Tournament solution

A tournament solution is a function that maps an oriented complete graph to a nonempty subset of its vertices. 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]

A tournament on 4 vertices:

A=\{1,2,3,4\}

,

\succ =\{(1,2),(1,4),(2,4),

(3,1),(3,2),(4,3)\}

Definition

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

If

h:A\rightarrow B

is a graph isomorphism between two tournaments

T=(A,\succ )

and

{\widetilde {T}}=(B,{\widetilde {\succ }})

, then

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 2 Laslier, J.-F. (1997). Tournament Solutions and Majority Voting. Springer Verlag.
1 2 3 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. Handbook of Computational Social Choice. Cambridge University Press. ISBN 978-1-316-48975-8.
↑ 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.
↑ Scott Moser. "Chapter 6: Majority rule and tournament solutions". In J. C. Heckelman; N. R. Miller. Handbook of Social Choice and Voting. Edgar Elgar.
↑ Fisher, D. C.; Ryan, J. (1995). "Tournament games and positive tournaments". Journal of Graph Theory. 19 (2): 217–236.
↑ Allesina, S.; Levine, J. M. (2011). "A competitive network theory of species diversity". Proceedings of the National Academy of Sciences. 108 (14): 5638–5642.
↑ 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.
↑ 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.
↑ 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.
↑ Fishburn, P. C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[:0-1] 1 2 Laslier, J.-F. (1997). Tournament Solutions and Majority Voting. Springer Verlag.

[BrandtConitzer2016-2] 1 2 3 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. 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. 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.

[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.

[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.

[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.