A Society Can Always Decide How to Decide: A Proof

Infinite regress lies within every democratic procedural choice. If society members try to select an appropriate rule [social choice correspondence (SCC)] entirely endogenously, they will need an appropriate rule to choose such a rule. However, this should also be selected by an appropriate rule to choose a rule to choose a rule, and so on. This paper explores how to solve this infinite regress. A preference profile over the set of alternatives is said to converge if, at a sufficiently high level, every feasible SCC in the menu ultimately results in the same alternative, and hence, further regress has no effective meaning. A menu is said to be convergent if all preference profiles converge under the menu (i.e., infinite regress can “always” be resolved). First, we characterize the convergent menus under a special case. Then, we prove two general possibility theorems: (1) there exists a menu of SCCs that is strongly convergent (i.e., the outcome is uniquely determined); (2) any set of scoring rules can be extended to a superset that is asymptotically convergent for a large society (i.e., the probability of a convergent profile occurring goes to one as the population goes to infinity). Therefore, such a large society can “almost always” resolve the infinite regress by adding multiple SCCs. These theorems are expected to build new ground for SCCs in a distinct way from the axiomatic characterizations of standard social choice theory.


Introduction
Infinite regress lies within every democratic procedural choice (PC).If members of society try to determine the appropriate social choice correspondence (SCC) ("how to choose") entirely endogenously, they would need an appropriate SCC to select SCC ("how to choose how to choose"); this should also be selected by an appropriate SCC to select SCC to select SCC ("how to choose how to choose how to choose"), and so on.When there is no ex-ante agreement on the appropriate SCCs at some level, this process continues forever, and they cannot justify the use of a particular SCC.A similar infinite regress of justification is a classic problem that arises in various academic disciplines in, for example, the search for knowledge (Klein 2007), constitutional choice (Buchanan and Tullock 1962;Eichberger and Pethig 1994), and choice of SCCs in voting theory (Kultti and Miettinen 2009;Almeida and Nurmi 2015).
This article aims to solve this infinite regress for PC in collective decision-making.Suzuki and Horita (2017a) proposed a solution concept called convergence. 1et X be the set of alternatives.Roughly speaking, a preference profile over X is said to converge to A ⊆ X if-at a sufficiently high level-every feasible SCC ulti- mately reaches A .Suppose that a society has a triple (the plurality f P , Borda count f B , and anti-plurality f A ) as the set of all feasible SCCs (the menu) and that each of them-as an SCC to choose SCC-ultimately designates {a} (Fig. 1).Once this con- vergence occurs, higher level PC has no effective meaning; this is because no matter which of f P , f B , f A is selected at such a high level, the ultimate outcome must be {a}.
When a menu of feasible SCCs is given, whether a convergence occurs depends on the preference profiles over X. 2 In some cases, convergence is not found at any level (e.g., Fig. 2 shows an extreme case called trivial deadlock (Suzuki and Horita 2017a).Nevertheless, Suzuki and Horita (2017a) prove the frequency of convergence in some situations: that given three alternatives, a large society (i.e., the population n goes to infinity) with f P , f B , f A finds convergence with probability 0.982 under the impartial culture model (IC) and 0.988 under the impartial anonymous culture (IAC) model.Suzuki and Horita (2017b) investigate other typical menus under IAC in a large society.
These results imply that convergence frequently occurs in a large society with a relatively small menu of familiar SCCs; however, they cannot deny the possibility that convergence fails to occur.Indeed, a large society equipped with f P , f B , f A could face a non-convergent preference profile at a small-but positive-probability of 1 − 0.982 [IC] and 1 − 0.988 [IAC].
If so, the biggest challenge should be how a society can "always" resolve the infinite regress by avoiding such failure.The main contribution of this paper is to give two positive answers to this challenge (Theorems 2 and 3).More formally, a menu F is said to be convergent if all preference profiles over X converge under menu F .

3
A Society Can Always Decide How to Decide: A Proof The goals of this paper are to show that 1) there is a menu of SCCs that is convergent in a strong sense (strongly convergent), under which the outcome is uniquely determined, and 2) any set of scoring rules can be extended to a superset that is convergent in an asymptotic sense for a large society (asymptotically convergent), under which the probability of a convergent profile occurring approaches one as the population approaches infinity.The results provide democratic societies with a reasonable solution to infinite regress.
In Sect.3, we first characterize the (strongly) convergent menus (Theorem 1) under some extra assumptions (such as "population |N| is odd", "each feasible SCC satisfies the Condorcet winner criterion", etc.) that relieve technical complexity.Since convergence (of a profile) means that all feasible SCCs in the menu at some level indicate the same alternative(s), it is likely to occur if the SCCs in the menu are homogeneous ("similar" to each other).Theorem 1 is a formal description of this observation.At the same time, however, this observation also implies the difficulty of achieving convergence when the menu includes heterogeneous SCCs.
In Sects.3.2 and 3.3, we remove the extra assumptions for Theorem 1-and prove two general possibility theorems.
One is the existence of strongly convergent menus (Theorem 2).As noted, the central issue of this paper is whether it is ever possible to always solve infinite regress.Theorem 2 is a positive answer for this: there exists a nontrivial3 menu that is strongly convergent.Compared with the previous studies (Suzuki and Horita 2017a;b), Theorem 2 advances the state of the literature in that it can apply to a finite society and guarantees convergence without exception.Particularly, the constructive proof of Theorem 2 states that there exist infinitely many menus that are strongly convergent.The most intuitive one shown in the proof would be the menu F 0 of Copeland's method, Borda count, and Black's procedure (with a proper tie- breaking rule).A society with such a menu can always justify an outcome through infinite regress (such an outcome does not depend on the choice of admissible metalevel preferences under strongly convergent menus).
If there is no restriction on the menus of feasible SCCs, the strongly convergent menus, such as F 0 , provide straightforward solutions for PC.On the other hand, this Fig. 1 Introductory example of convergence to a might not be so appealing if the society has some historical or institutional, or practical reason to use f P instead of Copeland's method.Then, the fact that the menu f P , f B , f A is not convergent remains a problem.Suzuki and Horita (2017b) provide a positive answer for this.They prove that by adding an extra SCC into the menu f P , f B , f A , the extended menu G = f P , f B , f A ∪ { } can be made asymptotically convergent (the probability of the occurrence of convergent profiles goes to 1 as n → ∞ ).Theorem 3 is a complete generalization of this observation: we prove that such an extension is possible for any menu consisting of scoring rules.This means that any large society with a menu of (only) scoring rules can asymptotically always find convergence by adding appropriate SCCs into the menus.
In democratic decision-making, PC is a public concern: its impact is argued in practical cases such as U.S. presidential elections (Riker 1982;Tabarrok and Spector 1999).In social choice theory, the closest to our motivation would be procedural autonomy by Dietrich (2005).This premise states that a "good" decision rule is one favored by the individuals within that society; whether it is a dictatorship or strategically manipulable rules, a decision rule is judged "good" if the members of society approve it.Once this premise is applied literally, however, society needs a rule to choose a rule, and thus faces infinite regress.Related literature on PC is reviewed based on three viewpoints.

