Informational requirements of social choice rules to avoid the Condorcet loser: A characterization of the plurality with a runoff

Mathematical Social Sciences 79 (2016) 11–19

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Informational requirements of social choice rules to avoid the Condorcet loser: A characterization of the plurality with a runoff Shin Sato Faculty of Economics, Fukuoka University, 8-19-1 Nanakuma, Jonan-ku, Fukuoka, 814-0180, Japan

highlights • We consider informational requirements of social choice rules. • We find properties of rules operating on the minimal informational requirements. • Depending on the number of agents and the number of alternatives, either the plurality rule or the plurality with a runoff is characterized.

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Article history: Received 3 October 2012 Received in revised form 9 August 2014 Accepted 23 October 2015 Available online 1 November 2015

abstract We consider informational requirements of social choice rules satisfying anonymity, neutrality, monotonicity, and efficiency, and never choosing the Condorcet loser. Among such rules, we establish the existence of a rule operating on the minimal informational requirement. Depending on the number of agents and the number of alternatives, either the plurality rule or the plurality with a runoff is characterized. In some cases, the plurality rule is the most selective rule among the rules operating on the minimal informational requirement. In the other cases, each rule operating on the minimal informational requirement is a two-stage rule, and among them, the plurality with a runoff is the rule whose choice at the first stage is most selective. These results not only clarify properties of the plurality rule and the plurality with a runoff, but also explain why they are widely used in real societies. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Imagine that you are the rule designer. Your intention is to choose the best alternatives. (Ties are allowed.) Along with standard axioms (anonymity, neutrality, monotonicity, and efficiency), assume that you want to avoid the Condorcet loser and to minimize the cost of informational processing. Moreover, you want to narrow the social outcome as much as possible, i.e., you want a selective rule. Then, what is the ‘‘right’’ rule? This paper gives the answer to this question. The answer is either the plurality rule or the plurality with a runoff; it is the plurality rule if and only if n = 2 or m = 2 or (n, m) = (4, 3), where n is the number of agents, and m is the number of alternatives; in all the other cases, the answer is the plurality with a runoff. Thus, in most cases, we have the plurality with a runoff as the answer to the designer’s question. The informational requirement is one of the most important features of each rule. The informational requirement of a rule tells us how practically useful the rule is. For example, if we ignore the

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.mathsocsci.2015.10.003 0165-4896/© 2015 Elsevier B.V. All rights reserved.

informational requirement of rules, it would be difficult to explain why the plurality rule is one of the most common rules in our lives. Sato (2009) shows that among the one-stage rules satisfying anonymity, neutrality, monotonicity, and efficiency, the plurality rule operates on the minimal informational requirement, and the plurality rule is more selective than each other such rule operating on the minimal informational requirement. Sato (2009)’s result theoretically supports our intuition and experience that, compared with other ‘‘reasonable’’ social choice rules, a main advantage of the plurality rule lies in its simplicity and selectivity. However, it is well known that the plurality rule can choose ‘‘bad’’ alternatives.1 Thus, in this paper, in addition to anonymity, neutrality, monotonicity, and efficiency, we require each rule not to choose the Condorcet loser. The Condorcet loser is the counterpart of the Condorcet winner, which wins a strict majority against each alternative in a pairwise comparison. The Condorcet loser is defeated by each other alternative. If you consider that the Condorcet winner is the best alternative when it exists, then you

1 For example, see Nurmi (1999, Section 3.1).

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S. Sato / Mathematical Social Sciences 79 (2016) 11–19

should be ready to accept the view that the Condorcet loser is the worst alternative. Even if you do not consider that the Condorcet winner and the loser are the best and the worst alternative,2 it is reasonable to exclude the Condorcet loser from the social choice. In any case, it would be widely accepted that the Condorcet loser is not the best alternative, and hence should not belong to the social choice. As we mentioned earlier, in most cases, the answer to the designer’s question stated at the beginning of this section is the plurality with a runoff. To the best of my knowledge, this is the first characterization of the plurality with a runoff.3 The plurality with a runoff is another widely used rule, especially in ‘‘serious’’ social choice problems such as elections of the president in some nations, the president of some political parties, the dean of faculties, and so on.4 This result is consistent with our intuition that although the plurality with a runoff uses more amount of information than the one-stage plurality rule, it is still simple, and more ‘‘correctly’’ reflects preferences than the plurality rule. Finally, we discuss related literature. Fishburn (1977) is closest to our work in the sense that we both study informational requirements and discriminability (or selectivity) as they would give us important insights on social choice rules. Conitzer and Sandholm (2005) study communication complexities of eleven widely used voting rules. Among many differences, the most significant and essential one between our approach and Conitzer and Sandholm (2005) is that we consider a minimal informational requirement as an ‘‘axiom’’ and hence measuring the informational size of specific social choice rules is not our objective while it is in Conitzer and Sandholm (2005). The plurality rule is axiomatically characterized by Richelson (1978); Roberts (1991); Ching (1996); Yeh (2008). Smith (1973) and Pattanaik (2002, Section 6.3) discuss rules with runoffs. This paper is organized as follows. Section 2 gives basic notation and definitions. Section 3 gives main results. Section 4 concludes the paper. Proofs are collected in the Appendix. 2. Basic notation and definitions For each real number a, let ⌈a⌉ denote the smallest integer greater than or equal to a. For example, ⌈2.5⌉ = 3 and ⌈2⌉ = 2. Let N = {1, . . . , n} be a finite set of agents with n ≥ 2, and X be a finite set of alternatives with |X | = m ≥ 2. Let X be the set of all nonempty subsets of X . For each x ∈ X , we write x for {x} when the omission of the braces does not cause confusion. For each A ∈ X, let L(A) denote the set of all linear orders (complete, transitive, and antisymmetric binary relations) on A. Let L denote L(X ). Each linear order can be represented as a ranking of the alternatives without ties. We use linear orders to represent agents’ preferences over each A ∈ X. Let R be our generic notation for a linear order. When a preference relation belongs to a particular agent i ∈ N, we write it as Ri . The strict parts of R and Ri are denoted by P and Pi , respectively. An n-tuple R = (R1 , . . . , Rn ) is a preference profile. For each R ∈ L(A) and each k ∈ {1, . . . , |A|}, let rk (R) denote the kth ranked alternative according to R in A. For each R ∈ L

2 I believe that choosing the Condorcet winner whenever it exists is not necessarily an appealing property of a rule. 3 Freeman et al. (2014) characterize several rules with runoff systems. They consider the model such that one alternative is eliminated in each step. Thus, neither the plurality rule nor the plurality with a runoff can be defined in their model. 4 Precisely speaking, the plurality with a runoff in this paper is slightly different from some of them. In this paper, if an alternative winning a majority in a weak sense (50 out of 100) in the first stage, then there is no second stage, and the alternative is the winner. In real societies, an alternative is often required to win a strict majority (51 out of 100) to be the winner without the second stage.

