Independence of Irrelevant Opinions and Symmetry Imply Liberalism

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1 Independence of Irrelevant Opinions and Symmetry Imply Liberalism Wonki Jo Cho and Biung-Ghi Ju March 15, 2014 Abstract There are at least three groups and each person has to be identied as a member of one of these groups. A social decision rule associates the memberships for the groups with each prole of individual opinions on who belong to what groups. We introduce an axiom termed independence of irrelevant opinions, sharing the spirit of Arrow's independence of irrelevant alternatives. It requires that the membership for each group should be decided based solely on the opinions on who belong to that group and who do not (that is, independently of opinions on who belong to the other groups). We show that this axiom, together with symmetry and non-degeneracy, implies the liberal decision rule (each person self-determines her own membership). JEL Classication Numbers: C0; D70; D71; D72 Key Words: group identication; independence of irrelevant opinions; symmetry; liberalism School of Social Sciences, University of Manchester, Oxford Road, Manchester, United Kingdom, M13 9PL; jo.cho@manchester.ac.uk Department of Economics, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, South Korea, ; bgju@snu.ac.kr 1

2 1 Introduction Group identication has drawn the attention of several authors in recent years since the two seminal contributions by Kasher and Rubinstein (1997) and Samet and Schmeidler (2003); e.g., Sung and Dimitrov (2005), Houy (2007), Miller (2008), Çengelci and Sanver (2010), Ju (2010, 2013), etc. Most authors in the literature focus on the issue of binary identication, identication of two groups, say, the group of J's and the group of non- J's. In practice, however, we often face the issue of classifying individuals into three or more groups. For example, in countries with diverse ethnic or racial backgrounds such as China, India, Russia, the United Kingdom, and the United States of America, ethnic or racial classications are used evaluate social institutions and public policies in terms of equal opportunity and anti-discrimination. In the group identication literature, to our knowledge, no study investigates the problem of classifying individuals into three or more groups. One may think that the binary model is general enough to deal with the multinary model; applying a binary identication rule to all groups will do. But the feasibility of such proposal is not guaranteed. For example, the majority rule cannot be extended, as proposed above, to handle three or more groups (a person may not belong to any group, e.g., none of J or K or non-j and non-k). We nd that there is a signicant qualitative dierence between the binary and multinary setups. This is somewhat reminiscent of the dierence between Arrovian preference aggregation with two alternatives and with three or more alternatives. We introduce an independence axiom that is similar to Arrow's independence of irrelevant alternatives (Arrow, 1951). It is vacuous in the binary model but turns out to be quite demanding in the multinary model. Combined with the basic condition of nondegeneracy (there should be no agent who is always put in one group), it characterizes a small family of rules. Further, when symmetry (the names of agents should not matter; Samet and Schmeidler, 2003) is imposed in addition, only the liberal rule (each person self-determines her identication) satises it. This result stands in contrast with Samet and Schmeidler (2003), where they characterize a large family of rules, including the liberal rule and other democratic rules that count everyone's vote equally. In our model, there are three or more groups, and each person needs to be identied 2

3 as a member of one of the groups. Taking as input individual opinions on who belong to what groups, a social decision rule determines memberships for the groups. We can formulate a variant of Arrow's independence of irrelevant alternatives as follows: the membership for each group should be decided based solely on opinions on who belong to that group and who do not (that is, independently of opinions on who belong to the other groups). We call this requirement independence of irrelevant opinions. It is vacuous in the binary identication model, as is Arrow's independence axiom when there are only two alternatives. Our main goal is to investigate implications of independence of irrelevant opinions together with non-degeneracy. We show that each rule satisfying independence of irrelevant opinions and non-degeneracy identies each person using only one vote (a single entry in the opinion prole) (Theorem 1). We call them one-vote rules, noting the connection with the one-vote rules in the binary identication model (Miller, 2008). Adding symmetry, we characterize the liberal rule (Theorem 2). We believe that this is a new and interesting case in Social Choice Theory where an independence axiom, similar to Arrow's independence of irrelevant alternatives, together with standard symmetry and non-degeneracy, implies liberalism. In the literature on preference aggregation, Arrow's independence axiom together with a few fairly mild axioms implies quite an unequal allocation of decision powers: only a single person or a single group has a decisive power (Arrow, 1951; Guha, 1972; Mas-Colell and Sonnenschein, 1972). Similar results are available in the literature on the aggregation of equivalence relations by Fishburn and Rubinstein (1986a, 1986b) and Dimitrov, Marchant and Mishra (2012), where they consider a variant of Arrow's independence axiom (Dimitrov, Marchant, and Mishra call it binary independence). Our independence axiom provides a somewhat dierent outcome since the rules satisfying it do not necessarily assign decision powers unequally; it is possible to avoid a dictator or an oligarch. In particular, in the case of the liberal rule, everyone has the equal power of self-determination. Closely related to our approach is Miller (2008). He studies binary identication problems in a model where the group under question can vary. He characterizes the same family of one-vote rules as we do here but his results are based on the axiom of consistent decision making across groups with regard to the conjunction and dis- 3

