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Single profile of preferences with variable societies: A characterization of simple majority rule
Economics Letters 107 (2010) 119–121
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Single profile of preferences with variable societies: A characterization of simple majority rule Yongsheng Xu a,b,⁎, Zhen Zhong c a b c
Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA 30303, USA School of Public Finance and Taxation, Southwestern University of Finance and Economics, Chengdu, China Risk Management Department, Bank of China, 1 Fuxingmen Nei Dajie, Beijing 100818, China
a r t i c l e
i n f o
Article history: Received 19 June 2009 Received in revised form 27 November 2009 Accepted 17 December 2009 Available online 4 January 2010
a b s t r a c t We study simple majority rule for a fixed profile of individuals' preferences and a variable society. Four properties each linking decisions by a group to decisions by its various subgroups are introduced, and are used for characterizing simple majority rule. © 2009 Elsevier B.V. All rights reserved.
JEL classification: D7 Keywords: Simple majority rule Single profile of preferences Variable societies
1. Introduction One of the best known voting procedures is simple majority rule: when a group of people deciding on two options and assuming that each individual casts one vote, the option that gets ‘more’ votes than the other emerges as the winner. Simple majority rule is fairly easy to understand and to implement, and has several attractive normative properties as studied in May (1952): it treats individuals symmetrically (the rule is anonymous), it is neutral with respect to options (there is no significance of names attached to the options), and it responds to individuals' preferences positively. In May's (1952) study of simple majority rule, he takes a group of individuals as fixed and allows their preferences to vary. This is the predominant approach in the literature studying normative properties of simple majority rule. In this paper, we propose a different framework in which a list of individuals' preferences is given while individuals may form different subgroups to study simple majority rule. Simple majority rule is thus investigated from a perspective that links the group's decision with decisions made by its subgroups. This is motivated by an observation that group decisions are essentially compromises among group members and/or among its various subgroups: a group's decision on two options depends on how its various subgroups decide on the two ⁎ Corresponding author. Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA 30303, U.S.A. Tel.: +1 404 413 0158; fax: +1 404 413 0145. E-mail address: [email protected] (Y. Xu). 0165-1765/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2009.12.042
options. How exactly are subgroups' decisions linked to the entire group's decision? For example, when an option x considered to be the winner for each of the two disjoint subgroups over another option y, would x continue to be considered the winner for the group joined by the two subgroups? This is one of the possible links between a group of individuals and its subgroups that we intend to explore in this paper. In particular, we show that simple majority rule is characterized by the following properties (see formal definitions of these properties in Section 3): (a)the decision by any subgroup with just one individual must rest on this individual's preferences; (b) if two disjoint subgroups each consider an option x as the winner over another option y, then when they join into a single group, the option x continues to be the winner over the option y by the enlarged group; (c) any subgroup of two individuals with opposite views over two options x and y should express an indifference between the two options; and (d) whenever a subgroup of a given group is indifferent between two options x and y, the group's decision on x and y is determined by the other group after ‘taking out’ this indifference-subgroup. In Section 2, we introduce our basic notation and definitions. Section 3 presents several attractive properties and gives an axiomatic characterization for simple majority rule.
2. Notation and definitions Let there be n ≥ 2 individuals and two alternatives x and y. The set of individuals is to be denoted by N. For each i ∈ N, Ri stands for individual
120
Y. Xu, Z. Zhong / Economics Letters 107 (2010) 119–121
i's preferences over x and y. Let Pi and Ii stand, respectively, for the asymmetric and symmetric part of Ri. Non-empty subsets of N are denoted by S, T, ⋯, and are called subgroups. For any subgroup S, #S denotes the cardinality of S. Let αN ≡ {R1, ⋯, Ri, …, Rn} denote a profile of individuals' preferences over x and y. In this paper, we consider αN as fixed. For any subgroup S, let αS denote the set {Ri ∈ α N : i ∈ S}. An aggregation rule f assigns, for each αS ∈ ⋃ T ∈ N αT, a complete binary relation R(αS) over x and y: R(αS) = f(αS). The asymmetric and symmetric part of R(αS) are denoted by P(αS) and I(αS), respectively. It may be noted that our domain in defining an aggregation rule is different from others in the literature that study simple majority rule. The prevailing domains studied in the literature are: (a) the set of all possible profiles of preferences of a given society (May (1952), and more recently Llamazares (2006)), and (b) the set of all possible profiles of preferences of various subgroups of a given society (Miroiu (2004), Woeginger (2003, 2005), and Yeh (2008)). It is therefore clear that our domain is just one element of the domain specified under (b) (making our problem even more difficult).
