Majority rule is often considered to be a pillar of democratic decision-making. Yet there are several problems with relying on majority rule as **Jac Heckelman** goes on to discuss.

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One problem is that a simple majority of people make the decision. This means that small gains to the majority outweigh large losses to the minority. Even if the policy being considered has negative net benefits overall, it will still pass if the *number* of winners exceeds the number of losers. Thus, majority outcomes can be inefficient.

Another problem is that majority rule can lead to what is known as “**tyranny by the majority**”. Consider for example the case of Albuquerque, New Mexico which is considered a “swing city” that has majority supported both Democratic and Republican candidates in various past elections, at the local, state, and federal levels. The current city council is comprised of five Democrats and four Republicans, consistent with the nearly equal split in partisan behavior of the Albuquerque electorate. Yet, on any legislative policy under consideration by the city council, the Democratic majority will always win whenever they are in agreement. Thus, despite the slimmest of majority by representation (5 – 4), Democratic policies should be expected to dominate the proceedings. Although on its face using majority rule to decide policy appears to be a fair and democratic (small ‘d’) process because each person’s vote would seem to be equal, in reality Republican council member votes are irrelevant whenever the Democrats are unified. As soon as a majority is achieved, the rest of the votes are meaningless. This gets overlooked because there is no order to the voting so everyone is counted at the same time. Yet if the Democrats were to vote first and announce they have five votes in favor (or in opposition), the rest of the council need not vote at all. Only when the Democrats are split amongst themselves are any of the Republican votes even necessary. This problem of tyranny becomes even more disconcerting when group decisions break-down along racial, religious, or gender lines. The majority subgroup can craft policies to benefit themselves at the expense of the minority and know it will always carry the day regardless of how many different policies it designs. Similarly, the majority subgroup can block any policy it opposes.

The problem of majority rule actually becomes even more acute when there is *not* majority cohesion. Consider the fanciful example where a panel of 15 mathematicians are trying to decide in whose honor a new award should be named. Their preferences are as follows:

# of mathematicians | 6 | 5 | 4 |

Top preference | Pythagoras | Archimedes | Euclid |

Middle preference | Archimedes | Euclid | Pythagoras |

Bottom preference | Euclid | Pythagoras | Archimedes |

The first problem the panel will run into is that no name receives a majority of votes. Relying upon strict majority rule will leave the panel in a deadlock. This is always possible when there are more than two alternatives under consideration. To avoid such uncertainty, an alternative method of plurality, also known as relative majority or first-past-the-post, is often utilized to decide elections. Plurality declares the winner to be whomever has the most votes, without having to reach any particular thresholds (such as a pure majority). The plurality winner in this example would be Pythagoras with 6 votes, despite having received only a minority of the total votes possible (40%), which may be unpalatable to some. This is always possible whenever there are more than two alternatives, and the more alternatives the lower percentage possible for the winner. Indeed, Paul LePage was elected governor of Maine in 2010 by plurality rule despite receiving less than 38% of the total vote.

An alternative to plurality, described by Marquis de Condorcet over 200 hundred years ago, requires every pair of candidates to be voted on separately, and whichever candidate wins all of its pairwise votes is to be declared the winner. Yet Condorcet also noted that no one may qualify as the winner. In the above example, in a head-to-head matchup between Pythagoras and Archimedes, Pythagoras would receive a majority of the vote (10 – 5) but lose against Euclid (9 – 6). Because Euclid defeats Pythagoras, and Pythagoras defeats Archimedes, it would be only natural to presume Euclid should defeat Archimedes as well. However, the reverse occurs. Archimedes defeats Euclid by the most lopsided of the votes, 11 – 4. Thus, a *majority cycle* occurs and Condorcet’s method does not solve the problem of majority indecision.

Another solution is to eliminate from consideration any candidate the first time it loses. So in the above example, Archimedes would be eliminated once he lost to Pythagoras, and when Euclid then defeats Pythagoras, everyone but Euclid has been eliminated leaving Euclid the winner for which the award will be named. Yet a complication arises. The first group of mathematicians who are most upset with the outcome could rightly claim that the outcome was arbitrarily decided by the random choice of how to order the pairwise voting. Had instead the first vote been between Euclid and Archimedes, and that winner to face Pythagoras, then Euclid would be eliminated by the first vote, and Pythagoras would then be eliminated by Archimedes, resulting in the decision to have the name be the Archimedes Award. But it also could have become the Pythagoras Award if the order of voting first paired Archimedes against Euclid. Then Euclid would be initially eliminated and Archimedes would advance to face Pythagoras resulting in the elimination of Archimedes and the award named for Pythagoras.

As no one order of voting is any fairer than another, each group of mathematicians would have their own preferences on ordering after realizing what would be the outcomes of each order. A majority cycle would then emerge on how to order the majority vote. The quagmire deepens. To rely on majority rule is to resign oneself to indecision, manipulation, or arbitrariness.

**Jac Heckelman** is Professor of Economics at Wake Forest University.

The first chapter of** Jac’s** new book *Handbook of Social Choice and Voting* can be downloaded for free on Elgaronline.

January 13, 2016

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