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Encyclopedia of Complexity and Systems Science pp 9942–9965Cite as

Voting Procedures, Complexity of

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Definition of the Subject

One main concern of voting theory is to determine a procedure (also called, according to the context orthe authors, rule, method, social choice function, social choice correspondence,system, scheme, count, rank aggregation, principle, solution and so on), for choosinga winner from among a set of candidates, based on the preferences of the voters. Each voter's preferencemay be expressed as the choice of a single individual candidate or, more ambitiously, a ranked list including allor some of the candidates. Such a situation occurs, obviously, in the field of social choice and welfare (fora broader presentation of the field of social choice and welfare, see for instance [4,7,8,9,14,17,67,75,97,101,136,143,154]) and especially ofelections (for more about voting theory, see [27,63,107,109,110,125,141,156,167,168,169]), but also in manyother fields: games, sports, artificial intelligence, spam detection, Web search engines, Internet applications,statistics, and so...

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Abbreviations

Condorcet winner :

A candidate is a Condorcet winner if he or she defeats any other candidate in a one-to-one matchup. Such a candidate may not exist; at most, there is only one. Though it could seem reasonable to adopt a Condorcet winner (if any) as the winner of an election, many common voting procedures bypass the Condorcet winner in favor of a winner chosen by other criteria.

Majority relation , strict majority relation:

In a pairwise comparison method , each candidate is compared to all others, one at a time. If a candidate x is preferred to a candidate y by at least \( { m/2 } \) voters (a majority), where m denotes the number of voters, x is said to be preferred to y according to the majority relation. The strict majority relation is defined in a similar way, but with \( { (m + 1)/2 } \) instead of \( { m/2 } \). If there is no tie, the strict majority relation is a tournament, i. e., a complete asymmetric binary relation, called the majority tournament.

Preference , preference aggregation :

A voter's preference is some relational structure defined over the set of candidates. Such a structure depends on the chosen voting procedure, and usually ranges between a binary relation on one extreme and a linear order on the other. Given a collection, called a profile, of individual preferences defined on a set of candidates, the aggregation problem consists in computing a collective preference summarizing the profile as well as possible (for a given criterion).

Profile :

A profile \( { \Pi =(R_{1}, R_{2},\dots, R_{m}) } \) is an ordered collection (or a multiset) of m relations \( { R_{i}\,(1\leqslant i\leqslant m) } \) for a given integer m. As the relations R i can be the same, another representation of a profile Π consists in specifying only the q relations R i which are different, for an appropriate integer q, and the number m i of occurrences of each relation \( R_{i}\,(1\leqslant i\leqslant q) \colon \Pi =(R_{1}, m_{1}; R_{2}, m_{2};\dots; R_{q}, m_{q}) \).

Social choice function , social choice correspondence :

A social choice function maps a collection of individual preferences specified on a set of candidates onto a unique candidate, while a social choice correspondence maps it onto a nonempty set of candidates. This provides a way to formalize what constitutes the most preferred choice for a group of agents.

Voting procedure , voting theory :

A voting procedure is a rule defining how to elect a winner (single‐winner election) or several winners (multiple‐winner election) or to rank the candidates from the individual preferences of the voters. Voting theory studies the (axiomatic, algorithmic, combinatorial, and so on) properties of the voting procedures designed in order to reach collective decisions.

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Acknowledgments

I would like to thank Ulle Endriss, Jérôme Lang and Bernard Monjardet for their help. Their commentswere very useful to improve the text.

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    Olivier Hudry

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Hudry, O. (2009). Voting Procedures, Complexity of. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_585

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