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IMAGINE THERE'S ... A DEMOCRATIC VOTING SYSTEM
by Danny Kleinman

Forget for a moment that in the United States we cannot vote for President but only for "electors" in an Electoral College; suppose this defect were to be fixed by providing direct voting for President.

Forget also that we cannot vote for President without also voting for  Vice President in a "package deal" of "candidate" and "running mate"; suppose this were also to be fixed, but without returning to this country's original system for electing a Vice President. That was simply to hold an election for President and Vice President together ("vote fortwo"), with the candidate who received the second-highest number of votes becoming Vice President. Doing so had some merit. There were no "tie-in sales" of "strong" Presidential candidates with "weak" Vice-Presidential candidates, and the Vice President thus chosen was presumably thought capable of being President by a large segment of the electorate. Indeed, since the main role of the Vice President is to be next in line in case the President dies in (resigns, or is removed by impeachment from) office, this is an important merit. However, the election of 1800 highlighted a terrible defect of that system: the tie between Aaron Burr and Thomas Jefferson (both "Democratic Republicans" as the party now known as the Democratic Party was called at that time), broken only by a vote in the House of Representatives.

Forget your own political preferences and convictions, as I will put aside mine for purposes of this essay. Though tomorrow's election prompted me to write this essay, I am not trying to convince you to vote for any particular candidate. Indeed, not all of you who will read it are Americans. The only things you have in common are that you play duplicate bridge, and have the intelligence necessary to understand what I am about to say.

Forget the kinds of voting systems that are familiar to you, and imagineinstead that we were to start from scratch to design an optimal voting system for electing the President and Vice President of the United States, or Governor and Lieutenant Governor of a particular state, or any single office-holder (with or without one or more "next-in-line" candidates). Here, I suggest, are some desirable criteria for such a voting system.

(1) Maximal Choice. Voters should not be limited to a choice of two candidates (e.g. Democrat and Republican). The ballot should include candidates of minor parties and independent candidates with no party affiliation at all. Indeed, the ballot should not be limited to just one candidate of each party. For example, in the current presidential election, the ballot might include two Democrats---Al Gore and Bill Bradley---and three Republicans---George Bush, John McCain, and Pat Buchanan (a lifelong Republican whose switch to the Reform Party stemmed not from a change in his political program but from a quirk in our election laws)---as well as candidates from minor parties.

(2) Every Voter Has a Say. If it turns out that the "race" is between two or three "front runners," with less popular candidates being "also rans," those voters who prefer some "irrelevant" unpopular candidate should have a say in which of the "front runners" wins.

(3) No Guesswork. It might be argued that under the current American voting system, every voter does have a say. That is, a supporter of Buchanan isn't required to vote for him. A voter who can predict how other voters will vote may realize that a vote for Buchanan will be "wasted" when the real contest is between Bush and Gore, and can have a say in the real contest by voting for that "front-runner" whom he views as the "lesser evil" candidate: in this example, presumably, Bush. If, however, the level of support for Buchanan approached the level of support for Gore and Bush, Buchanan supporters would face an insoluble dilemma: to vote for Buchanan only to have Gore defeat Bush by a narrow margin; or to vote for Bush when Buchanan would have defeated both rivals if only all who favored Buchanan had voted for him. Every voter should be able to vote his "conscience"; no voter should have to engage in such guesswork.

(4) No Tie-In Sales. The inclusion of Vice-Presidential candidates on a "ticket" with Presidential candidates creates other insoluble dilemmas.  In the 1988 election, for example, voters who favored the current George Bush's dad but were horrified at the thought that "running mate" Dan Quayle might ascend to the Presidency were in an impossible position.  Their votes might have depended on the accuracy of the available information about the elder Bush's health, and on the imponderable probability that the elder Bush would be assassinated. I wouldn't have known how to advise them. In the 2000 election, supporters of Green Party candidate Ralph Nader face not only the dilemmas cited here for supporters of the elder Bush in 1988 ("Who is this Winona Laduke that is Nader's running mate, and how qualified is she to be President?") and in

(3) for supporters of Buchanan, but a third dilemma. The public-financing law makes the funding of the Greens in the 2004 election dependent on how many votes Nader receives in the 2000 election, a "tie-in sale" that those who wrote and voted for the public-financing may not have considered.

