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Electoral rules are criteria to transform votes into collective decisions. They are generally inspired by one of three main principles: unanimity, which requires maximum consensus; majority, which makes a decision supported by half the voters valid; and proportionality, which gives every group a share of representation for further negotiations and broad agreements. Different electoral rules can be used for different purposes, involving both mass political elections and small-group and committee voting and decisions, because they induce the formation of different alternatives and different agendas and may produce diverse results, as will be discussed below.

Unanimity

Election of delegates and approval of proposals by unanimity are almost instinctive procedures in simple, homogeneous gatherings and assemblies with easily identifiable common interests and priorities. Families, groups of friends, urban gangs, neighborhood meetings, corporation partners, and club members tend to make collective decisions under conditions of general agreement. In the ancient world, the Justinian code of Rome established the principle that “what concerns similarly all ought to be approved by all,” which was adopted by the Christian Church in the fifth century as the following: “He who governs all should be elected by all.” In medieval Europe, consuls were elected by traders, bishops by priests and believers, magistrates by citizens, and so forth, on the basis of large consensus. Many medieval assembly regulations established that decisions should be made by “consensus and acclamation,” “approval and consent,” with “no discrepancy” or “no contradiction,” by “free veto,” and so on.

However, the requirement of unanimity made it difficult to reach many decisions, especially as communes and countries became more open and socially heterogeneous, which provoked conflicts and schisms. A variety of medieval institutions provided procedures to create unanimity where it did not exist, including silent acquiescence; shouts of commendation or acclamation; murmurs in favor of or cries against the proposer; explicit acceptance of the elected by the dissidents; preliminary voting followed by formal, public expression of the decision by all the community members; and acceptance of elections or decisions made by a qualified part of voters to whom the other voters would submit. Certain Italian communes adopted less-than-unanimity rules, such as those requiring two-thirds, four-sevenths, or other supermajorities or qualified majorities, together with indirect elections in several stages and other devices. The elections of the Christian pope since the twelfth century, as well as those of several central European kings and the German emperor from the fourteenth century on, were redesigned along similar lines. But certain assemblies, such as the Aragonese and Catalan parliaments and the Polish Diet, were reluctant to abandon the requirement of unanimity. As late as the early nineteenth century, even in the British House of Commons most decisions were still made by acclamation, which implied near-unanimous consent.

In both ancient and medieval political institutions and in modern private communities and companies, forcing explicit acquiescence of the dissidents to unanimous consent is a means to ensure that they will respect the elected, obey the collective decisions, or contribute with their duties in spite of previous disagreements. Similar features can be found in a number of international organizations, such as the United Nations General Assembly and the Council of the European Union, where it is assumed that each party is sovereign and has a veto right. Despite their many differences, all these institutions—whether they are ancient, medieval, or modern; private or public; local or international—have the following basic elements: corporate or government (not individual) suffrage, decision making that is limited to those issues in which a general common interest can be presumed, the search for near-unanimous consensus, distribution of burdens in proportion to contribution of resources, and offices that are held by turns or lots.

The unanimity rule has some good properties. Unanimous decisions correspond to the efficiency criterion associated with the name of Italian economist Vilfredo Pareto. A collective decision is said to be “Pareto-superior” if it improves the situation of some participants and does not worsen the situation of others. Also, decisions by the unanimity rule give a positive response to changes in voters’ preferences. Specifically, under the unanimity rule, an increase in voters’ support for the incumbent or the status quo will not result in its replacement. Similarly, a loss of support for an alternative candidate or proposal will not make it the winner—a property also called “monotonicity” that is not fulfilled, in contrast, by several procedures based on majority rule.

However, unanimity decisions may be impossible to make if voters’ preferences can be located along a single dimension, such as the left-right axis or any other issue or ideological dimension on which the participants have consistent preferences. Any voter can veto a move of the status quo away from his or her preference. Advantaged voters, or in spatial terms, those whose preferences are closer to the status quo, can consolidate their advantages. The collective outcome will remain stable independent of the existence of alternative candidates or proposals that are able to reduce the aggregated distance from all individual preferences and thus increase collective satisfaction.