3. Simple majority rule For each subgroup S, let N(x,y;αS) ≡ #{i ∈S:xRi y for some Ri ∈α S}. An aggregation rule f is said to be simple majority rule if, for all subgroup S, xf ðαS Þy⇔Nðx; y; αS Þ≥Nðy; x; αS Þ. How is a group's decision linked with decisions made by its members and its subgroups? We consider the following properties. Self-determination (SD): for all i ∈ N, xRðαfig Þy⇔xRi y. Simple equal treatment (SET): for all i, j ∈ N, if [xPi y and yPjx] then xI(α{i, j})y. Monotonicity (M): for all subgroup S and T with S ∩T = ∅, if [xP(α S)y and xP(αT)y] then xP(α S ∪αT)y. Independence of an unconcerned coalition (IUC): for all subgroups S and T with S ∩ T = ∅, if xI(α S)y, then xRðαT Þy⇔xRðαS ∪αT Þy. SD simply says that the decision by any subgroup with just one member must be determined by this member. In a very weak sense, SD reflects an idea of self-determination in making decisions. It may be noted that SD is a consequence of several other properties studied in Miroiu (2004) (Lemma 1). SET says that, for a subgroup consisting of two individuals, if these two individuals have opposite views over x and y (one prefers x to y and the other prefers y to x), then this subgroup should regard x and y as indifferent. SET reflects the idea that an aggregation rule should treat individuals equally in this simple situation. It may be noted that SET resembles an idea behind a simplification procedure discussed by Gaertner (1988) for reducing originally given profiles of preferences to their equivalent profiles of preferences. We also remark that the idea behind SET is similar to the axiom cancellation introduced in Llamazares (2006) for characterizing simple majority rule in a different context. Cancellation requires that, when two individuals change their opposite views over x and y to the same indifferent-view over x and y while all other individuals' views do not change, then the group's preferences over x and y should remain the same before and after the change. M stipulates that, whenever two disjoint subgroups, S and T, both regard x better than y, x must be regarded better than y by the group joined by S and T. The idea behind M is similar to that of the weak Pareto principle in the literature. Finally, IUC requires that, for any two disjoint subgroups S and T, whenever S is indifferent between x and y, the decision over x and y by the group, S ∪T, is determined by T. IUC may be regarded as similar to Pareto principles. The ideas behind M and IUC may also be found in the literature on characterizing simple majority rule. See, for example, Miroiu's (2004) Additive Positive Responsiveness (see also Woeginger (2005)), Woeginger's (2003) Pareto Optimality and Weak Pareto Optimality, and Yeh's (2008) Reinforcement for the discussion of plurality rule.