(5) No Vote-Splitting Risk. One reason the voters do not have as wide a choice as lies in the phenomenon known as "splitting" the vote, which is an artifact of our American voting system. The real choice of the American electorate in 2000 might be John McCain or Bill Bradley, but neither is on the ballot. It might be argued that both were rejected in the primaries and conventions of their parties, but that is irrelevant, for each might attract wide support from voters not in their parties. In most states, only voters who register in a particular party may vote for candidates of that party in the primaries: Democrats for McCain, Republicans for Bradley, and Independents (or voters registered in minor parties) have no say in the Republican and Democratic primaries, yet their votes might elect McCain or Bradley if either or both were on the November ballot. Under the present voting system, the Republican Party would be foolish to nominate both Bush and McCain for President, since the "split" Republican votes would guarantee victory for the lone Democratic candidate Gore. Likewise, the Democratic Party would be foolish to nominate both Gore and Bradley for President, since the "split" Democratic votes would guarantee victory for the lone Republican candidate Bush. McCain, whose political position is not far from Bush's, and Bradley, whose political position is close to Gore's, would be equally foolish to run as Independents, for they'd also split the vote of their parties. An optimal voting system would allow each party to nominate two or more candidates without diminishing the chance for a candidate of the party to win the election.

(6) Condorcet. Marie Jean Antoine Nicolas Caritat (1743-1794), the Marquis de Condorcet, was a great French philosopher and mathematician who formulated what came to be called the Condorcet Criterion. When you think about it, the Condorcet Criterion is obvious:

If, among a given set of candidates, there is a particular candidate who would outpoll each of his rivals head-to-head in a two-candidate election, that candidate must win.

Conceivably, there may not be any "Condorcet candidate" in an election. As bridge players, you have experienced "round robins" among three teams
in which Team A trounces Team B, Team B trounces Team C, yet Team C
trounces Team A. A similar phenomenon is conceivable in voting. It is conceivable, for example, that in two-candidate elections, McCain would defeat Gore, Gore would defeat Bradley, yet Bradley would defeat McCain.

A widely-quoted piece of advice from S.J. Simon's delightful book "Why You Lose at Bridge" urges bridge players to seek: "The best result possible. Not the best possible result." The placement of the word "possible" before or after the word "result" doesn't really distinguish the two meanings. I would rephrase Simon for clarity: "The best result attainable. Not the best result conceivable."

There is a wide gap between what can be conceived and what can occur realistically. Like the outcome cited for a round-robin team game in bridge, the outcome "McCain defeats Gore, Bradley defeats McCain, Gore defeats Bradley" is conceivable. Unlike the bridge outcome, however, that electoral outcome simply can't happen in any practical sense. When you think about it, you'll see why:

The results of the three head-to-head contests are not independent from each other. Rather, they reflect political views that can be mapped, roughly, along a "political spectrum" (generally conceived as running from "left" to "right" even if nobody can define these terms precisely).  I'm going to draw such a map for you. Besides the Presidential candidates already mentioned, I'll include Harry Browne (a Libertarian candidate who advocates a much smaller role for government than any of the others) on the right and David McReynolds (a Socialist candidate who doesn't appear on the ballot in some states) on the left, to provide a truly broad spectrum.

McReynolds Nader Gore Bradley McCain Bush Buchanan Browne 
Now each voter, besides having a most-preferred candidate, has a candidate he prefers next, plus a third choice, a fourth choice, and so forth, though there may be ties, especially for last and near-last choices (a voter might think McReynolds, Nader, Buchanan and Browne equally abhorrent, for example). Like the candidates, voters will have places on the political spectrum. Supposing that I've mapped the spectrum correctly (though if some other map were more accurate, trivial changes in my examples would be necessary), Democrats will lie to the right of Nader and the left of McCain, Republicans will lie to the right of Bradley and the left of Browne, and "independents" will lie to the right of Bradley and the left of McCain in the center of the political spectrum. With only rare exceptions, voters will prefer candidates roughly in the order of the "distance" between the candidates' positions on the political spectrum and their own.