With the introduction of new issues or new values of judgment creating a multidimensional space, several unanimous decisions within the Pareto set, or the set of decisions fulfilling the Pareto criterion, can be made available. While new candidates or proposals may be placed at a larger distance from some voters’ preferences than the status quo on one issue or value, they can also be closer to them on other issues or values and become globally more satisfactory and acceptable. But each one of the possible new winners by unanimity will give different voters different degrees of satisfaction of their preferences. Only when the initial status quo is very unsatisfactory, that is, very distant from the voters’ preferences, can a new socially efficient decision be made by unanimity. In contrast, if the initial status quo is relatively less unsatisfactory, some voters will veto alternative candidates or proposals, thus causing mediocrity to endure.

Majority

The majority principle was first introduced in medieval Germanic law and the Christian Church’s canon law as a consequence of failures in making decisions by unanimity. When dissident members or minority groups were sufficiently large or determined and could effectively resist the imposition of the dominant group’s will, there was a strong incentive to adopt a less-than-unanimity, typically majority, rule. With the formation of two or more fractions or parties, decisions made by acclamation were replaced with more formal procedures requiring counting votes and the achievement of a majority threshold. In contrast to the unanimity-based organizations mentioned above, majority rule usually requires individual suffrage and implies that the winner takes all.

It has been argued that the majority principle is the only one that satisfies these few reasonable criteria: (1) decisiveness, but only if there are no more than two alternatives (candidates, parties, or policy proposals) to choose from; (2) anonymity or voters’ equality; (3) neutrality with respect to issues, not giving advantage to the largest group or to the status quo (in contrast to unanimity rule, for instance, as discussed above); and (4) positive response to changes in voters’ preferences or monotonicity (but only if one alternative obtains an absolute majority support based on voters’ first preferences).

In practice, different procedures have been invented to try to make the majority principle viable in real elections. Two late-eighteenth-century French academics made sophisticated proposals. Marquis de Condorcet proposed that the winner in an election should be the alternative (candidate or proposal) preferred by a majority against every other alternative, which may require multiple rounds of voting or comparisons between pairs of alternatives. When the alternatives are located along a single issue or ideological dimension, exhaustive pairwise comparisons or the Condorcet voting procedure makes the median voter’s preference the winner. By definition, the median voter, that is, the voter whose preference is located in an intermediate position with less than half of voters on both sides, is always necessary to form a consistent majority on a single dimension. Since the median voter’s preference minimizes the sum of distances from all other individual preferences, it can be considered a socially efficient outcome. But in a more dispersed set of alternatives forming multiple dimensions, the Condorcet procedure may not produce a winner, thus lacking decisiveness. A variant gives the victory to the alternative that wins a higher number of times (as divulgated by the Catalan philosopher Ramon Llull in the Middle Ages).

In turn, another eighteenth-century French academic, JeanCharles de Borda, proposed a rank-order count procedure by which the voter should order preferences and give zero, one, two, and successive points to the alternatives; the winner should be the one with the highest sum of points (the German philosopher Nicolas of Cusa had also proposed this procedure a few centuries before). A more traditional procedure, also with medieval precedents, is approval voting, which allows voters to vote for all those alternatives that they consider acceptable, from a minimum of one to a maximum of all minus one; the alternative with the highest number of votes becomes the winner. There has been much discussion about how different results may be produced by these different procedures, depending on voters’ distribution of preferences and their degree of homogeneity. But as has been shown recently, in most real-world settings, exhaustive pairwise comparisons, rank-order count, and approval voting tend to select the same winner.

