Tournament solution

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results A tournament solution is a function that maps an oriented complete graph to a nonempty subset of its vertices.

Tournament solutions originate from social choice theory,[1][2][3][4] but have also been considered in sports competition, game theory,[5] multi-criteria decision analysis, biology,[6][7] webpage ranking,[8] and dueling bandit problems.

[9] In the context of social choice theory, tournament solutions are closely related to Fishburn's C1 social choice functions,[10] and thus seek to show who are the strongest candidates in some sense.

is a connex and asymmetric binary relation over the vertices.

In social choice theory, the binary relation typically represents the pairwise majority comparison between alternatives.

(called the choice set[2]) and does not distinguish between isomorphic tournaments: Common examples of tournament solutions are the:[1][2]