With the help of the above properties, we present the following proposition which constitutes our main result. Proposition 1. An aggregation rule f satisfies SD, SET, M and IUC if and only if it is simple majority rule. Proof. It can be checked that simple majority rule satisfies SD, SET, M and IUC. We now show that, if an aggregation rule f satisfies SD, SET, M and IUC, then it must be simple majority rule. Let f be an aggregation rule satisfying SD, SET, M and IUC. We first note that, by SD, it follows that for all S = fig; xRi y⇔xRðα Þy S
ð1Þ
Consider any S = {i, j} where i and j are distinct. We note that, for a, b ∈ {x, y}, if aPib and bPja, then, by SET, aI(α S)b follows immediately; if aPib and aPjb, then, from Eq. (1), we must have aP(α{i})b and aP(α{ j})b; a simple application of M gives us aP(α S)b; if aRib and aIjb, then, from Eq. (1), we have aR(α{i})b and aI(α{ j})b; by IUC, aRðαS Þb⇔aRðαfig Þb follows easily from IUC. Therefore, we obtain for all S = fi; jg and all a; b ∈fx; yg; if ½ðaPi b & aIj bÞ or ðaPi b & aPj bÞ S
S
then aPðα Þb; and if ½ðaIi b & aIj bÞ or ðaPi b & bPj aÞ then aIðα Þb:
ð2Þ
Suppose that, for all S with #S ≤ 2, f(α S) is given by simple majority rule. We show that for all T with #T = #S + 1, f(αT) is given by simple majority rule as well. Let T be a subgroup such that #T = #S + 1. We distinguish four cases: (i) for some i ∈ T, xIiy; (ii) for some j, k ∈ T, xPj y and yPk x; (iii) for all i ∈ T, xPi y; and (iv) for all i ∈ T, yPi x. Consider case (i) first. Note that in this case, for some i ∈T, xIi y. From Eq. (1), xI(α{i})y. By IUC, we then have xRðαT Þy⇔xRðαT−fig Þy. From our induction hypothesis, xRðαT−fig Þy⇔Nðx; y; αT−fig Þ≥Nðy; x; αT−fig Þ. Therefore, from xRðαT Þy⇔xRðαT−fig Þy and xI i y, it follows that xRðαT Þy⇔Nðx; y; αT Þ≥Nðy; x; αT Þ. Consider case (ii) in which for some j,k ∈T, xPj y and yPk x next. If xPj y and yPk x for some j,k ∈T, then, from Eq. (2), xI(α{ j, k})y. By IUC, we then have xRðαT Þy⇔xRðαT−f j;kg Þy. From our induction hypothesis, xRðαT−f j; kg Þy⇔Nðx; y; αT−f j; kg Þ≥ Nðy; x; αT−f j; kg Þ. Then, xRðαT Þy⇔Nðx; y; αT Þ≥ Nðy; x; αT Þ follows from xRðαT Þy⇔xRðhaT−f j; kg Þy and [xPj y and yPk x]. Thirdly, we consider case (iii) where xPi y for all i ∈ T. Note that, in this case, N(x, y; αT) = #T and N(y, x; αT) = 0. From our induction hypothesis, we must have xP(αT−{ j})y and xP(α{ j})y for some j ∈ T. Therefore, by M, it follows that xP(αT)y. Case (iv) in which for all i ∈ T, yPix, is similar to case (iii), and it can be shown that yP(αT)x. The above cases exhaust all possibilities. Therefore, combining Eqs. (1) and (2), we have shown that for all S; xRðα Þy⇔Nðx; y; α Þ≥Nðy; x; α Þ: S
S
S
ð3Þ
This completes the proof of Proposition 1. □ It may be noted that, in Proposition 1, M can be replaced by the following unanimity property: Unanimity (U): for all subgroup S, if [xPi y for all i ∈ S] then xP(α S)y. U says that if every individual in a subgroup prefers x to y, then the subgroup must rank x better than y. This is the weak Pareto principle in our context. As a final remark, we locate our contribution to the literature by grouping characterizations into the following categories: 1. A fixed society and multi profiles of preferences: Fishburn (1973), Llamazares (2006), and May (1952) work with two alternatives, and Campbell (1982, 1988), Campbell and Kelly (2000, 2003), Dasgupta
Y. Xu, Z. Zhong / Economics Letters 107 (2010) 119–121
and Maskin (2008), Maskin (1995), and Yi (2005) work with more than two alternatives. 2. Variable societies and multi profiles of preferences: Asan and Sanver (2002), Miroiu (2004), and Woeginger (2003, 2005) work with two alternatives. 3. Variable societies and a single profile of preferences: this paper works with two alternatives. References Asan, G., Sanver, M.R., 2002. Another characterization of the majority rule. Economics Letters 75, 409–413. Campbell, D.E., 1982. On the derivation of majority rule. Theory and Decision 14, 133–140. Campbell, D.E., 1988. A characterization of simple majority rule for restricted domains. Economics Letters 28, 307–310. Campbell, D.E., Kelly, J.S., 2000. A simple characterization of majority rule. Economic Theory 15, 689–700. Campbell, D.E., Kelly, J.S., 2003. A strategy–proofness characterization of majority rule. Economic Theory 22, 557–568.
