The greatest unhappiness of the least number Walter Bossert Centre Interuniversitaire de Recherche en Economie Quantitative (CIREQ) P.O. Box 6128, Station Downtown Montreal QC H3C 3J7 Canada e-mail: walter.bossert@videotron.ca and Kotaro Suzumura School of Political Science and Economics Waseda University 1-6-1 Nishi-Waseda Shinjuku-ku, Tokyo 169-8050 Japan e-mail: ktr.suzumura@gmail.com This version: January 27, 2016 Abstract. We propose an alternative articulation of the Benthamite greatest-happinessof-the-greatest-number principle. With ordinally measurable and interpersonally noncomparable utilities, the rule chooses those feasible alternatives that maximize the number of individuals who end up with their greatest element. This rule is tantamount to the plurality rule. Furthermore, in the spirit of Rawls s maximin principle, we propose the greatest-unhappiness-of-the-least-number principle. In analogy to the greatest-happiness principle, the least-unhappiness principle is formally equivalent to the anti-plurality rule. Our main result is a characterization of the least-unhappiness principle. Journal of Economic Literature Classification Nos.: D71, D72. We thank Thierry Marchant and several seminar audiences for their comments and suggestions. Financial support from a Grant-in-Aid for Specially Promoted Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan for the Project on Economic Analysis of Intergenerational Issues (grant number 22000001), the Fonds de Recherche sur la Société et la Culture of Québec, and the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.
1 Introduction In this paper, we consider an ordinal and interpersonally non-comparable interpretation of Bentham s well-known principle that It is the greatest happiness of the greatest number that is the measure of right and wrong, as formulated in the preface to the first edition of Bentham (1776); the phrase can be found in Bowring (1843, p. 227). In a setting that uses ordinal and interpersonally non-comparable information on individual well-being only, this principle can be expressed in terms of a collective choice correspondence that is equivalent to the plurality voting rule. According to the standard understanding, the Benthamite decision rule, which seeks to attain the greatest happiness of the greatest number, is construed to maximize the social sum-total of individual utilities. This decision rule is clearly based on the informational basis of cardinal and interpersonally unit-comparable individual utilities. In the wake of the ordinalist revolution in the 1930s, which was initiated in the positive branch of economics by Hicks (1939a/1946) and Samuelson (1938a,b, 1947/1983) and adopted in normative economics by Robbins (1932/1935), Hicks (1939b, 1940), Kaldor (1939) and Samuelson (1950), the conventional informational basis of the Benthamite utilitarianism received harsh criticism. This criticism resulted in the arrival of new welfare economics that is based on ordinal and interpersonally non-comparable individual utilities, initiated by Bergson (1938) and Samuelson (1947/1983, 1981). We propose an alternative articulation of the Benthamite decision rule, that is, the greatest-happiness-of-the-greatest-number rule, which is based on ordinal and interpersonally non-comparable utilities. For each feasible social alternative, say x, we count the number g(x) of individuals who rank x at the top of their preference rankings over all feasible social alternatives. The ordinalist Benthamite decision rule chooses those feasible alternatives x such that g(x) g(y) holds for all feasible alternatives y. It is true that the ordinalist Benthamite decision rule, thus defined, is tantamount to the well-known plurality voting rule; see, for instance, Richelson (1978), Ching (1996), Yeh (2008) and Sekiguchi (2012). Although the plurality voting rule per se has been discussed extensively in the literature, it has never been recognized as the ordinalist articulation of Bentham s greatest-happiness-of-the-greatest-number principle. Going one step further, and more in the spirit of the Rawlsian principle of maximin justice (Rawls, 1971), this paper also proposes an alternative social decision rule that may be labeled the greatest-unhappiness-of-the-least-number rule. Just as is the case for the greatest-happiness principle and the plurality rule, the least-unhappiness principle turns out to be formally equivalent to the anti-plurality (or inverse plurality) rule. The new result of this paper is a characterization of the least-unhappiness principle. The anti-plurality rule has been axiomatized in earlier contributions such as Baharad and Nitzan (2005) and Barberà and Dutta (1982) but the characterization presented here is quite different from those that are already available; a more detailed discussion can be found in the section following the statement and proof of our result. To the best of our knowledge, Bentham s greatest-happiness principle and its leastunhappiness counterpart have not been investigated so far in the context of ordinal measurability and interpersonal non-comparability. In contrast, theories of justice based on variants of Bentham s (1789) original formulation of utilitarian principles have been an- 1
