Cosine similarity and the Borda rule

Transcription

1 Cosine siilarity and the Borda rule Yoko Kawada Abstract Cosine siilarity is a coonly used siilarity easure in coputer science. We propose a voting rule based on cosine siilarity, naely, the cosine siilarity rule. The proposed voting rule selects a social ranking that axiizes cosine siilarity between the social ranking and a given preference profile. We show that the cosine siilarity rule in fact coincides with the Borda rule. We also discuss an analogous relation between the Borda rule and the Condorcet rule fro viewpoints of distance and siilarity. Keywords: The Borda rule, Cosine siilarity. JEL Codes: D71, D63. The author thanks Toyotaka Sakai for valuable coents and discussions. I also thank Toru Hokari, Noriaki Okaoto, and participants of the Matheatical Econoics seinar at Keio University for helpful coents. Graduate School of Econoics, Keio University, Tokyo, ; ykawada@keio.jp 1

2 1 Introduction Cosine siilarity is a coonly used siilarity easure in coputer science. It has a variety of applications such as data ining, text ining, and inforation retrieval. 1 We apply this siilarity easure to define a new voting rule in social choice theory, naely, the cosine siilarity rule. The rule selects a social ranking that axiizes cosine siilarity between the social ranking and a given preference profile. Our ain finding is that the cosine siilarity rule in fact coincides with the Borda rule. The Borda rule is one of the ost iportant voting rules in social choice theory, which is introduced by Jean-Charles de Borda (1784). It is known that Borda s choice ethod has any desirable properties such as axiization of the average share of votes in pairwise coparison (Black 1976, Coughlin 1979), closest proxiity to unanious agreeent (Farkas and Nitzan 1979; Sen 1977), avoidance of any paradoxes observed in positional rules (Saari 1989), and avoidance of the Condorcet loser (Fishburn and Gehrlein 1976, Okaoto and Sakai 2013). In particular, Young (1974) characterizes Borda s choice ethod by a set of desirable properties: neutrality, consistency, faithfulness and cancellation property. On the other hand, there are a few studies on Borda s ranking ethod. In this paper, we focus on the Borda rule as a ranking ethod. We provide a rationale for the use of the Borda rule fro a new viewpoint. It selects the ranking closest to a given preference profile when easured by cosine siilarity. More forally, for each preference profile, the Borda ranking axiizes the su of cosine siilarities between itself and each voter s preference. Keeny (1959) introduces this type of approach to search for desirable ranking ethods based on siilarity or distance fro a given preference profile. Keeny (1959) defines a etric that easures the distance between two rankings, so called Keeny distance 2. Then he proposes the ranking ethod that selects a ranking iniizing the su of Keeny distances between the social ranking and each voter s preference. Surprisingly, it is the unique ranking ethod that satisfies neutrality, consistency, and the Condorcet criterion (Young and Levenglick 1978). Moreover, Young 1 See, for exaple, Tan, Steinbach and Kuar (2005). 2 The ain part of his analysis is the axioatization of Keeny distance. He characterizes the distance by the three axios of etrics, neutrality, local independence, betweenness, and noralization (Keeny 1959; Keeny and Snell 1962). 2

3 (1988) finds that what Condorcet (1785) had in ind was in fact the axiu likelihood ethod, and he shows that Keeny s rule coincides with the axiu likelihood ethod. In short, Condorcet s rule selects a ranking that iniizes the su of Keeny distances fro voter s preferences. Borda or Condorcet, which is better? is the two century old question in social choice theory 3. In the context, it is natural to check if the Borda ranking also iniizes soe distance or axiize soe siilarity fro voter s preferences. If we know that, we can copare the Borda rule with the Condorcet rule fro the sae viewpoints of distance and siilarity. In fact, our result ensures that the Borda rule selects a ranking that axiizes the su of cosine siilarities fro voter s preferences. This result is analogous to the ain result of Young and Levenglick (1978). This paper is organized as follows. In Section 2 we introduce definitions. In Section 3 we show the equivalence between the Borda rule and the cosine siilarity rule. In Section 4 we restrict the range of ranking ethods to linear orderings and show another equivalence theore. Section 5 concludes our discussion. 2 Definitions Let I = {1,2,...,n} be the finite set of voters and A = {a 1,a 2,...,a } the finite set of alternatives. Let R A A be the set of coplete and transitive binary relations on A and P A A the set of coplete, transitive and anti-syetric binary relations on A. 4 Each voter i I has a preference i P on A. A preference profile is a list of preferences ( i ) P n. A ranking ethod is a function F that aps each preference profile P n to a social ranking F( ) R. For each voter i I, each alternative a A and each preference i P, let r a ( i ) {a A : a i a } 3 See, for exaple, Saari (2006). 4 A binary relation is coplete if for any a,b A, a b or b a. It is transitive if for any a,b,c A, [a b and b c] iplies a c. It is anti-syetric if for any a,b A, [a b and b a] iplies a = b. 3