4 junction of groups (J and K, J or K). 1 Extending our multinary model to deal with conjunction or disjunction of basic groups, we nd that his consistency axiom can easily be satised: rst, identify groups J and K; then identify other derived groups such as groups J and K and J or K consistently with conjunction and disjunction of the two basic groups. Another notable feature of Miller (2008) is that his binary decision rule, when applied to the multinary setup, determines the membership for each group based only on the opinions on that group. This property, which we call component-wise independence, is similar to and yet much stronger and less appealing than independence of irrelevant opinions. For example, when determining the membership for group J and K, component-wise independence requires us to ignore the opinions on group J, which is quite relevant to the decision on group J and K. On the other hand, independence of irrelevant opinions requires that decisions be made independently only across basic groups (such as groups J and K), not derived groups. Our multinary setup allows us to uncover component-wise independence from Miller's binary setup and show that it is, in fact, the key axiom that yields (even without consistency) the decisive votes featuring the one-vote rules. Applying our main results, we show in Section 5 that while there is a wide variety of well-behaved consistent rules, the family of component-wise independent rules is highly restricted and is somewhat close to the one-vote rules. One important reason why independence of irrelevant opinions in our results has as strong an implication as component-wise independence and consistency together is that decisions in our model identify each person as a member of exactly one basic group; thus, decisions partition the set of persons into basic groups (there is no one who does not belong to any group). When the partitioning property in our model is relaxed slightly (e.g., allowing for the possibility of not belonging to any basic groups), a large family of rules are independent of irrelevant opinions and non-degenerate. In Section 2, we introduce the model, denitions, and axioms for group identication. 1 He calls the axiom separability. Meet separability requires the equivalence of (i) the conjunction of the two decisions for group J and group K; and (ii) the decision for group J and K. Join separability requires the equivalence of (i) the disjunction of the two decisions for group J and group K; and (ii) the decision for group J or K. 4

5 We establish some preliminary results in Section 3 and the main characterization results in Section 4. In Section 5, we construct an extended model to deal with the identication of both basic and derived groups. We discuss some extensions and applications of the main results. Finally, we conclude with several remarks in Section 6. 2 The Model There are n persons, each of whom needs to be identied as a member of one among m groups. Let N {1,, n} be the set of persons and G {1,, m} the set of groups. We assume, unless noted otherwise, that n 2 and m 3. Each person i N has an opinion on who they believe to be members of these groups, represented by P i (P ij ) j N G N, where for all j N and all k G, P ij = k means that person i believes person j to be a member of group k. These individual opinions P 1,, P n constitute a (identication) problem P (P ij ) i,j N, an n n matrix. Let P G N N be the set of all identication problems. A decision is a prole x (x i ) i N G N, where, for all i N and all k G, x i = k means that person i belongs to group k. A social decision rule, briey a rule f : P G N associates with each problem in P a decision. For example, a plurality rule puts each person i in group k with the most votes; i.e., for all k G, {j N : P ji = k} {j N : P ji = k } and when the inequality holds with equality, the tie-breaking rule of k k is satised. Dierent tie-breaking rules lead to dierent plurality rules. The liberal rule puts each person in the group she classies herself into; i.e., for all P P and all i N, i belongs to group P ii. When m = 2, our model reduces to the standard, binary group identication model (Kasher and Rubinstein, 1997): in essence, there is only one group and each person has an opinion about who belongs to the group. Axioms We rst introduce an axiom that is unique to the multinary setup. Consider two problems, P and P, such that all persons agree on the membership for group k. They may dier on the membership for other groups but if this dierence is viewed as irrelevant 5

6 when making a decision on who belong to group k, it is natural to require that f(p ) and f(p ) agree on the group-k membership. Independence of Irrelevant Opinions. Let P, P P and k G. Suppose that for all i, j N, P ij = k if and only if P ij = k. Then for all i N, f i (P ) = k if and only if f i (P ) = k. We also consider the following axioms well studied in the binary identication literature. Given a permutation π : N N, for all P P, let P π (P π(i),π(j) ) i,j N and f π (P ) (f π(i) (P )) i N. Permutation π changes the names of persons and P π is the problem obtained by changing names through π. Name changes are only nominal and involve no shift in the fundamental contents. Thus, it is reasonable to require that the decision be unaected by such nominal changes (Samet and Schmeidler, 2003). Symmetry. For all P P and all permutations π : N N, f(p π ) = f π (P ). Our next axiom concerns the decision for unanimous opinion proles: if all persons consider all persons belonging to one group, say group k, then all persons should be classied into group k. Unanimity. For all k G, f(k n n ) = k 1 n. A rule may be degenerate for a person in that it always classies her into one group. We require that such degeneracy occur for no person. Clearly, this is weaker than unanimity. Non-Degeneracy. For all i N, there are P, P P such that f i (P ) f i (P ). 3 Independence of Irrelevant Opinions and Decomposability A problem contains binary information on the memberships for all groups. Thus, we may decompose the problem into multiple binary problems, obtain the binary decisions for the latter, and combine them into a single decision. The combined decision may or may not be the same as the decision a rule assigns to the initial problem. In 6

