Condorcet method

The Condorcet or majority-rule methods (English: ; French:) are a family of election systems that elect the majority-preferred (Condorcet) winner if one exists. A majority-preferred candidate is a candidate who would win a majority of votes in any head-to-head election against any opponent. In other words, they are a candidate who would win in any one-on-one race (one without spoilers). The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.

Some elections may not have a Condorcet winner because the majority's preferences can be cyclic: in other words, it is possible every candidate has an opponent that would defeat them in a two-candidate contest. This is similar to a game of rock, paper, scissors: a candidate "scissors" may beat "paper," but still lose to "rock". The possibility of such a cycle is known as Condorcet's paradox. (Empirically, however, such cycles are rare, with most large public elections having a Condorcet winner. As a result, all Condorcet methods reduce to simple majority rule in most cases.)

Under many models of voting, the Condorcet winner is typically the same as the socially optimal winner. However, this is not guaranteed to be the case; situations where the two disagree are commonly called tyranny of the majority scenarios (particularly if the disagreement is large).

Condorcet voting methods are named for the 18th-century French mathematician, the Marquis de Condorcet, who popularized the concept. Condorcet methods were first extensively analyzed by Ramon Llull, but the manuscripts of these works were lost from the Late Middle Ages until being rediscovered in the 20th century.

Condorcet methods are commonly used to vote on motions and amendments in assemblies and legislatures, as they allow replacing paper ballots with a series of simple majority votes, with prominent parliamentary procedure handbooks such as Jefferson's Manual and Robert's Rules of Order prescribing their use. However, public elections typically use ranked ballots to avoid the time and complexity of a full round-robin tournament with each voter's choice in a paired matchup being determined by checking which candidate has the higher rank or rating on a ballot.

In a contest between candidates A, B and C using the preferential-vote form of Condorcet method, a head-to-head race is conducted between each pair of candidates. A and B, B and C, and C and A. If one candidate is preferred over all others, they are the Condorcet Winner and winner of the election.

Because of the possibility of the Condorcet paradox, it is possible (but empirically rare) that a Condorcet winner does not exist in a specific election. This is sometimes called a Condorcet cycle or just cycle and can be thought of as Rock beating Scissors, Scissors beating Paper, and Paper beating Rock. Various Condorcet methods differ in how they resolve such a cycle. (Most elections do not have cycles. See Condorcet paradox#Likelihood for estimates.) If there is no cycle, all Condorcet methods elect the same candidate and are operationally equivalent.

• Each voter ranks the candidates in order of preference (top-to-bottom, or best-to-worst, or 1st, 2nd, 3rd, etc.). The voter may be allowed to rank candidates as equals and to express indifference (no preference) between them. Candidates omitted by a voter may be treated as if the voter ranked them at the bottom.
• For each pairing of candidates (as in a round-robin tournament) count how many votes rank each candidate over the other candidate. Thus each pairing will have two totals: the size of its majority and the size of its minority (or there will be a tie).

For most Condorcet methods, those counts usually suffice to determine the complete order of finish (i.e. who won, who came in 2nd place, etc.). They always suffice to determine whether there is a Condorcet winner.

Additional information may be needed in the event of ties. Ties can be pairings that have no majority, or they can be majorities that are the same size. Such ties will be rare when there are many voters. Some Condorcet methods may have other kinds of ties. For example, with Copeland's method, it would not be rare for two or more candidates to win the same number of pairings, when there is no Condorcet winner.