The Majority Rule Voting Paradox


Part 2 of 2

What does the Majority Rule Voting Paradox imply about the choice of Voting Techniques?

by Peter K Harrell

As stated [in the earlier] post Plurality Voting is the voting technique used in most elections in the United States. The question then arises which candidate would Plurality Voting elect given the existence of a Majority Rule Voting Paradox within the electorate.

Given the “opportunity” to express themselves using Plurality Voting Person 1 can be expected to vote for Candidate A, Person 2 would most certainly vote for Candidate A, and Person 3 would vote for Candidate B.

Given the existence a Majority Rule Voting Paradox situation Plurality Voting would elect Candidate A, since two out of three voters Person 1 and Person 2 prefer Candidate A to Candidate B. Plurality Voting essentially ignores consent and dissent, since there is no way for the voter to directly express approval or disapproval of any of the candidates on the ballot.

Given the “opportunity” to express themselves using Instant Runoff Voting Person 1 would rank the candidates as follows: Candidate A then Candidate B. Person 2 would rank the candidates: Candidate A then Candidate B. Person 3 would rank the candidates: Candidate B then Candidate A.

Given the existence of a Majority Rule Voting Paradox situation Instant Runoff Voting would also elect Candidate A, since two out of three voters Person 1 and Person 2 have ranked Candidate A first. Like Plurality Voting Instant Runoff Voting also ignores consent and dissent, Instant Runoff Voting provides no way for the voter to directly express approval or disapproval of any of the candidates on the ballot.

I will respectfully submit that Candidate A is not the candidate that should be elected. That is because candidate A is disapproved of by a 2 to 1 majority of the voters consisting of Person 2 and Person 3. To elect Candidate A is to violate the principle of the Consent of the Governed. What is more Candidate B is approved of by a 2 to 1 majority of the voters consisting of Person 1 and Person 3. To elect Candidate B is both consistent with the Consent of the Governed as well as a form of majority rule.

Determining what would happen if voters had the opportunity to express themselves using Yes No ‘Maybe So’ Voting is a bit more complex than for Plurality Voting or Instant Runoff Voting. The greater freedom of expression and political association provided to each voter by the Yes No ‘Maybe So’ Voting technique complicates the analysis. For the purposes of this analysis we will initially assume voting strategies that can be considered essentially sincere.

Person 1 might vote as follows:

______________Yes_____No
Candidate A_____X
Candidate B_____X

or as follows:

______________Yes_____No
Candidate A_____X
Candidate B_____X_____X

For Person A to vote in any other way would be an insincere denial of the fact that Person 1 approves of both Candidate A and Candidate B. While Person 1 could vote differently we have not included that possibility here in our initial assumption. We will look at that possibility later in this analysis. At any rate if there were additional candidates on the ballot, a Candidate C or a Candidate D for instance, whom Person 1 disapproved of as there might well be, then Person 1 would incur greater risk in voting insincerely.

Person 2 might vote as follows:

______________Yes_____No
Candidate A____________X
Candidate B____________X

Since Yes No ‘Maybe So’ Voting does not include the option of distinguishing between two candidates while opposing both, in order to express a preference for Candidate A over Candidate B Person 2 might choose to vote as follows even though this vote lacks explicit disapproval of Candidate A:

______________Yes_____No
Candidate A
Candidate B____________X

Person 3 might vote as follows:

______________Yes_____No
Candidate A____________X
Candidate B_____X

or as follows:

______________Yes_____No
Candidate A____________X
Candidate B_____X_____X

Certainly given the greater freedom of expression provided the voter by Yes No ‘Maybe So’ Voting there are other possible ways that any one of these three people might choose to vote, but each of these voters is highly likely to express themselves as assumed above and all of these ways are consistent with their preferences.

Unless these three voters depart significantly from the voting strategies chosen above Yes No ‘Maybe So’ Voting will result in the election of Candidate B, because Candidate B has received some degree of support (a Yes vote or a Yes/No vote) from two out of the three voters, while Candidate A has not.

However, even if Person 1 decides to vote insincerely ignoring his or her approval of Candidate B , since this is after all a two candidate election, and cast the following vote Candidate B will still win under Yes No ‘Maybe So’ Voting. That is because Yes No ‘Maybe So’ Voting permits a candidate to be elected providing that the candidate satisfies a plurality threshold of support and no other candidates do better.

Person 1 might vote as follows:

______________Yes_____No
Candidate A_____X
Candidate B____________X

There are other departures from sincere voting that could of course result in Candidate A being elected. Many of these departures are inconsistent with any conceivable political interest of these three potential voters and are therefore unlikely.

However, if Person 2 decides to vote insincerely ignoring his or her disapproval of Candidate A, Candidate A could win under Yes No ‘Maybe So’ Voting. This is essentially what happens in both Plurality Voting and Instant Runoff Voting since neither one of those voting techniques permits Person 2 to acknowledge and express both a preference for Candidate A and disapproval of Candidate A.

Person 2 would be more likely to chose this sort of insincere voting strategy as shown below if there are no other candidates on the ballot and/or Candidate B is for some reason much more intensely disapproved of than Candidate A.

