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Computational social choice

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Computational social choice is a field at the intersection of social choice theory, theoretical computer science, and the analysis of multi-agent systems.[1] It consists of the analysis of problems arising from the aggregation of preferences of a group of agents from a computational perspective. In particular, computational social choice is concerned with the efficient computation of outcomes of voting rules, with the computational complexity of various forms of manipulation, and issues arising from the problem of representing and eliciting preferences in combinatorial settings.

Winner determination

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The usefulness of a particular voting system can be severely limited if it takes a very long time to calculate the winner of an election. Therefore, it is important to design fast algorithms that can evaluate a voting rule when given ballots as input. As is common in computational complexity theory, an algorithm is thought to be efficient if it takes polynomial time. Many popular voting rules can be evaluated in polynomial time in a straightforward way (i.e., counting), such as the Borda count, approval voting, or the plurality rule. For rules such as the Schulze method or ranked pairs, more sophisticated algorithms can be used to show polynomial runtime.[2][3] Certain voting systems, however, are computationally difficult to evaluate.[4] In particular, winner determination for the Kemeny-Young method, Dodgson's method, and Young's method are all NP-hard problems.[4][5][6][7] This has led to the development of approximation algorithms and fixed-parameter tractable algorithms to improve the theoretical calculation of such problems.[8][9][10]

Preference Representation

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An important differentiation between voting rules is the format of ballots used by the voters to represent their preference. The two most common formats are approval ballots and ordinal ranks.

In approval ballots, each voter approves some candidates she likes. There is no further differentiation or hierarchy within the approved candidates. The same holds for the non-approved candidates. Thus, such ballots are also called dichotomous. Approval ballots are used for instance in satisfaction approval voting and proportional approval voting.

In contrast, ordinal ranks require the voter to rank all candidates from best to worst. This type of ballot is used for example in Borda's rule or in Bucklin voting.

There are many other types of ballot formats described in literature, such as truncated ranks, trichotomous ballots, or cardinal utility ballots.

Some research in computational social choice is focused on how representative ballot formats are, and on developing expressive, yet compact ballot formats. This is especially important in combinatorial settings, such as multiwinner voting.

Other topics

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Tournament solutions

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A tournament solution is a rule that assigns to every tournament a set of winners. Since a preference profile induces a tournament through its majority relation, every tournament solution can also be seen as a voting rule which only uses information about the outcomes of pairwise majority contests.[11] Many tournament solutions have been proposed,[12] and computational social choice theorists have studied the complexity of the associated winner determination problems.[13][1]

Preference restrictions

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Restricted preference domains, such as single-peaked or single-crossing preferences, are an important area of study in social choice theory, since preferences from these domains avoid the Condorcet paradox and thus can circumvent impossibility results like Arrow's theorem and the Gibbard-Satterthwaite theorem.[14][15][16][17] From a computational perspective, such domain restrictions are useful to speed up winner determination problems, both computationally hard single-winner and multi-winner rules can be computed in polynomial time when preferences are structured appropriately.[18][19][20][21] On the other hand, manipulation problem also tend to be easy on these domains, so complexity shields against manipulation are less effective.[19][22] Another computational problem associated with preference restrictions is that of recognizing when a given preference profile belongs to some restricted domain. This task is polynomial time solvable in many cases, including for single-peaked and single-crossing preferences, but can be hard for more general classes.[23][24][25]

Multiwinner elections

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While most traditional voting rules focus on selecting a single winner, many situations require selecting multiple winners. This is the case when a fixed-size parliament or a committee is to be elected, though multiwinner voting rules can also be used to select a set of recommendations or facilities or a shared bundle of items. Work in computational social choice has focused on defining such voting rules, understanding their properties, and studying the complexity of the associated winner determination problems. See multiwinner voting.