7 this section, we establish that independence of irrelevant opinions is almost equivalent to requiring that the two decisions be the same. More precisely, let B {0, 1} N N. Given P P, for all k G, let B P,k be the binary problem concerning group k derived from P ; i.e., for all i, j N, (i) if P ij = k, then B P,k ij = 1; and (ii) if P ij k, then B P,k ij = 0. By denition, for all P P, P = k G kbp,k. An (binary) approval function ϕ: B {0, 1} N associates with each binary problem B B a binary decision, namely, a prole of 0's and 1's, where for all i N, ϕ i (B) = 0 means the disapproval of i's membership and ϕ i (B) = 1 means the approval of i's membership. For all binary problems B B, let B 1 n n B be the dual problem of B. Likewise, for all binary decisions x {0, 1} N, let x 1 1 n x be the dual decision of x. B Using these denitions, each problem P P can be decomposed into m binary problems, B P,1,, B P,m. The next axiom requires that the decision for problem P be identical to the combination of the binary decisions for the m binary problems assigned by an approval function. Decomposability. There is an approval function ϕ such that for all P P, all i N, and all k G, f i (P ) = k if and only if ϕ i (B P,k ) = 1. In this case, we say that f is represented by ϕ. Note that by decomposability, for all P P, f(p ) = k G kϕ(bp,k ). We show that an approval function representing a decomposable rule satises the following properties. problems B 1,, B m B, Approval function ϕ is m-unit-additive if for all m binary k G B k = 1 n n implies k G ϕ(b k ) = 1 1 n. (1) It is unanimous if ϕ(0 n n ) = 0 1 n and ϕ(1 n n ) = 1 1 n. The dual of ϕ, denoted by ϕ d, is the approval function such that for all B B, ϕ d (B) = ϕ(b). We say that ϕ is self-dual if ϕ = ϕ d. Finally, ϕ is monotonic if for all B, B B such that B B, ϕ(b) ϕ(b ). Proposition 1. An approval function represents a decomposable rule if and only if it is 7

8 m-unit-additive. Also, if an approval function is m-unit-additive, then it is unanimous, self-dual, and monotonic. Proof. First, we prove the if and only if statement. Note that for all P P, k G BP,k = 1 n n and that for all B 1,, B m B with k G Bk = 1 n n, there is P P such that B 1 = B P,1,, B m = B P,m. This observation is enough to prove the if part. Next, suppose that an approval function ϕ can represent a decomposable rule. Then for all P P and all i N, there is exactly one k G with ϕ i (B P,k ) = 1. Since for all k G\{k}, ϕ i (B P,k ) = 0, k G ϕ i(b P,k ) = 1. Thus, ϕ is m-unit-additive. Assume, henceforth, that ϕ is m-unit-additive. To prove that ϕ is unanimous, let i N. Let B 1 1 n n, and for all k G\{1}, B k 0 n n. Let s ϕ i (1 n n ) and t ϕ i (0 n n ). Since k G Bk = 1 n n, 1 = k G ϕ i ( B k ) = s + (m 1)t. Since s, t {0, 1} and m 3, it follows that s = 1 and t = 0. To prove that ϕ is self-dual, let B B. Let B 1 B, B 2 B, and for all k G\{1, 2}, B k 0 n n. Since k G Bk = 1 n n, then by m-unit-additivity and unanimity, 1 1 n = k G ϕ ( B k) = ϕ(b) + ϕ(b). This gives ϕ(b) = ϕ(b). Finally, to prove that ϕ is monotonic, let B, B B be such that B B. Let i N. If ϕ i (B) = 0, then trivially, ϕ i (B) ϕ i (B ). Thus, assume that ϕ i (B) = 1. Let B 1 = B and B 2 = B. Let B 3,, B m B be such that k G Bk = 1 n n (such B 3,, B m exist because B B and B 1 + B 2 = B + B 1 n n ). Since k G ϕ i ( B k ) = 1 and ϕ i (B 1 ) = 1, 0 = ϕ i (B 2 ) = ϕ i (B ). Since ϕ is self-dual, ϕ i (B ) = ϕ i (B ) = 1. If f is independent of irrelevant opinions, then it can be represented by m approval functions (ϕ k ) k G. To see this, let k G. Dene an approval function ϕ k : B {0, 1} N as follows: for all B B and all i N, ϕ k i (B) = 1 if and only if for some P P, B P,k = B and f i (P ) = k. Clearly, by independence of irrelevant opinions, ϕ k is welldened, 2 and f is represented by (ϕ k ) k G ; i.e., for all P P, all i N, and all k G, f i (P ) = k if and only if ϕ k i (B P,k ) = 1. When all the approval functions are identical (ϕ 1 = = ϕ m ), f is decomposable. Therefore, decomposability implies independence of irrelevant opinions. The converse does not hold. As we show below, the essential 2 By independence of irrelevant opinions, for all P, P P such that B P,k = B P,k = B, f i (P ) = k if and only if f i (P ) = k. 8

9 dierence between the two axioms is non-degeneracy. To prove it, we use the following lemma. Lemma 1. Independence of irrelevant opinions and non-degeneracy together imply unanimity. Proof. Let f be a rule satisfying independence of irrelevant opinions and non-degeneracy. Then there are approval functions (ϕ k ) k G representing f. Now we proceed in three steps. Step 1: For all i N, all P P, and all l G\{f i (P )}, ϕ l i(0 n n ) = 0. Let i N and P P. Let k f i (P ). Let l, h G\{k} be distinct. Let P P be such that for all j, j N, (i) P jj = k if and only if P jj = k; and (ii) P jj = h if and only if P jj k. By independence of irrelevant opinions, f i (P ) = f i (P ) = k, so that f i (P ) l. Then ϕ l i(0 n n ) = ϕ l i(b P,l ) = 0. Step 2: For all i N and all k G, ϕ k i (0 n n ) = 0. Let i N. By non-degeneracy, there are P, P P such that f i (P ) f i (P ). Let k f i (P ) and l f i (P ). By Step 1, for all h G\{k}, ϕ h i (0 n n ) = 0. Similarly, for all h G\{l}, ϕ h i (0 n n ) = 0. Step 3: f is unanimous. Suppose, by contradiction, that for some i N and k G, f i (k n n ) k. Let l f i (k n n ). Then ϕ l i(0 n n ) = ϕ l i(b k n n,l ) = 1, contradicting Step 2. Now we prove the logical relation between independence of irrelevant opinions and decomposability. Proposition 2. The combination of independence of irrelevant opinions and nondegeneracy is equivalent to decomposability. Proof. We already noted that decomposability implies independence of irrelevant opinions. When a rule is decomposable, by Proposition 1, the approval function representing it is unanimous. Therefore, the rule is also unanimous, and so non-degenerate. To prove the converse, let f be a rule satisfying independence of irrelevant opinions and non-degeneracy. Then f is represented by a prole of m approval functions (ϕ k ) k G. By Lemma 1, f is unanimous. Now we proceed in two steps. 9