(Note, however, that I haven't attempted to draw the political spectrum to scale. There may, for example, be very little "distance" between Gore and Bradley. Moreover, any scale would be highly subjective. Ardent Nader supporters perceive very little "distance" between Gore and Bush but a wide chasm between Gore and Nader, while "left-wing" Democrats perceive only a short distance between Gore and Nader but a great gap between Gore and Bush. There being no objective measuring rod, I wouldn't attempt to say which scale, if either, is accurate.)

Thus (again, with only rare exceptions) you won't find Nader supporters preferring Bush to Gore, though you may find some (those strongly opposed to socialism) preferring Bush to McReynolds. Likewise, you won't find Bush supporters preferring Bradley to McCain, though you may find some preferring Browne to McCain.

Now imagine that the portions of the spectrum to the left of Gore and the right of McCain are devoid of candidates, leaving a three-candidate race. (a) For McCain to defeat Gore, more voters must lie to the right of Gore than to the left of McCain. (b) For Bradley to defeat McCain, more voters must lie to the left of McCain than to the right of Bradley.  (c) For Gore to defeat Bradley, more voters must lie to the left of Bradley than to the right of Gore. But since Bradley is to the left of McCain, the number of voters to the left of Bradley is smaller than the number of voters to the left of McCain. From (a), however, we know that the number of voters to the left of McCain is smaller than the number of voters to the right of Gore, and there the number of voters to the left of Bradley is also smaller than the number of voters to the right of Gore. Therefore (c) cannot occur if (a) and (b) do.l

The above paragraphs---those that follow the map of the political spectrum---constitute an informal proof of a mathematical theorem which says, "When a political spectrum exists, and voters vote consistently with their positions on the political spectrum, there is always a Condorcet candidate."

Long before Kenneth Arrow won his Nobel Prize in Economics, he proved Arrow's Theorem, which says that given certain reasonable requirements, no perfect voting system could be devised. However, we needn't find defects in Arrow's proof, nor argue that his theorem is false, to seek an optimal voting system. And there is an optimal voting system, a system that satisfies Criteria (1) through (5), satisfies Criterion (6) when there is a Condorcet candidate, and elects a candidate who may reasonably be considered "the most popular" in the rare circumstance that there is no Condorcet candidate.

What is that system?

You already know the heart of it: matchpoints. Suppose that each candidate I've mapped on the political spectrum were to play (with a partner of the same name) in an 8-table pair game. I shall show one possible travelling scoresheet for Board 1:

CONTESTANT SCORE MATCHPOINTS
McReynolds +110 6.0
Nader +140 7.0
Gore +100 4.5
Bradley +100 4.5
McCain +50 3.0
Bush -50 2.0
Buchanan -110 0.5
Browne -110 0.5

Now I shall show how one voter, call him Voter 1, a Nader supporter,
might mark his ballot if permitted to rank the candidates fully rather
than just "vote for one":

CANDIDATE RANK BORDA POINTS
McReynolds 2nd 6.0
Nader 1st 7.0
Gore 3rd 4.5
Bradley 3rd 4.5
McCain 4th 3.0
Bush 5th 2.0
Buchanan 6th 0.5
Browne 6th 0.5

The ranking of the candidates by Voter 1 is exactly the same as the
rankings of the contestants on Board 1. The Borda Points of the
candidates are the same as the matchpoints of the bridge players. From
now on, I'll simply right "matchpoints" instead of "Board Points" (the
name by which mathematicians and political scientists call them). Of
course, just as there is more than one board in a duplicate bridge game,
there is more than one ballot in a Presidential election. Voter 2, a
Bush supporter, might mark his ballot as follows:

CANDIDATE RANK MATCHPOINTS
McReynolds 0.5
Nader 0.5
Gore 99 2.5
Bradley 99 2.5
McCain 2 6.0
Bush 1 7.0
Buchanan 5 4.5
Browne 5 4.5

Notice that I've dropped off the "st," "nd," "rd" and "th" that
indicate order, leaving only the numbers. Notice also that it is not
necessary to rank candidates with consecutive numbers; indeed, to require
voters to do so would result in discarding large numbers of ballots as
invalid. Notice, finally, that candidates for whom no rank has been
marked are treated as tied for last in the voter's order of preference.
In this example, Voter 2 has left Nader and McReynolds unmarked because
he finds them both abhorrent. That's all right: we don't want to
require a voter to vote "for" a candidate, which he may think he'd be
doing even by ranking that candidate last.

Though an election for President would use millions of ballots, for
purposes of illustration I'll show only one more, for Voter 3, an
independent "centrist" voter:

CANDIDATE RANK MATCHPOINTS
McReynolds 8 0.0
Nader 5 3.0
Gore 3 5.0
Bradley 2 6.0
McCain 1 7.0
Bush 4 4.0
Buchanan 6 2.0
Browne 7 1.0

All right, now that all the ballots have been matchpointed, let's add
the matchpoints of all three ballots:

CANDIDATE MATCHPOINTS
McReynolds 6.5
Nader 10.5
Gore 12.0
Bradley 13.0
McCain 16.0
Bush 13.0
Buchanan 7.0
Browne 6.0

Some political scientists have proposed that we stop here, and declare
McCain the winner on the basis of his high matchpoint total. However, to
adopt that proposal would sometimes result in a violation of Criterion
(6), the Condorcet Criterion. In this essay, I won't bother showing an
example of how that could happen, but when I prove that the voting system
whose presentation I am about to complete always satisfies the Condorcet
Criterion, you will see that the proof doesn't work if we stop here.

The next step is simply to eliminate the lowest-scoring candidate from
the tallying process (in this case, Browne) and repeat our tally with one
fewer candidate, so that the computer which does the tallying creates a
revised set of ballots, and keeps repeating the process until only one
candidate (the winner) remains:

CANDIDATE RANK MATCHPOINTS
McReynolds 2nd 5.0
Nader 1st 6.0
Gore 3rd 3.5
Bradley 3rd 3.5
McCain 4th 2.0
Bush 5th 1.0
Buchanan 6th 0.0

CANDIDATE RANK MATCHPOINTS
McReynolds 0.5
Nader 0.5
Gore 99 2.5
Bradley 99 2.5
McCain 2 5.0
Bush 1 6.0
Buchanan 5 4.0

CANDIDATE RANK MATCHPOINTS
McReynolds 8 0.0
Nader 5 2.0
Gore 3 4.0
Bradley 2 5.0
McCain 1 6.0
Bush 4 3.0
Buchanan 6 1.0



The new totals:

CANDIDATE MATCHPOINTS
McReynolds 5.5
Nader 8.5
Gore 10.0
Bradley 11.0
McCain 13.0
Bush 10.0
Buchanan 5.0

Eliminating the low scorer (Buchanan):

CANDIDATE RANK MATCHPOINTS
McReynolds 2nd 4.0
Nader 1st 5.0
Gore 3rd 2.5
Bradley 3rd 2.5
McCain 4th 1.0
Bush 5th 0.0

CANDIDATE RANK MATCHPOINTS
McReynolds 0.5
Nader 0.5
Gore 99 2.5
Bradley 99 2.5
McCain 2 4.0
Bush 1 5.0

CANDIDATE RANK MATCHPOINTS
McReynolds 8 0.0
Nader 5 1.0
Gore 3 3.0
Bradley 2 4.0
McCain 1 5.0
Bush 4 2.0

The new totals:

CANDIDATE MATCHPOINTS
McReynolds 4.5
Nader 6.5
Gore 8.0
Bradley 9.0
McCain 10.0
Bush 7.0