Timing of PC
A natural classifier of the models of PC is the timing of PC: The model should differ by whether PC is considered before/after a particular decision-making problem arises.The "before" case is similar to the veil of ignorance by Rawls (2001).If the decision problem has not emerged, voters do not know their preferences or which SCC leads to which alternatives.So, voters' preferences for an SCC are supposed to be determined by the characteristics/nature of the SCC.Two important approaches are to judge SCCs by the expected utility they will provide in the future decision problem (Rae 1969;Barberà and Jackson 2004;Kultti and Miettinen 2009) and to consider voters' preferences about criteria for SCCs (Nurmi 2015;Suzuki and Horita 2020).This article and others-such as Koray (2000) and Koray and Slinko (2006)-deal with the "after" case; namely, PC after the decision-making problem emerges.A major difficulty of this situation is that the occurrence of a particular decision problem should reveal the conflicts between the voters' interests.A voter who usually prefers the use of Borda count for its theoretical superiority might not A Society Can Always Decide How to Decide: A Proof support it or prefer it when they know that Borda count will select the worst alternative for that voter in the decision-making problem at hand.4 2. The existence of the status-quo procedure When there is an existing rule to choose a rule (e.g., the amendment rule of a constitution), PC involves the transition between rules; namely, the current rule f selects g as the appropriate rule, but g can select h as the appropriate rule-and so on.A rule is sup- posed to be "stable" if it is a fixed point of this process, selecting itself as the appropriate one.Such a fixed-point approach is made by Barberà and Jackson (2004) and Koray (2000), who called it "self-stability" and "self-selectivity," respectively.These properties demand that an SCC choose itself against the rival SCCs at hand in the face of a preference profile over the SCCs. 5 Although attractive, self-stable rules are often difficult to find.For example, Koray proves an impossibility theorem by saying that for unanimous and neutral SCFs, universal self-selectivity is logically equivalent to dictatorship.Later, the properties are applied to the set of SCCs (instead of a single SCC).A set of SCCs is stable if for each profile there is exactly one rule that chooses itself (Houy 2004) or if there is at least one self-selective SCC (Diss and Merlin 2010;Diss et al. 2012).Using the probability calculation methods developed by Saari and Tataru (1999) and Merlin et al. (2000), Diss et al. (2012) estimate the likelihood that the set of f P , f B , f A is stable under the IC model for a large soci- ety.Diss et al. (2012) also determine corresponding probabilities under the IAC model.These estimations show that the set of f P , f B , f A is stable, with a prob- ability of 84.49% under IC and 84.10% under IAC for a large society.This article considers a situation where no rule is accepted as the status quo.In this case, the fixed point approach is not so attractive, because the fixed point usually depends on the initial rule, but the initial rule itself has not been accepted. 6ndeed, there can be more than one self-selective rule (Fig. 2) and the society can face further choice from the fixed points.

The form of meta-level procedures
This paper assumes that the menu does not change by level.One interpretation of this assumption is that the menu represents the set of all feasible decision procedures for society.Thus, this list is applied to any decision problem, including how to choose, how to choose how to choose, and so on.Just as Arrow's (1951) axiom of unrestricted domain demands that SWF can be applied to any preference profile, our assumption means the universality of the considered menu.In the authors' view, a similar assumption would be necessary, at least to some extent, for a successful theory of PC (if the procedures change completely randomly by level, it would be nearly impossible to argue the outcomes and/or grounds of the procedures).
The assumption specifies the context of PC by defining the meta-level procedures.Other types of meta-level procedures are also found in the literature.For example, Nurmi (2015), Palha et al. (2017), and Suzuki and Horita (2020) consider the meta-level choice of criteria for voting rules, and List (2002) examines the process of seeking agreement at a meta-level.More technically, defining a meta-level procedure as a hierarchical set of decision rules (e.g., a pair of one rule and another rule to select the rule) is also an alternative approach (Barberà and Jackson 2004;Kultti and Miettinen 2009).In summary, this paper considers a situation where (1) a decision-making problem has arisen, but (2) there is no status quo rule, and (3) society agrees only on the set of all feasible SCCs, but without any meta-level norm that can order these SCCs.This situation naturally induces infinite regress of how to choose how to choose (and so on).As we stated above, this infinite regress is difficult to solve within the existing models.The major contribution of this paper is to show how society can "always" solve the infinite regress in this situation.It is also worth noting that our result builds new ground for SCCs.In standard social choice theory, SCCs are usually justified by their axiomatic properties.By contrast, our result proves that SCCs can be justified based solely on their procedural judgments through infinite regress.The axiomatic study of SCCs has theoretical elegance, but may have little appeal for decision-makers when the plausible axioms turn out to be inconsistent, or there is a disagreement even on which axioms are normative in the given decision-making problem.Our result is expected to compensate for such problems and build a positive theory of PC.This paper is organized as follows.Section 2 describes the basic notations and definitions.Section 3 provides the formal results.Section 4 provides the concluding remarks.All proofs are in the appendices.

General Assumptions
Throughout the paper, we consider a situation where a set of voters N = {1, 2, … , n} ( 3 ≤ n < ∞ ), which is called society, collectively chooses from X-which is the set of all alternatives and 2 ≤ m 0 ∶= |X| < ∞ .The set of all feasible SCCs (see Sect. 2.2 for the formal definition) is called a menu.The menu should be a finite set with at least two elements.
Each voter i ∈ N is assumed to be consequential in the sense that they evaluate meta-level SCCs by their outcomes (the formal definition of the consequentiality is in Sect.3.3, using the following e i ).Each voter i ∈ N is supposed to be equipped with a preference extension system e i , which maps a linear order L i over X to a lin- ear order e i L i over 2 X ⧵ { }-satisfying conditions (i) (the extension rule) and (ii) below: 1 3 A Society Can Always Decide How to Decide: A Proof (i) for all a, b ∈ X and L i ∈ L(X) , we have aL i b ⟺ {a}e i L i {b} ; and.(ii) for all A ∈ 2 X ⧵ { } and b ∈ X ⧵ A , if bL i a for all a ∈ A , then A ∪ {b}e i L i A.

