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Star-like Voting


Paul Cohen
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In the previous article of this series, a comment from William Waugh directed attention to a proposal for a Star-Like voting system. No, that is not an election on some planet like Mars that looks star-like. Rather, it is intended

Mars Behind Dark Earth Limb
Mars Behind Dark Earth Limb
(Image by NASA's Marshall Space Flight Center from nasa)
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for a compound voting system which first interprets the ballots of an election using one system but when that result is a tie it interprets those same ballots in some different way to (one hopes) break that tie. The particular example begins by interpreting the ballots using the Condorcet criterion, which is a notion that is applicable to ballot designs that like BAV or approval voting or ranked choice voting that allows voters to make clear preferences between pairs of candidates.

For such voting systems, we could define the C-score for a candidate X as the number of candidates Y that X beats minus the number of candidates Y that beat X. The Condorcet criterion says that the winner should be a candidate with the largest possible C-score. Several candidates might share that largest C-score, so some procedure for resolving such ties is indicated - a coin-toss perhaps or maybe another election. But the star approach of breaking the tie using an alternative method for counting votes does seem appealing. Still, a coin-toss is the most certain to resolve a two-way tie (otherwise one might draw straws).

A familiar tale tells of a billionaire walking into a bar and the bar-tender remarking, quite accurately, that the average wealth of his patrons just increased by many hundreds of million dollars. This illustrates a curious feature of the statistical concept of the mean (or average). But more fundamentally, it seems to suggest that perhaps a single number cannot possibly summarize a large data-set in a way that is satisfactory for all conclusions. Applying this observation to voting, it is probably too much to ask that any voting system take account of every possible consideration. Condorcet is just another way to count votes and it has some appeal we really should judge voting systems on their promise, not so much on their intuitive appeal.

As an example, consider a BAV election to determine the winner among five candidates. We know that BAV is a balanced system, but what if we instead used the Condorcet scoring to tally those votes; would that remain a balanced voting system? With BAV, when a one voter supports a particular candidate and another voter opposes that candidate, those two votes simply cancel each-other; the vote tallies are no different than if both of those voters instead had instead abstained regarding that candidate. That is how a balanced system should behave, with a vote against a candidate simply canceling another in support of that candidate.

But with Condorcet scoring, the vote tally is not particularly sensitive to aggregate votes for or against a fixed candidate. The tallies pit pairs of candidates against each-other. It is hard to see why votes of different voters should be expected to cancel one-another so an experiment seems in order; just take a look at an example election. In fact, it seems likely that that this test of balance will fail even the first try at this experiment. Using a spreadsheet to randomly generate BAV ballots for five candidates and calculated both BAV net votes and Condorcet scores for the elections is a straightforward project. Modifying two entries of 0 for one of the candidates to become instead, -1 and 1 has no effect on BAV net votes (as expected) but many of the Condorcet scores change, in fact even for candidates not involved in those two changes. If you would like to try this, feel free to use the Open Office spreadsheet and perform the experiment for yourself, here is a link.

Consider that in a contest between only two candidates, it is unclear what advantage Condorcet voting might have, even over plurality voting. Moreover, there is little reason to think Condorcet voting would lead to a multi-party democracy. In fairness it might be argued that, to voters, Condorcet voting can be structured to appear superficially to be balanced and that illusion might the voting, but although possible, it seems challenging to argue that Condorcet scoring would not bias its elections in favor of the largest parties.

It seems quite likely that Condorcet scoring would only result in the continuation of two-party politics and for this reason, it seems ill-advised to reverse the order and use the BAV net-vote count only to break a tie when that results from Condorcet scoring of the ballots. It would be best to first make sure two-party politics is undermined (by letting BAV go first) and then perhaps using Condorcet scoring to take care of tie votes when necessary.

If you take a look at the spreadsheet you will find two kinds of Condorcet scores, namely as 3-way and as X-Zero. The 3-way results are for interpreting the ballots as allowing three different choices to a voter, namely -1, 0, and 1. But consider whether this is appropriate. In its original concept, BAV voters only express support or opposition (or abstain). Voters, when they mark neither "support" nor "oppose" probably think they are abstaining with regard to the candidate in question; they want to opt out of being counted (as happens in computing the BAV net vote). Should their wish not be honored in Condorcet-like counting of BAV ballots by ensuring abstentions to be ignored? In the spreadsheet, that (excluded-middle) approach is identified as X-Zero.

This issue of whether and how to deal with abstention matters whenever the Condorcet criterion is to be applied to ballots that permit abstentions. This is something that real-world voting systems often do.

Twinkling twigs
Twinkling twigs
(Image by Sylvia Sassen from flickr)
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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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