alyzed quite extensively in the context of cardinal measurability and interpersonal unit comparability; see, for example, Blackorby, Bossert and Donaldson (2002). Likewise, the discussion of Rawls s (1971) notion of maximin justice has been restricted to informational requirements that assume individual utilities to be ordinally measurable and interpersonally level-comparable. 2 Definitions There is a non-empty and finite set of alternatives X and we use X to denote the set of all non-empty subsets of X. The set of potential agents is the set N of positive integers. As is the case for much of the literature on the (anti-)plurality voting rule and similar group-decision mechanisms, we assume that a collective choice correspondence is capable of being applied to different populations of possibly different size; see also Smith (1973) for a variable-electorate approach to voting problems. Thus, the set of possible societies under consideration is the set N of all non-empty and finite subsets of N. For all i N, R i is the set of all reflexive, complete, transitive and antisymmetric relations on X with typical element R i. Likewise, for all N N, R N = i N R i is the set of all profiles of such relations for population N with typical element R N = (R i ) i N. Furthermore, we define R = N N R N. A collective choice correspondence is a mapping f: R X X such that, for all N N, for all R N R N and for all S X, f(r N, S) S. In order to formulate the classical Benthamite greatest-happiness principle and its leastunhappiness counterpart in an ordinal setting without interpersonal comparability, we define, for all N N, for all R N R N, for all S X and for all x S, and g(r N, S, x) = {i N xr i y for all y S} l(r N, S, x) = {i N yr i x for all y S}. Clearly, g(r N, S, x) is the number of individuals for whom x is the greatest element in S according to the profile R N, whereas l(r N, S, x) is the corresponding number of those who have x as their least element. As mentioned earlier, we consider an informational environment in which individual well-being is ordinally measurable and interpersonally non-comparable. In such a setting, the greatest-happiness-of-the-greatest-number principle f H is defined by letting, for all N N, for all R N R N and for all S X, f H (R N, S) = {x S g(r N, S, x) g(r N, S, y) for all y S}. Analogously, the greatest-unhappiness-of-the-least-number principle can be expressed by means of the collective choice correspondence f U defined by letting, for all N N, for all R N R N and for all S X, f U (R N, S) = {x S l(r N, S, x) l(r N, S, y) for all y S}. It is straightforward to verify that f H is the well-known plurality rule and f U is its counterpart the anti-plurality rule, also referred to as the inverse plurality rule. 2
We now introduce the axioms that we employ in our characterization of f U. Our first property is the well-known anonymity axiom. It requires that the collective choice correspondence f is impartial, paying no attention to the identities of the individuals. Let Π be the set of all permutations π: N N. For all π Π and for all N N, let N π = {j N i N such that j = π(i)}. Thus, we have R Nπ = ( R π(i) )i N for all π Π, for all N N and for all R N R N. Anonymity. For all N N, for all R N R N, for all S X and for all π Π, f(r Nπ, S) = f(r N, S). Next, we define the property of neutrality, which is similar in spirit to anonymity in that it demands that the alternatives are treated symmetrically by f in the sense that the label we attach to them are irrelevant. Let Σ be the set of all permutations σ: X X. For all i N, for all R i R i and for all σ Σ, define the relation σ i (R i ) by letting σ(x) σ i (R i ) σ(y) xr i y for all x, y X. For all N N and for all R N R N, let σ N (R N ) = (σ i (R i )) i N. Neutrality. For all N N, for all R N R N, for all S X and for all σ Σ, f (σ N (R N ), S) = σ (f(r N, S)). Reinforcement is yet another standard axiom in the context of collective choice correspondences. It imposes a restriction that involves different societies. Suppose that there are two disjoint groups of agents N and N with preference profiles R N and R N. Furthermore, suppose that, for a given feasible set S of alternatives, there are some alternatives that are chosen by both groups. The reinforcement axiom requires that if the two groups are merged, the choice from S for the expanded group coincide with the non-empty intersection that contains the common choices of the original smaller groups. Reinforcement. For all N, N N such that N N =, for all R N R N R N and for all S X, if f(r N, S) f(r N, S), then R N, for all f ((R N, R N ), S) = f(r N, S) f(r N, S). The following axiom ensures that if an agent i s preference ordering R i changes to a preference ordering R i without changing the alternatives that are chosen from a feasible 3