Eliminating the low scorer (McReynolds):

CANDIDATE RANK MATCHPOINTS
Nader 1st 4.0
Gore 3rd 2.5
Bradley 3rd 2.5
McCain 4th 1.0
Bush 5th 0.0

CANDIDATE RANK MATCHPOINTS
Nader 0.0
Gore 99 1.5
Bradley 99 1.5
McCain 2 3.0
Bush 1 4.0

CANDIDATE RANK MATCHPOINTS
Nader 5 0.0
Gore 3 2.0
Bradley 2 3.0
McCain 1 4.0
Bush 4 1.0

The new totals:

CANDIDATE MATCHPOINTS
Nader 4.0
Gore 6.0
Bradley 7.0
McCain 8.0
Bush 5.0

Eliminating the low scorer (Nader):

CANDIDATE RANK MATCHPOINTS
Gore 3rd 2.5
Bradley 3rd 2.5
McCain 4th 1.0
Bush 5th 0.0

CANDIDATE RANK MATCHPOINTS
Gore 99 0.5
Bradley 99 0.5
McCain 2 2.0
Bush 1 3.0

CANDIDATE RANK MATCHPOINTS
Gore 3 1.0
Bradley 2 2.0
McCain 1 3.0
Bush 4 0.0

The new totals:

CANDIDATE MATCHPOINTS
Gore 4.0
Bradley 5.0
McCain 6.0
Bush 3.0

Eliminating the low scorer (Bush):

CANDIDATE RANK MATCHPOINTS
Gore 3rd 1.5
Bradley 3rd 1.5
McCain 4th 0.0

CANDIDATE RANK MATCHPOINTS
Gore 99 0.5
Bradley 99 0.5
McCain 2 2.0

CANDIDATE RANK MATCHPOINTS
Gore 3 0.0
Bradley 2 1.0
McCain 1 2.0

The new totals:

CANDIDATE MATCHPOINTS
Gore 2.0
Bradley 3.0
McCain 4.0

Eliminating the low scorer (Gore):

CANDIDATE RANK MATCHPOINTS
Bradley 3rd 1.0
McCain 4th 0.0

CANDIDATE RANK MATCHPOINTS
Bradley 99 0.0
McCain 2 1.0

CANDIDATE RANK MATCHPOINTS
Bradley 2 0.0
McCain 1 1.0

The new totals:

CANDIDATE MATCHPOINTS
Bradley 1.0
McCain 2.0

At last we have a "head-to-head" race. Bradley, the low scorer, is
eliminated and McCain is elected President.

Though this procedure, which I call SOME (for "Single Office Majority Election"), could be used in a separate election for Vice President, there is a better method for electing a Vice President, who should really be conceived as a "Back-Up" in case the President dies in, or is removed from, office. Simply repeat SOME, with the computer creating a new set of ballots identical to the original set but eliminating the candidate who was elected President. The Vice President will then be the candidate whom the voters would have elected if the President hadn't run for the office. Of course the prospect of electing a President and Vice President with similar political programs will provide an incentive for each party, and each faction within a party, to nominate more candidates and give the voters a wider choice.

Because each step eliminates an "irrelevant" candidate, SOME fulfills Criteria (1), (2), (3) and (5), so nominating "also-ran" candidates cannot harm a party or faction, nor can marking preferences for "also-ran" candidates thwart the wishes of voters. (Though organized insincere "strategic" voting can alter outcomes, it cannot do so in predictable ways and to attempt it entails great risk of unforeseen outcomes opposite to the intentions of the organizers.)

How can I prove that SOME fulfills Criterion (6), the Condorcet
Criterion?

Very simply. A Condorcet candidate beats each of his rivals
head-to-head. Vis-a-vis each, he collects a majority of the available
matchpoints (1 per ballot). That means on each successive tally he gets
an above-average score, which cannot be the low score, and therefore he
cannot be eliminated on any tally.


Inquiries welcome!
Danny Kleinman (SimpleSimon on SWAN)

 

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