Arranology

Commentary on the Majority Rule Voting Paradox

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by Peter K Harrell

The “majority rule paradox” is probably not much of a paradox in the real world today. That “lesser of two evils” situation almost certainly describes a large majority of the voters.

The “yes/no/maybe” voting has all sorts of legal problems. Perhaps some parties may adopt it for primaries when there are large numbers of candidates to be thinned down. In elections, though, the presumption of preference voting, the “one man, one vote” rule, and all sorts of other legal issues would probably prevent it from being implemented.

Hal

Before I begin I would like to point out that formal analysis indicates that the choice of voting techniques can affect the stability of democratic government. You may be surprised to learn that both Plurality Voting and Instant Runoff Voting will elect extremist candidates when an electorate is sufficiently polarized, but that Yes No ‘Maybe So’ Voting promotes the election of moderate candidates, who are better able to represent all of the voters and less likely to further antagonize opponents. I will be glad to post this analysis if you are interested.

Now regarding the Majority Rule Voting Paradox:

Actually, a probability analysis of the Majority Rule Voting Paradox, which assumes that each of the 6 opinion types used in the analysis is equally likely, indicates that there is over a 10% chance that the Majority Rule Voting Paradox will occur. Considering the large number of elections in most election years, this probability analysis strongly suggests that the Majority Rule Voting Paradox is actually a significant problem in elections today.

The next step of course would be to analyze opinion polls in an effort to identify the existence of actual Majority Rule Voting Paradox situations in the electorate. That would require including appropriate questions in opinion polls to ensure that enough information was collected to enable the analysis to take place.

But the significance of the Majority Rule Voting Paradox extends beyond the actual paradox itself. The Majority Rule Voting Paradox reveals in a concise analysis that Plurality Voting cannot ensure that election outcomes will be consistent with the principle of the Consent of the Governed, which is by the way a fundamental principle of American democracy (see note below). This is true even if we ignore situations where a Majority Rule Voting Paradox actually exists.

In order to see this more clearly we simply need to alter slightly the way opinion is modeled, relax the high standards of the Majority Rule Voting Paradox, and to recall that Yes No ‘Maybe So’ Voting permits the election of a candidate who has received only a plurality of support from the voters.

In the original description of the Majority Rule Voting Paradox we ignored the possibility that people might be indifferent or equivocal in their opinions of the candidates. If we include this possibility we alter the results of our probability analysis, but we also greatly increase the number of combinations of opinions where Plurality Voting would elect Candidate A, even though Candidate A is either disapproved of by a majority of the electorate (which will be our First Case) or is disapproved of by a larger portion of the electorate than approves of Candidate A (which will be our Second Case). In this Second Case we would say that Candidate A is disapproved of by a plurality of the electorate.

An example of the First Case can be constructed if this slight but arbitrary change simply means that we assume that only 40 % of the electorate approves of Candidate B. A Majority Rule Voting Paradox no longer exists under these circumstances, but Plurality Voting would still elect Candidate A, who is still disapproved of by over 50 % (i.e. 66.6…%) of the electorate. In this case we have an explicit lack of consent for Candidate A and what political philosophers would call tacit consent for Candidate B. Once again I would argue that Candidate B (40 % approval) should be elected not Candidate A (33.3…% approval). However, when we make these assumptions the model of our electorate becomes more complex.

Here is a First Case sample electorate:

Group 1A (20%):
Prefers Candidate A to Candidate B. Approves of Candidate A and Approves of Candidate B.

Group 1B (13.3…%):
Prefers Candidate A to Candidate B. Approves of Candidate A and is Indifferent/Equivocal with respect to Candidate B.

Group 2 (33.3…%):
Prefers Candidate A to Candidate B. Disapproves of Candidate A and Disapproves of Candidate B.

Group 3A (20%):
Prefers Candidate B to Candidate A. Disapproves of Candidate A and Approves of Candidate B.

Group 3B (13.3…%):
Prefers Candidate B to Candidate A. Disapproves of Candidate A and is Indifferent/Equivocal with respect to Candidate B.

What’s more we can relax the high standards of the Majority Rule Voting Paradox even further in our effort to reveal the flaws in Plurality Voting.

For our next example illustrating the Second Case mentioned above let us continue with a situation where only 40% of the electorate approves of Candidate B. However, this time let us assume that only 40% of the electorate disapproves of Candidate A. Citizens, who make up the remaining 26.6…% of the electorate, either are indifferent with respect to Candidate A because they have no opinion about Candidate A or are equivocal with respect to Candidate A because they have a conflicting set of opinions about Candidate A. Naturally when we make this new assumption the model of our electorate becomes even more complex.

Here is a Second Case sample electorate:

Group 1A (20%):
Prefers Candidate A to Candidate B. Approves of Candidate A and Approves of Candidate B.

Group 1B (13.3…%):
Prefers Candidate A to Candidate B. Approves of Candidate A and is Indifferent/Equivocal with respect to Candidate B.

Group 2A (6.6…%):
Prefers Candidate A to Candidate B. Disapproves of Candidate A and Disapproves of Candidate B.

Group 2B (26.6…%):
Prefers Candidate A to Candidate B. Is Indifferent/Equivocal with respect to Candidate B and Disapproves of Candidate B.

Group 3A (20%):
Prefers Candidate B to Candidate A. Disapproves of Candidate A and Approves of Candidate B.

Group 3B (13.3…%):
Prefers Candidate B to Candidate A. Disapproves of Candidate A and is Indifferent/Equivocal with respect to Candidate B.

In this example Plurality Voting would elect Candidate A even though there exist more members of the electorate who disapprove of Candidate A (40%) than who approve of Candidate A (33.3…%). In other words a plurality of voters disapprove of Candidate A (40%), while at the same time a plurality of voters approve of Candidate B (also 40%). What is more a larger percentage of the electorate approves of Candidate B (40%) than approves of Candidate A (33.3…%).

In this example there is no explicit lack of consent for Candidate A (only 40% disapproval). Also both Candidate A (60%) and Candidate B (93.3…%) receive tacit consent from the electorate. But Plurality Voting is incapable of making any of these distinctions. I would argue once again that Candidate B clearly should be elected not Candidate A.

What I hope is obvious to you from the preceding discussion is that there are huge numbers of combination of opinions which fall into the First Case, Second Case and Majority Rule Voting Paradox categories, and that the probability the electorate will actually hold a set of opinions that fall into these categories is quite high.

What is more you yourself have expressed the opinion that the lesser of two evils dilemma is a serious problem.

Plurality Voting cannot resolve the lesser of two evils dilemma for precisely the same reason that Plurality Voting cannot properly resolve the Majority Rule Voting Paradox and the other voting problems presented here in our First Case and Second Case scenarios.

The reason is simple. Plurality Voting does not permit enough appropriate information about the opinions of the voters to be collected in order to return a result, which consistently respects the principle of the Consent of the Governed.

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Written by Mick

February 3, 2004 at 4:27 am

Posted in Voting

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