Approval Voting

June 4, 2012 3:40 pm

Some of the most interesting classes I took during college were the artificial intelligence courses.  These courses usually took concepts from psychology, sociology, political science, and evolutionary biology, and discussed them in the context of logic, mathematics, and algorithms.  It's absolutely fascinating stuff.

One of the most interesting topics from all of my education was about voting—discussed in the context of Arrow's impossibility theorem, the Gibbard–Satterthwaite theorem, and Condorcet's paradox.

The Wikipedia introductions in each of those articles are pretty easy to understand.  But in quick, simplified, summary:

Condorcet's paradox explains how it's possible for an election to have no meaningful winner because any choice can be argued against due to a cyclical ordering of choices (think rock-paper-scissors).

The Gibbard–Satterthwaite theorem shows that (for 3 or more choices/candidates) if voters order the candidates by preference and you try to choose a single winner from those preferences, then either (1) someone is a dictator and controls the outcome, (2) some candidate can never win, or (3) voters have an incentive to lie about their preferences in order to influence the outcome (people can game the system).

Arrow's impossibility theorem is similar, but deals with systems that attempt to find a preference order over all candidates rather than a single winner.  For a reasonable set of axioms that define a "fair" voting system, there can be no voting system which satisfies all of the fairness axioms simultaneously.

I think these ideas are simply enthralling.  We then went over a slew of different voting protocols (ways of casting and counting votes) and showed how they were bound by these concepts.

Our class discussion naturally led to which voting protocol was "most fair."  But, necessary in that discussion is also which voting protocol is most fair without being too complex to actually use.

Most of the time when we think about voting in the United States, we're thinking about plurality voting (first-past-the-post or winner-take-all).  This is when, trying to get a single winner out of a group of candidates, each voter casts one vote and the candidate with the plurality of votes wins.  It happens to be a very simple protocol, but, in the opinion of the class (which I agree with), one of the least fair protocols.  Without discussing the technical violations of Arrow's fairness axioms, the reasoning we used was that when there are many candidates with similar levels of support, a large part of the population ends up being unrepresented and, due to this, plurality voting tends to collapse to a two-party system (often where neither candidate is really liked, but only preferred over the other candidate).

In our discussion, we tended to favor approval voting for its simplicity and ability to stave off a collapse to the two-party divisiveness.  In approval voting, each voter simply votes for any/all candidates of which they approve.  So if there are 4 candidates and you like 3 of them, you vote for all 3.  Or if you only like 1, just vote for that one.  There now is no reason to collapse into a two-party system because I can vote for all candidates I feel are qualified instead of fearing that the "other person" will win and I therefore must vote for the "most electable" of my actual preferred candidates.

Approval voting, of course, has some of its own problems, but we felt it was certainly more fair than plurality voting and would help solve some of the problems we're experiencing in U.S. politics right now in terms of partisanship, divisive rhetoric, and inviability of third-party candidates.

4 thoughts on “Approval Voting”

  1. Nice article. You may also find this quote from Arrow interesting:

    It should be made clear that my impossibility theorem is really a theorem [showing that] the contradictions are possible, not that they are necessary. What I claim is that given any voting procedure, there will be some possible set of preference orders for individuals that will lead to a contradiction of one of these axioms.

    But you say, ‘Well, okay, since we can’t get perfection, let’s at least try to find a method that works well most of the time.’ Then when you do have a problem, you don’t notice it as much. So my theorem is not a completely destructive or negative feature any more than the second law of thermodynamics means that people don’t work on improving the efficiency of engines. We’re told you’ll never get 100% efficient engines. That’s a fact—and a law. It doesn’t mean you wouldn’t like to go from 40% to 50%. — (Excerpt from For All Practical Purposes, pg 357-358)

    You should check out or our Facebook page at if you'd like to keep up on voting systems. We (The Center for Election Science) explicitly advocate Approval Voting.

  2. What I don't like about approval voting is that you can't vote for your "compromise choice" without having that vote cause the defeat of your favorite choice. That's fine if you know your favorite is going to lose. But if you think your favorite might win, you then won't vote for your compromise choice. That's a lot like plurality voting.

    Runoffs or instant runoffs avoid that problem.

    1. Some definitions for those following along that may not be familiar with the various voting protocols:

      Runoff voting: Voters vote for a single candidate and the 2 candidates yielding the most votes proceed to a second round of voting (some variations exist).

      Instant-runoff voting: Voters rank candidates in order of preference. During each round the candidate with the fewest votes is eliminated until one candidate receives more than 50% of votes. In each round a ballot's vote goes to the highest-ranked candidate that has not been eliminated.


      Runoff voting suffers from the same problem as plurality voting in that you need to ensure that a candidate you find acceptable makes it to the "finals." In which case the issue is really just backed up one level. There's that issue along with the additional cost and complexity of running a two-round election.

      However, I think a runoff system that uses approval voting in the first round could be interesting (traditionally it's one-vote per round). Then you vote for all acceptable candidates and get to choose from whoever makes it to round 2.

      The other criticism of runoff voting is tied to the same criticism of instant-runoff voting: Generally speaking, people are really bad at ranking choices. Beyond that, ranking doesn't account for strength of preferences (I like A with 100 units and B with 25 units and C with 24 units becomes A > B > C); but asking for this information from a voter is even harder than ranking. In the end, a voter either approves of a candidate or they don't and that determination is easier than ranking or strength of preference.


      As discussed, no protocol is perfect. I think a runoff/approval protocol would be interesting, but the 2-round system needs to be balanced by the inconvenience and costs of running an election twice.

      California is using a runoff protocol starting this year for state offices and federal congressional positions. Hooking it into the existing primary election reduces/eliminates the criticized impacts. As far as I can tell, however, it's still one vote per person, rather than approval voting. So I don't think it's going to be as effective at eliciting change from the status quo as was desired.

    2. Runoffs and IRV come with their own issues--much worse ones.

      1. They split votes so that Centrist candidates get squeezed out.
      2. Both allow for a bizarre event where you can show preference for a candidate and have it hurt that candidate, and vice versa (nonmonotonicity).
      3. Neither allow you to comfortably choose your favorite. Indeed, choosing your favorite can cause your least favorite to win--quite a punishment.

      Approval Voting, on the other hand, has none of these flaws. And if by chance your compromise vote did beat your favorite, then at least you got your compromise. That's much better than your least favorite, which a runoff and IRV can get you. Further, in practice, Approval Voting behaves nothing like Plurality and there are large scale studies that back this.

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