Definition of SCCs
Let A be any finite nonempty set.The set of all linear orders (resp.weak orders) over A is denoted by L(A) (resp.W(A) ).A preference profile over A is an n-tuple of linear orders L 1 , L 2 , … , L n ∈ L(A) n , where L i is interpreted as a voter i 's preference.
A social choice correspondence (SCC) is a correspondence that selects a set of winning options f (L)7 with  ≠ f (L) ⊆ A , whenever it receives any finite nonempty set A (called the set of options) and preference profile L ∈ L(A) n .An SCC is called a social choice function (SCF) if f (⋅) is always a singleton.In such a case, with a slight abuse of notations, we often write Note that our definition of the SCC is "option-free": we do not specify the set of options A .Rather, an SCC is considered "universal" and can be applied to various sets of options.Here, only two special cases matter: the case when A = X (i.e., each option is an alternative) and the case when A is the menu (i.e., each option is a fea- sible SCC).Such an option-free definition of SCCs is typical in the research of PC (Koray 2000;Barberà and Jackson 2004;Koray and Slinko 2006).
An SCC is called neutral if the names of the options do not affect the winning set.Formally, let A and B be two sets-such that 2 A scoring rule f is an SCC that is characterized by what we call score assign- ment s m 1 , s m 2 , … , s m m m∈ℕ (where m is the number of options).When there are m options, it assigns s m j scores to the option if it is ranked at the jth position by some voter.Aggregating the points, the winning options f (L) is determined as the set of options with the highest scores.We assume that the score assignment is normalized as When a particular value of m matters, the score assignment is denoted without the superscripts-i.e., s 1 , s 2 , … , s m .The score assignments of typical scoring rules are as follows: Plurality assigns one point to the first position and none to the others, i.e., [1, 0, 0, … , 0] .Borda count assigns 1, m−2 m−1 , m−3 m−1 , … , 0 ; and anti-plurality assigns zero points to the worst and one point to the others, i.e., [1, 1, … , 1, , 0].

Definition of Convergence
Throughout this subsection, the menu is denoted by F .By assumption, F includes at least two SCCs.
The intuition of the infinite regress of PC is as follows.For the society N , the first level PC problem (level-1 issue) is which feasible SCC should be appropriate for aggregating the preference profile L 0 over X .Since the level-1 issue is a deci- sion-making problem, society must face another PC problem (level-2 issue) of which feasible SCC should be used for aggregating the preferences on the level-1 issue.Inductively, for each k ∈ ℕ , the level-(k + 1) issue is defined as the choice of appropriate SCC for resolving the level-k issue.This process can go on ad infinitum unless there is an ex-ante agreement on the appropriate SCC at a certain level.In order to find an appropriate choice for the level-k issue, the level-(k + 1) issue must be resolved in advance.Therefore, society must resolve infinitely many levels-(k + 1), (k + 2), (k + 3), … issues before level-k issues-which we call the infinite regress of PC.We will formally state this discussion and define the convergence below.
• Level-1 issue is which SCC in menu F is appropriate for aggregating the pref- erence profile over X .In this context, each SCC in F is called a level-1 SCC, and menu F is called a level-1 menu-denoted by F 1 .A preference profile over F 1 is called the level-1 profile.First, when meta-level concepts matter, the levels are often denoted by superscripts, such as F k (level-k menu), L k (level-k profile), etc. (recall Figs. 1 and 2).Second, our model assumes that the menu-the set of all feasible SCCs-does not vary by the levels, i.e., F 1 = F 2 = ⋯ (= F) .Nevertheless, we often distinguish the notations F 1 , F 2 , … , F k , … to make clear the level under discussion.Third, with a little abuse of notation, the set of alternatives X is often called a level-0 menu (often denoted by F 0 ); a preference profile over X is called a level-0 profile.Let us denote by ℕ the set of all positive integers.Then, for k ∈ ℕ , a sequence of k preference profiles L Definition 2 (Leaf and Root).Let k ∈ ℕ , F be the menu-and L 0 , L 1 , … , L k−1 be a meta-sequence.The leaf of f ∈ F at L 0 , L 1 , … , L k−1 , denoted by leaf f ;L 0 , L 1 , … , L k−1 , is defined as follows.
1 3 A Society Can Always Decide How to Decide: A Proof Conversely, for C ⊆ X , we define the root of C ⊆ X at L 0 , L 1 , … , L k−1 , denoted by root C;L 0 , L 1 , … , L k−1 -as follows.
Intuitively, leaf f ;L 0 , L 1 , … , L k−1 represents the set of alternatives in X that f ultimately reaches; root C;L 0 , L 1 , … , L k−1 represents the set of all level-k SCCs in F k -whose leaf is C. Definition 3 (CI sequence).Let k ∈ ℕ , F be the menu, and L 0 , L 1 , … , L k−1 be a meta-sequence.A level-k consequentially induced (CI) weak profile with respect to A level-k CI profile L k is interpreted as a level-k preference profile based on the assumption that each voter evaluates level-k SCCs "consequentially" by their leaves.The leaf of an SCC is not an element but a subset of X .Therefore, we require the preference extension systems e 1 , e 2 , … , e n to define meta-level prefer- ences.The preference extension systems extend voters' preferences L 0 1 , L 0 2 , … , L 0 n into preferences over the power set.
Example 1 (leaf, root, and CI sequence).Suppose X = {a, b, c} , F = f P , f B , f A ( f P : plurality, f B : Borda count, and f A : anti-plurality).Consider a meta-sequence L 0 , L 1 such that B (the situation illustrated in Fig. 1).Then, we have that Level-1 CI profile is determined by the leaves.Suppose that voter 1 has a level- 0 preference L 0 1 ∶ a, b, c (i.e., a is the best preferred, b is the second, and c is the worst).By both Definition 3 and (i) in Sect.2.1 the level-1 CI weak profile should demand that R 1 , voter 1 is indifferent over f 1 P and f 1 B and they are both strictly preferred to f 1 A ). Since CI profile L 1 is compatible with such R 1 , it follows that L 1 1 is obtained by breaking the tie between f 1 P and f 1 B .As a result, we have two possibilities on voter 1 's level-1 preference at unique (as in Example 1).This is because a weak order can have more than one compatible linear order.Second, if the menu F is made up of SCFs only, then the leaf of any meta-level SCC in the menu must be a singleton.In that case, the level-k CI weak profile is equivalent to the level-k CI profile-which is also identical with what Koray (2000) calls an "induced preference."In this sense, our definition of the CI profile is a generalization of Koray's induced preference.
Remark 1 (On conditions for the preference extension system).A preference extension system e i maps a preference over X to a preference over 2 X ⧵ { } .Among numerous such systems studied so far (see Barberà et al. 2004, for a good survey), the present paper assumes conditions (i) and (ii) in subsection 2.1-which the authors believe to be mild and natural enough for our purpose.Condition (i) is called the extension rule (Barberà et al. (2004).It states that if a voter prefers element x to element y , then such a voter must prefer singleton {x} to singleton {y} .In the ter- minology of our model, this simply says that if the leaf of f is {x} , that of g is {y} , and a voter i prefers x to y-then such a voter must prefer f to g .Condition (ii) is (a part of) what is known as the dominance or Gärdenfors principle (Gärdenfors 1976;Kannai and Peleg 1984;Barberà et al. 2004).It states that if an extra alternative b -which is preferred to any element in A-to the set A , then the new set A ∪ {b} is strictly preferred to A .Note that there surely exists a preference extension system that satisfies conditions (i) and (ii).Suppose that each voter has a utility function surely satisfies the two conditions.Second, our subsequent argument holds no matter which combination of preference extension systems e = e i i∈N is given.
8 Whether a profile L 0 weakly converges or not depends on what kind of menu F the society considers.Therefore, it is more precise to say " L 0 weakly converges with respect to the menu F ." However-in the subsequent argument-because the menu F is explicit from the context, we simply say " L 0 weakly converges to C ⊆ X ".Furthermore, we do not specify individuals' preference extension systems e i i∈N in the definition of convergence.Strictly speaking, a profile L 0 is defined as weakly converging to C ⊆ X if and only if-for combinations of all preference extension systems e i i∈N -the required sequence of CI profiles exists.