In mass political elections, relatively less-demanding procedures are more frequently used. With a simple plurality rule, the winner is the candidate supported by only a relative majority, that is, by a higher number of voters than any other candidate but not requiring any particular number, proportion, or threshold of votes. In practice, this makes it possible for generally binding decisions presumably decided by the majority to actually be won with the support of only a minority of voters. In fact, in mass parliamentary elections by plurality rule, a single party has received a majority of seats on the basis of a minority of votes in about two-thirds of the cases in the United Kingdom and some former British colonies (out of 126 democratic elections in Canada since 1878, the United Kingdom since 1885, New Zealand from 1890 to 1993, and India since 1953) as well as in about one-third of the cases in the U.S. House of Representatives since 1828. In presidential elections, plurality rule has given the victory to minority-vote candidates in about two-fifths of the cases in the United States (through the plurality-based electoral college) and in more than two-thirds of the cases in eight countries in Latin America (out of 54 democratic elections during several democratic periods from 1945 on in Argentina, Brazil, Chile, Colombia, Ecuador, Peru, Uruguay, and Venezuela).

In contrast, majority runoff requires an absolute majority (more than half) of votes at the first round, while in a second round of voting the choice can be reduced to the two most voted-for candidates to secure majority support for the winner. A variant requires the voters to rank all candidates and proceeds to several counts of votes (instead of several rounds of voting) until a candidate obtains the most preferences, as in the majority-preferential vote (also called “alternative vote” or “instant runoff ”).

With both plurality and majority-runoff or majority preferential voting, the median voter’s preference can be defeated or eliminated. The nonmedian winner by any of these procedures could be defeated by another, losing candidate by absolute majority if the choice between the two were available; that is, he or she might not be the winner by Condorcet procedure (preferred by a majority against every other alternative).This has been the case, for instance, in five of eight presidential elections by majority runoff in France since 1965. Under plurality rule, the winner can even be the Condorcet loser or the most-rejected candidate by a majority of voters, as has happened, for instance, in several presidential elections in Latin American countries, including Brazil, Chile, Ecuador, and Peru, with disastrous political consequences.

In general, the most usual procedures of majority rule just discussed are dependent on irrelevant alternatives; that is, they are highly vulnerable to manipulation since the winner may be an indirect consequence of the merge or split of other, nonwinning alternatives. If a new, nonwinning candidacy splits the votes of the winner, a different winner can be created. This may happen even if the new winner has not gained larger support (thus not fulfilling the monotonicity criterion mentioned above). Majority-rule elections thus encourage strategies aimed at altering the number of alternatives, such as divide and win and merge and win, as well as nonsincere or strategic votes in favor of a less preferred but more likely winning alternative.

Proportionality

Proportional representation rules allocate different numbers of seats to multiple parties competing in an election on the basis of the votes received. They were invented with the aim of reducing single-party sweeps and exclusionary victories and preventing actual minority winners with the previously existing rules. As mentioned, majoritarian electoral rules had been widely used in contexts of simple societies with rather homogeneous electorates dealing with local issues. But the expansion of suffrage rights, the emergence of new political demands, and the creation of new parties trying to politicize new issues in newly complex societies made traditional results with majoritarian rules increasingly dissatisfactory for both voters and candidates. The winner-takes-all character of majority rule and the frequency of actual minority winners were at odds with increasing political and social pluralism. In a number of countries the introduction of proportional representation rules in the early twentieth century ran parallel to the introduction of other regulations favoring citizens’ participation and fair competition, such as more reliable electoral censuses, the written ballot, secret vote, and an independent electoral authority validating the results.

