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The Dodgson ranking and the Borda count: a binary comparison
Mathematical Social Sciences 48 (2004) 103 – 108 www.elsevier.com/locate/econbase
The Dodgson ranking and the Borda count: a binary comparison Christian Klamler * Institute of Public Economics, University of Graz, Universitaetsstr. 15, Graz A-8010, Austria Received 1 April 2003; received in revised form 1 October 2003; accepted 1 November 2003
Abstract This paper provides a binary comparison of two preference aggregation rules, the Borda rule and Dodgson’s rule. Both of these rules guarantee a transitive ranking of the alternatives for every list of individual preferences and therefore avoid the problem of voting cycles. It will be shown that for certain lists of individual preferences the rankings derived from the Borda rule and Dodgson’s rule are antagonistic. D 2004 Elsevier B.V. All rights reserved. Keywords: Voting paradox; Voting rules; Borda rule; Dodgson rule; Distance functions JEL classification: D70
1. Introduction The purpose of this paper is to provide a binary comparison of two preference aggregation rules, Borda’s rule and Dodgson’s rule. Both of these rules guarantee a transitive ranking of the alternatives for every list of individual preferences. Hence, they avoid the problems of voting cycles which affect the simple majority rule and are therefore discussed in the literature as simple majority rule extensions (Fishburn, 1977). For the aggregation of individual preferences over a set of m alternatives, Borda (1781) suggested to assign m 1 points to the top alternative in an individual’s ranking, m 2 points to its second ranked alternative down to 0 points to an individual’s bottom ranked alternative.
*Tel.: +43-316-380-3465; fax: +43-316-380-9530. E-mail address: [email protected] (C. Klamler). 0165-4896/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2003.11.003
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C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
The alternatives are then ranked according to the sum of assigned points. Dodgson (1876) suggested to choose the alternative that is closest from being a majority winner, i.e. which needs the fewest pairwise switches in the individual preferences to have a strict majority over every other alternative. A binary extension of Dodgson’s idea essentially ranks the alternatives according to the number of switches necessary to make them majority winners. The relevance of investigating voting rules, which not only choose alternatives but rank them, can be seen in the existence of various voting situations that ask for rankings in addition to choices. In sports competitions, ranking candidates or teams is common. Recruitment committees rank candidates to avoid having to call for a new meeting in case higher ranked candidates drop out later. In general, one can see the necessity in ranking alternatives for cases where there is uncertainty about whether the best alternative will be available after the decision has been taken. In this paper it will be shown that for certain lists of individual preferences over a given number of alternatives, the rankings derived from Borda’s rule and Dodgson’s rule will be antagonistic. The paper follows Ratliff’s (2001, 2002) comparisons of the Dodgson rule with other binary rules. In contrast to this binary approach, previous comparisons of aggregation rules either use an axiomatic framework (Fishburn, 1977) or a choice functional framework (Laffond et al., 1995). The paper is structured as follows. Section 2 introduces notation and the general framework. Section 3 defines the two voting rules under investigation and states the main result. Section 4 contains the proof of the main theorem.
2. Formal framework Let X denote a finite set of alternatives where throughout the paper we assume AXA = m>4, and I denote a finite set of n individuals. A preference cpX 2 is a binary relation on X, where, for all x, yaX, xcy denotes the weak preference of x over y. The symmetric and asymmetric part of c will be written as f and , respectively. For any subset A of X, the restriction of c to A, denoted cAA is the binary relation c\A2. Let B be the set of all complete binary relations on X, WoB the set of all weak orders (complete and transitive binary relations) on X and LoW the set of all linear orders (complete, transitive and asymmetric binary relations) on X. Assuming only linear orders for individual preferences, preference profiles will be written as P = ( Pi)iaIaLn where PiaL is individual i’s preference on X. Given a preference profile PaLn, for all xj, xkaX, the majority margin of xj over xk in P P is denoted by aj,k = A{iaI:xjPi xk}A A{iaI:xk Pi xj}A. Finally, use will be made of concepts measuring the distances between binary relations and preference profiles, respectively. Let R+ be the set of all non-negative real numbers. The Kemeny (1959) distance function on B uses the symmetric difference between preference relations. It will be defined as: d:B B ! R+ such that for all c,cVa B,d(c,cV) = A(c cV)[(cV c)A/2.1 Distance on the set of preference profiles will 1
The division by 2 has been added for the convenience of being able to talk about distance values and numbers of pairwise switches interchangeably and hence the definition slightly differs from Kemeny (1959).
