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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,^[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 the best candidates are among all candidates.

A tournament on 4 vertices: ${\displaystyle A=\{1,2,3,4\}}$, ${\displaystyle \succ =\{(1,2),(1,4),(2,4),}$${\displaystyle (3,1),(3,2),(4,3)\}}$

Definition

A tournament (graph) ${\displaystyle T}$ is a tuple ${\displaystyle (A,\succ )}$ where ${\displaystyle A}$ is a set of vertices (called alternatives) and ${\displaystyle \succ }$ 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 ${\displaystyle f}$ that maps each tournament ${\displaystyle T=(A,\succ )}$ to a nonempty subset ${\displaystyle f(T)}$ of the alternatives ${\displaystyle A}$ (called the choice set^[2]) and does not distinguish between isomorphic tournaments:

If ${\displaystyle h:A\rightarrow B}$ is a graph isomorphism between two tournaments ${\displaystyle T=(A,\succ )}$ and ${\displaystyle {\widetilde {T}}=(B,{\widetilde {\succ }})}$, then ${\displaystyle a\in f(T)\Leftrightarrow h(a)\in f({\widetilde {T}})}$

Examples

Common examples of tournament solutions are:^[1][2]

• Copeland's method
• Top cycle
• Slater set
• Bipartisan set
• Uncovered set
• Banks set
• Minimal covering set
• Tournament equilibrium set

References