Approval Voting versus Runoffs

Introduction

Which reform is better: Approval Voting or Top-Two with Runoff?

Plurality Voting (aka First-Past-the-Post) is the common single-winner voting in which a voter may cast only one vote per race, and the winner is the candidate with the most votes, even if no candidate receives a majority.

This article is an attempt to briefly explore the relative value of two potential reforms to the Plurality Voting system.

Hold a second “runoff” election between the top two candidates, if no candidate receives more than 50%. This system (Plurality Voting followed by a runoff in the event no one receives a majority) is often called Top-Two Runoff or just TTR for short.

Allow voters to vote for as many candidates as they want. This system is called Approval Voting.

Note: These two reforms are not strictly in conflict, because they can be used simultaneously. In that case, the two most approved candidates would go to a runoff election in the event no candidate was approved by more than 50% of the voters. However, our purpose here is merely to consider which reform would have a greater pro-democratic effect if we could only choose one.

A scenario in which Top-Two Runoff is better

In this scenario, the preferences voters would have with “perfect” knowledge are:

  • 35% X>Z>Y
  • 35% Z>X>Y
  • 30% Y>X>Z

This notation means, for instance, that 30% of the voters prefer Y over X over Z.

Further, let’s say that the right winner in this scenario is X. That is, X is the candidate that produces the highest average voter satisfaction.

But these are not the preferences on which the voters mark their ballots. These are the preferences they would hold, if they were perfectly informed. Unfortunately, voters are not perfectly informed. They may not have watched the debates, or read the candidates’ voting records and/or positions on the issues. They may have even fallen for negative campaigning or negative gossip, without getting all the facts. And when there are many candidates, the available media coverage, and voter attention thresholds, can be diluted. It can be hard to know who’s who when 10 candidates are running for a single office, for instance.

In order to simulate this issue, we distort the above preferences to produce a set of hypothetical estimated preferences, like so.

  • 35% X>Z>Y
  • 35% Z>X>Y
  • 14% Y>X>Z
  • 16% Y>Z>X

This means that 16% of voters who believe that Y is better than Z, and Z is better than X, would actually prefer Y over X over Z if only they were better informed. This is is a simplistic simulation of voter ignorance.

Based on these estimated preferences, and simplistically assuming that 80% of voters will be strategic, we get the following results.

Top-Two Runoff - X wins

First round – X and Z go to a runoff

  • 35% of the voters cast a sincere vote for X.
  • 35% of the voters cast a sincere vote for Z.
  • 6% of the voters cast a sincere vote for Y. (20% of the 30% who prefer Y)
  • 11.2% of the voters cast a tactical vote for X.
  • 12.8% of the voters cast a tactical vote for Z.


Second round - X defeats Z approximately 65% to 35% (depending on turnout in the runoff)
During the time leading up to the runoff , voters get a chance to focus on just X and Z, ridding them of their ignorance and causing them to vote in the runoff based on their “perfect information” preferences.

Approval Voting - Z wins

  • 35% of the voters cast a sincere vote for X.
  • 35% of the voters cast a sincere vote for Z.
  • 6% of the voters cast a sincere vote only for Y.
  • 11.2% of the voters cast a tactical vote for X, plus a sincere vote for Y.
  • 12.8% of the voters cast a tactical vote for Z, plus a sincere vote for Y.

Z is approved by 47.8% of the voters, and wins.

Result

By decreasing voter ignorance (actually, eliminating it), TTR produced a better outcome than Approval Voting.

A scenario in which Top-Two Runoff is better

Perfect-knowledge preferences:

  • 45% X>Y>Z
  • 45% Z>Y>X
  • 10% Y>X>Z

Best candidate is Y.

Estimated preferences:

…work in progress

Polling Assumption

(Filler for when article is finished)

Utility Assumption for Hypotheticals

In these cases, let’s assume you hate Candidate Awful, are okay with Candidate Better, and love Candidate Classy. Let’s give them honest utility values (we’re rating them on a 0-10 scale):

  • Awful: 0
  • Better: 6
  • Classy: 10

Approval Voting Example #1

If approval polls:

  • Awful: 50%
  • Better: 50%
  • Classy: 30%

You want to vote for Better and Classy here. You vote for Better because you want Better to beat Awful. Classy doesn’t have a shot, but you vote for her anyway to show your support and give her ideas more legitimacy.


Approval Voting Example #2

If approval polls:

  • Awful: 50%
  • Better: 50%
  • Classy: 50%

You still vote for Better and Classy. You don’t vote for Classy alone because you have a strong preference for Better against Awful. By only voting for Better or Classy, you risk Awful winning against both of them.

Approval Voting Example #3

If approval polls:

  • Awful: 30%
  • Better: 50%
  • Classy: 50%

You actually only vote for Classy here. When Awful is enough out of the race, you can narrow your sights against Better and show your support for Classy.

When exactly do you only vote for Classy? It depends on how far out of the competition Better is. And it depends on how much you dislike Better along with how likable Better is compared to Awful. If Awful and Better are similarly unlikable (you’re indifferent to which one wins), a voter may be more inclined to vote for Classy alone when she is closer to winning.


Approval Voting Example #4

If approval polls:

  • Awful: 50%
  • Better: 30%
  • Classy: 50%

Again, your only vote is for Classy here. It’s not Better that’s giving competition to Awful anymore; it’s Classy competing against Awful. Whether you include Better in the vote would depend on how much you actually supported Better's views. Like in the first example where Classy had 30% and was a token vote, support for Better in this case is also a token vote because it likely won’t change the outcome. So, if you wanted to give support for Better because of some view he had that you liked, then you could get away with supporting him and Classy.


Summary of Approach

Step 1: Who is likely to win? Consider the relative utility of each. Of those candidates, approve all whom you prefer. You may end up voting for more than one candidate within this group depending on whom is challenging your preferred candidate(s).

Step 2: Who is less likely to win? Of those candidates, approve of all you wish to give support.


Conclusion

These examples remove the argument that Approval Voting regresses to Plurality Voting (via bullet voting). There are numerous scenarios (as shown above) when bullet voting simply makes no strategic sense. But notice that when you do only vote for one candidate, it’s done in a way that’s not damaging to the outcome. Also, factoring in who is likely to win is something we do anyway when considering what to do under Plurality Voting. But with Approval Voting, we just have more options on what we can do with that information. Also note that it was always to your advantage to vote your favorite. That will ALWAYS be true with Approval Voting.

Also, when there are more candidates, there are more variations on what to do. Though the concepts are the same. Expectantly, with more candidates, voters will also approve of more candidates on average.  There may also be cross-support from multiple independents/third parties that share certain views.

Finally, even with “tactical” voting, Approval Voting will nearly always choose the candidate that can beat everyone in a head-to-head race. This is called a Condorcet winner. Approval voting does not achieve this flawlessly, but it does an excellent job nonetheless. It is also argued that when Approval Voting doesn’t select the Condorcet winner, it does so for good reason. More on this topic here.

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