C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
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n L n ! R+ such that for all P, PVa be measured by Pnthe distance function: d:L n L ; dðP; PVÞ ¼ i¼1 dðPi ; Pi VÞ.
3. Dodgson’s rule and Borda’s rule To be able to provide a binary extension of Dodgson’s idea2 some prerequisites are necessary. Following Laslier (1997), we call, for any caB, the subset YoX retentive for c if Y is non-empty and if there exists no xaX \Y and yaY such that xcy. The subset Y is further called minimal retentive for c if Y is retentive for c and contains no strict subset ZoY retentive for c. Such a minimal retentive subset will be called the top set of c and denoted by TS (c). For all xa X, let U(x)oLn denote the non-empty set of all preference profiles for which x is the strict majority winner. Then, for any preference profile PaLn , the Dodgson score of xa X is given by sðxÞ ¼ min dðP; PVÞ. PV aU ðxÞ
n
of X such that, Definition 3.1: Given PaL , let { YiAi = 1,. . .,k}be the partition for the simple majority relation cMaB, Y1 = TS (cM) and Yi ¼ TS cM AX [ Yj for i = 2, j
. . .,k. Then, cDaW is the Dodgson ranking if and only if for all x,yaX, xcDyZ[xaYj Z ygYi, i < j and x, yaYj Z s(x) V s( y)]. The Borda ranking will be derived using a method suggested in Young (1974). For P any xja X, its Borda count is given by bðxj Þ ¼ kpj aj;k . Definition 3.2: Given PaLn,cBaW is the Borda ranking if and only if for all x,yaX, xcB yZb(x) z b( y). For the rest of the paper we will call two rankings c,cVaW antagonistic if and only if for all x,yaX, xcyZycVx and for some x,yaX, xyZy Vx.3 The following theorem states that for some preference profiles, the Borda ranking and the Dodgson ranking will be antagonistic. Theorem 3.3: If there are more than four alternatives, then there exist preference profiles such that the Borda ranking and the Dodgson ranking are antagonistic.
4. Proof For the actual proof of Theorem 3.3 we will first introduce a result (Lemma 4.2) about the existence of preference profiles that satisfy certain conditions. The following Definition 4.1 introduces a property of preference profiles that requires that for some 2
For a further discussion on the reasonableness of the Dodgson ranking see Klamler (2002), for a general discussion on the idea of distance minimization see Baigent (1983). 3 There can be different (usually stronger) forms of antagonism if one uses choice functions. I am grateful to an anonymous referee for pointing this out to me.
