Abstract
Cosine similarity is a commonly used similarity measure in computer science. We apply this similarity measure to define a voting rule, namely, the cosine similarity rule. This rule selects a social ranking that maximizes cosine similarity between the social ranking and a given preference profile. Our main finding is that the cosine similarity rule in fact coincides with the Borda rule.
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Notes
Singhal (2001) briefly explains how to apply cosine similarity to measure the similarity between two text documents. Cosine similarity is also used as a basis of search engines (see, e.g., Bayardo et al. 2007). An application for face verification system is a recent interesting example of applications of cosine similarity (Nguyen and Bai 2010).
Saari and Merlin (2000) conduct a fairly complete analysis of the Kemeny ranking rule. Particularly, they reveal a deep relationship between the Borda ranking and the Kemeny ranking.
A binary relation \(\succsim \) is complete if for any \(a,b \in A\), \(a \succsim b\) or \(b \succsim a\). It is transitive if for any \(a,b,c \in A\), \([a \succsim b\) and \(b \succsim c]\) implies \(a \succsim c\). It is anti-symmetric if for any \(a,b \in A\), \([a \succsim b\) and \(b \succsim a]\) implies \(a=b\).
The cosine similarity rule is a well-defined (single-valued) function. Indeed, a vector x is not uniquely determined by \(\succsim \) but we will show that R(x) is uniquely determined by \(\succsim \) in the proof of Theorem 1.
In the definition of the cosine similarity rule, we use vector expression x. This point is different from that of Young and Levenglick (1978), but it is not essential for our results. In Section 4.1, we define a cosine similarity rule without using vector expression and consider only linear orderings, as Young and Levenglick (1978) do.
This is because \(\cos (x,y)=1\) if and only if the angle between x and y is 0, that is, their directions are identical.
The author thanks an anonymous referee for kindly informing him this embedding technique and the related literature.
It is worth noting that there are a small related literature discussing “generalized scoring rules” that include Borda, Condorcet, approval voting, and other standard voting rules. Generalized scoring rules score rankings to find a “desirable” social ranking. For example, see Conitzer et al. (2009) and Zwicker (2008).
References
Bayardo RJ, Ma Y, Srikant R (2007) Scaling up all pairs similarity search. In: Proceedings of the 16th international conference on World Wide Web, pp 131–140
Black D (1976) Partial justification of the Borda count. Publ Choice 28:1–15
Borda J-C (1784) de Mémoire sur les élections au scrutin. Hist de l’Acad R des Sci 1981:657–664
Condorcet M (1785) de Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, Reprinted by Chelfea Publishing Compeach (1972)
Conitzer V, Rognlie M, Xia L (2009) Preference functions that score rankings and maximum likelihood estimation. In: Proceedings of the twenty-first international joint conference on artificial intelligence (IJCAI-09), pp 109–115
Coughlin P (1979) A direct characterization of Black’s first Borda count. Econ Lett 4:131–133
Farkas D, Nitzan S (1979) The Borda rule and Pareto stability: a comment. Econometrica 47:1305–1306
Fishburn PC, Gehrlein WV (1976) Borda’s rule, positional voting, and Condorcet’s simple majority principle. Publ Choice 28:79–88
Hardy GH, Littlewood JE, Pólya G (1952) Inequalities, 2nd edn. Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK
Kemeny JG (1959) Mathematics without numbers. Deadalus 88:577–591
Nguyen HV, Bai L (2010) Cosine similarity metric learning for face verification. In: Asian conference on computer vision, pp 709–720
Okamoto N, Sakai T (2013) The Borda rule and the pairwise-majority loser revisited. Keio University, (unpublished manuscript)
Saari DG (1989) A dictionary for voting paradoxes. J Econ Theory 48:443–475
Saari DG, Merlin VR (2000) A geometric examination of Kemeny’s rule. Soc Choice Welf 17:403–438
Saari DG (2006) Which is better: the Condorcet and Borda winner? Soc Choice Welf 26:107–129
Sen AK (1977) Social choice theory: a re-examination. Econometrica 45:53–89
Singhal A (2001) Modern information retrieval: a brief overview. Bull IEEE Comput Soc Tech Comm Data Eng 24:35–43
Young HP (1974) An axiomatization of Borda’s rule. J Econ Theory 9:43–52
Young HP (1988) Condorcet’s theory of voting. Am Polit Sci Rev 82:1231–1244
Young HP, Levenglick A (1978) A consistent extension of Condorcet’s election principle. SIAM J Appl Math 35:285–300
Zwicker WS (2008) Consistency without neutrality in voting rules: when is a vote an average? Math Comput Model 48:1357–1373
Zwicker WS (2014) Universal and symmetric scoring rules for binary relations. In: Fifth international workshop on computational social choice, Pittsburgh, Pennsylvania. http://www.cs.cmu.edu/~arielpro/comsoc-14/papers/Zwicker2014.pdf. Accessed 20 July 2017
Acknowledgements
The author thanks Toyotaka Sakai for valuable comments and discussions. I also thank Takako Fujiwara-Greve, Toru Hokari, Noriaki Okamoto, Satoshi Nakada, and participants of the Annual Meeting of Japanese Economic Association in October 2015 at Sophia University, the 21st Decentralized Conference at Keio University, and the Summer Meeting on Game Theory 2015 in Shizuoka for insightful comments. I especially thank two anonymous referees for helpful comments. All remaining mistakes are my own. This research is financially supported by Keio Economic Association.
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Kawada, Y. Cosine similarity and the Borda rule. Soc Choice Welf 51, 1–11 (2018). https://doi.org/10.1007/s00355-017-1104-2
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DOI: https://doi.org/10.1007/s00355-017-1104-2