and each A ∈ X, let R|A ∈ L(A) be the restriction of R to A. Let R |A = (Ri |A)i∈N . For each A ∈ X, each x ∈ A, and each R ∈ L(A)N , let N (R , x) be the set of agents who most prefer x at R; formally, N (R , x) = {i ∈ N | r1 (Ri ) = x}. Let n(R , x) = |N (R , x)|. We often write N (x) and n(x) for N (R , x) and n(R , x), respectively, when the preference profile under consideration is obvious. We consider a rule consisting of at most two stages.5 In our definition of a rule, we use partitions of the set of linear orders as domains of social choice rules in each stage. The following is the idea behind the formulation. Because we are interested in informational requirements, we have to explicitly account for the kind of information each social choice rule uses. As an example, consider the plurality rule. It chooses the most preferred alternatives by the largest number of agents. Then, Ri and R′i such that r1 (Ri ) = r1 (R′i ) are ‘‘equivalent’’ from the viewpoint of the plurality rule. This is because the plurality rule uses information only on what the top ranked alternatives are. In other words, Ri and R′i send the same ‘‘message’’ to the plurality rule. Based on this observation, for each x ∈ X , let M (x) be the collection of all R ∈ L such that r1 (R) = x. For each i ∈ N, let Mi = {M (x) | x ∈ X }. Then, Mi is a partition of L, and each Mi ∈ Mi tells us what is the top ranked alternative by agent i. Such Mi and Mi are called a message and a message space, respectively. For the plurality rule, each message profile M = (M1 , . . . , Mn ) contains enough information to make a social choice. Next, consider the Borda rule. Intuitively speaking, the Borda rule needs much more information on agents’ preferences than the plurality rule. A finer message space reflects this fact; for the Borda rule, no Ri and R′i with Ri ̸= R′i are ‘‘equivalent’’. Thus, the message space for the Borda rule is Mi′ = {{Ri } | Ri ∈ L}. Two distinct preference relations serve as distinct messages. In general, the further amount of information is required, the more pairs of preference relations should be distinguished from the viewpoint of the rule. This observation motivates our definition of the informational size of a rule. What is important at this point is that coarseness of each message space tells us the informational requirement of each rule, and message spaces are explicitly incorporated into the definition of a rule. Compared with the standard model of social choice, our rule consists of many components: messages in the first stage, a social choice rule in the first stage, a tie-breaking rule between the stages, messages in the second stage, social choice rules in the second stage. Before the formal definition of a rule, we see a procedure of making a social choice according to a rule. Each notation will be formally defined below. In the first stage, agents send a message profile M = (M1 , . . . , Mn ) ∈ M to the social choice rule f in the first stage. The message profile contains information on preferences over X . Although we call f a social choice rule, it does more than usual ones. The value of f at M consists of two components f1 (M ) and f2 (M ), i.e., f (M ) = (f1 (M ), f2 (M )). The first component f1 (M ) is the social choice in the first stage. The second component f2 (M ) can be 0 or some member of X ∪ {∅}. If f2 (M ) is 0, it means ‘‘stop’’, and we do not proceed to the second stage. In this case, the social choice as a whole is f1 (M ). If f2 (M ) takes a value in X ∪ {∅}, then we proceed to the second stage, and we interpret f2 (M ) as the set of alternatives subject to tie-breaking before the second stage. To be consistent with this interpretation, we assume f2 (M ) ⊂ f1 (M ). When f2 (M ) = ∅, then each alternative in f1 (M ) proceeds to the second stage. The other extreme is the case f2 (M ) = f1 (M ). In this case, each alternative in f1 (M ) is subject to tie-breaking before the second stage. In sum, each alternative in f1 (M ) \ f2 (M ) surely proceeds to the second stage, but some alternatives in f2 (M ) might be dropped by a tie-breaking rule T . As a result, T [f (M )] is the set

5 This is for simplicity. With more than two stages, our results do not change.

S. Sato / Mathematical Social Sciences 79 (2016) 11–19

of alternatives at the beginning of the second stage. Because each alternative in f1 (M ) \ f2 (M ) proceeds to the second stage, f1 (M ) \ f2 (M ) ⊂ T [f (M )]. Because the tie-breaking rule does not add alternatives to f1 (M ), T [f (M )] ⊂ f1 (M ). Let A = T [f (M )]. In the second stage, agents send a message profile in M (A) to a social choice rule fA . The message profile contains information on preferences over A. Finally, fA makes a social choice based on the message profile. Definition 2.1. A rule consists of the following components:

• (A set of message profiles) For each i ∈ N, Mi is a partition of L and M = M1 × · · · × Mn . • (A social choice rule in the first stage) f is a function from M into X × (X ∪ {∅} ∪ {0}) such that f1 (M ) ⊃ f2 (M ) for each M ∈ M with f2 (M ) ̸= 0, where f1 (M ) and f2 (M ) are the first and the second component of f (M ), respectively. The following components are necessary if and only if I ≡ {f (M ) | M ∈ M and f2 (M ) ̸= 0} ̸= ∅, i.e., there is the second stage at some message profile in the first stage.

• (A tie-breaking rule) T is a function from I into X such that for each B = (B1 , B2 ) ∈ I, B1 \ B2 ⊂ T (B) ⊂ B1 . Let A ≡ {T (B) | B ∈ I}, i.e., the collection of possible sets of alternatives at the beginning of the second stage.

• (A set of message profiles in the second stage) For each i ∈ N and each A ∈ A, M (A)i is a partition of L(A), and M (A) = M(A)1 × · · · × M(A)n . • (A social choice rule in the second stage) For each A ∈ A, fA is a function from M (A) into X such that fA (M ) ⊂ A for each M ∈ M (A). If I = ∅, then the rule is called a one-stage rule. The statement ‘‘I = ∅’’ is equivalent to ‘‘f2 (M ) = 0 for each M ∈ M ’’, i.e., ‘‘stop’’ for each message profile. To describe a one-stage rule, it is necessary and sufficient to specify M and f1 . If I ̸= ∅, then the rule is called a two-stage rule. Note that it is possible that f2 (M ) = 0 for some M ∈ M in a two-stage rule. We often use F to represent a rule. Let (M , f , T , (M (A))A∈A , (fA )A∈A ) be our generic notation for a two stage rule, and (M, f1 ) for a one-stage rule. For convenience, unless clearly specified, F = (M , f , T , M (A), fA ) represents either a one-stage or twostage rule.6 (Whenever it does not cause confusion, we drop the subscript ‘‘A∈A ’’ from the presentation.) Although social choice rules f and fA are defined on partitions of the set of preferences, they are essentially functions on agents’ preferences, and each domain of social choice rules, such as M and M(A), represents an invariance condition of f and fA . Formally, for each R ∈ LN , let ϕ(R ) = (ϕ(R )1 , . . . , ϕ(R )n ) ∈ M be the message profile such that Ri ∈ ϕ(R )i for each i ∈ N. Then, f [ϕ(R )] = f [ϕ(R ′ )] for each pair R , R ′ ∈ LN such that ϕ(R ) = ϕ(R ′ ), i.e., R and R ′ belong to the same message profile. In this sense, M can be considered as an invariance condition. We often consider a relationship between preferences and social choices. In such cases, for simplicity, we drop ‘‘ϕ ’’, and write f (R ) for f [ϕ(R )]. Similar arguments apply to fA on M (A). In other words, we frequently describe a rule f on M as if it were a function on LN respecting the invariance condition M , and fA on M (A) as if it were a function on L(A)N respecting the invariance condition M (A). Let R ∈ LN . For each one-stage rule F , f1 (R ) is the social choice of F at R. For each two-stage rule F , the social choice of F at R is

• f1 (R ) if f2 (R ) = 0, and • fA (R |A) if f2 (R ) ̸= 0, where A = T [f (R )].

