In the study of electoral systems, the Droop quota is the minimum number of supporters a party or candidate needs to receive in a district to guarantee they will win at least one seat in a legislature.
The positive response, monotonicity, or nonperversitycriterion is a principle of social choice that says increasing a candidate's ranking or rating should not cause them to lose. Positive response rules out cases where a candidate loses an election as a result of receiving too much support from voters. Rules that violate positive response are said to show perverseresponse.
The highest averages, divisor, or divide-and-round methods are a family of apportionment algorithms that aim to fairly divide a legislature between several groups, such as political parties or states. More generally, divisor methods can be used to round shares of a total, e.g. percentage points.
The Gibbard–Satterthwaite theorem is a theorem in voting theory. It was first conjectured by the philosopher Michael Dummett and the mathematician Robin Farquharson in 1961 and then proved independently by the philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner, and shows that for every voting rule of this form, at least one of the following three things must hold:
- The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
- The rule limits the possible outcomes to two alternatives only; or
- The rule is not straightforward, i.e. there is no single always-best strategy.
The quota methods are a family of apportionment rules, i.e. algorithms for distributing the seats in a legislative body among a number of administrative divisions. The quota methods are based on calculating a fixed electoral quota, i.e. a given number of votes needed to win a seat. This is used to calculate each party's seat entitlement. Every party is assigned the integer portion of this entitlement, and any seats left over are distributed according to a specified rule.
A random ballot or random dictatorship is a randomized electoral system where the election is decided on the basis of a single randomly-selected ballot. A closely-related variant is called random serialdictatorship, which repeats the procedure and draws another ballot if multiple candidates are tied on the first ballot.
In social choice theory, the majority rule (MR) is a social choice rule that says that, when comparing two options, the option preferred by more than half of the voters should win.
In social choice theory, May's theorem, also called the general possibility theorem, says that majority vote is the unique ranked social choice function between two candidates that satisfies the following criteria:
In the study of apportionment, the Harequota is the number of voters represented by each legislator under an idealized system of proportional representation, where every legislator represents an equal number of voters. The Hare quota is the total number of votes divided by the number of seats to be filled. The Hare quota was used in the original proposal for a single transferable vote system, and is still occasionally used, although it has since been largely supplanted by the Droop quota.
The single transferable vote (STV) is a proportional representation system that elects multiple winners. It is one of several ways of choosing winners from ballots that rank candidates by preference. Under STV, an elector's vote is initially allocated to their first-ranked candidate. Candidates are elected (winners) if their vote tally reaches quota. After the winners in the first count are determined, if seats are still open, surplus votes — those in excess of an electoral quota— are transferred from winners to the remaining candidates (hopefuls) according to the surplus ballots' next usable back-up preference.
Bullet, single-shot, or plump voting is when a voter supports only a single candidate, typically to show strong support for a single favorite.
In social choice theory, the best-is-worst paradox occurs when a candidate finishes simultaneously in first- and last- place according to the same voting method. The worst candidate can be identified by reversing all ballots before rerunning the algorithm to find a single worst candidate. Such paradoxes can occur in ranked-choice runoff voting (RCV) and minimax; a well-known example is the 2022 Alaska special election, where candidate Mary Peltola simultaneously finished in first and last place.
In social choice theory, a dictatorship mechanism is a degenerate voting rule or mechanism where the result depends on only one person's preferences, without considering any other voters. A serial dictatorship is similar, but also designates a series of "backup dictators", who break ties in the original dictator's choices when the dictator is indifferent.
A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.
In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It states that for any deterministic process of collective decision, at least one of the following three properties must hold:
- The process is dictatorial, i.e. there is a single voter whose vote chooses the outcome.
- The process limits the possible outcomes to two options only.
- The process is not straightforward; the optimal ballot for a voter "requires strategic voting", i.e. it depends on their beliefs about other voters' ballots.
Fractional, stochastic, or weighted social choice is a branch of social choice theory in which the collective decision is not a single alternative, but rather a weighted sum of two or more alternatives. For example, if society has to choose between three candidates, then in standard social choice exactly one of these candidates is chosen. By contrast, in fractional social choice it is possible to choose any linear combination of these, e.g. "2/3 of A and 1/3 of B".
In fractional social choice, fractional approval voting refers to a class of electoral systems using approval ballots, in which the outcome is fractional: for each alternative j there is a fraction pj between 0 and 1, such that the sum of pj is 1. It can be seen as a generalization of approval voting: in the latter, one candidate wins and the other candidates lose. The fractions pj can be interpreted in various ways, depending on the setting. Examples are:
The mixed ballot transferable vote (MBTV) refers to a type of vote linkage-based mixed-member electoral system where a group of members are elected on local (lower) tier, for example in single-member districts (SMDs). Other members are elected on a compensatory national (upper) tier from a list and voters cast a single ballot where they may indicate their preferences separately.
Vote-ratio, weight-ratio, or population-ratio monotonicity is a property of some apportionment methods. It says that if the entitlement for grows at a faster rate than , should not lose a seat to . More formally, if the ratio of votes or populations increases, then should not lose a seat while gains a seat. Apportionments violating this rule are called population paradoxes.
In mechanism design, a regret-free truth-telling mechanism is a mechanism in which each player who reveals his true private information does not feel regret after seeing the mechanism outcome. A regret-free mechanism incentivizes agents who want to avoid regret to report their preferences truthfully.
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