Tournament solution

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:[1][2]

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