set S in the single-agent society composed of individual i, then the choice from S of any society N that contains i is unchanged as well. Individual-equality independence. For all N N, for all R N R N, for all S X, for all i N and for all R i R i, if f(r i, S) = f(r i, S), then f ( (R N\{i}, R i), S ) = f(r N, S). The remaining two properties are very limited in scope because they apply to singleagent societies only. The first of these two requires that, in choice situations involving a single agent and two alternatives, the better of the two alternatives is chosen and the worse alternative is not. Single-agent monotonicity. For all i N, for all R i R i, for all distinct x, y X, if xr i y, then f(r i, {x, y}) = {x}. All of the above axioms are rather un-controversial and are satisfied by a variety of collective choice correspondences. In contrast, our last property is what sets the greatestunhappiness-of-the-least-number principle f U apart from all others. Again, it is silent when considering societies with more than one agent and, thus, is limited in scope. If there is a single agent and there are at least two alternatives, the axiom requires that if an alternative x is added that is preferred to all alternatives in the original (smaller) set S, then this new alternative x should be chosen from S {x} in addition to the alternatives that are chosen from the smaller set S. Clearly, this axiom imposes an additional restriction for example, the greatest-happiness-of-the-greatest-number rule f H fails to satisfy it. Single-agent expansion. For all i N, for all R i R i, for all S X such that S > 1 and for all x X \ S, if xr i y for all y S, then f(r i, S {x}) = f(r i, S) {x}. We conclude this section with a statement of an axiom employed by Ching (1996) in his characterization of the greatest-happiness principle f H. The property demands that if an alternative y in a feasible set S is dominated by another alternative x in S in the sense that all individuals prefer x to y, then the set of chosen alternatives is unchanged if y is removed from S. Unlike the greatest-happiness principle, the least-unhappiness principle f U does not satisfy this axiom. Independence of dominated alternatives. For all N N, for all R N R N, for all S X and for all y S, if there exists x S \ {y} such that xr i y for all i N, then f(r N, S \ {y}) = f(r N, S). 4
3 Greatest unhappiness of the least number In order to facilitate a comparison with our result, we first state Ching s (1996) characterization of the plurality rule that is, the greatest-happiness principle f H. Ching s (1996) theorem generalizes a result by Richelson (1978) by dropping one of the axioms in the original characterization. The reason why the additional property a continuity requirement is needed in Richelson s (1978) result is that his proof proceeds by employing a well-known earlier theorem on scoring functions established by Young (1975). Other characterizations of the plurality rule include that of Yeh (2008) and its strengthening by Sekiguchi (2012). The results of Yeh (2008) and Sekiguchi (2012) and that of Ching (1996) are independent. Theorem 1 A collective choice correspondence f satisfies anonymity, neutrality, reinforcement and independence of dominated alternatives if and only if f = f H. Proof. See Ching (1996). We now characterize f U by means of the requisite axioms introduced in the previous section. That the axioms are independent is shown in the appendix. Theorem 2 A collective choice correspondence f satisfies anonymity, neutrality, reinforcement, individual-equality independence, single-agent monotonicity and single-agent expansion if and only if f = f U. Proof. If. That f U satisfies anonymity and neutrality is immediate because neither the identities of the individuals nor the labels of the alternatives play any role in the definition of the rule. To establish reinforcement, suppose that N, N N, R N R N, R N R N and S X are such that N N = and f U (R N, S) f U (R N, S). For future reference, note that l ((R N, R N ), S, z) = l(r N, S, z) + l(r N, S, z) (1) for all z S because N N =. Suppose first that x f U (R N, S) f U (R N, S). By definition of f U, we have l(r N, S, x) l(r N, S, y) and l(r N, S, x) l(r N, S, y) for all y S. Combined with (1), it follows that l ((R N, R N ), S, x) l ((R N, R N ), S, y) for all y S. Using the definition of f U again, we obtain x f U ((R N, R N ), S). Now suppose that x f U ((R N, R N ), S). Thus, for all y S and, using (1), it follows that l ((R N, R N ), S, x) l ((R N, R N ), S, y) l(r N, S, x) + l(r N, S, x) l(r N, S, y) + l(r N, S, y) (2) 5