10 Step 1: For all i N and all P P, f i (P ) is one of the entries of P. Suppose, by contradiction, that for some i N and P P, f i (P ) is not one of the entries of P ; i.e., for some k such that B P,k = 0 n n, f i (P ) = k. Let l G be one of the entries of P and consider l n n P. Then B P,k = 0 n n = B ln n,k. Thus applying independence of irrelevant opinions to P and l n n, f i (P ) = k implies f i (l n n ) = k, which contradicts unanimity. Step 2: ϕ 1 = ϕ 2 = = ϕ m. Suppose, by contradiction, that there are k, l G such that ϕ k ϕ l. Then there are B B and i N such that ϕ k i (B) ϕ l i(b). Without loss of generality, assume that ϕ k i (B) = 0 and ϕ l i(b) = 1. Let h G\{k, l}. Let P P be such that for all j, j N, (i) P jj = h if and only if B jj = 0; and (ii) P jj = k if and only if B jj = 1. Similarly, let P P be such that for all j, j N, (i) P jj = h if and only if B jj = 0; and (ii) P jj = l if and only if B jj = 1. By construction, BP,k = B P,l = B. Since ϕ k i (B) = 0 and ϕ l i(b) = 1, it follows that f i (P ) k and f i (P ) = l. By Step 1, f i (P ) k implies f i (P ) = h. Note that B P,h = B P,h. Hence, applying independence of irrelevant opinions to P and P, f i (P ) = h implies f i (P ) = h, which contradicts f i (P ) = l. Remark 1. In Proposition 2, non-degeneracy cannot be weakened any further without compromising the equivalence; i.e., for any axiom A weaker than non-degeneracy, the combination of independence of irrelevant opinions and A is not equivalent to decomposability. To see this, suppose, by contradiction, that for some axiom A weaker than non-degeneracy, the combination of independence of irrelevant opinions and A is equivalent to decomposability. For all i N, let f i be a rule such that (i) person i is always classied into group 1; and (ii) for all other persons, f i is liberal (i.e., for all P P, (i) fi i (P ) = 1; and (ii) for all j N\{i}, fj(p i ) = P jj ). Thus, f i violates non-degeneracy only for person i. Now consider the collection of rules (f i ) i N. Since axiom A is weaker than non-degeneracy, there is i N such that f i satises A. It is easy to check that f i is independent of irrelevant opinions but not decomposable, a contradiction. Note that by Proposition 2 and Lemma 1, decomposability also implies unanimity. 10

11 4 Main Results In this section, we present our main characterization results. We rst characterize the rules satisfying independence of irrelevant opinions and non-degeneracy. These rules are represented by the one-vote approval functions that Miller (2008) introduces in the binary identication model. An approval function ϕ is a one-vote approval function if for all i N, there are j, h N such that for all B B, ϕ i (B) = 1 if and only if B jh = 1. A rule f is a one-vote rule if for all i N, there are j, h N such that for all P P, f i (P ) = P jh. The one-vote rules are decomposable, represented by one-vote approval functions; moreover, they are the only decomposable rules. Theorem 1. The following are equivalent. (i) A rule is independent of irrelevant opinions and non-degenerate; (ii) A rule is decomposable; (iii) A rule is a one-vote rule. Proof. By Proposition 2, we only have to show the equivalence between (ii) and (iii). We only prove the non-trivial implication, (ii) implies (iii). Consider a decomposable rule represented by an approval function ϕ. It suces to show that ϕ is a one-vote approval function. We use the following notation in the proof. Let B B. Let B i,j N B ij be the number of 1's in B. Also, B is a unit binary problem if B = 1. For all i, j N, let U ij B be the unit binary problem such that U ij ij We proceed in two steps. Step 1: There are j, h N such that ϕ i ( U jh ) = 1. = 1. Now let i N. By Proposition 1, ϕ is m-unit-additive, unanimous, self-dual, and monotonic. We distinguish two cases. Case 1: m n 2. Consider m binary problems, consisting of n 2 unit problems (i.e., {U jh : j, h N}) and m n 2 zero problems 0 n n. The sum of these m problems equals 1 n n. Thus, by m-unit-additivity and unanimity, j,h N ϕ i(u jh ) = 1, which implies the desired conclusion. Case 2: m < n 2. 11

