Tournament solution

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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 are the strongest candidates in some sense.

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 ${\displaystyle a\in f(T)\Leftrightarrow h(a)\in f({\widetilde {T}})}$

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

• Copeland set
• Smith set (aka Top cycle)
• Slater set^{[clarification needed]}
• Bipartisan set
• Landau set
• Banks set
• Minimal covering set
• Tournament equilibrium set^{[clarification needed]}