That's enough to elect both F and B. So the 3 winners are D, F, and B.
Now note that C, one of the 3 losers, was preferred by the voters (by large margins) over all of the winners: C beats D by 59-40, beats F by 59-40, and beats B by 59-40! (If a candidate is not ranked by a voter, I assume that those candidates are coequally-bottom-ranked.) Indeed, the eliminated candidate C falls just short of being a "Condorcet (beats-all) winner," in that C beats all other candidates except A, who barely beats C (40-39). Also here is an election example where a genuine beats-all winner is rejected by PR-STV.
It is also interesting that every candidate is involved in at least one cycle; cycles include A>C>F>A, A>B>E>F>A, and C>D>A>C. Hence another problem is that the immediately-eliminated E is preferred over the winner F by a 40-20 majority.
This election also incentivized voter-dishonesty. For example, the voters in the top line would have been better off dishonestly ranking C top ahead of their true favorites A and B. That way the winners would have been C, D, and B (an outcome they would have preferred). This is also an example of the wasted vote problem in action: these voters "wasted" their honest vote for A and thus got a worse election outcome than if they'd voted dishonestly. STV proponents sometimes wrongly claim STV "solves" the wasted vote problem because honest votes for losers are not "lost" but rather transferred. Not necessarily so! (Indeed if the voters in the top line had simply refused to vote at all, then the winners would still have been D, B, and F, confirming that their honest A-votes indeed were a waste.)
Another example of voter-dishonesty paying were the voters in the middle line. If they'd dishonestly voted A top instead of their true favorite F, then A, B, and C would have won (probably an improved result in their view). These voters also could have (even-more-dishonestly) voted C top, in which case the winners would have been C, A, and D (also probably an improved result in their view).
This election also illustrates at least one "no-show paradox." Suppose 2 of the 20 F>A>C voters in the middle line had simply refused to vote. In that case E then F would have been eliminated, with the 18 F-votes going to A. Then A would have won with 38 votes (13 above quota). These surplus votes would have transferred 7 to B and 6 to C, who then both would have won. I.e. the new winners would be A, B, and C, an improved result in the view of these voters. In other words for these two voters not voting was a better strategy than voting honestly! Their act of casting honest votes actually hurt them!
So STV does not look very good to me for PR elections (no better than IRV for single-winner elections) at least based on this one example. STV is also very hard to tabulate, requiring, as does IRV, central tabulation.
I suspect that range/score voting and approval voting could be redone to adapt them to PR with fewer problems of all kinds. Has this been done?
By Warren D. Smith
Yes, it has been done, see asset voting and reweighted range voting (RRV).
Unger's example also suggests an interesting new "multiwinner Condorcet property" that multiwinner methods might (or might not) enjoy:
There should never exist a loser preferred by a voter-majority versus every winner.
As (a slight modification of) the example above shows, STV disobeys this property. RRV satisfies the score-ballot-based analogue of this property in which "preferred by a voter majority versus" is replaced by "has a higher average score than". Majority-preference (as we just saw) can exhibit self-contradictory "cycles," but "higher average score" always yields a self-consistent ordering.