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Runoff in a Star Election


Paul Cohen
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The human mind certainly is a marvel, but too often it is mysterious and perplexing as well. It is quite capable of holding two beliefs and, even after admiring how they appear to contradict one another. Though quite capable of clear logical thinking and of great cleverness, usually the mind is content to simply recall familiar nostrums and with undeserved confidence, continue to trust them.

We know that plurality voting produces questionable outcomes whenever faced with more than two candidates, but we nevertheless trust this unreliable system for most elections, even elections with great abundance of candidates. Foolishly, perhaps thinking there is no alternative, we take this approach even for our very important primary elections, which so often require selecting just one from among many more than two candidates.

While voters do deserve to have a substantial collection of candidates to choose from, we foolishly allow this to happen without providing them a voting system (such as balanced approval voting) that can sensibly negotiate the more complex interactions that a wide selection of candidates unavoidably introduces. Instead, we use plurality voting which cannot even cope with three candidates. Ranked-choice voting offers no more than a minor improvement. Even approval voting, though a much better approach, is not quite up to the job.

With three or more candidates, no voting system can guarantee that the winning candidate will gain majority support; at most, a plurality can be assured and even for that we must assume there is not a tie. But influential people dearly want to brag about the support of a majority of voters. So, to facilitate the illusion of a majority win, we tolerate a distortion of the notion of a majority. No longer insisting it should mean a majority of all voters, we accept that well, maybe it means only a majority of the few voters whose ballots survive to be significant in the very last phase of vote tallying.

Ranked-choice voting provides one example of this. It adopts a series of simulated plurality votes, removing just a single candidate with each iteration. But we conveniently ignore that voters can also be removed from participation in subsequent iterations. Supporters of this voting system will emphasize that the final step will be to choose between only two candidates. Not so surprisingly, this may not always be true either, but we let that pass. But even then a skeptic would observe, quite rightly, that many in that claimed majority would likely have much preferred one or another of the candidates who were eliminated prior to that last step in the tally. Support only at the very last might, in reality, be less than enthusiastic, merely an expression of opposition to the other even worse alternatives. And even then, the claimed majority would be only a majority of the voters who submitted ballots which chanced to survive for participation in the very last round of tallying ballots.

An analogy to this would be a slick salesman who urges a prospective new car buyer to judge a car only on last job that was completed at the end of the assembly line. The salesman insists that (while

Look at that fantastic paint job!!
Look at that fantastic paint job!!
(Image by misterbisson from flickr)
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ignoring that the brakes fail or that the engine runs poorly) that the customer should take account of only the outstanding paint job. While this seems to be quite foolish in the context of automobiles, it seems for some it is easy to accept in the context of elections.

Outcomes of plurality elections should be viewed with great skepticism whenever there are three or more candidates. Nonetheless, in the special case of an election between only two candidates, this traditional voting system is a perfectly good way to achieve a quite satisfactory election. In fact, for this special case, it is hard to imagine the plurality outcome could ever be improved. And aside from the possibility of a tie, with only two candidates, plurality voting ensures that one of the two candidates will win with a majority of votes. So, it is tempting to somehow fit a plurality vote into a better voting scheme in the way that ranked voting does. So long as the last step in the scheme resembles a plurality election there will be the rational for claiming the winner to have majority support. This serves as a sketch of the plan for star voting.

A star voting election uses the same ballots as for score voting (a.k.a. range voting}. Typically, score voting makes five scores available for voters to assign. Although the score election (provided there is not an inconvenient tie) is perfectly capable of choosing a single winner, it seems problematic because it fails to provide a rational for the winner to claim having majority support. To remedy that perceived inadequacy, the star voting system first conducts a score-voting election, not for choosing a single winner, but merely to reduce the count of candidates to two. Star voting then simulates a final runoff election using plurality voting to decide between the two finalists from the earlier score election.

For its use in the final (simulated) plurality runoff, the star-voting ballots must be re-interpreted as plurality votes for one or the other candidate. When a ballot shows A with a higher score than B, that ballot is interpreted as a (plurality} vote for A. Likewise, if a ballot rates B higher than A then that counts as a vote for B. Superficially this may seem quite reasonable, but unfortunately it fails to stand up under more careful examination.