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Several remarks are in order. First, a social choice as a whole can be either single-valued or multi-valued. Second, in a two-stage rule, by setting f2 (M ) = ∅ for each M ∈ M , we have a rule without any tie-breaking. Third, we explicitly consider tie-breaking between the first and the second stage. The purpose of this is to concentrate on the ‘‘essence’’ of social choice rules.7 Fourth, we do not consider evolution of preferences from the first to the second stage. This postulate is plausible when the first and the second stages are organized consecutively. Fifth, at least one important voting procedure, approval voting, cannot be defined in this model. In this model, each preference relation determines the message uniquely, whereas in approval voting, preference relations do not determine a message uniquely. We define two rules. Each rule plays an important role in later sections. Definition 2.2 (Plurality Rule). For each x ∈ X , let M (x) = {R ∈

L | r1 (R) = x}. For each i ∈ N, let Mi = {M (x) | x ∈ X }. For each M ∈ M and each x ∈ X , let n(x) = |{i ∈ N | M (x) = Mi }|. The

integer n(x) is the number of agents who rank x at the top of their p preferences. Then, for each M ∈ M , let f1 (M ) = {x ∈ X | n(x) ≥ p n(y), ∀y ∈ X }. The function f1 chooses the alternatives which are most preferred by the largest number of agents. The one-stage rule (M, f1p ) is called the plurality rule. p

For each R ∈ LN and each x ∈ f1 (R ),8 m × n( R , x ) ≥



n( R , y ) = n,

y∈X n ⌉. This fact is crucial for our results because and hence n(R , x) ≥ ⌈ m p n n when ⌈ m ⌉ = ⌈ 2 ⌉, for each R ∈ LN , each x ∈ f1 (R ) is not the Condorcet loser. See Section 3 for more discussion on this condition.

Definition 2.3 (Plurality with a Runoff). Before a formal definition, we explain the idea. In the first stage, each agent reports the most preferred alternative. If some alternative wins a majority in a weak sense (henceforth, simply ‘‘a majority’’),9 the set of the alternatives winning a majority is the social choice as a whole. (This set contains at most two alternatives.) Otherwise, we choose two alternatives most preferred by the agents in the first stage, and proceed to the second stage. If we cannot choose exactly two alternatives based on the number of votes by the agents, we use a tie-breaking rule. In the second stage, the social outcome is determined by a simple majority between the two alternatives. We define the plurality with a runoff formally. For each i ∈ N, let Mi be the one in the plurality rule. For each M ∈ M , let p V1 (M ) = f1 (M ), and V2 (M ) = {x ∈ X \ V1 (M ) | n(x) ≥ n(y), ∀y ∈ X \ V1 (M )}, where n(x) is the number of the agents whose message

7 Some problem is caused due to tie breaking. For example, consider the plurality with a runoff. As we define later, the social choice proceeding to the second stage should consist of two alternatives. Assume N = {1, 2, 3}, X = {x, y, z }, xR1 yR1 z , yR2 zR2 x, and zR3 xR3 y. This is a typical situation leading to the voting paradox. Without tie-breaking, {x, y, z } is chosen in the first stage. However, to proceed to the second stage, one alternative should be dropped. Then, regardless of which alternative is dropped and how that alternative is chosen, the choice rule before the second stage does not satisfy at least one of anonymity and neutrality. This violation is due to tie-breaking, and is not an essential problem. By dividing the procedure into a choice based on preferences and tie-breaking, we can consider anonymous and neutral social choice rules. 8 Remember that f p (R ) stands for f p (ϕ(R )), where ϕ(R ) is the message profile 1

6 Precisely speaking, this is an abuse of notation because T , M (A), and f should A be absent in the description of a one-stage rule.

1

such that R ∈ ϕ(R ). 9 An alternative wins a majority in a weak sense if he gets 50 out of 100, in a strict

sense if he gets 51 out of 100.

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S. Sato / Mathematical Social Sciences 79 (2016) 11–19

is M (x). V1 (M ) and V2 (M ) are sets of alternatives most preferred by the largest and the second largest number of agents, respectively. Then, let f r be a social choice rule in the first stage defined by for each M ∈ M ,

 n (V1 (M ), 0), if n(x) ≥ for each x ∈ V1 (M )    2    (V1 (M ), V1 (M )), n r if |V1 (M )| ≥ 2 and n(x) < for each x ∈ V1 (M ) f (M ) =  2  (V1 (M ) ∪ V2 (M ), V2 (M )),    n  if |V1 (M )| = 1 and n(x) < for each x ∈ V1 (M ). 2

Let T be a tie-breaking rule on I such that |T (B)| = 2 for each B ∈ I. For each A ∈ A, let M (A)i = {{R} | R ∈ L(A)}, and fAr be a simple majority rule, i.e., for each M ∈ M (A), x ∈ fAr (M ) if and only if at least half of the agents prefer x to the alternative in A \ x. The rule (M , f r , T , M (A), fAr ) is the plurality with a runoff. The three cases in the definition of f r are exhaustive, but some cases do not occur under some conditions. For example, when n = 2, only the first case is relevant. When n = 3, the third case never occurs. In the above definition, the only requirement for a tie-breaking rule is that the set of alternatives at the beginning of the second stage contains two alternatives. Ties might be broken alphabetically such as T ({x, y, z }, {y, z }) = {x, y} and T ({x, y, z }, {x, y}) = {x, z }. Ties might be broken ‘‘inconsistently’’ such as T ({x, y, z }, {x, y}) = {x, z } and T ({x, y, z }, {x, y, z }) = {y, z }. We define properties of a rule. For each rule F , a social choice rule f (for a one-stage rule, consider only f1 ) in the first stage satisfies

• anonymity if for each permutation σ of N and each R ∈ LN , f (R ) = f (R σ ), where R σ ∈ LN is defined by for each i ∈ N , Rσi = Rσ (i) . • neutrality if for each permutation ρ of X and each R ∈ LN , ρ[f1 (R )] = f1 [ρ(R )]10 and ρ[f2 (R )] = f2 [ρ(R )],11 where ρ(R ) = (ρ(R1 ), . . . , ρ(Rn )) ∈ LN is defined by for each i ∈ N , ρ(Ri ) = {(x, y) ∈ X 2 | (ρ −1 (x), ρ −1 (y)) ∈ Ri }. • monotonicity if for each pair R , R ′ ∈ LN such that x ∈ f1 (R ), R |(X \ x) = R ′ |(X \ x), and x = rk (Ri ) = rk′ (R′i ) with k′ ≤ k for each i ∈ N, we have x ∈ f1 (R ′ ). • efficiency if for each R ∈ L and each pair x, y ∈ X such that xPi y for each i ∈ N , y ̸∈ f1 (R ). Anonymity and neutrality mean a symmetric treatment of agents and alternatives, respectively. The symmetric treatments are not only at the choice of alternatives before tie-breaking, but also at the choice of alternatives subject to tie-breaking. Monotonicity means that when x ∈ f1 (R ) and the position of x improves through the change from R to R ′ while the relative comparison of each other pair of alternatives is unchanged, then x ∈ f1 (R ′ ). Efficiency means that when y ∈ X is dominated by some alternative, then y cannot be chosen. For each A ∈ A, anonymity, neutrality, monotonicity, and efficiency of fA are defined in a similar manner. (In the above definition, ignore the condition on f2 , replace f and f1 by fA , LN by L(A)N , and X by A.) A one-stage rule satisfies anonymity if f1 satisfies anonymity. A two-stage rule satisfies anonymity if f and fA for each A ∈ A satisfy anonymity. We use neutrality, monotonicity, and efficiency of a rule in a similar manner.