for all y S. Let z f U (R N, S) f U (R N, S). Such an alternative z exists because this intersection is non-empty. Thus, l(r N, S, z) l(r N, S, y) and l(r N, S, z) l(r N, S, y) (3) for all y S. Clearly, (3) is satisfied for the special case in which y = x. Thus, l(r N, S, z) + l(r N, S, z) l(r N, S, x) + l(r N, S, x). (4) Analogously, (2) applies when y = z and, thus, Combined with (4), it follows that l(r N, S, x) + l(r N, S, x) l(r N, S, z) + l(r N, S, z). l(r N, S, x) + l(r N, S, x) = l(r N, S, z) + l(r N, S, z). (5) If l(r N, S, z) < l(r N, S, x), (5) implies l(r N, S, z) > l(r N, S, x), contradicting (3) with y = x. Analogously, if l(r N, S, z) < l(r N, S, x), (5) implies l(r N, S, z) > l(r N, S, x), again contradicting (3) with y = x. Therefore, l(r N, S, x) l(r N, S, z) l(r N, S, y) and l(r N, S, x) l(r N, S, z) l(r N, S, y) for all y S, hence x f U (R N, S) and x f U (R N, S) and thus x f U (R N, S) f U (R N, S). Next, we prove that individual-equality independence is satisfied by f U. Let N N, R N R N, S X, i N and R i R i be such that f U (R i, S) = f U (R i, S). Thus, for all x S, x f U ( (R N\{i}, R i), S ) l(r N\{i}, S, x) + l(r i, S, x) l(r N\{i}, S, y) + l(r i, S, y) for all y S l(r N\{i}, S, x) + l(r i, S, x) l(r N\{i}, S, y) + l(r i, S, y) for all y S x f U (R N, S). Single-agent monotonicity follows because, by definition, f U (R i, {x, y}) = {x} whenever i N, R i R i and x, y X with x y and xr i y. Finally, single-agent expansion is satisfied because, by definition, f U (R i, S {x}) = f U (R i, S) {x} for all i N, for all R i R i, for all S X such that S > 1 and for all x X \ S such that xr i y for all y S. Only if. Now suppose that f is a collective choice correspondence that satisfies the axioms. Let N N, R N R N and S X. First, consider the case N = 1 where N = {i} with i N. If S = 1 with S = {x}, we obtain f(r i, {x}) = {x} = f U (R i, {x}) because f(r i, {x}) is non-empty. If S = 2, suppose that S = {x, y} and xr i y. By single-agent monotonicity, it follows that f(r i, S) = f(r i, {x, y}) = {x} = f U (R i, S). 6
If S > 2, let S = {x 1,..., x k 1, x k } be such that x 1 R i... R i x k 1 R i x k. By single-agent monotonicity, we obtain f(r i, {x k 1, x k }) = {x k 1 }. By (repeated if necessary) application of single-agent expansion, it follows that f(r i, S) = {x 1,..., x k 1 } = {x S l(r i, S, x) l(r i, S, y) for all y S} = f U (R i, S). This completes the proof in the single-agent case. Now suppose that N > 1. If there exists x S such that, for all i N, there exists y i S \ {x} such that xr i y i, it follows that l(r N, S, x) = 0 and, thus, l(r N, S, x) l(r N, S, y) for all y S. Using the argument employed in the case N = 1, it follows that f(r i, S) = {z S l(r i, S, z) l(r i, S, y) for all y S} = f U (R i, S) for all i N. By assumption, there exists at least one alternative x that is in f(r i, S) for all i N and, thus, it follows that i N f(r i, S). Furthermore, all z S such that, for all i N, there exists y i S \ {z} such that zr i y i are elements of this intersection. By (repeated if necessary) application of reinforcement, f(r N, S) = i N f(r i, S) = i N {z S l(r i, S, z) l(r i, S, y) for all y S} = {z S l(r N, S, z) l(r N, S, y) for all y S} = f U (R N, S). To complete the proof, consider the case in which N > 1 and, for all x S, there exists i N such that yr i x for all y S \ {x} that is, every alternative in S is located at the bottom of at least one individual ranking which, in turn, implies that l(r N, S, x) > 0 for all x S. Furthermore, note that this implies that we must have at least S agents that is, N S. Select a subset N 1 of S individuals from N such that, for each x S, there is exactly one i x N 1 such that yr ix x for all y S \ {x}. Define a profile R 1 N = (R 1 1 i ) i N 1 so that the restriction of this profile to S has the following properties. For all x S, let yri 1 x x for all y S \ {x} and, moreover, let the profile be fully symmetric in the sense that each element of S appears in each position exactly once that is, for each x S and for each rank k {1,..., S }, there exists an individual j x N 1 such that x appears in the k th position in Rj 1 x. By the argument used in the N = 1 case, it follows that f(ri 1, S) = f(r i, S) = f U (R i, S) for all i N 1. By repeated application of individualequality independence, we obtain f(r N 1, S) = f(r 1 N 1, S). (6) Because of the full symmetry in R 1 N 1, anonymity and neutrality together imply that f(r 1 N 1, S) = S 7