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C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
Table 1 Dodgson scores and Borda counts x1
ðm 2Þ 2r þ 1 (m-2)c + q r + 2
s( ) b( )
xaS m2 q
c 2þ1 þ2þ1 2 lc+2
xm m2 2 ðl þ 4Þ r m2 2 ð2l
þ q2 þ 1 þ 4Þ q 2
pair xj, xkaX, all individuals that rank xj above kk, rank xj so high above xk that each individual needs a certain minimum number of pairwise switches to invert its ranking of this pair to xk above xj. Definition 4.1. For all xj,xka X, let Lk, j={caL:xkxj}. A preference profile PaLn, satisfies j– k-agreement if and only if for the pair xj, xka X, biaI:xjPixk Z d( Pi,c) z AXA 2, for all caLk,j. Let Z be the set of all integers. Given preference profile PaLn , the vector ofpairwise m
majority margins will be denoted by wP ¼ ðaP1;2 ; . . . ; aPj;k ; . . . ; aPm1;m ÞaZ 2 where j,ka{1,2,. . .,m},j < k. 4 The following lemma extends Debord’s (1987) theorem , which states that for any m
maZ 2 with all entries having the same parity, there exists a preference profile PaLn such that wP = m. Lemma m 4.2. (Klamler, 2002): If there are more than three alternatives, then for any maZ 2 , with all entries having the same parity, there exists a preference profile PaLn and a pair xj,xkaX such that P satisfies j –k-agreement and wP = m. The proof will now be based on two lemmas showing that the theorem is true for even and odd numbers of alternatives. Lemma 4.3: If AXA z 5 and even, then there exist preference profiles such that the Dodgson ranking and the Borda ranking are antagonistic. Proof: Consider X={x1,x2,. . .,xm}, where m is an even number larger than 5. Let a1,2 = c + q + 2, a1, j = c for all ja{3,4,. . .,m 1} and a1,m = r. Furthermore, let a2,m = l + 4 and am 1,m = l + q + 2. For all ja{3,4,. . .,m 2}, let aj,m = l whenever j is odd, and aj,m = l + 4 whenever j is even. Let a3,4 = a5,6 = a7,8 = . . . = am 3,m 2 = 2q + 2. Let all other pairwise margins be aj,k = q whenever j < k and k has the opposite parity of j, and aj,k = q whenever j < k and k and j have the same parity. Those margins imply a strict majority cycle x1x2x3. . .xmx1 and by Lemma 4.2 we know that a preference profile with exactly the above margins exists and satisfies m1-agreement. The Dodgson scores and the Borda counts for the alternatives are given in Table 1, where S = X \{x1,xm}. 4
Saari (1995) and Ratliff (2001) prove equivalent statements and provide extensions.
C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
107
Table 2 Dodgson scores and Borda counts x1
s( ) b( )
ðm 2Þ 2r þ 1 (m 2)c r
xaS m3 q 2
lc
c 2þ1 þ2þ1
xm
ðm 2Þ 2l þ 1 r (m 2)l
To establish the desired result it is now sufficient to solve the following system of inequalities: q c I ðm 2Þ 2r þ 1 > m2 2 2þ1 þ2þ1 q q c m2 II m2 2 2 þ 1 þ 2 þ 1 > 2 ðl þ 4Þ þ 2 þ 1 III (m2)c + qr + 2>lc + 2 IV l c þ 2 > r m2 2 ð2l þ 4Þ q 2 As one solution to this system of inequalities for m>5, m even, is l = 2, r = 8, q = 10 and c = 12, this proves the lemma. Lemma 4.4: If AXA z 5 and odd, then there exist preference profiles such that the Dodgson ranking and the Borda ranking are antagonistic. Proof: Consider X={x1,x2,. . .,xm}, with at least five alternatives. Let a1,m = r and, for all ja{2,3,. . .,m 1}, let a1, j = c and aj,m = l. Furthermore, for all ja{2,3,. . .,m 1} let aj, k = q whenever j < k and k has the opposite parity of j, and aj,k = q whenever j < k and k and j have the same parity. Those margins imply a strict majority cycle x1x2x3. . . xmx1 and by Lemma 4.2 we know that a preference profile with exactly those margins exists and satisfies m1-agreement. The Dodgson scores and Borda counts for the alternatives are given in Table 2, where S = X \{x1,xm}. 5 To establish the desired result it is now sufficient to solve the following system of inequalities: q c I ðm 2Þ 2r þ 1 > m3 2 2þ1 þ l 2 þ 1 q c II m3 2 2 þ 1 þ 2 þ 1 > ðm 2Þ 2 þ 1 III (m2)cr>lc IV lc>r(m2)l As one solution to this system of inequalities for m z 5, m odd, is c = l = 2, r = 4 and q = 10, this proves the lemma. Proof of Theorem 3.3: Lemma 4.3 proves all cases for m even, and Lemma 4.4 proves all cases for m odd. Hence the theorem is true. 5
Acknowledgements I am very grateful to Daniel Eckert, Nick Baigent and an anonymous referee for valuable comments.