10 For each A ∈ X, ρ(A) ≡ {ρ(x) | x ∈ A}. 11 Define ρ(∅) = ∅ and ρ(0) = 0. (Remember that f (R ) can be ∅ and 0.) 2

For each R ∈ LN , an alternative x ∈ X is the Condorcet loser at R if |{i ∈ N | xRi y}| < |{i ∈ N | yRi x}| for each y ∈ X \ x. When x is the Condorcet loser, x is defeated by each other alternative in a pairwise comparison. A rule avoids the Condorcet loser if for each R ∈ LN , the Condorcet loser at R does not belong to the social choice of a rule at R. Definition 2.4. Let R denote the set of rules avoiding the Condorcet loser and satisfying anonymity, neutrality, monotonicity, and efficiency. It is easy to see that the plurality with a runoff belongs to R. Whether the plurality rule belongs to R or not depends on n and m. For example, if n = 2, then the plurality rule avoids the Condorcet loser. On the other hand, if n = m = 3, there is R ∈ LN such p that f1 (R ) contains the Condorcet loser. Our theorems in the next section specify the condition on n and m under which the plurality rule belongs to R. As we discussed before Definition 2.1, the number of messages in each message space Mi is considered as the informational requirement of a rule. Definition 2.5. For each rule F and each R ∈ L, the informational size of F at R is defined by

 • i∈N |Mi |, if f 2 (R ) = 0, • i∈N |Mi | + i∈N |M(T [f (R )])i |, if f2 (R ) ̸= 0.  The integer i∈N |Mi | is the informational size of the first  stage. Thus, if there is no second stage, i.e., f2 (R ) = 0, then i∈N |Mi | is the informational size of a rule at R. If f2 (R ) ̸= 0, the set of alternatives at the beginning of the second stage is T [f (R )], and the message space of agent i in the second stage is M (T [f (R )])i .  Thus, i∈N |M (T [f (R )])i | is the informational size of the second stage. The sum of the informational size of the first and that of the second stage is the informational size of a rule at R. For a one-stage rule, the informational size at R and that at R ′ are the same for each pair R , R ′ ∈ LN , while not necessarily the same for a two-stage rule. Thus, for a two-stage rule, we have some degree of freedom in the choice of the informational size. Fortunately, for our purpose, the following definition of the minimal informational requirement, which seems uncontroversial, is sufficient. Definition 2.6. A rule F operates on the minimal informational requirement in R if for each rule F ′ ∈ R and each R ∈ LN , the informational size of F at R is less than or equal to the informational size of F ′ at R. When a rule F operates on the minimal informational requirement in R, for each rule F ′ ∈ R, whichever preference profile we have, the informational size of F at the profile is not greater than the informational size of F ′ at the profile. Note that the existence of a rule operating on the minimal informational requirement is not trivial at all. In the next section, we will establish the existence as well as properties of each rule operating on the minimal informational requirement. 3. Results p

In this section, let F p = (M p , f1 ) denote the plurality rule, and F = (M r , f r , T r , (M (A)r )A∈Ar , (fAr )A∈Ar ) denote the plurality with a runoff.12 r

12 By definition, M p = M r .

S. Sato / Mathematical Social Sciences 79 (2016) 11–19 n Theorem 3.1. Assume ⌈ m ⌉ = ⌈ 2n ⌉; that is, n = 2 or m = 2 or (n, m) = (4, 3).

(i) The plurality rule operates on the minimal informational requirement in R. (ii) If a rule F operates on the minimal informational requirement in R, then F is a one-stage rule (M, f1 ) such that p (a) Mi = Mi for each i ∈ N, and p (b) f1 (R ) ⊃ f1 (R ) for each R ∈ LN . Three remarks are in order. First, the theorem says that each rule operating on the minimal informational requirement in R is necessarily a one-stage rule, and it is a supercorrespondence of the plurality rule. This can be considered as a characterization of the plurality rule; it is the most selective rule among the ones operating on the minimal informational requirement in R. Second, Sato (2009) shows that among the rules which operate on the minimal informational requirements in the class of rules satisfying anonymity, neutrality, monotonicity, and efficiency, the plurality is the most selective. Therefore, in Theorem 3.1, the requirement for rules not to choose the Condorcet loser is redundant. Third, we explain why the plurality rule avoids the Condorcet n loser when ⌈ m ⌉ = ⌈ 2n ⌉. Let R ∈ LN and x ∈ f1p (R ). Then, as we n ⌉ = ⌈ 2n ⌉. Thus, for each showed below Definition 2.2, n(R , x) ≥ ⌈ m y ∈ X , at least half of the agents prefer x to y. Therefore, x is not the Condorcet loser.

violated.13 Also, Smith (1973) shows that when n = 37 and m = 3, monotonicity is violated.14 (Moulin, 1988, Chapter 9 contains other examples.) In this sense, axioms are weak when they are considered within each stage in two-stage rules. Fifth, we have a characterization of the plurality with a runoff. Assume that you are the rule designer, and that you want to choose a rule in R. Also, you want a rule with the least informational requirements. Moreover, in the first stage, you want the rule to be as selective as possible. In other words, you do not want to depend on a tie-breaking rule as much as possible. Then, the answer is the plurality with a runoff. 4. Concluding remarks What we learn through this paper and Sato (2009) concerning the cost of information processing can be summarized as follows:





Anonymity   Neutrality    Monotonicity  + minimal information + selectivity   Efficiency No Condorcet loser plurality,



= or

plurality with a runoff,

and

 Theorem 3.2. Assume ⌈ ⌉ < ⌈ ⌉. n m

n 2

(i) The plurality with a runoff operates on the minimal informational requirement in R. (ii) If a rule F operates on the minimal informational requirement in R, then F is a two-stage rule (M , f , T , (M(A))A∈A , (fA )A∈A ) such that (a) Mi = Mir for each i ∈ N, (b) for each R ∈ LN , f2 (R ) = 0 if and only if f2r (R ) = 0, (c) f1 (R ) ⊃ f1r (R ) for each R ∈ LN , (d) each A ∈ A contains two alternatives, (e) for each i ∈ N and each A ∈ A, M (A)i = {{R} | R ∈ L(A)}, and (f) for each A ∈ A, fA is the simple majority rule. Several remarks are in order. n ⌉ = ⌈ 2n ⌉ and ⌈ mn ⌉ < ⌈ 2n ⌉, First, because m ≥ 2, the two cases ⌈ m considered in Theorems 3.1 and 3.2, respectively, are exhaustive. Second, Theorem 3.2 says that each rule operating on the minimal informational requirement in R is necessarily a two-stage rule, and the social choice rule in the first stage is a supercorrespondence of the one in the plurality with a runoff. Moreover, only two alternatives proceed to the second stage, and the second stage is the simple majority rule between them. Third, it is more common to require a majority in a strict sense instead of a weak sense for an alternative to be a winner without the second stage. Let F r +1 denote such a rule. If n is odd, there is no difference between F r and F r +1 . Assume that n is even. Then, at some preference profile, the informational size of F r +1 is larger than F r . Nevertheless, it would be safe to say that the difference is practically negligible in such ‘‘big’’ social choice problems (for example, presidential elections by a people) that the cost of information processing matters. Fourth, we impose axioms (anonymity, neutrality, monotonicity, efficiency) within each stage. This is essential for the theorem. If we consider the correspondence from the preference profile over X to the final social outcome of the plurality with a runoff, the correspondence violates some axioms in some situations. For example, if n = m = 3, at least one of anonymity and neutrality is

15



Anonymity  Neutrality  Monotonicity + minimal information + selectivity Efficiency