3
A Society Can Always Decide How to Decide: A Proof Intuitively, " L 0 converges to C " means that by going up to a sufficiently high level k ∈ ℕ , every level-k SCC in the menu has the same leaf C .As we argued in Sect. 1, once this phenomenon occurs, further regress has no effective meaning because the leaf of any higher-level SCC in the menu must be C-no matter what higher level profiles are considered.
Our definition of convergence is close to self-selectivity (Koray 2000;Koray and Slinko 2006) in that convergence (self-selectivity) requires only that at least one CI profile (induced profile) exists such that the menu (or SCF) satisfies the required condition.As Koray and Slinko (2006) noted, requiring the property in all induced profiles makes the concept empty, and this is also the case for convergence.Indeed, this definition is even more plausible in convergence because its target is the infinite regress of PC.The existence of a single CI profile (a scenario) enables the justification of the SCC, which is, at first glance, impossible because of the troublesome infinite regress.
Technically speaking, it is possible for a single profile L 0 to converge to two different subsets C and C ′ via different CI sequences (Example 2).The concept of strong convergence will be introduced (Definition 5) That avoids this multiplicity.Strong convergence, once it occurs, designates the single outcome without the ambiguity of CI sequences.
On the other hand, suppose L 1 such that: Definition 5: Strong Convergence A profile L 0 ∈ L(X) n is said to strongly converge to C ⊆ X if and only if it weakly converges to C ⊆ X and does not weakly converge to any other C ′ ≠ C. With a little abuse of terminology, we say that a level-0 profile L 0 is convergent (resp.strongly convergent) if it converges (resp.strongly converges) to some subset of X.

General Assumption (Regular Probability Model)
Some of our results (only Sect.3.3) refer to the probability of convergence, which requires a certain probability model on the voting behavior.Regarding this point, the present paper stands from an axiomatic standpoint.We do not assume a particular probability model, such as the impartial culture model (IC), impartial anonymous culture model (IAC), etc. Rather, we give a set of mild conditions that a probability model should satisfy (these conditions are generalizations of our natural intuition that "ties" are unlikely when the population is large).Probability models that satisfy these conditions are called regular.Our theorems apply to any regular probability model.In this subsection, we explain the meaning of regularity in detail.
Let F be the menu.As noted in Sect.2.1, we assume that m 0 = |X| and n = |N| .For = 1, 2, … , m 0 !, let n represent the number of voters whose preference over X is the th linear order9 of X .Certain occurrence of n 1 , n 2 , … , n m 0 ! is called a (vot- ing) situation.For instance, (n, 0, 0, … , 0) represents a case where every voter has the first linear order of X as their preference over X .A probability model (for voting behavior on X ) specifies the probability for each voting event (e.g., the probability that a voter has a certain linear order as their preference, a probability that a certain voting situation occurs, etc.).Definition 6 (SC, NTP, DVR) Let G be the menu.A preference profile (over X ) L 0 ∈ L(X) n is said to satisfy.
• Singleton Condition (SC) (with respect to menu G ) if f L 0 is a singleton for all f ∈ G.
• No tie by plurality (NTP) if for all Y ⊆ X and for all x, y ∈ Y , we have 1 3 A Society Can Always Decide How to Decide: A Proof • Diverse (DVR) (with respect to menu G ) if for all Y ⊆ X and for all = 1, 2, … , |Y|! , we have Assumption 1 (Regular Probability Model).A probability model for voting behavior on X is called regular if (the probability that the level-0 preference profile L 0 satisfies all SC, NTP, and DVR with respect to a given menu G ) goes to 1 as n → ∞ .Throughout this paper, we assume that the probability model is regular.
Each SC, NTP, and DVR rules out certain types of "ties" (or such extreme cases).SC demands that each feasible SCC in the menu selects one, and only one, alternative (no ties by feasible SCCs).NTP requires that there is no tie by the Y -plurality scores.DVR demands that at least c , a constant number of voters exist who have each possible type of preference.
Several comments are in order.First, among many probability models studied so far (see Gehrlein and Lepelley [2010] for a survey), IC (for each voter, the m 0 !linear orders of X are equally likely as their preferences over X ) and IAC (each voting situation is equally likely to occur) would be the most familiar ones.In these models, it is often explicitly/implicitly argued that "ties" by a certain rule are unlikely to happen in a large society ( n → ∞ ) (Marchant 2001;Diss and Merlin 2010;Diss et al. 2012).Both IC and IAC are typical examples of regular probability models.The authors wish to emphasize that the three conditions (SC, NTP, and DVR) are not claimed to be normatively preferable per se; rather, they are technical conditions obtained by generalizing the unlikeliness of the ties for our theoretical purpose.Our results hold under any probability model that satisfies the three conditions (IC and IAC are the typical cases).
Second, if the menu G is made up of SCFs only, any level-0 preference profile sat- isfies SC.In general, DVR and NTP are logically independent of each other (there exists a profile L 0 that is DVR but not NTP [resp.NTP but not DVR]).
The third and most important comment is that the probability model is relevant only in Sect.3.3 (when the probability of convergent profiles is considered).The results in Sects.3.1 and 3.2 holds under any, even any non-regular, probability model.