121
Dasgupta, P., Maskin, E., 2008. On the robustness of majority rule. Journal of European Economic Association 6 (5), 949–973. Fishburn, P.C., 1973. The Theory of Social Choice. Princeton University Press, Princeton. Gaertner, W., 1988. Binary inversions and transitive majorities. In: Eichhorn, W. (Ed.), Measurement in Economics. Physica-Verlag, Heidelberg, pp. 253–267. Llamazares, B., 2006. The forgotten decision rules: majority rules based on difference of votes. Mathematical Social Sciences 51, 311–326. Maskin, E.S., 1995. Majority rule, social welfare functions and game forms. In: Basu, K., Pattanaik, P.K., Suzumura, K. (Eds.), Choice Welfare and Development. The Clarendon Press, Oxford, pp. 100–109. May, K.O., 1952. A set of independent necessary and sufficient conditions for simple majority decisions. Econometrica 20 (4), 680–684. Miroiu, A., 2004. Characterizing majority rule: from profiles to societies. Economics Letters 85, 359–363. Woeginger, G.J., 2003. A new characterization of the majority rule. Economics Letters 81, 89–94. Woeginger, G.J., 2005. More on the majority rule: profiles, societies, and responsiveness. Economics Letters 88, 7–11. Yeh, C., 2008. An efficiency characterization of plurality rule in collective choice problems. Economic Theory 34, 575–583. Yi, J., 2005. A complete characterization of majority rules. Economics Letters 87, 109–112.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Single profile of preferences with variable societies: A characterization of simple majority rule Yongsheng Xu a,b,⁎, Zhen Zhong c a b c
Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA 30303, USA School of Public Finance and Taxation, Southwestern University of Finance and Economics, Chengdu, China Risk Management Department, Bank of China, 1 Fuxingmen Nei Dajie, Beijing 100818, China
a r t i c l e
i n f o
Article history: Received 19 June 2009 Received in revised form 27 November 2009 Accepted 17 December 2009 Available online 4 January 2010
a b s t r a c t We study simple majority rule for a fixed profile of individuals' preferences and a variable society. Four properties each linking decisions by a group to decisions by its various subgroups are introduced, and are used for characterizing simple majority rule. © 2009 Elsevier B.V. All rights reserved.
JEL classification: D7 Keywords: Simple majority rule Single profile of preferences Variable societies
1. Introduction One of the best known voting procedures is simple majority rule: when a group of people deciding on two options and assuming that each individual casts one vote, the option that gets ‘more’ votes than the other emerges as the winner. Simple majority rule is fairly easy to understand and to implement, and has several attractive normative properties as studied in May (1952): it treats individuals symmetrically (the rule is anonymous), it is neutral with respect to options (there is no significance of names attached to the options), and it responds to individuals' preferences positively. In May's (1952) study of simple majority rule, he takes a group of individuals as fixed and allows their preferences to vary. This is the predominant approach in the literature studying normative properties of simple majority rule. In this paper, we propose a different framework in which a list of individuals' preferences is given while individuals may form different subgroups to study simple majority rule. Simple majority rule is thus investigated from a perspective that links the group's decision with decisions made by its subgroups. This is motivated by an observation that group decisions are essentially compromises among group members and/or among its various subgroups: a group's decision on two options depends on how its various subgroups decide on the two ⁎ Corresponding author. Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA 30303, U.S.A. Tel.: +1 404 413 0158; fax: +1 404 413 0145. E-mail address: [email protected] (Y. Xu). 0165-1765/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2009.12.042
options. How exactly are subgroups' decisions linked to the entire group's decision? For example, when an option x considered to be the winner for each of the two disjoint subgroups over another option y, would x continue to be considered the winner for the group joined by the two subgroups? This is one of the possible links between a group of individuals and its subgroups that we intend to explore in this paper. In particular, we show that simple majority rule is characterized by the following properties (see formal definitions of these properties in Section 3): (a)the decision by any subgroup with just one individual must rest on this individual's preferences; (b) if two disjoint subgroups each consider an option x as the winner over another option y, then when they join into a single group, the option x continues to be the winner over the option y by the enlarged group; (c) any subgroup of two individuals with opposite views over two options x and y should express an indifference between the two options; and (d) whenever a subgroup of a given group is indifferent between two options x and y, the group's decision on x and y is determined by the other group after ‘taking out’ this indifference-subgroup. In Section 2, we introduce our basic notation and definitions. Section 3 presents several attractive properties and gives an axiomatic characterization for simple majority rule.