= plurality. Unlike usual ‘‘+’’, the above ‘‘+’’ are not commutative. If we first imposed minimality of informational requirements, then constant rules would be the answer. Thus, the above formulas should be read from left to right. The difference between the left sides of the two formulas is ‘‘No n Condorcet loser’’. If ⌈ m ⌉ = ⌈ 2n ⌉, this difference does not make a n difference in the ‘‘answer’’. However, if ⌈ m ⌉ < ⌈ 2n ⌉, the existence of ‘‘No Condorcet loser’’ is crucial and changes the conclusion. Like other axiomatic analyses in social choice theory, these results clarify properties of the plurality rule and the plurality with a runoff. In addition to this, the results explain why these two rules are widely used in real societies. Their distinctive characteristic is informational efficiency, which is a practically important feature of rules. Appendix. Proofs We use many permutations of N and X . For simplicity, when we describe a permutation, we do not specify the part on which the permutation is the identity function. For example, when we say that σ is the permutation of N exchanging i and j, then it should be understood that σ is the identity function on N \ {i, j}. As we mentioned in the last paragraph in Section 2, the existence of a rule operating on the minimal informational requirement is nontrivial. Thus, we cannot start with a statement such as

13 We can establish this by arguments similar to those in footnote 7. 14 Precisely speaking, Smith (1973) considers a function mapping each preference profile to a social preference relation. His Theorem 2 shows that each ‘‘point runoff system’’ violates monotonicity. It can be seen that his arguments apply to the plurality with a runoff in our framework.

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S. Sato / Mathematical Social Sciences 79 (2016) 11–19

‘‘Let F ≡ (M , f , T , (M (A))A∈A , (fA )A∈A ) be a rule which operates on the minimal informational requirement in R’’. We will introduce the concept of informational efficiency, and show that each informationally efficient rule operates on the minimal informational requirement. In the process of showing this, we find properties of each informationally efficient rule. Those are what we want to show as properties of each rule operating on the minimal informational requirement in R. This is the outline of the proof. For each rule F and each R ∈ LN , let s(F , R ) denote the informational size of F at R. For each F , F ′ ∈ R, F is informationally more efficient than F ′ is (i) for each R ∈ LN , s(F , R ) ≤ s(F ′ , R ), and (ii) for some R ∈ LN , s(F , R ) < s(F ′ , R ). A rule F ∈ R is informationally efficient in R (simply, informationally efficient) if there is no F ′ ∈ R which is informationally more efficient than F . p In the following, as in Section 3, let F p = (M p , f1 ) denote r r r r r the plurality rule, and F = (M , f , T , (M (A) )A∈Ar , (fAr )A∈Ar ) denote the plurality with a runoff. Without proof, we will use the n ⌉ = ⌈ 2n ⌉, and F r ∈ R. fact that F p ∈ R when ⌈ m A.1. Lemmas Let F ′ ∈ R. Let F ≡ (M , f , T , (M (A))A∈A , (fA )A∈A ) ∈ R be an informationally efficient rule such that for each R ∈ LN , s(F , R ) ≤ s(F ′ , R ).15 These F ′ and F are fixed in the rest of the Appendix. The rule F is either a one-stage or two-stage rule. (See footnote 6.) However, whenever we talk about A, M (A), and fA , then it should be understood that the rule under consideration is a two-stage rule. Among the following lemmas, Lemmas A.1, A.2 and A.5 are based on Sato (2009)’s Lemmas 4.1, 4.2, and arguments in Sato (2009, p.196–197), respectively. We give proofs of them for completeness. Lemma A.1 (Message Spaces are Identical Across Agents). (i) For each pair i, j ∈ N , Mi = Mj . (ii) For each pair i, j ∈ N and each A ∈ A, M (A)i = M (A)j . Proof. (i) Suppose Mi ̸= Mj for some pair i, j ∈ N. Step 1: Either

¯ i ∈ Mi such that M ¯ i ∩ Mj1 ̸= ∅ and M ¯ i ∩ Mj2 ̸= ∅ for (a) there is M some distinct Mj1 , Mj2 ∈ Mj , or

¯ j ∈ Mj such that Mi1 ∩ M ¯ j ̸= ∅ and Mi2 ∩ M ¯ j ̸= ∅ for (b) there is M 1 2 some distinct Mi , Mi ∈ Mi . Suppose neither one of (a) and (b) holds. First, we prove Mi ⊂ Mj . Let Mi ∈ Mi . Then, there is Mj ∈ Mj such that Mi ⊂ Mj . For Mj , there is Mi′ ∈ Mi such that Mj ⊂ Mi′ . Thus, Mi ⊂ Mj ⊂ Mi′ . Because Mi is a partition of L, Mi = Mi′ . This implies Mi = Mj , and hence Mi ∈ Mj . Because Mi was arbitrary in Mi , we have Mi ⊂ Mj . Symmetric arguments show Mi ⊃ Mj . Thus, Mi = Mj . This is a contradiction to the assumption Mi ̸= Mj . Without loss of generality, assume (a).

(R1j , R−j )σ and (R2j , R−j )σ belong to the same message profile, we have f [(R1j , R−j )σ ] = f [(R2j , R−j )σ ]. Because f satisfies anonymity, f ([(R1j , R−j )σ ]σ ) = f ([(R2j , R−j )σ ]σ ). Because [(R1j , R−j )σ ]σ = (R1j , R−j ) and [(R2j , R−j )σ ]σ = (R2j , R−j ), we have f (R1j , R−j ) = f (R2j , R−j ). This implies that f (Mj1 , M−j ) = f (Mj2 , M−j ). Step 2 shows that Mj1 and Mj2 in Mj do not serve as ‘‘distinct’’ messages. Let Mj∗ = Mj1 ∪ Mj2 . By merging Mj1 and Mj2 into one message Mj∗ , we have the message space Mj∗ = {Mj | Mj ∈

Mj \ {Mj1 , Mj2 } or Mj = Mj∗ }. Replace Mj in M by Mj∗ . Let M∗ denote the resulting profile of message spaces. Define f ∗ on M∗ by f ∗ (R ) = f (R ) for each R ∈ LN .16 Then, the rule F ∗ = (M∗ , f ∗ , T , (M(A))A∈A , (fA )A∈A ) is in R. Also, at each R ∈ LN , F ∗