4 be the rank of a in i. For each i P, define r( i ) (r a1 ( i ),r a2 ( i ),...,r a ( i )) N. We call r( i ) the rank expression of i. A preference i and its rank expression r( i ) have the sae inforation. The Borda score of a in is given by S(a, ) r a ( i ). The following ranking ethod is proposed by Borda (1784). Definition 1 (Borda ranking ethod). The Borda ranking ethod is the ranking ethod F B such that for each P n and each a,b A, a F B ( ) b if and only if S(a, ) S(b, ). Next, we define a siilarity easure that plays a key role in our analysis. Definition 2 (Cosine siilarity). For each vector x,y R ++, the cosine siilarity between x and y is C(x,y) x y x y, where x is the Euclidean nor of x, and x y denotes the inner product between x and y. The cosine siilarity is a coonly used siilarity easure between two vectors. By definition, C(x,y) = cosθ x,y, where θ x,y is the angle between x and y. Therefore, C(x,x) = cos0 = 1 and C(x,y) 1 for all x,y R ++. If two vectors x and y are siilar, the value of C(x,y) is close to 1. For exaple, consider the following four vectors x,y,z and w x = 2 3, y = 2 3, z = 1 1, w =

5 We have C(x,y) = C(x,z) = C(x,w) = , , = 1. 2 Note that cosine siilarity easures the siilarity of orientations of two vectors and is independent fro their lengths. Using cosine siilarity and rank expressions of preferences, we can calculate the siilarity between two preferences. Definition 3 (Cosine siilarity between two preferences). For each i and j P, the cosine siilarity between i and j is C(r( i ),r( j )). A social ranking can be a weak ordering that cannot be associated with rank expression r, so we introduce the following notation. For each vector x R ++, we call R(x) the ranking expressed by x if for each k,l {1,2,...,}, a k R(x) a l x k x l. The following ranking ethod is proposed in this paper. Definition 4 (Cosine siilarity ranking ethod). The cosine siilarity ranking ethod is the ranking ethod F C such that for each P n, F C ( ) = R(x), where x axiizes C(x,r( i )). This rule selects a social ranking that axiizes the su of cosine siilarities between the social ranking and each voter s ranking 5. 5 In the definition, we use vector expression x. That point is different fro Young and Levenglick (1978) but it is not essential for our results. In Section 4, we consider only linear orderings sae as Young and Levenglick (1978) and define another cosine siilarity ranking ethod without using vector expression x. 5

6 3 Equivalence theore We are now in a position to state our ain result. We show that the cosine siilarity ranking coincides with the Borda ranking. Theore 1. For each P n, F C ( ) = F B ( ). In other words, for each preference profile, the Borda rule selects the social ranking closest to the preference profile when easured by cosine siilarity. As before entioned, the Keeny rule, which selects the social ranking closest to a given preference profile when easured by Keeny distance, coincides with the Condorcet rule. Figure 1 illustrates the relations between the Keeny rule, the Condorcet rule, the cosine siilarity rule, and the Borda rule. While Condorcet s rule iniizes Keeny distance, Borda s rule axiizes cosine siilarity. Figure 1: Borda s rule and Condorcet s rule 6

7 Proof. Take any P n. We show that if x N axiizes C(x,r( i )), then there exists α R ++ such that x k = α S(a k, ) for all k = 1,2,...,. Assue that x axiizes C(x,r( i )). By the definition of cosine siilarity, C(x,r( i )) = = x r( i ) x r( i ) j=1 x jr a j ( i ) x r( i ). Let r r( i ) = for all i I. By the coutative property of addition, Since r a j ( i ) = S(a j, ), 1 r j=1 x jr a j ( i ) x r( i ) = 1 r x j r a j ( i ) = 1 j=1 x r By the definition of cosine siilarity, = 1 r j=1 x j S(a j, ) j=1 x = (S(a j, )) j=1 r = (S(a j, )) j=1 r j=1 x jr a j ( i ) x x j r a j ( i ). x j=1 x js(a j, ) x (S(a j, )) j=1 x (S(a j, )) j=1 x (S(a j, )) j=1. x (S(a j, )) j=1 x (S(a j, )) j=1 = C(x,S(a j, )) j=1 ) 1. 7