and, by (6), it follows that f(r N 1, S) = S. If N 1 = N, it follows immediately that f(r N, S) = f(r N 1, S) = S = f U (R N, S) and we are done. If N 1 N, consider the set N \ N 1. If, for all x S, there exists i N \ N 1 such that yr i x for all y S \ {x}, select a subset N 2 of N \ N 1 in the same way N 1 was selected from N, and define a corresponding profile R 2 N = (R 2 2 i ) i N 2 analogously. Again, we obtain f(r N 2, S) = S by applying individual-equality independence, anonymity and neutrality as required. Because N is finite, this process can be repeated until we reach a number K N \ {1} and pairwise disjoint sets N 1,..., N K N such that f(r N 1, S) =... = f(r N K, S) = S and not all alternatives in S appear at the bottom of some individual preference R i with i N \ K k=1 N k. If K k=1 N k = N, it follows that K k=1 f(r N k, S) = S and, by repeated application of reinforcement, we obtain f(r N, S) = K k=1f(r N k, S) = S = f U (R N, S) and we are done. If K k=1 N k N, the above procedure can be applied as many times as necessary with successively smaller subsets of S to arrive at a number M N and sets N K+1,..., N K+M such that K+M k=1 N k = N and S = f(r N 1, S)... f(r N K+M, S) = K+M k=1 f(r N k, S) and, by repeated application of reinforcement, it follows that which completes the proof. f(r N, S) = f(r N K+M, S) = K+M k=1 f(r N k, S) = f U (R N, S) Considering the complexity of the last case in this proof (where N > 1 and l(r N, S, x) > 0 for all x S), it may be useful to illustrate the requisite steps by means of an example. Example 1 Suppose that S = {x, y, z, w}, N = {1,..., 10} and the restriction of the profile R N to S is such that individuals 1, 2 and 9 have x at the bottom of their ranking; individuals 3 and 4 have y at the bottom of their ranking; individuals 5 and 6 have z at the bottom of their ranking; individuals 7, 8 and 10 have w at the bottom of their ranking. Clearly, in this example, every element of S appears at least once at the bottom of an individual preference. In accordance with the above proof, we define N 1 = {1, 3, 5, 7} so that each alternative in S appears exactly once at the bottom of one of the preferences in R N 1. Define R 1 N = (R 1 1 i ) i N 1 to be fully symmetric; for instance, let yp 1 1 zp 1 1 wp 1 1 x; zp 1 3 wp 1 3 xp 1 3 y; wp 1 5 xp 1 5 yp 1 5 z; xp 1 7 yp 1 7 zp 1 7 w. 8
By anonymity and neutrality, it follows that f(r 1 N 1, S) = S = {x, y, z, w}. Furthermore, using the result for the one-agent case, we have f(r i, S) = f(r 1 i, S) = f U (R i, S) for all i N 1. By repeated application of individual-equality independence, it follows that f(r N 1, S) = f(r 1 N 1, S) = S. (7) The set N \N 1 is given by {2, 4, 6, 8, 9, 10} and, again, every element in S appears at least once at the bottom of one of the requisite individual relations. Letting N 2 = {2, 4, 6, 8}, the same argument as above can be employed to conclude that f(r N 2, S) = f(r 2 N 2, S) = S. (8) We obtain N \ (N 1 N 2 ) = {9, 10} and, this time, there exist two alternatives (namely, y and z) that do not appear at the bottom of R 9 or of R 10. Let N 3 = {9, 10}. By definition, and l(r N 3, S, y) = l(r N 3, S, z) = 0 l(r N 3, S, x) = l(r N 3, S, w) = 1. Furthermore, f(r 9, S) = {y, z, w} and f(r 10, S) = {x, y, z} by the one-agent result. Therefore, f(r 9, S) f(r 10, S) = {y, z} and, by reinforcement, it follows that We have N = N 1 N 2 N 3 and, combining (7), (8) and (9), f(r N 3, S) = {y, z}. (9) f(r N 1, S) f(r N 2, S) f(r N 3, S) = S S {y, z} = {y, z}. Repeated application of reinforcement yields as desired. f(r N, S) = f(r N 1, S) f(r N 2, S) f(r N 3, S) = {y, z} = f U (R N, S) 4 Discussion The primary objective of this paper is to formulate the Benthamite greatest-happiness principle in an informational framework that involves ordinal and interpersonally noncomparable utilities only. Moreover, by phrasing its least-unhappiness counterpart in the same informational setting, we are able to allow for a clear-cut comparison of these two fundamentally conflicting principles. Note that, according to the traditional interpretation of Bentham s phrase, utilitarianism results as a consequence of the greatest-happiness principle, whereas the closest analogue to the least-unhappiness principle amounts to the Rawlsian maximin criterion. The reason why a comparison of these two principles is by no means a straightforward task is that they are based on rather different assumptions regarding the measurability and interpersonal comparability of individual utilities. Utilitarianism 9
is well-defined if utilities are cardinally measurable and interpersonally unit-comparable, whereas the maximin criterion is well-defined in the presence of ordinal level comparability. The common informational assumption of ordinal measurability and interpersonal non-comparability that we employ in this paper allows us to phrase both principles in the same informational setting. From a formal perspective, the greatest-happiness principle and its least-unhappiness counterparts are not new; they correspond to the well-known plurality rule and the antiplurality rule. However, the road that we take to arrive at these rules is quite different from what has been done in the earlier literature. In addition, while there are some formal similarities, both our characterization of f U and its proof are new and rather different from what