An obvious question to ask is whether, using star voting, it is even possible for the winner of the runoff election to differ from the winner of the score voting election. That does turn out to be possible, so we might then ask whether that change is truly an improvement.

Both questions can be settled with an example. The table below shows that it is possible for the two outcomes to differ.

Example Election
Example Election
(Image by Paul Cohen, screen capture from Word Document)
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This example is of a star election with 11,300 voters in which voters are asked to assign one of the scores 0, 1, 2, 3, or 4 for each candidate. The vote-counts for the top two contenders are not widely different because 10,000 voters choose to assign the maximum score of 4 to each. The remaining 1300 voters can only play a minor role in the selection of the top two candidates. It seems surprising then, that in the simulated plurality runoff, the remaining 1300 votes become the only votes that matter. The winner, A, of the initial score election receives a score of 3 on 1000 of these remaining 1300 ballots and the second-place finalist, B, receives a score of 4 on these ballots. The remaining 300 ballots rate A with a score of 4 but B with a score of 0.

In the simulated runoff, candidate B is promoted to become the winner of the star election. It seems troubling that such a large majority of voters might be simply ignored in the runoff, but what else could be done? The 10,000-vote majority are ignored because their ballots rated A and B equally, showing no preference between A and B. Much as with ranked-choice voting the final round results in a majority for the winner but it is a majority of only the 1300 voters whose ballots survived to participate in the final runoff election.

What this example illustrates is that the approach described above for how star voting simulates the runoff election, despite appearing quite sensible, is nevertheless ill-conceived. The larger lesson is that we should be very cautious about simulating runoff elections. In this example, it seems clear that this voting pattern would never have happened had there been a genuine plurality runoff election with voters submitting a second round of ballots. In that case, surely the 10,000 voters would split roughly evenly between the two runoff candidates; that simply is what happens in a plurality election with two similar candidates; voters are coerced into choosing arbitrarily between them. It is a phenomenon called vote splitting or sometimes, the spoiler effect.

It is thought-provoking to consider what clever innovations in strategic voting might be offered to avoid the anomalies that simulated runoff elections cause.

The appeal of star voting rests largely on the fact that the runoff election is simulated, introducing no significant costs. But it would seem very difficult to justify calling for an extra election using plurality ballots; that proposal would involve considerable trouble and expense and likely it would lead to a public discussion likely revealing an absence of significant benefit to the public. The benefits would seem to be limited to providing the winner a dubious claim to majority support from the voters; it would seem better to simply choose the winner based on score voting.

In the United States, anyone is free to write letters to their newspapers, to legislators and even to the President. The Constitution even guarantees the right to hold protest demonstrations to demand a redress of grievances. But typically, voters communicate their concerns only by voting; unfortunately, many potential voters fail to even do that.

Pundits tell us that our elected leaders are the ones that the voters have chosen. And it is true that voters have a significant voice in choosing those leaders. But there are other factors that distort our election outcomes, and which constrain the choices available to voters. Gerrymandering, voter suppression, ballot access restrictions, misleading propaganda and archaic structural flaws like the Electoral College and with disproportionate political strength of low-population states all conspire to distribute the electoral power of voters unequally. And quite likely, these issues contribute to the low voter turnout.

Significant also is that we continue using an antiquated voting system which provides voters too restricted a range of choices. Plurality voting is a large factor in perpetuating a two-party duopoly. For many decades, ordinary citizens have been lulled into complacency, convinced, despite its apparent flaws, that we have the world's greatest democracy. But today there seems to be a renewed awareness of a need to strengthen that democracy. How do we do that?

I have tried in this series of articles to make the case that widespread adoption of balanced approval voting would contribute significantly to improving democracy. It could not eliminate every flaw in our elections, but it would provide voters with more freedom of expression, and it would provide more choices for voters by aggressively undermining the two-party duopoly. It could be a good first step.

In turn, the diminishing power of two dominant political parties would surely dampen the toxic polarization that we now suffer, and it seems quite possible that this would also improve our dismal record of voter participation. Moreover, reduced polarization should make progress on solving the other important issues much more feasible.

(Article changed on Jul 04, 2024 at 11:41 AM EDT)

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Attended college thanks to the generous state support of education in 1960's America. Earned a Ph.D. in mathematics at the University of Illinois followed by post doctoral research positions at the Institute for Advanced Study in Princeton. (more...)
 

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