operates on less informational requirements than F . This is a contradiction to the assumption that F is informationally efficient. (ii) Let A ∈ A. Suppose M (A)i ̸= M (A)j for some i, j ∈ N. By similar arguments in (i), we have M (A)∗ and fA∗ on M (A)∗ such  that fA∗ (R ) = fA (R ) for each R ∈ L(A)N and i∈N |M (A)i | >  ∗ ∗ ∗ ∗ i∈N |M (A)i |. In F , replace M (A) by M (A) , and fA by fA . Let F be the resulting rule. Then, F ∗ is in R. Let R ∈ LN be such that T [f (R )] = A. Then, s(F , R ) > s(F ∗ , R ) and for each R ′ ∈ LN , s(F , R ) ≥ s(F ∗ , R ′ ). This is a contradiction to the assumption that F is informationally efficient.  Lemma A.2 (The Image of Each Message is Also a Message). For each i ∈ N, each M ∈ Mi , and each permutation ρ of X , ρ(M ) ≡ {ρ(R) | R ∈ M } ∈ Mi . Proof. Suppose ρ(M ) ̸∈ Mi . There are two cases. Case 1: ρ(M ) ( M ′ for some M ′ ∈ Mi . Because ρ(M ) ( M ′ , we have M ( ρ −1 (M ′ ).17 Because Mi is a partition of L, there is M ∗ ∈ Mi such that M ∗ ̸= M and M ∗ ∩ ρ −1 (M ′ ) ̸= ∅. Let R1i ∈ M and R2i ∈ M ∗ ∩ ρ −1 (M ′ ). We claim that for each R−i ∈ LN \{i} , f (R1i , R−i ) = f (R2i , R−i ). Suppose not. Then, ρ[f (R1i , R−i )] ̸= ρ[f (R2i , R−i )], where ρ[f (R1i , R−i )] = (ρ[f1 (R1i , R−i )], ρ[f2 (R1i , R−i )]) and ρ[f (R2i , R−i )] = (ρ[f1 (R2i , R−i )], ρ[f2 (R2i , R−i )]). By neutrality, f [ρ(R1i , R−i )] ̸= f [ρ(R2i , R−i )]. However, because ρ(R1i ) and ρ(R2i ) belong to the same message M ′ , we have f [ρ(R1i , R−i )] = f [ρ(R2i , R−i )]. This is a contradiction. Therefore, f (R1i , R−i ) = f (R2i , R−i ). This implies that M and M ∗ can be integrated into one message without any substantial change. (See the arguments after Claim 2 in the proof of Lemma A.1.) This is a contradiction to the assumption that F is informationally efficient. Case 2: ρ(M )∩ M 1 ̸= ∅ and ρ(M )∩ M 2 ̸= ∅ for some M 1 , M 2 ∈ Mi with M 1 ̸= M 2 . Let R1i ∈ ρ(M ) ∩ M 1 and R2i ∈ ρ(M ) ∩ M 2 . In this case, by using the fact ρ −1 (R1i ), ρ −1 (R2i ) ∈ M, it can be seen that f (R1i , R−i ) = f (R2i , R−i ) for each R−i ∈ LN \{i} . (See arguments in Case 1.) Then, two messages M 1 and M 2 can be integrated into one message without any substantial change. This is a contradiction to the informational efficiency of F .  Lemma A.3 (From Each Message, The Rule Knows the Top Ranked Alternative).

Step 2: For each M−j ∈ M−j , f (Mj1 , M−j ) = f (Mj2 , M−j ).

¯ i ∩ Mj1 and R2j ∈ M ¯ i ∩ Mj2 . Let M−j ∈ M−j and R−j ∈ Let R1j ∈ M M−j . Let σ be the permutation of N exchanging i and j. Because

15 We can find such F as follows. If F ′ is informationally efficient, then F = F ′ . If F ′ is not informationally efficient, then there is a rule F ′′ ∈ R which is informationally more efficient than F ′ . If F ′′ is informationally efficient, then F = F ′′ . If F ′′ is not informationally efficient, there is a rule F ′′′ ∈ R. By continuing this process, we can find F .

16 The function f ∗ on M ∗ can be defined in this manner because f respects an invariance condition M ∗ . 17 We prove formally M ( ρ −1 (M ′ ). First, we prove M ⊂ ρ −1 (M ′ ). Let R ∈ M.

Because ρ(M ) ⊂ M ′ , ρ(R) ∈ M ′ . Then, by definition of ρ −1 (M ′ ), ρ −1 [ρ(R)] ∈ ρ −1 (M ′ ). Because ρ −1 [ρ(R)] = R, we have R ∈ ρ −1 (M ′ ). Thus, M ⊂ ρ −1 (M ′ ). Next, we prove M ̸= ρ −1 (M ′ ). Because ρ(M ) ( M ′ , there exists R′ ∈ M ′ such that R′ ̸∈ ρ(M ). By definition, ρ −1 (R′ ) ∈ ρ −1 (M ′ ). We claim ρ −1 (R′ ) ̸∈ M. Suppose ρ −1 (R′ ) ∈ M. Then, ρ[ρ −1 (R′ )] ∈ ρ(M ). Because ρ[ρ −1 (R′ )] = R′ , we have R′ ∈ ρ(M ), which is a contradiction.

S. Sato / Mathematical Social Sciences 79 (2016) 11–19

17

(i) For each i ∈ N and each M ∈ Mi , there is x ∈ X such that M ⊂ M (x) ≡ {R ∈ L | r1 (R) = x}. (ii) For each i ∈ N, each A ∈ A, and each M ∈ M (A)i , there is x ∈ A such that M ⊂ MA (x) ≡ {R ∈ L(A) | r1 (R) = x}.

A.2. A proof of Theorem 3.1

Proof. (i) Let i ∈ N and M ∈ Mi . Suppose M ̸⊂ M (x) for each x ∈ X . Let M ∈ M be such that Mj = M for each j ∈ N. (By Lemma A.1, such M exists in M .) Then, there are R, R′ ∈ M such that r1 (R) ̸= r1 (R′ ). Let x = r1 (R) and y = r1 (R′ ). Let R ∈ M be such that Ri = R for each i ∈ N. Let R ′ ∈ M be such that R′i = R′ for each i ∈ N. Because R , R ′ ∈ M , f1 (R ) = f1 (R ′ ). Because f satisfies efficiency, for each z ∈ X \x, we have z ̸∈ f1 (R ). By efficiency, we also have x ̸∈ f1 (R ′ ). Thus, f1 (R ) = f1 (R ′ ) = ∅, which is a contradiction. (ii) In the proof of (i), replace Mi by M (A)i , X by A, M by M (A), and f1 by fA . 

Step 2: For each i ∈ N , Mi = Mi . This is established in Lemma A.4.

p

Lemma A.4. For each i ∈ N , Mi = Mi = Mir . Proof. The second equality holds by definition. Thus, we prove Mi = Mir . Step 1: Mi ⊂ Mir . Let M ∈ Mi . Let x ∈ X be such that M ⊂ M (x). (By Lemma A.3, such x exists.) We claim M = M (x) ∈ Mir . Suppose M ( M (x). Because Mi and Mir are partitions of L, there is M ′ ∈ Mi such that M ′ ⊂ M (x) and M ̸= M ′ . Let y ∈ X , and ρ be the permutation of X exchanging x and y. Then, ρ(M ) and ρ(M ′ ) are subsets of M (y). By Lemma A.2, ρ(M ), ρ(M ′ ) ∈ Mi . Because y was arbitrary, for each z ∈ X , M (z ) contains at least two messages in Mi . Thus, |Mi | ≥ 2m. By Lemma A.1, i∈N |Mi | ≥ 2mn. Therefore, for each R ∈ LN , the informational size of F at R , s(F , R ), is at least 2mn. n There are two cases to consider. First, assume ⌈ m ⌉ = ⌈ 2n ⌉. Then, N p for each R ∈ L , s(F , R ) = nm < s(F , R ). This is a contradiction to the assumption that F is informationally efficient. Next, assume ⌈ mn ⌉ < ⌈ 2n ⌉. Then, at each R ∈ LN , s(F r , R ) ≤ nm + 2n < 2mn. n ⌉ < ⌈ 2n ⌉ implies m ≥ 3.) This is a contradiction (Remember that ⌈ m to the assumption that F is informationally efficient. Step 2: Mi ⊃ Mir . Let M (x) ∈ Mir . Because Mir and Mi are partitions of L, there is M ∈ Mi such that M (x) ∩ M ̸= ∅. By Lemma A.3, M ⊂ M (x). By Step 1, M ∈ Mir . Thus, M = M (y) for some y ∈ X . Since M ⊂ M (x), x = y. Thus, M = M (x), and hence M (x) ∈ Mi .  p