Person 2 might vote as follows:

______________Yes_____No
Candidate A_____X_____X
Candidate B____________X

or as follows:
______________Yes_____No
Candidate A_____X
Candidate B____________X

As an example Person 2 would be much more likely to chose this sort of strategy if Candidate B were some sort of anathema candidate, a David Duke or even an Adolph Hitler type. Of course in this latter instance the election of Candidate B would be an undesirable outcome despite the fact that a majority of the electorate disapprove of Candidate A.

Also worth mentioning is that a Majority Rule Voting Paradox relationship between Candidate A and Candidate B would be less likely to exist in the electorate if Candidate B were an anathema candidate. That is because potential voters would be less likely to approve of both Candidate A and Candidate B and hold opinions consistent with Person 1.

Another possibility is that Person 2 will decide not to vote at all since Person 2 disapproves of both Candidate A and Candidate B. The abstention of Person 2 is inconsistent with his or her political interests so long as Person 2 does in fact prefer Candidate A to Candidate B is some significant way. Still given the intense dissatisfaction that Person 2 might well have for both the candidates on the ballot, Person 2 seems reasonably likely to abstain. If Person 2 abstains given the sorts of scenarios already discussed above all three voting techniques would produce a tie or in the case of a larger electorate a very close election.

But both Instant Runoff Voting and Yes No ‘Maybe So Voting allow greater freedom of expression and greater freedom of political association than Plurality Voting. When either one of these two techniques are used, voters with opinions consistent with Person 2 would be less likely to abstain since they would be able to express their opinions about any other candidates that might be on the ballot, while still expressing their preference for Candidate A over Candidate B. Yes No ‘Maybe So’ Voting, of course, also gives the voter the additional satisfaction of directly expressing disapproval of one or both of these two candidates.

Voting is both a means of communicating the opinions of voters to society at large and permitting voters to exercise political power. As a means of communication voting has the potential to reveal the will of the people in all its complexity and to provide a basis for legitimizing democratic government. As a means of exercising political power voting has the potential to provide a countervailing force to the great influence of well connected and well healed political elites, and to ensure that ordinary people have some form of representation in the important decisions made by government.

The act of consent is an act that involves the expression of agreement or of acquiescence. The expression of that consent, which can be either tacit or explicit, arises out of the opinions of each and every voter and takes form according to whatever means for expressing those opinions and exercising political power is available to them.

As the Majority Rule Voting Paradox demonstrates both Plurality Voting and Instant Runoff Voting are unable to determine the Consent of the Government and may frequently in fact result in the election of a candidate who is disapproved of by a majority of the electorate despite the fact that there exists some other candidate on the ballot who is approved of by a majority of that same electorate.

Yes No ‘Maybe So’ Voting can determine the Consent of the Governed should any such consent exist whether that consent is tacit or explicit. Given both a Majority Rule Voting Paradox situation and voters intent on expressing their opinions and effectively exercising political power Yes No ‘Maybe So’ Voting will result in the election of the candidate that best reflects the will of the people.

As we are beginning to understand, neither plurality voting nor instant-runoff voting allow us as voters to express “consent or dissent” directly for any candidate. In a situation where we are forced–as we have been so often lately–into a “lesser of two evils” choice, both assume that our choice is positive (iow, that we are voting for someone) when in fact we may not have been all that thrilled with the candidate we voted for and voted as we did because we were dead set against the opposing candidate. Put in those terms, PV and IR are bot lousy ways to run a democracy.

What Y/N/M voting does is restore the kind of direct, personal consent that allowed democracy to flourish in Greece. Once again we have a voice, once again we are able to consent or dissent to our leaders in clear terms. We’ve taken out the guesswork (“What did the voters mean when they did X?”) and replaced it with solid evidence of what we really wanted to say.

In the phrase “the consent of the governed”, the Framers of the consitution were clearly looking more to the example of Greece than to the Republic of Rome. From the time of Julius Ceasar, Rome was a Republic in name only. The Emperor held Supreme power–and proved it on numerous occasions when the Senate disagreed with him and he ran roughshod over them without a second thought.

In Greece, leaders were chosen in a Public Forum (the origin of the word)–an exercise that we would recognize as a “town meeting”–where every elegible voter had a say. But more importantly, how they expressed themselves mattered. Just like in a town meeting, if a majority of people expressed mild approval of a particular policy but a significant minority expressed very strong disapproval, the majority would probably be swayed by the strength of the minority feeling. As Mr Harrell points out, that sort of clarity of viewpoint simply can’t exist in our present system. The mild approval of the majority would win, leaving a great many people dissatisfied by a weak choice, and the minority would lose, leaving a significant number of people angry and disaffected because their strong feelings had been ignored.

In short, as long as we are stuck with plurality voting or some weak, nearly vapid alternative like instant-runoff voting, we do not have a democracy in any important sense of the word. We don’t even have a democratic republic. We have a plutocracy as in Rome after Julius Ceasar, where the rich bought their seats in the Senate and used them to further their own economic interests at the expense of everyone else’s. (Ring a bell?)

“Voting is both a means of communicating the opinions of voters to society at large and permitting voters to exercise political power,” Mr Harrell correctly says, but it is neither when there is no real mechanism for those shades of meaning and intent to be expressed.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s