12 Suppose, by contradiction, that for all B B, B = 1 implies ϕ i (B) = 0. (2) We prove by induction that for all B B, ϕ i (B) = 0. Let l < n 2 and assume that for all B B, B l implies ϕ i (B) = 0. (3) Let B B be such that B = l + 1. Then B = n 2 l 1 and there are B 1 and B 2 such that B 1 = 1, B 2 = l and B 1 + B 2 + B = 1 n n. By m-unit-additivity and unanimity, ϕ i (B 1 ) + ϕ i (B 2 ) + ϕ i ( B) = 1. Since by the induction hypothesis (3), ϕ i (B 1 ) = ϕ i (B 2 ) = 0, we obtain ϕ i ( B) = 1. By self-duality, ϕ i (B) = 0. Hence, for all B B, B l + 1 implies ϕ i (B) = 0. Therefore, (2) and the induction argument prove that for all B B, ϕ i (B) = 0. In particular, ϕ i (1 n n ) = 0, which contradicts unanimity of ϕ. Step 2: For all B B, ϕ i (B) = 1 if and only if B jh = 1. Let j, h N be such that ϕ i (U jh ) = 1. Let B B. If B jh = 1, then since B U jh, monotonicity implies that ϕ i (B) ϕ i ( U jh ) = 1. If B jh = 0, then since B U jh, monotonicity and self-duality imply that ϕ i (B) ϕ i ( U jh ) = 0. rule. When n 3, among the one-vote rules, there is only one symmetric rule: the liberal Theorem 2. Assume that there are at least three persons (n 3). Then the following are equivalent. (i) A rule is independent of irrelevant opinions, non-degenerate, and symmetric; (ii) A rule is decomposable and symmetric; (iii) A rule is the liberal rule. Proof. We prove that (ii) implies (iii). Let f be a rule satisfying decomposability and symmetry. By Theorem 1, it is a one-vote rule. Then there is a function h: N N N 12

13 such that for all P P and all i N, f i (P ) = P h(i). Now symmetry implies that h satises the following: for all permutations π : N N and all i N, h(π(i)) = (π(h 1 (i)), π(h 2 (i))). (4) It is enough to show that for all i N, h(i) = (i, i). Now suppose, by contradiction, that there is i N such that h(i) (i, i). Let (j, k) h(i). Without loss of generality, assume that k i. Since n 3, there is l N\{i, k}. Let π : N N be the transposition of k and l. Then h(π(i)) = h(i) = (j, k) but (π(h 1 (i)), π(h 2 (i))) = (π(j), π(k)) = (j, l), contradicting (4). Remark 2. When n = 2, there are other, non-liberal one-vote rules satisfying the axioms in Theorem 2. In fact, parts (i) and (ii) of Theorem 2 are equivalent to the following statement: (iii ) the rule f is (a) the liberal rule; or (b) such that for all P P, f(p ) = (P 21, P 12 ); or (c) such that for all P P, f(p ) = (P 12, P 21 ); or (d) such that for all P P, f(p ) = (P 22, P 11 ). Therefore, when there are only two persons, four rules satisfy the axioms in parts (i) or (ii). Proof of Remark 2. Let n = 2. As in the proof of Theorem 2, we can obtain a function h : N N N. By symmetry, h satises equation (4). Since n = 2, h(1) determines h(2) as well: letting π : N N be the transposition of 1 and 2, it follows that h(2) = h(π(1)) = (π(h 1 (1)), π(h 2 (1))). For instance, if h(1) = (1, 2), then h(2) = (2, 1). Since h(1) {(1, 1), (1, 2), (2, 1), (2, 2)}, we can dene h in four dierent ways, thus obtaining the four rules in Remark 2. 5 Extended Model and Applications We now consider the extended framework where the issues arise not only of identifying who are J and who are K, but also of simultaneously identifying who are J and K, who are J or K, who are non-j and K, who are J and non-k, and who are non-j and non-k. A reduced form of this model focusing on binary identication is studied by Miller (2008). Here we extend his model to deal with multinary identication. 13

14 In what follows, we call G the set of basic groups. Basic groups are not necessarily mutually exclusive (i.e., a person can belong to two or more groups). Let G be the Boolean algebra generated by the set of basic groups G through conjunction ( ), disjunction ( ), and negation ( ). Let 1 G be the group everyone and 0 G the group no one. We call the groups in G\G derived groups. A derived group is complete if it is a conjunction of qualication or disqualication for all basic groups in G. For example, when G = {a, b}, there are four complete groups, which are a b, a b, a b, and a b. Let G c be the set of complete groups. An extended problem B (B g ) g G B G is a prole of binary problems for all groups in G satisfying the following property: for all g, g G, B g g = B g B g, B g g = B g B g, B 1 = 1 n n, and B 0 = 0 n n (so B g = 1 n n B g ). Call this property opinion-consistency. Let B be the set of all extended problems. Note that by opinion-consistency, a prole (B g ) g G of binary problems for all basic groups generates an (unique) extended problem. An extended decision x (x g ) g G ({0, 1} N ) G is a prole of decisions for all groups in G with x 1 = 1 1 n and x 0 = 0 1 n. An extended decision x (x g ) g G satises meet-consistency if for all g, g G, x g g = x g x g. It satises join-consistency if for all g, g G, x g g = x g x g. Let (x k ) k G be a prole of decisions for all basic groups and call it a basic group decision. Then (x k ) k G generates an (unique) extended decision satisfying the two consistency properties. An extended rule F : B ( {0, 1} N)G maps each extended problem into an extended decision. For all B B and all g G, let F g (B) be the group-g decision for B assigned by the extended rule F. The extended rule F satises meet-consistency (or join-consistency, respectively) if it always produces a meet-consistent (or joinconsistent) decision; i.e., for all B B and all g, g G, F g g (B) = F g (B) F g (B) (or F g g (B) = F g (B) F g (B)). 3 It satises consistency if it satises both meetconsistency and join-consistency. It satises non-degeneracy if for all g G\{1, 0} and all i N, there are B, B B such that F g g i (B) = 1 and Fi (B ) = 0. An extended rule F is an extended one-vote rule if for all i N, there are j, h N such that for all B B and all g G, F g i (B) = Bg jh. Clearly, all extended one-vote rules are consistent. In fact, they are simple outliers of the extremely rich 3 Miller (2008) calls these two properties meet separability and join separability. 14