can be found in the literature on the anti-plurality rule. To illustrate how our approach differs from the earlier contributions, we conclude this paper by analyzing the features that set our observations apart from those that are currently available. Baharad and Nitzan (2005) use anonymity, neutrality, reinforcement, a continuity property employed by Richelson (1978) and a minimal-veto condition in order to arrive at the anti-plurality (or inverse plurality) rule which corresponds to our rule f U. The continuity property is required because Baharad and Nitzan (2005) invoke Young s (1975) theorem on scoring functions, as does Richelson (1978) in his result; see also our earlier discussion of Ching s (1996) improvement upon Richelson s (1978) characterization. We proceed without continuity and, moreover, the remaining axioms we use differ from the minimal-veto property. Barberà and Dutta (1982) also employ anonymity, neutrality and reinforcement in a characterization of the anti-plurality rule. Their result is quite different in nature from ours in that they focus on additional properties that are related to the implementation of collective choice correspondences. In addition to anonymity, neutrality and reinforcement, they use a monotonicity axiom akin to Maskin monotonicity (Maskin, 1999) and a topinvariance property to arrive at f U. As observed by Merlin (2003), a parallel result for f H is obtained by Merlin and Naeve (1999) who use a bottom-invariance property to characterize the plurality rule; see Merlin (2003) for details. Thus, the common axioms of the two results are anonymity, neutrality, reinforcement and monotonicity. Adding topinvariance characterizes the anti-plurality rule, whereas adding bottom-invariance to the common set of axioms leads to the plurality rule. We chose our specific sets of axioms to illustrate what we think of as an intuitively appealing partition of the constituent parts of the least-unhappiness rule. Our proof technique, which differs from those employed in earlier contributions, may be useful in characterizing alternative collective choice rules in a similar setting; this is a task that we suggest for further work in this area. Appendix In this appendix, we show that the axioms of our theorem are independent. For each of the six properties, we define a collective choice rule that violates this property and satisfies the remaining five. 10
Anonymity. Define, for all N N, for all R N R N and for all S X, { f f(r N, S) = U (R 1, S) if N = {1, 2} and S = 2, f U (R N, S) otherwise. To see that this collective choice correspondence violates anonymity, consider the following example. Let S = {x, y} with x y and let N = {1, 2}. The preferences R 1 and R 2 restricted to S are xr 1 y and yr 2 x. Let π Π be such that π(2) = 3, π(3) = 2 and π(i) = i for all i N \ {2, 3}. Thus, we have N π = {1, 3}, R π(1) = R 1 and R π(2) = R 3. We obtain f(r N, S) = f U (R 1, S) = {x} and f(r Nπ, S) = f U ((R 1, R 3 ), S) = {x, y} {x}, a contradiction to anonymity. Neutrality is satisfied because the labels attached to the alternatives are irrelevant. To show that reinforcement is satisfied, suppose that N, N N, R N R N, R N R N and S X are such that N N = and f(r N, S) f(r N, S). If N and N both differ from {1, 2} or if S 2, the conclusion follows immediately because f U satisfies reinforcement. In all remaining cases, one of the sets N and N is equal to {1, 2}, the other set has an empty intersection with {1, 2}, and S = 2. Without loss of generality, suppose that N = {1, 2}, N N = and S = {x, y}. Furthermore, without loss of generality, suppose that xr 1 y so that f(r N, {x, y}) = f U (R 1, {x, y}) = {x}. It follows that l(r N, {x, y}, x) l(r N, {x, y}, y) (10) because agent 1 considers x to be better than y and there are only two agents in N. Also, we must have f(r N, {x, y}) f(r N, {x, y}) = {x} because this intersection is non-empty. This, in turn, implies that x f(r N, {x, y}) and thus l(r N, {x, y}, x) l(r N, {x, y}, y) (11) because f(r N, {x, y}) = f U (R N, {x, y}) by definition of f. Combining (10) and (11) and using the assumption that the intersection of N and N is empty, it follows that l ((R N, R N ), {x, y}, x) l ((R N, R N ), {x, y}, y) and, because f ((R N, R N ), {x, y}) = f U ((R N, R N ), {x, y}) by definition of f, we obtain f ((R N, R N ), {x, y}) = f U ((R N, R N ), {x, y}) = {x} = f(r N, {x, y}) f(r N, {x, y}) as was to be proven. Individual-equality independence is satisfied because all relevant comparisons according to f involve the respective comparisons according to f U. Finally, single-agent monotonicity and single-agent expansion are satisfied because f coincides with f U for single-agent societies. 11