See also

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References

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  1. ^ a b Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016-04-25). Handbook of Computational Social Choice. Cambridge University Press. ISBN 9781107060432.
  2. ^ Schulze, Markus (2010-07-11). "A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method". Social Choice and Welfare. 36 (2): 267–303. doi:10.1007/s00355-010-0475-4. S2CID 1927244.
  3. ^ Brill, Markus; Fischer, Felix (2012-01-01). "The Price of Neutrality for the Ranked Pairs Method". Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence. AAAI'12: 1299–1305.
  4. ^ a b Bartholdi III, J.; Tovey, C. A.; Trick, M. A. (1989-04-01). "Voting schemes for which it can be difficult to tell who won the election". Social Choice and Welfare. 6 (2): 157–165. doi:10.1007/BF00303169. S2CID 154114517.
  5. ^ Hemaspaandra, Edith; Spakowski, Holger; Vogel, Jörg (2005-12-16). "The complexity of Kemeny elections". Theoretical Computer Science. 349 (3): 382–391. doi:10.1016/j.tcs.2005.08.031.
  6. ^ Hemaspaandra, Edith; Hemaspaandra, Lane A.; Rothe, Jörg (1997). "Exact Analysis of Dodgson Elections: Lewis Carroll's 1876 Voting System is Complete for Parallel Access to NP". J. ACM. 44 (6): 806–825. arXiv:cs/9907036. doi:10.1145/268999.269002. S2CID 367623.
  7. ^ Rothe, Jörg; Spakowski, Holger; Vogel, Jörg (2003-06-06). "Exact Complexity of the Winner Problem for Young Elections". Theory of Computing Systems. 36 (4): 375–386. arXiv:cs/0112021. doi:10.1007/s00224-002-1093-z. S2CID 3205730.
  8. ^ Caragiannis, Ioannis; Covey, Jason A.; Feldman, Michal; Homan, Christopher M.; Kaklamanis, Christos; Karanikolas, Nikos; Procaccia, Ariel D.; Rosenschein, Jeffrey S. (2012-08-01). "On the approximability of Dodgson and Young elections". Artificial Intelligence. 187: 31–51. doi:10.1016/j.artint.2012.04.004.
  9. ^ Ailon, Nir; Charikar, Moses; Newman, Alantha (2008-11-01). "Aggregating Inconsistent Information: Ranking and Clustering". J. ACM. 55 (5): 23:1–23:27. doi:10.1145/1411509.1411513. S2CID 5674305.
  10. ^ Betzler, Nadja; Fellows, Michael R.; Guo, Jiong; Niedermeier, Rolf; Rosamond, Frances A. (2008-06-23). "Fixed-Parameter Algorithms for Kemeny Scores". In Fleischer, Rudolf; Xu, Jinhui (eds.). Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science. Vol. 5034. Springer Berlin Heidelberg. pp. 60–71. CiteSeerX 10.1.1.145.9310. doi:10.1007/978-3-540-68880-8_8. ISBN 9783540688655.
  11. ^ Fishburn, P. (1977-11-01). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030.
  12. ^ Laslier, Jean-François (1997). Tournament Solutions and Majority Voting. Springer Verlag.
  13. ^ Moon, John W. (1968-01-01). Topics on tournaments. Holt, Rinehart and Winston.
  14. ^ Black, Duncan (1948-01-01). "On the Rationale of Group Decision-making". Journal of Political Economy. 56 (1): 23–34. doi:10.1086/256633. JSTOR 1825026. S2CID 153953456.
  15. ^ Rothstein, P. (1990-12-01). "Order restricted preferences and majority rule". Social Choice and Welfare. 7 (4): 331–342. doi:10.1007/BF01376281. S2CID 153683957.
  16. ^ Arrow, Kenneth J. (2012-06-26). Social Choice and Individual Values. Yale University Press. ISBN 978-0300186987.
  17. ^ Sen, Amartya; Pattanaik, Prasanta K (1969-08-01). "Necessary and sufficient conditions for rational choice under majority decision". Journal of Economic Theory. 1 (2): 178–202. doi:10.1016/0022-0531(69)90020-9.
  18. ^ Elkind, Edith; Lackner, Martin; Peters, Dominik (2016-07-01). "Preference Restrictions in Computational Social Choice: Recent Progress" (PDF). Proceedings of the 25th International Conference on Artificial Intelligence. IJCAI'16: 4062–4065.
  19. ^ a b Brandt, Felix; Brill, Markus; Hemaspaandra, Edith; Hemaspaandra, Lane (2015-01-01). "Bypassing Combinatorial Protections: Polynomial-Time Algorithms for Single-Peaked Electorates". Journal of Artificial Intelligence Research. 53: 439–496. doi:10.1613/jair.4647. hdl:1802/10425.
  20. ^ N., Betzler; A., Slinko; J., Uhlmann (2013). "On the Computation of Fully Proportional Representation". Journal of Artificial Intelligence Research. 47 (2013): 475–519. arXiv:1402.0580. Bibcode:2014arXiv1402.0580B. doi:10.1613/jair.3896. S2CID 2839179.
  21. ^ Skowron, Piotr; Yu, Lan; Faliszewski, Piotr; Elkind, Edith (2015-03-02). "The complexity of fully proportional representation for single-crossing electorates". Theoretical Computer Science. 569: 43–57. arXiv:1307.1252. doi:10.1016/j.tcs.2014.12.012. S2CID 5348844.
  22. ^ Faliszewski, Piotr; Hemaspaandra, Edith; Hemaspaandra, Lane A.; Rothe, Jörg (2011-02-01). "The shield that never was: Societies with single-peaked preferences are more open to manipulation and control". Information and Computation. 209 (2): 89–107. arXiv:0909.3257. doi:10.1016/j.ic.2010.09.001.
  23. ^ Peters, Dominik (2016-02-25). "Recognising Multidimensional Euclidean Preferences". arXiv:1602.08109 [cs.GT].
  24. ^ Doignon, J. P.; Falmagne, J. C. (1994-03-01). "A Polynomial Time Algorithm for Unidimensional Unfolding Representations" (PDF). Journal of Algorithms. 16 (2): 218–233. doi:10.1006/jagm.1994.1010.
  25. ^ Escoffier, Bruno; Lang, Jérôme; Öztürk, Meltem (2008-01-01). "Single-peaked Consistency and Its Complexity". Proceedings of the 2008 Conference on ECAI 2008: 18th European Conference on Artificial Intelligence: 366–370. ISBN 9781586038915.
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  • The COMSOC website, offering a collection of materials related to computational social choice, such as academic workshops, PhD theses, and a mailing list.
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