An Analysis of Approval Voting as Compared to Plurality Voting

When searching for the optimal voting system, two presented options are plurality rule and the approval voting rule. For the purposes stated here, I will define plurality voting as a system where all alternatives are voted on at once, and the alternative that gets the most votes, regardless of a majority, is selected. I will assume no runoffs are held. I will define approval voting as a system where voters can approve or not approve all candidates, and they can approve or disapprove as many people as they want. I will operate under the assumption that people must either approve or disapprove. They cannot refuse to answer for an individual person. In the process of this brief paper, I will evaluate both systems in regards to Condorcet winners. Then, I will proceed to evaluate them in terms “reverse relevance” and polarization management.

A Condorcet winner is one that can beat any other option when put in a two alternative contest with that other option. Although such a winner does not always exist, when it does, it is usually socially desirable for it to be chosen. As such, it is a valid criterion to begin evaluating the two methods of voting. I start with plurality voting. In plurality voting under my assumption, the Condorcet winner can lose. In order to prove this, it is only necessary to show one counter example that is feasible. Such a counter example is the following voter profile chart:

VOTERS
Preferences P1 P2 P3 P4 P5  P6  P7
B B A A D C B
C C C C C B C
A A D D B D A
D D B B A A D

In the chart, C is clearly the Condorcet winner, because C would beat every other candidate, even B, in a one-on-one matchup. However, in the plurality system under sincere voting, B would be chosen. This is because we assume all candidates are admitted and that all people vote for their most preferred candidate. However, let us also consider a realistic situation of strategic voting. Let us assume that the voting block composed of {P1, P2,P7} who all most prefer B, are very vocal about their support, so much so that everyone is aware that they will all certainly vote for B. If everyone has this information, is it clear that enough people will strategically vote for C? In this case it is: B is the least preferred outcome of voting block {P3,P4} and the second to last preferred outcome for P5. The only other feasible strategic vote block would be for A, but as the non-B coalition is polarized on that choice, such a coalition seems not to be likely. Therefore under strategic voting assumptions there do appear to be instances where plurality voting can overcome the elimination of the Condorcet winner.

In the same way, I now move on to approval voting. In the case of approval voting, if I take the Condorcet winner to be the person that has the most number of “yes” or “approve” votes, then by definition the Condorcet winner will always win. However, given that people rarely ever have binary opinions of anything, it is almost certainly the case that the approval voting system is really a simplification of the underlying preference listings of each voter for the purpose of public choice. With this in mind, it is quite obvious that depending on which arbitrary point a voter places their standard of approval, the Condorcet winner can either always be selected or sometimes be selected. So a Condorcet analysis of the approval voting system is actually paradoxically illogical, because it is difficult to say whether people have some absolute threshold for competence or approval (ie, whether someone would mark No for everyone if they did not meet the criterion, or yes for everyone if they all did) or whether they only approve or disapprove relative to other candidates. It is my general belief that they likely have a mixture of both systems, but will always approve someone and likely never approve no-one, for the simple reason that doing so makes a vote worthless.

So the Condorcet analysis is inconclusive on both fronts. However, there are other ways to compare the two methods. One is to examine whether each method allows certain bad outcomes to occur. Returning to approval voting, it can be shown that approval voting can allow someone who would receive no support in a non-strategic plurality system to win:

Preference P1 P2 P3 P4 P5
A A B B C
D D D D D
B B C C B
C C C C C

If it is assumed that in the above example all voters approve their top two alternatives and disapprove all the others, then the Condorcet winner, D, will win. However, D would win no votes in a nonstrategic plurality vote. This property, which I will name “reverse relevance”, places value on plurality voting over approval voting, as plurality voting has reverse relevance with approval voting, as the winner of plurality voting in a nonstrategic framework must by definition be approved by at least one voter. (If my assumption that no one will vote for everyone or no one holds true).

Now on to polarization management, which I define as the ability to recognize polarized preferences as well as manage or reduce polarized results. Let us consider this example of a polarized electorate:

Preference P1 P2 P3 P4 P5
A A B B B
C C C C C
D D A A D
B B D D A

In this group of voters, the Condorcet winner is B, and the plurality system would pick B the winner. However, it is also true that B is tied for the least preferred candidate. The plurality system even under strategic voting allows no ability to express extreme support or extreme dislike for a candidate, and it has no likelihood of selecting a moderate candidate under nonstrategic conditions. In a less obvious way the approval system does allow voters to express degrees of support, and it also allows for a moderate winner. If in the above example at least 4 of the players approve C along with their top choice, then C , the moderate choice, will be chosen.  As for expressing degree of preference, take, for example, voter P1 and P2. Those voters can express extreme dislike for B by choosing to approve every candidate except B. If they do so, it does not change the vote outcome, but it does in a way express relative degree of dislike, in that the person was only not satisfied with B. Alternatively, the voter can choose to convert his vote into essentially a plurality system vote. In doing so, he can fail to approve all but his top choice.

This analysis forgoes any empirical evidence, and as a result is mainly just a thought exercise. It mainly reveals the well-hashed out conclusion that even with the most innovative of voting systems, there are always drawbacks. This is a sad result of democratic choice, and it even applies to systems that choose the Condorcet winner every time. However, I believe that a crude evaluation would conclude that approval voting is preferable to plurality voting, just due to voter flexibility and polarization management.

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2 thoughts on “An Analysis of Approval Voting as Compared to Plurality Voting

  1. It is mathematically proven that the Condorcet winner is not necessarily the group’s preferred candidate. See Arrow’s Theorem. So I wouldn’t put much focus on this. Although it is convenient that Approval Voting tends to elect the Condorcet winner when there is one.

    http://ScoreVoting.net/ArrowThm.html

    The correct measure of voting method performance is Bayesian regret. Approval Voting is good, and Score Voting (aka Range Voting) is best.

    http://ScoreVoting.net/BayRegsFig.html

    Clay Shentrup
    Co-founder, The Center for Election Science

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