2. Notation and definitions Let there be n ≥ 2 individuals and two alternatives x and y. The set of individuals is to be denoted by N. For each i ∈ N, Ri stands for individual
120
Y. Xu, Z. Zhong / Economics Letters 107 (2010) 119–121
i's preferences over x and y. Let Pi and Ii stand, respectively, for the asymmetric and symmetric part of Ri. Non-empty subsets of N are denoted by S, T, ⋯, and are called subgroups. For any subgroup S, #S denotes the cardinality of S. Let αN ≡ {R1, ⋯, Ri, …, Rn} denote a profile of individuals' preferences over x and y. In this paper, we consider αN as fixed. For any subgroup S, let αS denote the set {Ri ∈ α N : i ∈ S}. An aggregation rule f assigns, for each αS ∈ ⋃ T ∈ N αT, a complete binary relation R(αS) over x and y: R(αS) = f(αS). The asymmetric and symmetric part of R(αS) are denoted by P(αS) and I(αS), respectively. It may be noted that our domain in defining an aggregation rule is different from others in the literature that study simple majority rule. The prevailing domains studied in the literature are: (a) the set of all possible profiles of preferences of a given society (May (1952), and more recently Llamazares (2006)), and (b) the set of all possible profiles of preferences of various subgroups of a given society (Miroiu (2004), Woeginger (2003, 2005), and Yeh (2008)). It is therefore clear that our domain is just one element of the domain specified under (b) (making our problem even more difficult).
3. Simple majority rule For each subgroup S, let N(x,y;αS) ≡ #{i ∈S:xRi y for some Ri ∈α S}. An aggregation rule f is said to be simple majority rule if, for all subgroup S, xf ðαS Þy⇔Nðx; y; αS Þ≥Nðy; x; αS Þ. How is a group's decision linked with decisions made by its members and its subgroups? We consider the following properties. Self-determination (SD): for all i ∈ N, xRðαfig Þy⇔xRi y. Simple equal treatment (SET): for all i, j ∈ N, if [xPi y and yPjx] then xI(α{i, j})y. Monotonicity (M): for all subgroup S and T with S ∩T = ∅, if [xP(α S)y and xP(αT)y] then xP(α S ∪αT)y. Independence of an unconcerned coalition (IUC): for all subgroups S and T with S ∩ T = ∅, if xI(α S)y, then xRðαT Þy⇔xRðαS ∪αT Þy. SD simply says that the decision by any subgroup with just one member must be determined by this member. In a very weak sense, SD reflects an idea of self-determination in making decisions. It may be noted that SD is a consequence of several other properties studied in Miroiu (2004) (Lemma 1). SET says that, for a subgroup consisting of two individuals, if these two individuals have opposite views over x and y (one prefers x to y and the other prefers y to x), then this subgroup should regard x and y as indifferent. SET reflects the idea that an aggregation rule should treat individuals equally in this simple situation. It may be noted that SET resembles an idea behind a simplification procedure discussed by Gaertner (1988) for reducing originally given profiles of preferences to their equivalent profiles of preferences. We also remark that the idea behind SET is similar to the axiom cancellation introduced in Llamazares (2006) for characterizing simple majority rule in a different context. Cancellation requires that, when two individuals change their opposite views over x and y to the same indifferent-view over x and y while all