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C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
References Baigent, N., 1983. Some reservation on strictly majoritarian social choice. Discussion Paper Series. University of Essex. Borda, J.C., 1781. Memoire sur les Elections au Scrutin. Memoires de l’Academie Royale des Sciences. Debord, B., 1987. Caracte´risation des matrices de pre´fe´rences nettes et me´thodes d’agre´gation associe´es. Mathe´matiques et Sciences Humaines 97, 5 – 17. Dodgson, C., 1876. A method of taking votes on more than two issues. Black, 1958. The Theory of Committees and Elections. Cambridge Univ. Press, London. Fishburn, P., 1977. Condorcet social choice functions. SIAM Journal on Applied Mathematics 33, 469 – 489. Kemeny, J., 1959. Mathematics without numbers. Daedalus 88, 577 – 591. Klamler, C., 2002. The Dodgson ranking and its relation to Kemeny’s method and Slater’s rule. Forthcoming: Social Choice and Welfare. Laffond, G., et al., 1995. Condorcet choice correspondences: a set-theoretical comparison. Mathematical Social Sciences 30, 23 – 35. Laslier, J.-F., 1997. Tournament Solutions and Majority Voting. Springer-Verlag, Berlin. Ratliff, T.C., 2001. A comparison of Dodgson’s method and Kemeny’s rule. Social Choice and Welfare 18, 79 – 90. Ratliff, T.C., 2002. A comparison of Dodgson’s method and the Borda count. Economic Theory 20, 357 – 372. Saari, D.G., 1995. Basic Geometry of Voting. Springer-Verlag, Berlin. Young, H.P., 1974. An axiomatisation of Borda’s rule. Journal of Economic Theory 9, 43 – 52.
The Dodgson ranking and the Borda count: a binary comparison Christian Klamler * Institute of Public Economics, University of Graz, Universitaetsstr. 15, Graz A-8010, Austria Received 1 April 2003; received in revised form 1 October 2003; accepted 1 November 2003
Abstract This paper provides a binary comparison of two preference aggregation rules, the Borda rule and Dodgson’s rule. Both of these rules guarantee a transitive ranking of the alternatives for every list of individual preferences and therefore avoid the problem of voting cycles. It will be shown that for certain lists of individual preferences the rankings derived from the Borda rule and Dodgson’s rule are antagonistic. D 2004 Elsevier B.V. All rights reserved. Keywords: Voting paradox; Voting rules; Borda rule; Dodgson rule; Distance functions JEL classification: D70
1. Introduction The purpose of this paper is to provide a binary comparison of two preference aggregation rules, Borda’s rule and Dodgson’s rule. Both of these rules guarantee a transitive ranking of the alternatives for every list of individual preferences. Hence, they avoid the problems of voting cycles which affect the simple majority rule and are therefore discussed in the literature as simple majority rule extensions (Fishburn, 1977). For the aggregation of individual preferences over a set of m alternatives, Borda (1781) suggested to assign m 1 points to the top alternative in an individual’s ranking, m 2 points to its second ranked alternative down to 0 points to an individual’s bottom ranked alternative.