The basic mathematical formulas that would make the principle of proportional representation operable were invented in the late eighteenth century for apportioning seats in the U.S. House of Representatives among the differently populated states. But they were reinvented in Europe in the late nineteenth century for the allocation of parliamentary seats to political parties with different numbers of votes. A proportional representation formula defines a quota of inhabitants or votes worth a seat. The “simple” quota (as devised by both eighteenth-century U.S. politician Alexander Hamilton and nineteenth-century English lawyer Thomas Hare) is the divisor between the total number of inhabitants or votes and the total number of seats. But since inhabitants or votes are not distributed in exact multiples of the quota, it usually requires an additional criterion to allocate some of the seats, most commonly to the largest remainders after the quota is used. In contrast, the smaller “highest average” or “distributive number” (as devised by both eighteenth-century U.S. politician and president Thomas Jefferson and nineteenth-century Belgian law professor Victor d’Hondt) is sufficient to allocate all seats. This quota can be calculated after the election by several procedures, including using trial and error, using a series of divisors, or lowering the simple quota until fitting all the seats to be allocated. Variants include the so-called major-fractions formula (proposed by both nineteenth-century U.S. politician Daniel Webster and twentieth-century French mathematician André Sainte-Laguë) and others. The “fixed” quota is an absolute number of votes established a priori as worthy of a seat (as proposed separately by nineteenth-century French mathematician Joseph-Diaz Gergonne and nineteenth-century U.S. activist Thomas Gilpin). Although rarely used in mass political elections, it may encourage turnout and work with uniform criteria in all districts, even if it does not permit the establishment of a previously set number of seats.

Proportional representation rules can be used with closed party lists, permitting the voter to choose categorically only one alternative. But they are also used with other ballot formulas, including open lists or preferential votes, permitting the voter to select one or a few candidates within a party list (as used in Scandinavian countries); the double vote, requiring voters to choose both a closed-party list and one individual candidate (as used, e.g., in Germany); the open ballot, permitting the voter to vote for individual candidates from different parties (as in Switzerland); and the single-transferable vote, requiring voters to rank all individual candidates (as used in relatively small districts in Ireland).

In comparison with the majoritarian rules discussed above, electoral systems with proportional representation rules are more inclusive of several groups. They encourage multiple parties to run separately according to their own profiles, that is, not to withdraw or merge. They tend to facilitate the election of members of ethnic minorities and women representatives. Political and ideological minorities can be included in the system and have an influence on collective decision making according to their popular support to form an actual majority for institutional decision making. Proportional representation, by placing electoral contests in large districts, may also encourage the development of political parties promoting broad interests and the provision of large-scale public goods, in contrast to more frequent focuses on narrow local interests, private goods, and clientelism in individual elections in single member districts. With proportional representation, since most votes count to elect seats, voters are encouraged to promote a more sincere revelation of preferences. Electoral participation tends to be higher in elections with proportional representation rules than in single-member districts.

The degree of proportionality between votes and seats for each party produced by different rules can be measured with several indexes, which are all strongly correlated. With conventional measures, the deviation from proportionality may take values from as low as less than 2 percent (as in Germany, with a simple-quota proportional system) to up to 20 percent overrepresentation in favor of the larger parties (as in the plurality-rule system in the United Kingdom).

While plurality rule may fabricate a single party’s absolute majority of seats on the basis of a minority of popular votes (not necessarily including the median voter, as previously mentioned), multiparty parliaments based on proportional representation tend to produce multiparty coalition governments based on a majority of seats and popular votes. In practice, there is a paradox: majoritarian electoral systems often create governments with minority electoral support, while proportional representation rules, which are praised for the inclusion of minorities, tend to produce governments with majority electoral support. In plurality-rule electoral systems, a small change in the total of popular votes can provoke a complete alternation of the party in government. With proportional representation, since some parties may have opportunities to share power with different partners, in the long term we should expect relatively more policy stability.

Nowadays, plurality rule is used for mass political elections, mostly in a number of old democratic regimes in former British colonies (as well as in most fake noncompetitive elections in authoritarian regimes). Proportional representation rules began to be used in the early twentieth century in relatively small but socially or ethnically complex countries in Western Europe, when they introduced new regulations of universal male suffrage, including Belgium, the Netherlands, Switzerland, and the Scandinavian states. Today they are used in most democratic regimes across the world.

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