8 Therefore, C(x,r( i )) (S(a j, )) j=1. (1) r Since C((S(a j, )) j=1,(s(a j, )) j=1 ) = 1, by scale invariance property of cosine siilarity, 6 C(x,r( i )) = (S(a j, )) j=1 x = α (S(a j, )) j=1 r, where α is a positive real nuber. Thus, if x axiizes the left hand side of (1), then there exists α > 0 such that x k = α S(a k, ) for all k = 1,2,...,. Assue that x axiizes the left hand side of (1). By the definition of R(x), x k x j S(a k, ) S(a j, ) a k R(x) a j. (2) Therefore, the ranking R(x) is uniquely deterined by, which iplies that F C ( ) = R(x) is uniquely deterined by. In addition, by (2) and the definition of the Borda ranking, F B ( ) = R(x). Therefore, F C ( ) = F B ( ). 4 Linear orderings and another equivalence Even though voter s preferences are linear orderings, there ay be cases in which two alternatives get the sae scores. Therefore, it is natural to assue that the range of ranking ethods is the set of weak orderings. Indeed, in the previous section, we establish the equivalence between the weak ordering Borda rankings and the weak ordering cosine siilarity rankings. However, Young and Levenglick (1978) show the equivalence between the Keeny ranking and the Condorcet rankings on the restricted range of linear orderings. By considering only linear orderings, we can clarify analogous relation between Young s and Levenglick s (1978) analysis and our analysis In this section, we show the equivalence between the linear ordering Borda rankings and 6 Cosine siilarity is scale invariant, that is, for each x,y R ++ and for each α > 0, C(α x,y) = C(x,y). Other properties of cosine siilarity are discussed in Section 5. 8

9 the linear ordering cosine siilarity rankings. Definition 5 (Linear Borda ranking ethod). The linear Borda ranking ethod is a correspondence F LB : P n P such that for each P n, F LB ( ) = {P P : a j P a k = S(a j, ) S(a k, ) a j,a k A}. (3) For exaple, suppose S(a, ) = 11,S(b, ) = 14,S(c, ) = 11, and S(d, ) = 4. Then the set of linear Borda rankings is F LB ( ) = {P,P }, where b P c P a P d and b P a P c P d. Definition 6 (Linear cosine siilarity ranking ethod). The linear cosine siilarity ranking ethod is a correspondence F LC : P n P such that for each P n, F LC ( ) = {P P : P axiizes C(r(P),r( i ))}. The linear cosine siilarity rule can be defined without using vector expressions, so its definition is sipler than that of the weak ordering cosine siilarity rule. Theore 2. For each P n, F LB ( ) = F LC ( ). Before giving the proof, we offer a lea. Lea 1. For any x 1 x 2 x, y 1 y 2 y and any perutation σ on {1,2,...,}, x k y k x k y σ(k). Proof. Consider any x,y R with x 1 x 2 x and y 1 y 2 y. Consider any perutation σ on {1,2,...,}. We prove this lea by atheatical induction. 9

10 First, we shall show that if σ(), then there exists a perutation π on {1,2,...,} such that x k y π(k) x k y σ(k). (4) Assue that σ(). Then, there exist i, j < such that σ() = i and σ( j) =. Since x x j and y y i, (x x j )(y y i ) 0. Therefore, x y + x j y i x y i + x j y. (5) Let π be the perutation on {1,2,...,} such that By (5), that is, we have (4). if k =, π(k) = i if k = j, σ(k) if k / {, j}. x k y π(k) = x k y σ(k) + x y + x j y i k, j k, j x k y σ(k) + x y i + x j y x k y σ(k), Next, we show that if σ() = and σ( 1) 1, then there exists a perutation π such that x k y π(k) x k y σ(k). (6) Assue that σ() = and σ( 1) 1. Then, there exist i, j < 1 such that σ( 1) = i and σ( j) = 1. Since x 1 x j and y 1 y i, we have (x 1 x j )(y 1 y i ) 0. Therefore, x 1 y 1 + x j y i x 1 y i + x j y 1. (7) 10