Lemma A.5. For each R ∈ LN , f1 (R ) ⊃ f1 (R ). p

p

Proof. Suppose f1 (R ) ̸⊃ f1 (R ) for some R ∈ LN . Let x ∈ f1 (R ) \ f1 (R ). p First, we claim f1 (R ) ∩ f1 (R ) = ∅. Suppose that there is y ∈ p f1 (R ) ∩ f1 (R ). Let σ be a permutation of N such that σ ({i ∈ N | r1 (Ri ) = x}) = {i ∈ N | r1 (Ri ) = y} and σ ({i ∈ N | r1 (Ri ) = y}) = {i ∈ N | r1 (Ri ) = x}. By anonymity, y ∈ f1 (R σ ). Let ρ be the permutation of X exchanging x and y. By neutrality, x ∈ f1 [ρ(R σ )]. Note that the two n-tuples of top ranked alternatives in R and ρ(R σ ) are the same. Therefore, by Lemma A.4, R and ρ(R σ ) belong to the same message profile. Thus, x ∈ f1 (R ), which is a contradiction. p Let z ∈ f1 (R ). By the above argument, z ̸∈ f1 (R ). Let Nx be a subset of {i ∈ N | r1 (Ri ) = x} such that |Nx | = |{i ∈ N | r1 (Ri ) = z }|. Let σ ′ be a permutation of N such that σ ′ (Nx ) = {i ∈ N | r1 (Ri ) = z } and σ ′ ({i ∈ N | r1 (Ri ) = z }) = Nx . By anonymity, ′ z ∈ f1 (R σ ). Let ρ ′ be the permutation exchanging x and z. By ′ neutrality, x ∈ f1 [ρ ′ (R σ )]. For each i ∈ {j ∈ N | r1 (Rj ) = x} \ Nx , lift x to the top in ρ ′ (Rσi ). Let R ′′ ∈ LN be the resulting preference profile. By monotonicity, x ∈ f1 (R ′′ ). The two n-tuples of top ranked alternatives in R and R ′′ are the same. Therefore, R and R ′′ belong to the same message profile. Because x ∈ f1 (R ′′ ), it follows that x ∈ p f1 (R ). This is a contradiction to the choice of x ∈ f1 (R ) \ f1 (R ).  ′

Step 1: The plurality rule is in R. We omit the proof. p

Step 3: F is a one-stage rule, and its informational size at each R ∈ LN is nm. If F is a two-stage rule, by Step 2, F p is informationally more efficient than F . This is a contradiction to the assumption that F is informationally efficient. Thus, F is a one-stage rule, and by Step 2, its informational size at each R ∈ LN is nm. p

Step 4: For each R ∈ LN , f1 (R ) ⊃ f1 (R ). This is established in Lemma A.5. Step 5: The plurality rule and F operates on the minimal informational requirement in R. By Step 3, for each R ∈ RN , s(F ′ , R ) ≥ s(F , R ) = s(F p , R ) = nm. (Remember that F ′ was arbitrary chosen from R at the beginning of Appendix A.1.) This implies that F p operates on the minimal informational requirement in R. Moreover, because s(F , R ) = s(F p , R ) for each R ∈ LN , F also operates on the minimal informational requirement in R. Step 6: If a rule operates on the minimal informational requirement in R, it satisfies the properties listed in (ii) in Theorem 3.1. If a rule operates on the minimal informational requirement in R, it is informationally efficient. Therefore, by Steps 2, 3, and 4, the rule satisfies the properties listed in (ii). A.3. A proof of Theorem 3.2 We omit the proof of F r ∈ R. Step 1: For each i ∈ N , Mi = Mir . This is established in Lemma A.4. Step 2 F is not a 1-stage rule. Suppose that F is a 1-stage rule. Let x ∈ X . There is R ∈ LN such p n 18 that x ∈ f1 (R ) and n(R , x) = ⌈ m ⌉. Because f1 (R ) ⊃ f1p (R ) (by Lemma A.5), we have x ∈ f1 (R ). For each i ∈ N \ N (R , x), take x to the bottom of Ri . Let R ′ be the resulting preference profile. Because Mi = Mir for each i ∈ N (by Step 1), R and R ′ belong to the same n ⌉ < ⌈ 2n ⌉, a message profile. Thus, x ∈ f1 (R ′ ). Because n(R , x) = ⌈ m ′ strict majority puts x at the bottom of preferences at R . Therefore, x ∈ f1 (R ′ ) is a contradiction to the assumption that F avoids the Condorcet loser. Step 3: For each R ∈ LN , f2 (R ) = 0 implies f2r (R ) = 0. Let R ∈ LN . Assume f2 (R ) = 0. Then, f1 (R ) is the social choice of p F at R. By Lemma A.5, f1 (R ) contains f1 (R ). It can be seen that each p 19 alternative in f1 (R ) wins a majority. By definition, f2r (R ) = 0.

18 We show this formally. Let α and β be nonnegative integers such that n = α m + β with β < m. If β = 0, then let R ∈ LN be such that for each y ∈ X , n(R , y) = α . If β > 0, let A ⊂ X be such that |A| = β and x ∈ A, and let R ∈ LN be such that for each y ∈ X ,  α, if y ∈ X \ A n (R , y ) = α + 1, if y ∈ A. p

n In either case (β = 0 or β > 0), x ∈ f1 (R ) and n(R , x) = ⌈ m ⌉. 19 To see this, suppose that some alternative in f p (R ) does not win a majority. Let p

1

x ∈ f1 (R ) be such an alternative. For each j ∈ N \ N (R , x), take x to the bottom of Rj . Let R ′ be the resulting preference profile. Because |N \N (R , x)| > 2n , a strict majority puts x at the bottom of preferences. Thus, x is the Condorcet loser. By Step 1, R and R ′ belong to the same message profile. Thus, f (R ) = f (R ′ ). Therefore, x ∈ f1 (R ′ ). Because f2 (R ′ ) = 0, x belongs to the social choice of F at R ′ . This is a contradiction to the assumption that F avoids the Condorcet loser.