other individuals' views do not change, then the group's preferences over x and y should remain the same before and after the change. M stipulates that, whenever two disjoint subgroups, S and T, both regard x better than y, x must be regarded better than y by the group joined by S and T. The idea behind M is similar to that of the weak Pareto principle in the literature. Finally, IUC requires that, for any two disjoint subgroups S and T, whenever S is indifferent between x and y, the decision over x and y by the group, S ∪T, is determined by T. IUC may be regarded as similar to Pareto principles. The ideas behind M and IUC may also be found in the literature on characterizing simple majority rule. See, for example, Miroiu's (2004) Additive Positive Responsiveness (see also Woeginger (2005)), Woeginger's (2003) Pareto Optimality and Weak Pareto Optimality, and Yeh's (2008) Reinforcement for the discussion of plurality rule.
With the help of the above properties, we present the following proposition which constitutes our main result. Proposition 1. An aggregation rule f satisfies SD, SET, M and IUC if and only if it is simple majority rule. Proof. It can be checked that simple majority rule satisfies SD, SET, M and IUC. We now show that, if an aggregation rule f satisfies SD, SET, M and IUC, then it must be simple majority rule. Let f be an aggregation rule satisfying SD, SET, M and IUC. We first note that, by SD, it follows that for all S = fig; xRi y⇔xRðα Þy S
ð1Þ
Consider any S = {i, j} where i and j are distinct. We note that, for a, b ∈ {x, y}, if aPib and bPja, then, by SET, aI(α S)b follows immediately; if aPib and aPjb, then, from Eq. (1), we must have aP(α{i})b and aP(α{ j})b; a simple application of M gives us aP(α S)b; if aRib and aIjb, then, from Eq. (1), we have aR(α{i})b and aI(α{ j})b; by IUC, aRðαS Þb⇔aRðαfig Þb follows easily from IUC. Therefore, we obtain for all S = fi; jg and all a; b ∈fx; yg; if ½ðaPi b & aIj bÞ or ðaPi b & aPj bÞ S
S
then aPðα Þb; and if ½ðaIi b & aIj bÞ or ðaPi b & bPj aÞ then aIðα Þb:
ð2Þ
Suppose that, for all S with #S ≤ 2, f(α S) is given by simple majority rule. We show that for all T with #T = #S + 1, f(αT) is given by simple majority rule as well. Let T be a subgroup such that #T = #S + 1. We distinguish four cases: (i) for some i ∈ T, xIiy; (ii) for some j, k ∈ T, xPj y and yPk x; (iii) for all i ∈ T, xPi y; and (iv) for all i ∈ T, yPi x. Consider case (i) first. Note that in this case, for some i ∈T, xIi y. From Eq. (1), xI(α{i})y. By IUC, we then have xRðαT Þy⇔xRðαT−fig Þy. From our induction hypothesis, xRðαT−fig Þy⇔Nðx; y; αT−fig Þ≥Nðy; x; αT−fig Þ. Therefore, from xRðαT Þy⇔xRðαT−fig Þy and xI i y, it follows that xRðαT Þy⇔Nðx; y; αT Þ≥Nðy; x; αT Þ. Consider case (ii) in which for some j,k ∈T, xPj y and yPk x next. If xPj y and yPk x for some j,k ∈T, then, from Eq. (2), xI(α{ j, k})y. By IUC, we then have xRðαT Þy⇔xRðαT−f j;kg Þy. From our induction hypothesis, xRðαT−f j; kg Þy⇔Nðx; y; αT−f j; kg Þ≥ Nðy; x; αT−f j; kg Þ. Then, xRðαT Þy⇔Nðx; y; αT Þ≥ Nðy; x; αT Þ follows from xRðαT Þy⇔xRðhaT−f j; kg Þy and [xPj y and yPk x]. Thirdly, we consider case (iii) where xPi y for all i ∈ T. Note that, in this case, N(x, y; αT) = #T and N(y, x; αT) = 0. From our induction hypothesis, we must have xP(αT−{ j})y and xP(α{ j})y for some j ∈ T. Therefore, by M, it follows that xP(αT)y. Case (iv) in which for all i ∈ T, yPix, is similar to case (iii), and it can be shown that yP(αT)x. The above cases exhaust all possibilities. Therefore, combining Eqs. (1) and (2), we have shown that for all S; xRðα Þy⇔Nðx; y; α Þ≥Nðy; x; α Þ: S