Convergent Properties of a Menu
Finally, we introduce some properties saying that society can always find convergence with the menu.Let F be the menu.
Definition 7: Convergent Properties (1) A menu F is said to be convergent (resp.strongly convergent) if every L 0 ∈ L(X) n converges (strongly converges).
(2) Let p WC (resp.p SC ) be the probability that the level-0 preference profile is convergent (resp.strongly convergent).A menu F is said to be asymptotically con- vergent (asymptotically strongly convergent), if p WC → 1 (resp.p SC → 1 ) as n → ∞.
The strongly convergent property of a menu is logically the strongest of the four properties, and the asymptotically convergent property is the weakest.

Example 3: Menus of Two Neutral SCFs
Let F = {f , g} be a menu of two neutral SCFs and |X| = 2 .If L 0 ∈ L(X) n exists such that f L 0 ≠ g L 0 , then F is not weakly convergent because one can verify that such a profile L 0 causes a trivial deadlock (Suzuki and Horita 2017a); hence, it never weakly converges.Suzuki and Horita (2017b) show that the following 10 menus are asymptotically convergent (under IAC):

Results
This section is made up of three parts.Sect.3.1 characterizes (strongly) convergent menus under some technical conditions that relieve technical complexity; Sects.3.2 and 3.3 state general results without assuming such technical conditions.

Characterization of Convergent Properties in a Miniature Model
We say that a menu F satisfies.
• neutrality (N)/Condorcet winner criterion (CW) if each f ∈ F satisfies neutral- ity/Condorcet winner criterion, respectively (see, e.g., Nurmi 2002, for precise definitions), • singleton-valued (SV) if each f ∈ F is an SCF, • similar if for all preference profiles L 0 ∈ L(X) n , there is a pair f , g ∈ F of two distinct elements such that f L 0 = g L 0 .
Theorem 1 (Characterization of Convergent Properties in a Miniature Model).
Suppose that |X| = |F| = 3 (there are three alternatives and three feasible SCCs), F satisfies N, CW, and SV, and n is odd.
A Society Can Always Decide How to Decide: A Proof Then, the following are equivalent to each other: Albeit under a couple of conditions, Theorem 1 gives a characterization of (strongly) convergent properties of a menu.The intuition behind this is as follows.
The definition of convergence demands that every level-k SCF has the same leaves; however, this is likely to occur if these SCFs give a "similar" judgment (a particular case is when every level-(k + 1) SCF selects the same level-k SCF).Therefore, a menu F is likely to be convergent if the ingredients make "similar" judgments in some way.
Let us comment on the conditions in the theorem.From [ n is odd]: In each case of the proof of Theorem 1, we soon find the Condorcet winner.This is because we have assumed that n is odd.When n is even, it is possible that exactly half of the vot- ers prefer one alternative-while the other half prefer the other.In this case, there is no Condorcet winner, so further refinement of the proof is required.At the same time, however, it should be noted that such a case (a tie between some alternatives) becomes less likely as the population becomes larger (based on a regular probability model in Assumption 1).
As of [ |X| = 3 ]: The number of alternatives is insignificant in the proof.Indeed, suppose that there are more than three alternatives.We say that a menu F is similar with respect to X if for all L 0 ∈ L(X) n , there exists L 1 ∈ L L 0 such that at least two elements of F 2 have the same leaf.Similarly (like Theorem 1), one can verify that if n is odd; |F| = 3 ; and that F satisfies N, CW, and SV; it is (strongly) convergent if and only if it is similar with respect to X.
As of [ |F| = 3 ]: When there are more than three SCFs in the menu, the proposi- tion does not hold as it is.In general, convergent menus are similar (Proposition 1)-but not vice versa.Indeed, there exists a menu that satisfies both N and CW and is similar with respect to X-but is not convergent (Example 5).
Proposition 1 If a menu F satisfies N and SV, it is convergent only if it is similar with respect to X.

Strongly Convergent Menu
This subsection presents a general possibility result showing the existence of strongly convergent menus.Note that the strong convergent property is the strongest of the four in Definition 7. Theorem 2 no longer requires the technical conditions in Theorem 1 (such as the population n or the number of alternatives |X| ), and so it is much more general.

Theorem 2: Strongly Convergent Menu
There is a set F of neutral SCCs that is strongly convergent.
Theorem 2 says that there exists a strongly convergent menu F with which any society can reach the strong convergence no matter which level-0 preference profile is given.If a society is equipped with such a menu, then infinite regress can "always" (at any level-0 preference profile) be resolved.
Several comments are in order.First, our proof is constructive in that it contains an example of such strongly convergent menus-as well as the recipe of the appropriate CI sequence.In the proof, we show that a menu of three SCCs f 1 , f 2 , f 3 is convergent: for f 1 : Copeland's method, f 2 : any scoring rule with score assignment [1, s, 0] , and f 3 : a hybrid rule that chooses the Condorcet winner if it exists and oth- erwise chooses the winner of f 2 (all ties are broken by anti-plurality in each f 1 , f 2 , f 3 , so that they become SCFs).The menu is also shown to be strongly convergent if s ≥ 1∕2 .Since f 2 is any scoring rule with s ≥ 1∕2 , there are an infinite amount of strongly convergent menus.It is also worth noting that if f 2 is Borda count ( s = 0.5 ), f 3 is known as Black's procedure.Thus, the menu F 0 ∶= {Copeland, Borda, Black}-with a proper tie-breaking rule-is shown to be strongly convergent.
Second, our proof demonstrates how the unique alternative is determined as the convergent outcome through strong convergence.By the definition of convergence, the alternative is selected by at least one SCC in the menu; e.g., in the case of F 0 , the alternative is selected either by Copeland's method, Borda count, or Black's procedure.Table 1 is the summary of convergent outcomes by F 0 in each possi- ble case (the case of f 1 L 0 = f 2 L 0 = f 3 L 0 is straightforward; the other rows follow directly from (a)-(d) in the proof).If each f 1 , f 2 , f 3 selects the same winner, that is the unique convergent outcome.Otherwise, when the judgments of Copeland's method and Borda count disagree, the convergent outcome should be the one that the majority of individuals prefer or else the one that the majority of the three SCCs in the menu selects.In other words, the convergent winner reflects (at least in this case) the judgment of the majority of individuals/SCCs.This intuitive result has something to do with the definition of convergence.If each of the SCCs in the menu 1 3 A Society Can Always Decide How to Decide: A Proof satisfies a particular intra-profile axiom (Fishburn 1973;Geist and Peters 2016) such as the Pareto principle, then so does the convergent outcome.In general, the system that yields the convergent outcome retains some properties of the SCCs in the menu.
Third, let us introduce another axiom of a menu.
Definition 8: Difference A menu F satisfies difference if, for all f , g ∈ F and set X of alternatives with |X| ≥ 3 , there exists a profile L ∈ L(X) n such that f (L) ≠ g(L).
This axiom was introduced by Houy ( 2004) and is quite a weak condition: it demands only that F should not include more than two identical SCCs.Houy (2004) shows that there is no set F that satisfies neutrality (i.e., each f ∈ F is neutral), dif- ference, and strong first-level stability (i.e., exactly one self-selective SCC exists at every L ∈ L(X) n ).Although using a different notion of stability, this shows the dif- ficulty of determining a unique outcome.Our Theorem 2, however, proves the existence of a menu that satisfies strong convergence; that is, uniqueness of the outcome, as well as both neutrality and difference (one can easily find that the menu F 0 satis- fies difference).
The last remark refers to the implication of Theorem 2. Its proof demonstrates infinite strongly convergent menus.If we start from the premise that the appropriate rule should be determined endogenously (as in procedural autonomy in Dietrich 2005), the abundance of such menus is preferable, because it means a greater variety of solutions for the infinite regress of PC.At the same time, however, this may not be sufficient if societies for some reason favor other (possibly nonconvergent) menus, be they conventional, historically practical, etc.For example, some people might refuse the use of F 0 because the SCCs in F 0 are highly technical.In summary, Theorem 2 deals with cases when there are no menu constraints, and it successfully figures out how a society can always resolve the infinite regress.The remaining (and seemingly most difficult) case is when the society is concerned with a particular menu of SCCs-e.g.f P , f B , f A .This will be positively solved in the next subsection.