15 family of consistent rules. A simple way of constructing a consistent extended rule is rst to dene a decision rule for basic groupslet us call it a basic group ruleand second to extend the basic group decisions to an extended decision using consistency. For example, we may use the family of consent approval functions for basic group rules (Samet and Schmeidler, 2003). Let s,t {1,, n} with s + t n + 2. The basic group consent rule with (s, t) satises the following: for all B B, all k G, and all i N, (i) when B k ii = 1, Fi k (B) = 1 if and only if {j N : B k ji = 1} s, and (ii) when B k ii = 0, Fi k (B) = 0 if and only if {j N : B k ji = 0} t. An extended consent rule is the consistent extension of a basic group consent rule. construction, all extended consent rules are consistent. 4 Thus, by The extended consent rules show by example that in the extended setup, consistency and non-degeneracy do not characterize the family of extended one-vote rules. This may appear to be in conict with Miller (2008). However, in Miller's (2008) model, another independence axiom, which we dene below, is implicit. 5 Once the independence axiom is imposed in addition, the same characterization as in Miller (2008) holds. Yet the independence axiom is stronger than independence of irrelevant opinions, and we show that in fact, a weaker set of axioms suce. Now we introduce axioms for the extended setup. First is a straightforward extension of independence of irrelevant opinions : Independence of Irrelevant Opinions. For all B, B B and all k G, if B k = B k, then F k (B) = F k (B ). Note that this axiom does not require independence across derived groups and so the decision on a b may depend on B a and B b as well as on B a b. It is evident that all extended consent rules are independent of irrelevant opinions. Non-degeneracy in the multinary setup corresponds to the following axiom in the extended setup. Basic Group Non-Degeneracy. For all k G and all i N, there exist B, B B 4 Alternatively, given (s, t), one may dene a rule by applying conditions (i) and (ii) in the denition of the basic group consent rule to all groups g G, basic and derived. However, this rule is not consistent unless s = t = 1, in which case, it coincides with the extended consent rule with s = t = 1, namely, the liberal rule. 5 We will elaborate more on this claim after we dene component-wise independence. 15

16 such that F k i (B) = 0 and F k i (B ) = 1. In our model, a decision associates with each person a membership for exactly one basic group. In the extended model, this can be stated as follows. Basic Group Partitioning. For all B B, ( {i N : Fi k (B) = 1} ) is a partition k G of N. This axiom together with consistency implies that for all k, l G, F k l ( ) is degenerate; it takes the constant value of 0 1 n. This partitioning requirement may be too strong in this extended model and a weaker version may be formulated by requiring basic group partitioning only under the condition that all persons agree with basic group partitioning. Unanimous Basic Group Partitioning. For all B B, if each i N partitions N into basic groups at B (i.e., for all i N, ({j N : B k ij = 1}) k G is a partition of N), then ( {i N : Fi k (B) = 1} ) is a partition of N. k G In our model, all persons view each person as a member of exactly one group in G. Thus, when viewed within the extended model, our model corresponds to the restricted domain B consisting of all problems B B such that k G Bk = 1 n n. Then for all B B and all distinct pairs k, k G, B k k = 0 n n and B 1 m = 1 n n. Note that on this restricted domain, all persons agree with partitioning N into basic groups and so basic group partitioning is equivalent to unanimous basic group partitioning. Now we establish a result that corresponds to Theorem 1. Due to independence of irrelevant opinions, our result holds both on the restricted domain B and the whole domain B. Proposition 3. (i) If an extended rule F on B (or B) satises independence of irrelevant opinions, basic group non-degeneracy, and unanimous basic group partitioning, then there is an extended one-vote rule ˆF such that for all B B (or B) and all k G, F k (B) = ˆF k (B). (ii) An extended rule F on B (or B) satises independence of irrelevant opinions, basic group non-degeneracy, unanimous basic group partitioning, and consistency if and only if it is an extended one-vote rule. 16

17 Proof. We prove part (i) for the domain B and skip the simpler proof for B. Let F be an extended rule on B satisfying the three axioms. Let f be the rule in our model dened from F as follows: for all P P, all k G, and all i N, f i (P ) = k F k i (B) = 1, (5) where B is such that for all l G, B l = B P,l. By unanimous basic group partitioning, f is well-dened. Independence of irrelevant opinions and basic group non-degeneracy of F imply independence of irrelevant opinions and non-degeneracy of f, respectively. Thus by Theorem 1, f is a one-vote rule; there is h: N N N such that for all P P, f i (P ) = P h(i). Let ˆF be the extended one-vote rule such that for all B B, all g g G\{1, 0}, and all i N, ˆF i (B) = Bg h(i). For all B B, there is P P such that for all k G, B k = B P,k. Since f i (P ) = P h(i), then by (5), for all k G, F k (B) = ˆF k (B). When B / B, for each k G, there is B such that the extended problem B generated by (B k, B k ) is in B and B k = B k. Then as shown earlier, F k ( B) = ˆF k ( B) and by independence of irrelevant opinions, F k (B) = ˆF k (B). Part (ii) follows directly from part (i) and consistency. Remark 3. Even if consistency is dropped in part (ii), we cannot get much far away from the one-vote rules because, as part (i) shows, any rule satisfying the other axioms coincides with an extended one-vote rule in its decisions on basic groups. On the other hand, if independence of irrelevant opinions is dropped in part (ii), one can nd a rich family of rules, quite dierent from the one-vote rules, satisfying the other axioms; examples are obtained by modifying the extended consent rules. We, therefore, nd that independence of irrelevant opinions plays a much more signicant role than consistency here. Also, the extended one-vote rules do not satisfy basic group partitioning on B. Thus, unanimous basic group partitioning in the proposition cannot be replaced by basic group partitioning. In Miller's (2008) model, a decision rule takes as an argument only one component of B, say B g for some g G. Thus, the identication of group g only depends on B g. 17