11 Let π be such that 1 if k = 1, π(k) = i if k = j, σ(k) if k / { 1, j}. By (7), that is, we have (6). x k y π(k) = x k y σ(k) + x 1 y 1 + x j y i k 1, j k 1, j x k y σ(k), x k y σ(k) + x 1 y i + x j y 1 In general, take any l {1,2,...,} and assue that σ() =,σ( 1) = 1,...,σ(l + 1) = l + 1, and σ(l) l. Let π be identical to σ except that π(l) = l and π(σ 1 (l)) = σ(l). Then, Therefore, x ky k x ky σ(k). x k y π(k) x k y σ(k). Proof of Theore 2. Consider any P n. Let us show F LC ( ) F LB ( ). Take any P F LC ( ). Let s N be such that s j = S(a j, ) for each a j A. By an arguent siilar to the proof of Theore 1, C(r(P),r( i )) = s r(p) s r r(p) s Since P is a linear order, r(p) = = r 1(P)s 1 + r 2 (P)s r (P)s. r r(p) 1 6 ( + 1)(2 + 1), which is independent of the choice of P. Thus, P axiizes C(r(P),r( i )) if and only if P axiizes 11

12 r 1 (P)s 1 + r 2 (P)s r (P)s. Since P F LC ( ), P axiizes C(r(P),r( i )). By Lea 1, r j (P) > r k (P) = s j s k (8) for each j,k = 1,2,...,. Since r j (P) > r k (P) iplies a j P a k, we have (3). Therefore P F LB ( ), which in turn iplies F LC ( ) F LB ( ). Proving F LB ( ) F LC ( ) is siilar because if P F LB ( ), then P satisfies (8). It iplies that P axiizes C(r(P),r( i )) by Lea 1. 5 Conclusion We have proposed a new voting rule, the cosine siilarity rule. We have shown that the cosine siilarity rule coincides with the Borda rule. Our analysis provides a rationale for the use of the Borda rule fro the viewpoint of siilarity. Because the Borda rule selects a social ranking that is closest to voters preferences, we can say that it is suitable for deocratic decision akings. Moreover, we have found an analogous relation between Borda s rule and Condorcet s rule. While the consistently-extended Condorcet rule coincides with the Keeny rule, the Borda rule coincides with the cosine siilarity rule. By our analysis, we can copare the fro the sae perspectives of distance and siilarity. Finally, we exaine properties of the cosine siilarity operator C. Exaining properties of it is helpful to investigate properties of the Borda rule. Syetry For all x,y R ++, C(x,y) = C(y,x). Noralization For all x,y R ++, 0 C(x,y) 1 and C(x,x) = 1. Scale invariance For all x,y R ++ and each k > 0, C(kx,y) = C(x,y). Neutrality For all x,y R ++ and any perutation π on {1,2,...,}, C(πx,πy) = C(x,y), where πx = (x π( j) ) j=1. These properties are not sufficient to characterize the cosine siilarity operator. Characterizations of cosine siilarity easure reains future research. 12

13 References Black, D. (1976) Partial justification of the Borda count, Public Choice, Vol. 28, pp Borda, J.-C. de (1784) Méoire sur les élections au scrutin, Histoire de l Acadéie Royal des Sciences 1981, pp Condorcet, M. de (1785) Essai sur l application de l analyse à la probabilité des décisions rendues à la pluralité des voix. Reprinted by Chelfea Publishing Copeach (1972). Coughlin, P. (1979) A direct characterization of Black s first Borda count, Econoics Letters, Vol. 4, pp Farkas, D. and Nitzan, S. (1979) The Borda Rule and Pareto Stability: A Coent, Econoetrica, Vol. 47, pp Fishburn, P. C. and Gehrlein, W. V. (1976) Borda s rule, positional voting, and Condorcet s siple ajority principle, Public Choice, Vol. 28, pp Keeny, J. G. (1959) Matheatics without nubers, Deadalus, Vol. 88, pp Keeny, J. G., and Snell, J. L. Matheatical odels in the social sciences. New York: Ginn. Okaoto, N. and Sakai, T. (2013) The Borda rule and the pairwise-ajority loser revisited, unpublished anuscript, Keio University. Saari, D. G. (1989) A dictionary for voting paradoxes, Journal of Econoic Theory, Vol. 48, pp Saari, D. G. (2006) Which is better: the Condorcet and Borda winner?, Social Choice and Welfare, Vol. 26, pp Sen, A. K. (1977) Social choice theory: A re-exaination, Econoetrica, Vol. 45, pp

14 Young, H. P. (1974) An axioatization of Borda s rule, Journal of Econoic Theory, Vol. 9, pp Young, H. P. (1988) Condorcet s theory of voting, Aerican Political Science Review, Vol. 82, pp Young, H. P. and Levenglick, A. (1978) A consistent extension of Condorcet s election principle, SIAM Journal on Applied Matheatics, Vol. 35, pp