S
S
ð3Þ
This completes the proof of Proposition 1. □ It may be noted that, in Proposition 1, M can be replaced by the following unanimity property: Unanimity (U): for all subgroup S, if [xPi y for all i ∈ S] then xP(α S)y. U says that if every individual in a subgroup prefers x to y, then the subgroup must rank x better than y. This is the weak Pareto principle in our context. As a final remark, we locate our contribution to the literature by grouping characterizations into the following categories: 1. A fixed society and multi profiles of preferences: Fishburn (1973), Llamazares (2006), and May (1952) work with two alternatives, and Campbell (1982, 1988), Campbell and Kelly (2000, 2003), Dasgupta
Y. Xu, Z. Zhong / Economics Letters 107 (2010) 119–121
and Maskin (2008), Maskin (1995), and Yi (2005) work with more than two alternatives. 2. Variable societies and multi profiles of preferences: Asan and Sanver (2002), Miroiu (2004), and Woeginger (2003, 2005) work with two alternatives. 3. Variable societies and a single profile of preferences: this paper works with two alternatives. References Asan, G., Sanver, M.R., 2002. Another characterization of the majority rule. Economics Letters 75, 409–413. Campbell, D.E., 1982. On the derivation of majority rule. Theory and Decision 14, 133–140. Campbell, D.E., 1988. A characterization of simple majority rule for restricted domains. Economics Letters 28, 307–310. Campbell, D.E., Kelly, J.S., 2000. A simple characterization of majority rule. Economic Theory 15, 689–700. Campbell, D.E., Kelly, J.S., 2003. A strategy–proofness characterization of majority rule. Economic Theory 22, 557–568.
121
Dasgupta, P., Maskin, E., 2008. On the robustness of majority rule. Journal of European Economic Association 6 (5), 949–973. Fishburn, P.C., 1973. The Theory of Social Choice. Princeton University Press, Princeton. Gaertner, W., 1988. Binary inversions and transitive majorities. In: Eichhorn, W. (Ed.), Measurement in Economics. Physica-Verlag, Heidelberg, pp. 253–267. Llamazares, B., 2006. The forgotten decision rules: majority rules based on difference of votes. Mathematical Social Sciences 51, 311–326. Maskin, E.S., 1995. Majority rule, social welfare functions and game forms. In: Basu, K., Pattanaik, P.K., Suzumura, K. (Eds.), Choice Welfare and Development. The Clarendon Press, Oxford, pp. 100–109. May, K.O., 1952. A set of independent necessary and sufficient conditions for simple majority decisions. Econometrica 20 (4), 680–684. Miroiu, A., 2004. Characterizing majority rule: from profiles to societies. Economics Letters 85, 359–363. Woeginger, G.J., 2003. A new characterization of the majority rule. Economics Letters 81, 89–94. Woeginger, G.J., 2005. More on the majority rule: profiles, societies, and responsiveness. Economics Letters 88, 7–11. Yeh, C., 2008. An efficiency characterization of plurality rule in collective choice problems. Economic Theory 34, 575–583. Yi, J., 2005. A complete characterization of majority rules. Economics Letters 87, 109–112.