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Step 4: For each R ∈ LN , f2r (R ) ̸= 0 implies that T [f (R )] contains at least two alternatives. Assume f2r (R ) ̸= 0. By Step 3, f2 (R ) ̸= 0. By the definition of p p r f2 , no alternative in f1 (R ) wins a majority. Because f1 (R ) contains alternatives which are most preferred by the largest number of agents, no alternative in f1 (R ) wins a majority. Suppose that T [f (R )] = {x} for some x ∈ X . Then, for each j ∈ N \ N (R , x), take x to the bottom of Rj . Let R ′ be the resulting preference profile. Because |N \ N (R , x)| > 2n , a strict majority puts x at the bottom of preferences. Thus, x is the Condorcet loser. By Step 1, R and R ′ belong to the same message profile. Thus, f (R ) = f (R ′ ), and T [f (R ′ )] = {x}. Thus, the social choice of F at R ′ is x, which is a contradiction to the assumption that F avoids the Condorcet loser. Step 5: For each R ∈ LN , f2r (R ) = 0 implies f2 (R ) = 0. Let R ∈ LN . If f2r (R ) ̸= 0, by Step 3, the rule F has the second stage, and by Step 4 and Lemma A.3, each M (T [f (R )])i contains at least two messages. Thus, s(F , R ) ≥ s(F r , R ). If f2r (R ) = 0, by Step 1, s(F , R ) ≥ s(F r , R ). Suppose that there is R ′ ∈ LN such that f2r (R ′ ) = 0 and f2 (R ′ ) ̸= 0. Then, the last inequality in the last paragraph holds strictly with R ′ . Then, F r is informationally more efficient than F . This is a contradiction to the assumption that F is informationally efficient. Step 6: For each R ∈ LN , f1 (R ) ⊃ f1r (R ). p Let R ∈ LN . By Lemma A.5, f1 (R ) ⊃ f1 (R ). By definition of p r p F , f1 (R ) = f1 (R ) in the first and the second case in the definition of f r . (See Definition 2.3.) Thus, it suffices to consider the remaining p p case where |f1 (R )| = 1 and n(R , x) < 2n for each x ∈ f1 (R ). In other words, the set of winners under the plurality rule is a singleton, and the alternative does not win a majority. Let {x} ≡ p f1 (R ). Claim 6.1. f1 (R ) ̸= {x}. Proof of Claim 6.1. Because x does not win a majority at R, the agents in N \ N (R , x) form a strict majority. Then, for each i ∈ N \ N (R , x), take x down to the bottom of Ri . Let R ′ be the resulting preference profile. Because R and R ′ belong to the same message profile, f (R ) = f (R ′ ). Thus, if {x} = f1 (R ), then {x} = f1 (R ′ ). Because the social choice at each preference profile cannot be ∅, x is the social choice of F at R ′ . Because x is the Condorcet loser at R ′ , this is a contradiction.  It suffices to show f1 (R ) ⊃ V2 (R ). (See Definition 2.3 for the definition of V2 (R ).) Suppose f1 (R ) ̸⊃ V2 (R ). Let y ∈ V2 (R ) \ f1 (R ). Claim 6.2. f1 (R ) ∩ V2 (R ) = ∅. Proof of Claim 6.2. Apply the arguments of the second paragraph p in the proof of Lemma A.5. (Replace f1 (R ) by V2 (R ), y by w , and x by y.)  Let z ∈ f1 (R ) \ x. By Claim 6.1, such z exists. By Claim 6.2, z ̸∈ V2 (R ). Because z ̸∈ (V1 (R ) ∪ V2 (R )), n(R , y) > n(R , z ). Then, by applying the arguments of the last paragraph in the proof of p Lemma A.5, we have a contradiction. (Replace x by y, and f1 (R ) by V2 (R ).) Step 7: For each A ∈ A, |A| = 2. Let R ∈ LN be such that f2 (R ) ̸= 0. By Step 5, f2r (R ) ̸= 0. By Step 4, T [f (R )] contains at least two alternatives. Thus, there is no singleton in A. Suppose that there is A ∈ A such that |A| ≥ 3. By Lemma A.3(ii), |M(A)i | ≥ 3 for each i ∈ N. Thus, the informational size of the second stage is at least 3n. By Steps 3 and 5, f2 (R ) = 0 if and only if f2r (R ) = 0. Also, the informational size of F r ’s second stage is 2n, and the informational sizes of the first stage of F and F r are

the same. Thus, F r is informationally more efficient than F . This is a contradiction to the assumption that F is informationally efficient. Step 8: For each i ∈ N and each A ∈ A, M (A)i = {{R} | R ∈ L(A)}. Let A ∈ A. By Step 7, A consists of two alternatives. Let A = {x, y}. There are only two linear orders over A; x is preferred to y, or y is preferred to x. By Lemma A.3(ii), these two preferences belong to different messages. Thus, M (A)i = {{R} | R ∈ L(A)}. Step 9: For each A ∈ A, fA is the simple majority rule between the two alternatives in A. p Let A ∈ A. By Step 7, A consists of two alternatives. Let fA be the simple majority rule between the two alternatives. We prove p fA (R A ) = fA (R A ) for each R A ∈ L(A)N . (In this step, we put a superscript ‘‘A’’ to a preference profile over A.) Let R A ∈ L(A)N . p

Claim 9.1. fA (R A ) ⊃ fA (R A ). Proof of Claim 9.1. This can be established by similar arguments in Lemma A.5.  p

Claim 9.2. fA (R A ) ⊂ fA (R A ). Proof of Claim 9.2. We prove the contrapositive; for each x ∈ p A, x ̸∈ fA (R A ) implies x ̸∈ fA (R A ). Thus, let x ∈ A be such that p A x ̸∈ fA (R ). Let y = A \ x. Let R ∈ LN be such that T [f (R )] = A. By Step 5, both x and y do not win a majority at R, i.e., n(R , x) < ⌈ 2n ⌉ and n(R , y) < ⌈ 2n ⌉. Then, there is R ′ ∈ LN such that r1 (Ri ) = r1 (R′i ) for each i ∈ N , x is the Condorcet loser at R ′ , and at R ′ |A, x does not win a majority in A by one vote. Such R ′ can be obtained as follows; let N ′ ⊂ N be such that N ′ ⊃ N (R , x), N ′ ∩ N (R , y) = ∅, and |N ′ | = ⌈ 2n ⌉− 1. Such N ′ exists because neither N (R , x) nor N (R , y) forms a majority at R. Modify R as follows; for each i ∈ N ′ , take y to the bottom of Ri , and for each i ∈ N \ N ′ , take x to the bottom of Ri . Let R ′ be the resulting preference profile. Then, R ′ is the desired one. Because R and R ′ belong to the same message profile, T [f (R ′ )] = A. Because x is the Condorcet loser at R ′ , x ̸∈ fA (R ′ |A). At R ′ |A, ⌈ 2n ⌉ − 1 agents prefer x to y. Remember that at R A , at most ⌈ 2n ⌉ − 1 agents prefer x to y. By anonymity and monotonicity, it can be seen that x ̸∈ fA (R A ).  p

By Claims 9.1 and 9.2, we have fA (R A ) = fA (R A ). r

Step 10: F operates on the minimal informational requirement in R. By Steps 1, 3, 5, and 8, for each R ∈ LN , s(F , R ) = s(F r , R ). Thus, for each R ∈ LN , s(F ′ , R ) ≥ s(F r , R ). Because F ′ was arbitrary in R, F r operates on the minimal informational requirement in R. Step 11: Each rule operating on the minimal informational requirement in R satisfies the properties listed in (ii) of Theorem 3.2 If a rule F ∗ operates on the minimal informational requirement in R, it is informationally efficient. Because F can be considered as an arbitrary informationally efficient rule, F ∗ satisfies the properties described in Steps 1 through 9. The properties are what we described in the theorem. References Ching, Stephen, 1996. A simple characterization of plurality rule. J. Econom. Theory 71, 298–302. Conitzer, Vincent, Sandholm, Tuomas, 2005. Communication complexity of common voting rules. In: Proceedings of the ACM Conference on Electronic Commerce, pp. 78–87. Fishburn, Peter C., 1977. Condorcet social choice functions. SIAM J. Appl. Math. 33, 469–489. Freeman, Rupert, Brill, Markus, Conitzer, Vincent, 2014. On the axiomatic characterization of runoff voting rules. In: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence (AAAI-14). Moulin, Hervé, 1988. Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge.

S. Sato / Mathematical Social Sciences 79 (2016) 11–19 Nurmi, Hannu, 1999. Voting Paradoxes and How to Deal With Them. Springer, Berlin. Pattanaik, Prasanta K., 2002. Positional rules of collective decision-making. In: Arrow, Kenneth J., Sen, Amartya, Suzumura, Kotaro (Eds.), Handbook of Social Choice and Welfare, Vol. 1. North-Holland, Amsterdam, pp. 361–423. chapter 7. Richelson, Jeffrey T., 1978. A characterization result for the plurality rule. J. Econom. Theory 19, 548–550.

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Roberts, Fred S., 1991. Characterization of the plurality function. Math. Social Sci. 21, 101–127. Sato, Shin, 2009. Informational requirements of social choice rules. Math. Social Sci. 57, 188–198. Smith, H.John, 1973. Aggregation of preferences with variable electorate. Econometrica 41, 1027–1041. Yeh, Chun-Hsien, 2008. An efficiency characterization of plurality rule in collective choice problems. Econom. Theory 34, 575–583.