Neutrality. Let x X and define, for all N N, for all R N R N and for all S X, {x} if S = {x, x} for some x X \ {x } and N 2 and l(r f(r N, S) = N, {x, x}, x ) = l(r N, {x, x}, x) > 0, f U (R N, S) otherwise. To see that this correspondence does not satisfy neutrality, let N = {1, 2}, S = {x, x} for some x X \ {x }, x R 1 x and xr 2 x. Furthermore, let y X \ {x, x} and consider the permutation σ Σ such that σ(x ) = y, σ(y) = x and σ(z) = z for all z X \ {x, x}. By definition, we have y = σ(x ) σ 1 (R 1 ) σ(x) = x and x = σ(x) σ 2 (R 2 ) σ(y) = x. It follows that f(r N, {x, x}) = {x} and f(r N, {x, y}) = f U (R N, {x, y}) = {x, y}, in contradiction to neutrality. Anonymity is satisfied because the identities of the individuals are irrelevant in the definition of this collective choice correspondence. To prove that f satisfies reinforcement, note first that we only have to consider cases in which S = {x, x} for some x X \ {x }; in all other cases, only the second branch of the above definition applies and the conclusion of reinforcement follows because the axiom is satisfied by f U. Moreover, we can restrict attention to cases in which the first branch of the definition applies for N or for N. So suppose that S = {x, x} for some x X \ {x } and, moreover, that N, N N, R N R N and R N R N are such that N N = and f(r N, S) f(r N, S). Up to exchanging the roles of N and N, we have the following two cases left to examine. Case 1. In this case, it follows that N 2 and l(r N, {x, x}, x ) = l(r N, {x, x}, x) > 0 N 2 and l(r N, {x, x}, x ) = l(r N, {x, x}, x) > 0. f(r N, {x, x}) = f(r N, {x, x}) = f(r N, {x, x}) f(r N, {x, x}) = {x} by definition of f. Because l(r N, {x, x}, x ) = l(r N, {x, x}, x) > 0 and l(r N, {x, x}, x ) = l(r N, {x, x}, x) > 0, it follows that and l ((R N, R N ), {x, x}, x) = l(r N, {x, x}, x) + l(r N, {x, x}, x) = l(r N, {x, x}, x ) + l(r N, {x, x}, x ) = l ((R N, R N ), {x, x}, x ) > 0 (recall that N N = ). Thus, because we also have N N 2, the first branch of the definition of f applies and we obtain f ((R N, R N ), {x, x}, x ) = {x} = f(r N, {x, x}) f(r N, {x, x}). 12
Case 2. N 2 and l(r N, {x, x}, x ) = l(r N, {x, x}, x) > 0 N = 1 or l(r N, {x, x}, x ) l(r N, {x, x}, x) or l(r N, {x, x}, x ) = 0. We now obtain and f(r N, {x, x}) = {x} = f(r N, {x, x}) f(r N, {x, x}) = {x} by definition of f and because the above intersection is assumed to be non-empty. Thus, x f(r N, {x, x}) and it follows that l(r N, {x, x}, x) l(r N, {x, }, x ). Hence, l ((R N, R N ), {x, x}, x) = l(r N, {x, x}, x) + l(r N, {x, x}, x) l(r N, {x, x}, x ) + l(r N, {x, x}, x ) = l ((R N, R N ), {x, x}, x ). If l(r N, {x, x}, x) = l(r N, {x, x}, x ), it follows that f ((R N, R N ), {x, x}) = {x} = f(r N, {x, x}) f(r N, {x, x}) irrespective of whether the first or the second branch applies to (R N, R N ). If l(r N, {x, x}, x) < l(r N, {x, x}, x ), it follows that the first branch of the definition of f applies and, as in the previous case, we obtain f ((R N, R N ), {x, x}) = {x} = f(r N, {x, x}) f(r N, {x, x}). Individual-equality independence is satisfied because the axiom only imposes restrictions when there are at least three alternatives and, thus, f coincides with f U in all relevant cases. Single-agent monotonicity and single-agent expansion are satisfied because f coincides with f U whenever only a single agent is present. Reinforcement. Define, for all N N, for all R N R N and for all S X, { f f(r N, S) = U (R N, S) if N = 1 or S = 1, i N f U (R i, S) otherwise. This correspondence does not satisfy reinforcement. Consider the following example. Let S = {x, y, z}, N = {1}, N = {2}, xr 1 yr 1 z and xr 2 zr 2 y. By definition of f, we obtain and f(r N, S) = f(r 1, S) = f U (R 1, S) = {x, y} f(r N, S) = f(r 2, S) = f U (R N, S) = {x, z}. Thus, N N = and f(r N, S) f(r N, S) = {x}. By definition of f, we obtain f ((R N, R N ), S) = f ((R 1, R 2 ), S) = f(r 1, S) f(r 2, S) = {x, y, z} {x} = f(r N, S) f(r N, S), 13
contradicting reinforcement. That anonymity and neutrality are satisfied follows because neither the identities of the individuals nor the labels of the alternatives are relevant in the definition of f. Individualequality independence is satisfied because the order of non-worst alternatives in the profiles is irrelevant. Single-agent monotonicity and single-agent expansion are satisfied because f coincides with f U in the single-agent case. Individual-equality independence. For our next example, some further definitions are required. For all N N, for all R N R N and for all S X, let L 0 (R N, S) = S and define the following S sets recursively. For all j {1,..., S }, let L j (R N, S) = {x L j 1 (R N, S) l(r N, L j 1 (R N, S), x) l(r N, L j 1 (R N, S), y) for all y L j 1 (R N, S)}. That is, L 1 (R N, S) contains the alternatives that minimize the number of agents i who end up with their worst alternative in S according to R i. Furthermore, if S > 1, L j (R N, S) identifies the set of alternatives that minimize the number of agents i who end up with their j-worst alternatives in L j 1 (R N, S) according to R i for all j {2,..., S }. Note that, in