*Tel.: +43-316-380-3465; fax: +43-316-380-9530. E-mail address: [email protected] (C. Klamler). 0165-4896/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2003.11.003
104
C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
The alternatives are then ranked according to the sum of assigned points. Dodgson (1876) suggested to choose the alternative that is closest from being a majority winner, i.e. which needs the fewest pairwise switches in the individual preferences to have a strict majority over every other alternative. A binary extension of Dodgson’s idea essentially ranks the alternatives according to the number of switches necessary to make them majority winners. The relevance of investigating voting rules, which not only choose alternatives but rank them, can be seen in the existence of various voting situations that ask for rankings in addition to choices. In sports competitions, ranking candidates or teams is common. Recruitment committees rank candidates to avoid having to call for a new meeting in case higher ranked candidates drop out later. In general, one can see the necessity in ranking alternatives for cases where there is uncertainty about whether the best alternative will be available after the decision has been taken. In this paper it will be shown that for certain lists of individual preferences over a given number of alternatives, the rankings derived from Borda’s rule and Dodgson’s rule will be antagonistic. The paper follows Ratliff’s (2001, 2002) comparisons of the Dodgson rule with other binary rules. In contrast to this binary approach, previous comparisons of aggregation rules either use an axiomatic framework (Fishburn, 1977) or a choice functional framework (Laffond et al., 1995). The paper is structured as follows. Section 2 introduces notation and the general framework. Section 3 defines the two voting rules under investigation and states the main result. Section 4 contains the proof of the main theorem.
2. Formal framework Let X denote a finite set of alternatives where throughout the paper we assume AXA = m>4, and I denote a finite set of n individuals. A preference cpX 2 is a binary relation on X, where, for all x, yaX, xcy denotes the weak preference of x over y. The symmetric and asymmetric part of c will be written as f and , respectively. For any subset A of X, the restriction of c to A, denoted cAA is the binary relation c\A2. Let B be the set of all complete binary relations on X, WoB the set of all weak orders (complete and transitive binary relations) on X and LoW the set of all linear orders (complete, transitive and asymmetric binary relations) on X. Assuming only linear orders for individual preferences, preference profiles will be written as P = ( Pi)iaIaLn where PiaL is individual i’s preference on X. Given a preference profile PaLn, for all xj, xkaX, the majority margin of xj over xk in P P is denoted by aj,k = A{iaI:xjPi xk}A A{iaI:xk Pi xj}A. Finally, use will be made of concepts measuring the distances between binary relations and preference profiles, respectively. Let R+ be the set of all non-negative real numbers. The Kemeny (1959) distance function on B uses the symmetric difference between preference relations. It will be defined as: d:B B ! R+ such that for all c,cVa B,d(c,cV) = A(c cV)[(cV c)A/2.1 Distance on the set of preference profiles will 1
The division by 2 has been added for the convenience of being able to talk about distance values and numbers of pairwise switches interchangeably and hence the definition slightly differs from Kemeny (1959).
C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
105
n L n ! R+ such that for all P, PVa be measured by Pnthe distance function: d:L n L ; dðP; PVÞ ¼ i¼1 dðPi ; Pi VÞ.
3. Dodgson’s rule and Borda’s rule To be able to provide a binary extension of Dodgson’s idea2 some prerequisites are necessary. Following Laslier (1997), we call, for any caB, the subset YoX retentive for c if Y is non-empty and if there exists no xaX \Y and yaY such that xcy. The subset Y is further called minimal retentive for c if Y is retentive for c and contains no strict subset ZoY retentive for c. Such a minimal retentive subset will be called the top set of c and denoted by TS (c). For all xa X, let U(x)oLn denote the non-empty set of all preference profiles for which x is the strict majority winner. Then, for any preference profile PaLn , the Dodgson score of xa X is given by sðxÞ ¼ min dðP; PVÞ. PV aU ðxÞ
n
of X such that, Definition 3.1: Given PaL , let { YiAi = 1,. . .,k}be the partition for the simple majority relation cMaB, Y1 = TS (cM) and Yi ¼ TS cM AX [ Yj for i = 2, j
. . .,k. Then, cDaW is the Dodgson ranking if and only if for all x,yaX, xcDyZ[xaYj Z ygYi, i < j and x, yaYj Z s(x) V s( y)]. The Borda ranking will be derived using a method suggested in Young (1974). For P any xja X, its Borda count is given by bðxj Þ ¼ kpj aj;k . Definition 3.2: Given PaLn,cBaW is the Borda ranking if and only if for all x,yaX, xcB yZb(x) z b( y). For the rest of the paper we will call two rankings c,cVaW antagonistic if and only if for all x,yaX, xcyZycVx and for some x,yaX, xyZy Vx.3 The following theorem states that for some preference profiles, the Borda ranking and the Dodgson ranking will be antagonistic. Theorem 3.3: If there are more than four alternatives, then there exist preference profiles such that the Borda ranking and the Dodgson ranking are antagonistic.
