Chocolate and Gold Coins

Tuesday, September 20, 2005

Voting Schemes

Crooked Timber had a very interesting post on schemes to improve voting in U.S. style elections. The main idea is to eliminate the need for a special runoff election. But maybe more importantly, any election reform should strive to reduce, to the greatest possible extent, the problems that multiple-choice elections create.

If there are only two choices, democracy is easy: the majority choice should win. But if there are three or more choices, it gets complicated. Suppose there are three candidates and one is arch-conservative, one is moderate, and one is extreme-leftist. Maybe the extreme leftist will not get more than one or two percent of the vote, but he might draw votes away from the moderate and help elected the arch-conservative. This probably happened in 2000 when Ralph Nader helped elect George Bush in a close election with Al Gore. A runoff would have prevented this problem, but a better ballot could have produced enough information to determine the result of the runoff without bothering with a whole extra election.

There has been a lot theoretical thinking about elections and unfortunately there is no way to run an election without some potential pathology. However, there are ways to make minimizes these pathologies. One pathology that an election could produce is that it might eliminate a candidate who could beat all other candidates in a head-to-head election. Such a candidate is called a Condorcet winner. If there is a Condorcet winner, he should be the ultimate victor, but most voting schemes don’t guarantee this result.

A practical problem is to make the election simple enough for people to participate in and not be terribly irritated by it. Here is a list of election requirements:
1. It must always produce a result
2. It must be able to handle any number of candidates
3. It should be able to use your vote even if you do not bother to list a full list of preferences
4. It should result in the Condorcet winner being elected if one exists

Here is an important point: if I don’t want to force people to rank all 1000 candidates for an election, I need to allow people to voice each extreme of their preferences. This means that I need to allow people to cast negative votes: a vote that is specifically cast to negate the vote of someone else’s positive vote. This is necessary because otherwise, the voter cannot convey the message that candidate number 456 is someone she cannot live with and should be eliminated straightaway.

Here is how the election would work: The ballot would give you a limit number of choices. For each choice, you could either approve or reject a candidate. So maybe first choice is to reject H. Then second choice is to approve J, and so on. You need not give any more choices if you really don’t care to.

The computer would take in all of this information and look at all of the head-to-head elections among the candidates. If there is a Condorcet winner, that’s the result. If there is no Condorcet winner, a candidate is eliminated and we proceed as if that candidate never existed. After n – 1 steps, the computer gives a winner.

The person who is eliminated is the one who came closest to being the anti-Condorcet: the one who came closest to losing to everyone in a head-to-head election. The computer figures out the minimum number of vote changes necessary for each candidate to be the anti-Condorcet and eliminates that person.

The advantage of this system is that it isn’t so hard for people to use: most people know whom they like and dislike and can list the first five or six preferences easily. People will no longer have to vote strategically: “I prefer A but he has no chance, so I’ll vote for B and make sure C cannot win.” This would be a major benefit for U.S. style elections. I am not exactly sure how parliamentary elections work (I believe they vary from nation to nation), but I would guess that similar improvements could be used there as well.

It occurred to me that an easy way to implement this method is to simply tabulate all of the head-to-head contests involving candidate X and against all others and add up X's votes. The low vote getter could never be the Condorcet winner and would be the obivious one to eliminate. It is actually very easy.


  • Michael,

    A very interesting post, but I noted one error. You state that:

    "If there are only two choices, democracy is easy: the majority choice should win."

    If we are seeking to maximize social utility, a binary outcome is an insufficient condition to assure that democracy will reach the efficient outcome.

    Consider an election with two candidates (call them 1 and 2) and 3 voters (call them A, B, and C). Suppose the values for each voter of each candidate is as follows:

    1 2

    A $10 0

    B $10 0

    C $0 $1,000

    Clearly, 2 is the socially optimal candidate, as he generates $1,000 of value. But majority-rule voting will elect candidate 1, with a mere $20 of value.

    Maybe the moral of the story is that we should decide things by vote as seldom as possible.


    By Anonymous Abel Winn, at 8:05 AM  

  • I see that my little chart underwent a format change when I posted it. So allow me to restate. The values of A, B, and C of candidate 1 winning the election are $10, $10, and 0 respectively. Their respective values for candidate 2 winning are $0, $0, and $1,000.

    By Anonymous Abel, at 8:07 AM  

  • Hi Abel:
    I agree that any election might produce undesirable politics. But atleast if a candidate win a clear majority, he can rightfully claim to be the "will of the people". He might still be a fool, though.

    Although, in your example, we cannot rule out the possiblity that some "horsetrading" will allow C to get his wish. In politics, there are always compromises that enable every politician to get his pet project. That might be a big part of the problem, actually.

    By Blogger Michael Higgins, at 11:08 PM  

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