18 This means that the following property is implicitly assumed. Component-wise Independence. For all B, B B and all g G, if B g = B g, then F g (B) = F g (B ). This axiom requires independent decision making across derived groups as well as across basic groups. It is quite stronger than independence of irrelevant opinions. In fact, most extended consent rules violate it. For example, consider the extended majority consent rule with s = t = n+1. Then for all i N and all k G, person i belongs 2 to group k if and only if a majority believes her to be a member of group k, that is, {j N : B k ji = 1} n+1. Let i N. Consider B B such that (i) a majority 2 identies i as a member of group k; (ii) a majority identies i as a member of group l, and (iii) only a minority (less than n+1 ) identies i as a member of group k l. Then for 2 B, the extended majority consent rule determines i as a member of groups k (by (i)), l (by (ii)), and k l (by consistency). Now consider B B such that (i ) a minority identies i as a member of group k; and (ii ) B k l = B k l. For B, the extended majority consent rule determines i as a member of neither k (by (i )) nor k l (by consistency). Thus, although B and B agree on the group-k l membership, the extended majority consent rule assigns dierent decisions to them, violating component-wise independence. It can be shown that only one extended consent rule is component-wise independent : the extended consent rule with s = t = 1, namely the liberal rule. Adding component-wise independence, the characterization of the extended one-vote rules as in Miller (2008) can be obtained directly from his result. Proposition 4 (Miller 2008). An extended rule on B satises component-wise independence, non-degeneracy and (meet and join) consistency if and only if it is an extended one-vote rule. Proof. To prove the nontrivial direction, let F be an extended rule satisfying the three axioms. Dene φ: G B {0, 1} N such that for all g G and all B B, φ(g, B) F g (B) for some B B with B g = B. Then by By component-wise independence, φ is well dened and for all B B and all g G, F g (B) = φ(g, B g ). Now applying Theorem 2.5 in Miller (2008) to φ( ), we conclude that F is an extended one-vote rule. 18

19 In fact, we can obtain results that are stronger than Proposition 4 as corollaries to Theorem 1. Further, it turns out that the decisive votes that feature the extended onevote rules emerge even in the absence of consistency. To this end, we introduce three axioms that applies to complete groups. The following axiom, weaker than componentwise independence, requires decisions to be independent across complete groups. Complete Group Independence. For all B, B B and all g G c, if B g = B g, then F g (B) = F g (B ). Next is the restriction of non-degeneracy to complete groups. Complete Group Non-Degeneracy. For all g G c and all i N, there exist B, B B such that F g g i (B)=0 and Fi (B ) = 1. Finally, since complete groups are mutually exclusive, we require that the decisions on complete groups partition N. This is weaker than consistency. Complete Group Partitioning. partition of N. For all B B, ({i N : F g i (B) = 1}) g G c is a Applying Theorem 1, we obtain the following result. Proposition 5. (i) If an extended rule F on B satises complete group independence, complete group non-degeneracy, and complete group partitioning, then there is an extended one-vote rule ˆF such that for all B B and all g G c, F g (B) = ˆF g (B). (ii) An extended rule F on B satises complete group independence, complete group non-degeneracy and consistency if and only if it is an extended one-vote rule. Proof. Part (i). Let F be a rule satisfying the three axioms. Let K {1,, 2 m } and x a one-to-one correspondence θ : K G c. For each B B, by opinion-consistency, there is P K N N such that for all i, j N and all k K, P ij = k B θ(k) ij = 1. (6) Conversely, for each P K N N, there is B B satisfying (6). In fact, (6) denes a one-to-one correspondence Θ: K N N B. Let f : K N N K N be dened as follows: for all P K N N and all i N, f i (P ) = k if F θ(k) i (Θ(P )) = 1. By complete group 19

20 partitioning, f is well-dened. Complete group independence of F implies independence of irrelevant opinions of f. Complete group non-degeneracy of F implies non-degeneracy of f. Finally, applying Theorem 1, we conclude that f is a one-vote rule, which gives the desired conclusion. Part (ii). To prove the non-trivial direction, let F be a rule satisfying complete group independence, complete group non-degeneracy, and consistency. Since consistency implies complete group partitioning, part (i) holds and the decision on any complete group by F coincides with the decision by an extended one-vote rule. Then by consistency, we conclude that the decision on any other group by F also coincide with the decision by the extended one-vote rule. Remark 4. This proposition can be used to prove Theorem 2.5 in Miller (2008) as follows. Consider a rule φ: G B {0, 1} N satisfying the axioms of consistency (separability as Miller calls it) and non-degeneracy in his theorem. Now use this rule φ to dene an extended rule F such that for all B B, all g G, and all i N, F g i (B) φ i(g, B g ). Then by construction, F satises component-wise independence, and the axioms of consistency and non-degeneracy of φ imply consistency and nondegeneracy of F, respectively. Hence by part (ii) of the proposition, we conclude that F is an extended one-vote rule and so is φ. Remark 5. Part (ii) of the proposition strengthens Proposition 4 by weakening componentwise independence to complete group independence. Also, part (i) indicates that even if consistency is dropped in part (ii) (while complete group partitioning is retained), we cannot get that far way from the extended one-vote rules. Any rule satisfying the other axioms coincides with an extended one-vote rule in its decisions on complete groups. However, if complete group independence is dropped, all extended consent rules satisfy the other axioms in part (ii). As in Remark 3, the independence axiom here, complete group independence, plays a much more signicant role than consistency. Note that on the restricted domain B, independence of irrelevant opinions is equivalent to complete group independence. Also, under a basic property of unanimity (if B k = 0 n n, F k (B) = 0 1 n ), (unanimous) basic group partitioning is equivalent to complete group partitioning. Thus on the restricted domain B, we get pretty much the same results using the two sets of axioms. 20