Convergent Expansion
When there are no menu requirements, using a strongly convergent menu is the most convenient way to solve the infinite regress of PC.But what can be done if the society is concerned about some non-convergent menus-such as f P , f B , f A , and thus Fig. 3 How the menu f 1 , f 2 , f 3 , f 4 fails to find a convergence convergence is not always found?An interesting answer is presented in Suzuki and Horita (2017b).They prove that by adding an extra SCC, denoted by , the extended menu f P , f B , f A , turns out to be asymptotically convergent.This section presents the full generalization of this fact: we prove that for any menu F , which is made up of scoring rules only, there exists an extended menu G ⊇ F that is asymptotically convergent.
Definition 9 A menu of SCCs G is said to be an asymptotically-convergent (AC) expansion of F if (i) F ⊆ G and (ii) G is asymptotically convergent.
The menu F = f P , f B , f A is already shown to have an AC expansion (Suzuki  and Horita 2017b).Our result shows that any menu of scoring rules also has an AC expansion.

Theorem 3 If a menu F is made up of scoring rules only, then F has an AC expansion.
Several remarks are given for Theorem 3. First, our proof is constructive in that it demonstrates the recipe for an AC expansion G F for an arbitrary menu F of scoring rules.The main idea of the proof is that if "most" of the SCCs in the menu select the alternative with the greatest B-plurality score, then the level-0 profile converges to such an alternative for a large society (except when the level-0 profile violates Assumption 1).Suppose that the leaves of level-1 SCCs are x 1 , x 2 , x 3 with In such a case, we will prove that the most SCCs ( Δ SCCs) in the menu G F must choose x 1 ; every level-2 SCC in the menu must have the leaf x 1 at a proper CI sequence L 0 , L 1 .Note that this process is far from trivial; this is because we have no information on the score assignments by SCCs in F (recall that F is an arbitrary set of scoring rules).This is partly why we need a slightly complicated procedure (Sects.C.1 to C.7) to prove Theorem 3.
Second, we would like to stress that our proof of Theorem 3 finds the convergence within two levels of arguments (how to choose how to choose), and this level does not depend on the size of the menu F .If F is huge, level-1 menus might have Table 1 Convergent outcomes under F 0 (i.e., {Copeland ( f 1 ), Borda ( f 2 ), Black ( f 3 )}) with anti-plurality for tie-breaking

Possible cases
Conv. outcome

Same as above
1 3 A Society Can Always Decide How to Decide: A Proof hundreds of different leaves in general; however, all level-2 SCCs in the menu G F are shown to have the same leaf under a proper CI sequence.It is interesting to compare this fact with the proof of Theorem 2 by Suzuki and Horita (2017b), which proves that F 1 ∶= f P , f B , f A , is asymptotically convergent (and thus, an AC expansion of F 1 ∶= f P , f B , f A ) via three levels of arguments (how to choose how to choose how to choose).These are two comparative cases: the latter makes F 1 asymptotically con- vergent by the minimum addition of SCCs (adding only one SCC ) with higher levels, and the former, Theorem 3, makes F 1 asymptotically convergent based on a huge 10 menu G F with the minimum level (note that the main significance of the lat- ter lies in its universal applicability to any set of scoring rules).These two results demonstrate two comparative routes by which a society can reach an asymptotically convergent menu.