4. Proof For the actual proof of Theorem 3.3 we will first introduce a result (Lemma 4.2) about the existence of preference profiles that satisfy certain conditions. The following Definition 4.1 introduces a property of preference profiles that requires that for some 2
For a further discussion on the reasonableness of the Dodgson ranking see Klamler (2002), for a general discussion on the idea of distance minimization see Baigent (1983). 3 There can be different (usually stronger) forms of antagonism if one uses choice functions. I am grateful to an anonymous referee for pointing this out to me.
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C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
Table 1 Dodgson scores and Borda counts x1
ðm 2Þ 2r þ 1 (m-2)c + q r + 2
s( ) b( )
xaS m2 q
c 2þ1 þ2þ1 2 lc+2
xm m2 2 ðl þ 4Þ r m2 2 ð2l
þ q2 þ 1 þ 4Þ q 2
pair xj, xkaX, all individuals that rank xj above kk, rank xj so high above xk that each individual needs a certain minimum number of pairwise switches to invert its ranking of this pair to xk above xj. Definition 4.1. For all xj,xka X, let Lk, j={caL:xkxj}. A preference profile PaLn, satisfies j– k-agreement if and only if for the pair xj, xka X, biaI:xjPixk Z d( Pi,c) z AXA 2, for all caLk,j. Let Z be the set of all integers. Given preference profile PaLn , the vector ofpairwise m
majority margins will be denoted by wP ¼ ðaP1;2 ; . . . ; aPj;k ; . . . ; aPm1;m ÞaZ 2 where j,ka{1,2,. . .,m},j < k. 4 The following lemma extends Debord’s (1987) theorem , which states that for any m
maZ 2 with all entries having the same parity, there exists a preference profile PaLn such that wP = m. Lemma m 4.2. (Klamler, 2002): If there are more than three alternatives, then for any maZ 2 , with all entries having the same parity, there exists a preference profile PaLn and a pair xj,xkaX such that P satisfies j –k-agreement and wP = m. The proof will now be based on two lemmas showing that the theorem is true for even and odd numbers of alternatives. Lemma 4.3: If AXA z 5 and even, then there exist preference profiles such that the Dodgson ranking and the Borda ranking are antagonistic. Proof: Consider X={x1,x2,. . .,xm}, where m is an even number larger than 5. Let a1,2 = c + q + 2, a1, j = c for all ja{3,4,. . .,m 1} and a1,m = r. Furthermore, let a2,m = l + 4 and am 1,m = l + q + 2. For all ja{3,4,. . .,m 2}, let aj,m = l whenever j is odd, and aj,m = l + 4 whenever j is even. Let a3,4 = a5,6 = a7,8 = . . . = am 3,m 2 = 2q + 2. Let all other pairwise margins be aj,k = q whenever j < k and k has the opposite parity of j, and aj,k = q whenever j < k and k and j have the same parity. Those margins imply a strict majority cycle x1x2x3. . .xmx1 and by Lemma 4.2 we know that a preference profile with exactly the above margins exists and satisfies m1-agreement. The Dodgson scores and the Borda counts for the alternatives are given in Table 1, where S = X \{x1,xm}. 4
Saari (1995) and Ratliff (2001) prove equivalent statements and provide extensions.
