A tournament solution is a function that maps an oriented complete graph to a nonempty subset of its vertices. It can informally be thought of as a way to find the "best" alternatives among all of the alternatives that are "competing" against each other in the tournament. Tournament solutions originate from social choice theory, but have also been considered in sports competition, game theory, multi-criteria decision analysis, biology, webpage ranking, and dueling bandit problems.
In the context of social choice theory, tournament solutions are closely related to Fishburn's C1 social choice functions, and thus seek to show who the best candidates are among all candidates.
Definition
A tournament (graph) is a tuple where is a set of vertices (called alternatives) and 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.
A tournament solution is a function that maps each tournament to a nonempty subset of the alternatives (called the choice set) and does not distinguish between isomorphic tournaments: If is a graph isomorphism between two tournaments and , then
Examples
Common examples of tournament solutions are: Copeland's method Top cycle Slater set Bipartisan set Uncovered set Banks set Minimal covering set Tournament equilibrium set == References ==