Conclusions
This article has provided solutions to the infinite regress of PC.The most important contribution is the two general possibility theorems: Theorems 2 and 3.Both show how society can "always" (i.e., regardless of the preference profile over the set of alternatives) find convergence.When there is no constraint on the menu of SCCs the society should have, Theorem 2 gives a complete solution; it demonstrates menus with which any finite size of society can find a unique convergent outcome, regardless of the preference profile over the set of alternatives.On the other hand, Theorem 3 considers a situation where a large society is tackling the choice between a fixed set of scoring rules-like f P , f B , f A .Even then, Theorem 3 guarantees that the society can almost always resolve the infinite regress of PC by adding a proper set of extra SCCs into the menu (i.e., an AC expansion).
Theorems 2 and 3 represent a philosophical achievement in that they provide a positive answer to the infinite regress of justification.They show and prove how society can always justify the use of a certain SCC within finite levels of reasoning when there is no ex-ante agreement on the meta-level decision procedures.Ever since Arrow's (1951) seminal work, axiomatic studies in social choice theory have disclosed many (im)possibilities of SCCs.Given that no rule satisfies all the appealing axioms, many scholars have investigated the escape routes from the impossibilities within the framework of an axiomatic approach (by weakening/dropping some conditions, etc.).Our convergence-based approach provides a new way of justifying SCCs based on society members' procedural judgments (instead of "normative" axioms that might not appeal to some decision-making problems).This approach is beneficial in practice, especially when the voters seek an appropriate decision rule but the existing axiomatic results do not fully match voters' views-for instance, when voters support an inconsistent set of axioms or when voters have their own 10 For a menu F with M SCCs, our proof constructs an expanded menu G F ⊇ F with We have that g(2) = 1024 , g(3) = 4.5 × 10 13 , g(4) = 2.7 × 10 39 , g(5) = 2.4 × 10 88 , and so on.competing standards/criteria that are not studied in the literature.In such a case, our theorems guarantee how such societies can reach an appropriate decision based on the infinite chain of grounds.
Some complementary remarks are in order.The first remark concerns the assumption that voters are consequentialist (as stated in both Sect.2.1 and Definition 3).Taken literally, this assumption requires that voters' preferences for SCCs at the meta-level are fully specified by their outcomes.Consider the choice of SCCs for the election of the group leader.Some voters prefer the plurality because of its simplicity, while other voters prefer the Borda count, perhaps because of its theoretical superiority.In such a case, the consequentialism assumption in Definition 3 does not hold, because preferences for SCCs are not determined by their outcomes (but by other reasons specific to feasible SCCs).Is our theory unhelpful in such a case?
The answer depends on meta-level reasoning.Suppose that voters are consequential on the SCCs; that is, voters who prefer plurality (or Borda count) as a level-1 SCC should prefer those level-2 SCCs that choose plurality (or Borda count).This situation is surely within the scope of our theory because we can apply Theorem 2 and 3 by assuming that plurality and Borda count are the "alternatives."Similarly, if there exists a finite level k such that every voter is consequential at that level or above (i.e., each level-(k + 1), (k + 2), … SCCs are evaluated based on which level-k SCCs they choose), then we can apply our theory by properly transforming the set X .Thus, the limitation of our theory concerns whether voters could ever have non-consequential preferences over infinitely many levels in some way; this may be rather difficult within the finite cognitive capacity of human beings.
The second remark is on the probability calculation.As we noted in Sect.2.4, the present paper assumes a regular probability model (roughly speaking, a model where ties are unlikely to occur in a large population).However, the conditions of SC, NTP, and DVR (Definition 6) formally state which profile is disregarded.Therefore, if we can obtain the probability of such disregarded events, we can evaluate the significance of Theorem 3 (i.e., the probability of convergence under the G F )-even for a finite-sized society, although Theorem 3 argues only an infinite population.In the literature, mathematical tools to estimate the probability of various voting events have been developed (see, e.g., Lepelley et al. 2007;Wilson and Pritchard 2007;Brandt et al. 2016 for probability calculation under IAC model).Most of them use computational software, such as Normaliz11 and barvinok12 (implemented by Verdoolaege et al. 2004); however, it is often computationally hard to deal with many alternatives/rules, because the number of variables grows rapidly.In this view, it is worth noting that Theorem 3 successfully evaluates the probability no matter how many alternatives/rules there are.
The third and last remark is on the style of Theorems 2 and 3. Social choice theorists might doubt the significance of the theorems because they do not determine the 1 3 A Society Can Always Decide How to Decide: A Proof "best" menu(s); rather, they both demonstrate the existence of (extended) menus that are convergent in a certain sense.This criticism should be strongly concerned with the heart of the theory of PC.Recall that the infinite regress of PC is introduced from the premise of procedural autonomy (Dietrich 2005).Based on this premise, no theory could determine "the best" menu; no matter how many normative grounds-including the theory of PC-support the use of a certain menu, it might not be appropriate if members of society would not favor it.This is considered a big dilemma for PC theory: if the theory confines the class of good-performing menus, it reduces the applicability of the theory to those decision-making problems where other menus or SCCs are considered important.In this view, a good theory of PC should have broad applicability rather than specifying only a few good-performing menus.The greatest contribution of Theorem 3 is that it gives a universal solution that is applicable to any menus of scoring rules.Future research should explore a better way of expansion, such as expanding the original menu into an asymptotically strongly convergent menu or determining the minimum number of SCCs required for an AC expansion.

Appendix A: Proofs/Notes on Section 3.1
Proof of Theorem 1: Throughout the proof, we denote the by F = {f , g, h} and the set of alternatives by X = x 1 , x 2 , x 3 .

1⇒3 (as well as 2⇒3):
If F is not similar, there exists L 0 ∈ L(X) n such that each SCF in F selects dis- tinct alternatives.Let f L 0 = x 1 , g L 0 = x 2 , and h L 0 = x 3 .In this case, the level-1 CI profile is unique: as L 1 is obtained from L 0 by simply replacing x 1 , x 2 , x 3 with f , g, h , respectively.Since F is assumed to satisfy N, it follows that f L 1 = f , g L 1 = g, h L 1 = h .In the same way, it holds that the level-k CI pro- file L k is uniquely determined, and 3⇒2 (as well as 3⇒1): Fix any L 0 ∈ L(X) n .Since F is similar, we have either (1) f L 0 = g L 0 = h L 0 or (2) exactly two of f L 0 , g L 0 , h L 0 are the same.In the former case, L 0 strongly converges to f L 0 .In the latter, without loss of generality, we assume f L 0 = g L 0 = x 1 and h L 0 = x 2 .Fix any CI profile By the definition of CI profile, we have that [ fL 1 i h and gL 1 i h for all i ∈ A ], and [ hL 1 i f and Otherwise, a > n∕2 ( a cannot be n∕2 because n is assumed to be odd).Since n is odd, one of f , g wins the other in majority voting at L 1 .Whichever wins the other, the winner must be the Condorcet winner at L 1 ; thus, the leaf of each f L 1 , g L 1 , h L 1 is x 1 .
Since L 1 is assumed to be any level-1 CI profile, it follows that L 0 strongly converges to x 1 .■ Proof of Proposition 1 Suppose that F is not similar with respect to X .Then, there exists L 0 ∈ L(X) n and L 1 ∈ L L 0 ; such that each element of F 2 has a distinct leaf at L 0 , L 1 .Since F is SV, each level-2 SCF's leaf is a singleton; it is also a leaf of some level-1 SCF.Therefore, the leaves of level-1 SCFs must also be distinct. Let By the above observa- tion, each x 1 , x 2 , … , x M ∈ X is distinct; in addition, there exists a bijection . By definition of CI profile, the level-2 CI profile is unique: it is obtained from L 1 by replacing each f 1 j with f 2 (j) .Since F satisfies N, the fact that f 2 (j) chooses f 1 j at L 1 implies that f 3 (j) chooses f 2 (j) at L 2 .In this way, the level-k ≥ 2 CI profile is unique and f k+1 j L k = f k j for all j = 1, 2, … , M .Therefore, L 0 never weakly converges.■