21 It should be noted that both component-wise independence and complete group independence are quite more demanding than independence of irrelevant opinions. For simplicity, suppose that G = {a, b}. Component-wise independence and complete group independence require that for a decision on who are a and b (a b), we should only pay attention to opinions on who are a and b (a b) and ignore any other opinions, including the opinions on who are a and on who are b, which do not seem irrelevant to the decision on a and b. For example, there are many cases of B a and B b which give rise to the same opinions on a b, B a b = 0 n n, such as when ( B a = 0 n n and B b = ) 0 n n, 1k k 0 when B a = 1 n n and B b = 0 n n, and when B a k (n k) = and 0 (n k) k 1 (n k) (n k) ( ) 0k k 1 B b k (n k) =. The two independence axioms require that in all 1 (n k) k 0 (n k) (n k) these dierent cases, there should be the same decision on a b, which seems too strong. Such an independence is not demanded by independence of irrelevant opinions since it requires independent decisions only across basic groups. 6 Concluding Remarks In the binary group identication model, independence of irrelevant opinions is vacuous; decomposability is also mild since it coincides with self-duality. However, with three or more groups, the two axioms become extremely demanding as shown by Theorems 1 and 2. The contrasting consequences of these axioms in the binary and multinary setups are similar to those in Arrovian preference aggregation with two alternatives and with three or more alternatives. 6 We can extend our model to allow for status-quo memberships, and modify independence accordingly. Suppose that each person i N has an initial membership for group σ i G. To take account of the initial memberships, independence of irrelevant opinions can be relaxed by requiring the same independence only for those groups 6 When there are three or more alternatives, independence of irrelevant alternatives, transitivity, and unanimity (or Pareto principle) imply dictatorship (Arrow's Impossibility Theorem; Arrow, 1951). When there are two alternatives, independence of irrelevant alternatives and transitivity are vacuous, and there are numerous non-dictatorial aggregation rules that perform well in terms of, e.g., monotonicity and anonymity. 21

22 to which persons do not initially belong. Thus, only when a person is assigned to a new group, the membership decision for that group should be independent of opinions on the other groups. Regrouping Independence. Let P, P P and k G. Suppose that for all i, j N, P ij = k if and only if P ij = k. Then for all i N with σ i k, f i (P ) = k if and only if f i (P ) = k. Regrouping independence seems to relax independence of irrelevant opinions only slightly. But interestingly, a substantially larger family of rules satises it as well as non-degeneracy and symmetry. In particular, this family contains rules similar to the consent rules by Samet and Schmeidler (2003). For all k G, let q k {1,, n}. The consent rule with quotas q (q k ) k G, denoted f q, is dened as follows: for all P P and all i N, (i) if there is k G\{σ i } such that P ii = k and {j N : P ji = k} q k, then f q i (P ) = k; otherwise, f q i (P ) = σ i. Under the consent rules, each person i N can belong to only two groups: the group of her own decision (P ii ) or the status-quo group (σ i ). It is simple to show that the consent rules satisfy regrouping independence. Also, the consent rules satisfy unanimity, and hence, non-degeneracy. When q 1 = = q m = 1, f q is the liberal rule. References [1] Arrow, K.J. (1951), Social Choice and Individual Values, Wiley, New York.Çengelci, M.A. and R. Sanver (2010), Simple collective identity functions, Theory and Decision, 68, [2] Dimitrov, D., T. Marchant, and D. Mishra (2012), Separability and aggregation of equivalence relations, Economic Theory, 51, [3] Fishburn, P.C., and A. Rubinstein (1986a), Algebraic aggregation theory, Journal of Economic Theory, 38, [4] Fishburn, P.C., and A. Rubinstein (1986b), Aggregation of equivalence relations, Journal of Classication, 3,

23 [5] Guha, A.S. (1972), Neutrality, monotonicity,a nd the right of veto, Econometrica, 40, [6] Houy, N (2007), I want to be a J!: Liberalism in group identication problems, Mathematical Social Science, 54, [7] Ju, B.-G. (2010), Individual powers and social consent: an axiomatic approach, Social Choice and Welfare, 34, [8] Ju, B.-G. (2013), On the characterization of liberalism by Samet and Schmeidler, Social Choice and Welfare, 40, [9] Kasher, A. and A. Rubinstein (1997), On the question `Who is a j', a social choice approach, Logique et Analyse, 40, [10] Mas-Colell, A. and H. Sonnenschein (1972), General possibility theorem for group decisions, Review of Economic Studies, 39, [11] Miller, A.D. (2008), Group identication, Games and Economic Behavior, 63, [12] Samet, D. and D. Schmeidler (2003), Between liberalism and democracy, Journal of Economic Theory, 110, [13] Sung, S.C. and D. Dimitrov (2005), On the axiomatic characterization of Who is a J?, Logique et Analyse, 48,