C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
107
Table 2 Dodgson scores and Borda counts x1
s( ) b( )
ðm 2Þ 2r þ 1 (m 2)c r
xaS m3 q 2
lc
c 2þ1 þ2þ1
xm
ðm 2Þ 2l þ 1 r (m 2)l
To establish the desired result it is now sufficient to solve the following system of inequalities: q c I ðm 2Þ 2r þ 1 > m2 2 2þ1 þ2þ1 q q c m2 II m2 2 2 þ 1 þ 2 þ 1 > 2 ðl þ 4Þ þ 2 þ 1 III (m2)c + qr + 2>lc + 2 IV l c þ 2 > r m2 2 ð2l þ 4Þ q 2 As one solution to this system of inequalities for m>5, m even, is l = 2, r = 8, q = 10 and c = 12, this proves the lemma. Lemma 4.4: If AXA z 5 and odd, then there exist preference profiles such that the Dodgson ranking and the Borda ranking are antagonistic. Proof: Consider X={x1,x2,. . .,xm}, with at least five alternatives. Let a1,m = r and, for all ja{2,3,. . .,m 1}, let a1, j = c and aj,m = l. Furthermore, for all ja{2,3,. . .,m 1} let aj, k = q whenever j < k and k has the opposite parity of j, and aj,k = q whenever j < k and k and j have the same parity. Those margins imply a strict majority cycle x1x2x3. . . xmx1 and by Lemma 4.2 we know that a preference profile with exactly those margins exists and satisfies m1-agreement. The Dodgson scores and Borda counts for the alternatives are given in Table 2, where S = X \{x1,xm}. 5 To establish the desired result it is now sufficient to solve the following system of inequalities: q c I ðm 2Þ 2r þ 1 > m3 2 2þ1 þ l 2 þ 1 q c II m3 2 2 þ 1 þ 2 þ 1 > ðm 2Þ 2 þ 1 III (m2)cr>lc IV lc>r(m2)l As one solution to this system of inequalities for m z 5, m odd, is c = l = 2, r = 4 and q = 10, this proves the lemma. Proof of Theorem 3.3: Lemma 4.3 proves all cases for m even, and Lemma 4.4 proves all cases for m odd. Hence the theorem is true. 5
Acknowledgements I am very grateful to Daniel Eckert, Nick Baigent and an anonymous referee for valuable comments.
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C. Klamler / Mathematical Social Sciences 48 (2004) 103–108
References Baigent, N., 1983. Some reservation on strictly majoritarian social choice. Discussion Paper Series. University of Essex. Borda, J.C., 1781. Memoire sur les Elections au Scrutin. Memoires de l’Academie Royale des Sciences. Debord, B., 1987. Caracte´risation des matrices de pre´fe´rences nettes et me´thodes d’agre´gation associe´es. Mathe´matiques et Sciences Humaines 97, 5 – 17. Dodgson, C., 1876. A method of taking votes on more than two issues. Black, 1958. The Theory of Committees and Elections. Cambridge Univ. Press, London. Fishburn, P., 1977. Condorcet social choice functions. SIAM Journal on Applied Mathematics 33, 469 – 489. Kemeny, J., 1959. Mathematics without numbers. Daedalus 88, 577 – 591. Klamler, C., 2002. The Dodgson ranking and its relation to Kemeny’s method and Slater’s rule. Forthcoming: Social Choice and Welfare. Laffond, G., et al., 1995. Condorcet choice correspondences: a set-theoretical comparison. Mathematical Social Sciences 30, 23 – 35. Laslier, J.-F., 1997. Tournament Solutions and Majority Voting. Springer-Verlag, Berlin. Ratliff, T.C., 2001. A comparison of Dodgson’s method and Kemeny’s rule. Social Choice and Welfare 18, 79 – 90. Ratliff, T.C., 2002. A comparison of Dodgson’s method and the Borda count. Economic Theory 20, 357 – 372. Saari, D.G., 1995. Basic Geometry of Voting. Springer-Verlag, Berlin. Young, H.P., 1974. An axiomatisation of Borda’s rule. Journal of Economic Theory 9, 43 – 52.