this iteration, alternatives that do not appear as minimizers at some stage are eliminated when moving to the following stage. Analogously, for all N N, for all R N R N, for all S X, for all z S and for all j {1,..., S }, define l j (R N, S, z) = {i N yr i z for all y L j 1 (R N, S)}. For future reference, note that, for all N, N N, for all R N R N, for all R N R N, for all S X such that N N = and f(r N, S) f(r N, S), and for all j {1,..., S }, l j ((R N, R N ), S, z) = l j (R N, S, z) + l j (R N, S, z) (12) for all z S because N N =. To define the collective choice correspondence f relevant here, we proceed as follows. (a) If there exists j {1,..., S } such that L j (R N, S) = 1, let f(r N, S) = L j (R N, S), where j = min{j {1,..., S } L j (R N, S) = 1}. (b) If L j (R N, S) > 1 for all j {1,..., S }, let f(r N, S) = L S (R N, S). This correspondence f violates individual-equality independence. Suppose S = {x, y, z}, N = {1, 2, 3}, xr 1 yr 1 z, zr 2 xr 2 y, yr 3 zr 3 x, 14
and yr 1xR 1z. By definition, f(r 1, S) = f(r 1, S) = {x, y}. Furthermore, we have and, thus, L 1 ((R 1, R 2, R 3 ), S) = L 2 ((R 1, R 2, R 3 ), S) = L 3 ((R 1, R 2, R 3 ), S) = {x, y, z} by definition of f. Furthermore, f ((R 1, R 2, R 3 ), S) = L 3 ((R 1, R 2, R 3 ), S) = {x, y, z} L 1 ((R 1, R 2, R 3 ), S) = {x, y, z} and L 2 ((R 1, R 2, R 3 ), S) = {y}. Thus, L 2 ((R 1, R 2, R 3 ), S) = 1 and it follows that f ((R 1, R 2, R 3 ), S) = {y} {x, y, z} = f ((R 1, R 2, R 3 ), S), contradicting individual-equality independence. That anonymity and neutrality are satisfied follows because neither the identities of the individuals nor the labels of the alternatives are relevant in the definition of f. To prove that reinforcement is satisfied, suppose that N, N N, R N R N, R N R N and S X are such that N N = and f(r N, S) f(r N, S). Up to exchanging the roles of N and N, the following three cases exhaust all possibilities. Case 1. There exist j, j {1,..., S } such that L j (R N, S) = 1 and L j (R N, S) = 1. By definition of f, it follows that f(r N, S) and f(r N, S) are singletons and, because the intersection of these two sets is non-empty, each of the two sets must have some x S as its only member. Let and j = min{j {1,..., S } L j (R N, S) = 1} j = min{j {1,..., S } L j (R N, S) = 1}. Without loss of generality, suppose that j j. It follows that f(r N, S) = L j (R N, S) = L j (R N, S) = f(r N, S) = f(r N, S) f(r N, S) = {x}. By definition of f, we have for all j {1,..., j } \ {j }, and l j (R N, S, x) l j (R N, S, y) for all y L j 1 (R N, S), (13) l j (R N, S, x) l j (R N, S, y) for all y L j 1 (R N, S) l j (R N, S, x) < l j (R N, S, y) for all y L j 1 (R N, S) \ {x}, l j (R N, S, x) l j (R N, S, y) for all y L j 1 (R N, S). Thus, using (12), we obtain l j ((R N, R N ), S, x) = l j (R N, S, x) + l j (R N, S, x), l j (R N, S, y) + l j (R N, S, y) = l j ((R N, R N ), S, y) 15
for all j {1,..., j }\{j } and for all y [L j 1 (R N, S) L j 1 (R N, S)]\{x} and, moreover, l j ((R N, R N ), S, x) = l j (R N, S, x) + l j (R N, S, x), < l j (R N, S, y) + l j (R N, S, y) (14) = l j ((R N, R N ), S, y) for all y [ L j 1 (R N, S) L j (R N, S) ] \ {x}. This implies that x f ((R N, R N ), S) by definition. Furthermore, because of the strict inequality in (14), no other element of S can be chosen; see also the argument employed in the if part of the proof of Theorem 2. Thus, as required. f ((R N, R N ), S) = {x} = f(r N, S) f(r N, S) Case 2. There exists j {1,..., S } such that L j (R N, S) = 1 and L j (R N, S) > 1 for all j {1,..., S }. By definition of f, it follows that there exists x S such that f(r N, S) = L j (R N, S) = x, where j = min{j {1,..., S } L j (R N, S) = 1}. Moreover, f(r N, S) = L S (R N, S) and, because f(r N, S) f(r N, S), we have x f(r N, S). Starting with (13), the remainder of the proof is identical to that of case 1. Case 3. L j (R N, S) > 1 for all j {1,..., S } and L j (R N, S) > 1 for all j {1,..., S }. By definition of f, f(r N, S) = L S (R N, S) and f(r N, S) = L S (R N, S). By assumption, f(r N, S) f(r N, S). (15) Now a slightly modified version of the argument employed in case 1 can be used; the only change required is that if the intersection in (15) contains more than one element, the strict-inequality argument applies to all members of the intersection versus all alternatives that are not in this intersection. See again the proof of Theorem 2 for details. Single-agent monotonicity and single-agent expansion are satisfied because f coincides with f U in the single-agent case. Single-agent monotonicity. Let, for all N N, for all R N R N and for all S X, f(r N, S) = S. This collective choice correspondence does not satisfy single-agent monotonicity. Let x, y X be distinct alternatives such that xr 1 y. By definition, we have f(r 1, {x, y}) = {x, y} {x}. That all other axioms are satisfied is immediate. 16
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