Tournament solution

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

1. Laslier, J.-F. (1997). Tournament Solutions and Majority Voting. Springer Verlag.
2. Felix Brandt; Markus Brill; Paul Harrenstein (28 April 2016). "Chapter 3: Tournament Solutions" (PDF). In Felix Brandt; Vincent Conitzer; Ulle Endriss; Jérôme Lang; Ariel D. Procaccia (eds.). Handbook of Computational Social Choice. Cambridge University Press. ISBN 978-1-316-48975-8.
3. Brandt, F. (2009). Tournament Solutions - Extensions of Maximality and Their Applications to Decision-Making. Habilitation Thesis, Faculty for Mathematics, Computer Science, and Statistics, University of Munich.
4. Scott Moser. "Chapter 6: Majority rule and tournament solutions". In J. C. Heckelman; N. R. Miller (eds.). Handbook of Social Choice and Voting. Edgar Elgar.
5. Fisher, D. C.; Ryan, J. (1995). "Tournament games and positive tournaments". Journal of Graph Theory. 19 (2): 217–236. doi:10.1002/jgt.3190190208.
6. Allesina, S.; Levine, J. M. (2011). "A competitive network theory of species diversity". Proceedings of the National Academy of Sciences. 108 (14): 5638–5642. doi:10.1073/pnas.1014428108.
7. Landau, H. G. (1951). "On dominance relations and the structure of animal societies: I. Effect of inherent characteristics". Bulletin of Mathematical Biophysics. 13 (1): 1–19. doi:10.1007/bf02478336.
8. Felix Brandt; Felix Fischer (2007). "PageRank as a Weak Tournament Solution" (PDF). Lecture Notes in Computer Science (LNCS). 3rd International Workshop on Internet and Network Economics (WINE). 4858. San Diego, USA: Springer. pp. 300–305.
9. Siddartha Ramamohan; Arun Rajkumar; Shivani Agarwal (2016). Dueling Bandits: Beyond Condorcet Winners to General Tournament Solutions (PDF). 29th Conference on Neural Information Processing Systems (NIPS 2016). Barcelona, Spain.
10. Fishburn, P. C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030.