Appendix B: Proof of Theorem 2
We prove the existence of required menus by constructing a series of examples.Let f 1 be Copeland's method and f 2 be any scoring rule (its score assignment for three options is denoted by [1, s, 0] ), where all ties are broken by anti-plurality so that both f 1 and f 2 are SCFs (if ties occur, even under anti-plurality, such a tie is supposed to be broken in favor of voter 1 .However, this extra tie-breaking rule is arbitrary as long as it is neutral.The following argument holds irrespective of this extra tiebreaking rule).We define SCF f 3 as follows.
We will prove that the menu 1) holds, L 0 strongly con- verges to f 1 L 0 .From now on, we assume (2).Without the loss of generality, let if the Condorcet winner exists, and f 2 (L) otherwise.
1 3 A Society Can Always Decide How to Decide: A Proof (c) If a = n∕2 , L 0 weakly converges to 1.
(d) If a = n∕2 and s ≥ 1∕2 , L 0 strongly converges to 1.These four statements together prove that f 1 , f 2 , f 3 is convergent, and is strongly convergent if s ≥ 1∕2.
Let s f ∶ L 1 be the score of f ∈ F 1 at L 1 ∈ L 1 L 0 evaluated by [1, s, 0] .Fur- thermore, let s c f ∶ L 1 be the Copeland's score of f ∈ F 1 at L 1 , i.e., the number of the other options that f defeats in majority rule at L 1 .
Proof of (a): for all i ∈ N ⧵ A (one can easily verify that this is certainly a CI profile).Since a = |A| > n∕2 , f 1 is the Condorcet winner and any scoring rule selects f 1 .There- fore, This says that L 0 weakly converges to x 1 .We next prove that L 0 does not converge to other alternative(s).At any level-1 CI pro- file, a voters in A (i.e., a majority of voters) must rank f and f 3 above g .So, Copeland's method would not select g .This means that leaf f 2 1 ;L 0 , . Since the level is not essential in the subsequent proof, one can inductively say that L 0 never converges to other alternative(s) than x 1 . If is the Condorcet winner.Therefore, we have and leaf f 2 2 ;L 0 , L 1 = x 2 , we can confirm again that the level-3 Copeland's method would not select f 2 2 ; and so (b) For all L 1 ∈ L L 0 , n − a(> n∕2) voters rank g 1 above f 1 and f 1 3 .So, g 1 is the Condorcet winner and f 2 1 L 0 = f 2 3 L 0 = h .This means that F L 0 , L 1 belongs to either [1 ∶ 2] or [0 ∶ 3] .One can verify that L 0 strongly converges to x 2 similarly as in (a) (swapping x 1 and x 2 , mutatis mutandis).
(c) Let L 1 ∈ L L 0 be such that f 1 L 1 i f 1 3 L 1 i g for all i ∈ A and gL 1 i f 1 L 1 i f 1 3 for all i ∈ N ⧵ A (the same as the first line of [a]).In this case, there is a tie between f 1 and g 1 .According to anti-plurality (tie-breaking rule), f 1 wins g 1 .So, f 2 2 L 1 = f 1 .In any case, both f 1 1 and f 1 2 select f 1 at L 1 .Accordingly, f 2 3 also selects it.Therefore, L 0 weakly converges to 1.(d) Fix any L 1 ∈ L L 0 .By the definition of CI profile, a voters rank f 1 and f 1 3 above g 1 , while the other a voters rank g 1 above f 1 and f , f 1 is the unique winner by Copeland's method at L 1 -and so f 2 1 L 1 = f 1 (resp.f 1 3 ).This means that leaf f 2 1 ;L 0 , L 1 = x 1 .If b < n∕2 , we can similarly verify that f 2 1 L 1 = f 1 3 , and so leaf f 2 1 ;L 0 , L 1 = x 1 .Otherwise; i.e., if b = n∕2 , there is a tie in the sense of Copeland's method; and 1 3 A Society Can Always Decide How to Decide: A Proof For instance, if Ỹ = {{x}, {y}, {z}} , then Y = {x, y, z} .In addition, let Intuitively, Y is the set of alternatives that is selected by at least one SCC in the menu G at L 0 ; m is the cardinality of Y .By NTP, plscore x, Y, L 0 differs by each x ∈ Y .Let x j ∈ Y be the element whose plscore ⋅, Y, L 0 is the j th largest among Y: Now, for = 1, 2, … , m! and j = 1, 2, … , m , let Definition 10 (Standard CI profile).Let G be the menu and L 0 ∈ L(X) n be a level- 0 profile satisfying SC and NTP.Let us follow basic symbols (3) with respect to G, L 0 .Then, the level-1 CI profile L 1 ∈ L(G) n is called standard with respect to L 0 , if L 1 ∈ L L 0 and.
(i) For all i, i � ∈ N, where ⌊r⌋ is the greatest integer below or equal to r.
For L 0 ∈ L(X) n satisfying SC and NTP, let L L 0 be the set of all level-1 CI stand- ard profiles.We say that a CI sequence L Remark 3 (Construction of a Standard CI Profile).Note that L L 0 ≠  whenever L 0 is SC and NTP.Let us demonstrate this by constructing a standard CI profile.For where p < q ⇒ plscore x p , Y, L 0 > plscore x q , Y, L 0 .
By the definition of the CI profile, a voter in N should rank G 1 , G 2 , … , G m according to the th linear order of Y , but his/her preferences inside each G 1 , G 2 , … , G m are undetermined.Therefore, we will stipulate the latter.Define i ,p 's CI preference as follows: And for j = 2, 3, … , m, where mod(p, q) denotes the remainder of p divided by q .For instance, if have the th linear order of G j as their preference over G j .One can verify that ( 10) and ( 11) prove ( 8) and ( 9), respectively.■

C.2 An Evaluation of the Scores
Lemma 1 (upper/lower bounds of the scores of level-1 rules).Let G be the menu and L 0 ∈ L(X) n be a level-0 profile that satisfies SC and NTP.Let us follow basic sym- bols (3) with respect to G, L 0 .Fix any L 1 ∈ L L 0 and let f be any scoring rule with score assignment s 1 , s 2 , … , s |G| .Let s(g) be the score of g ∈ G at L 1 evaluated by f .For j = 1, 2, … , m, let Then, the following holds: For all j = 2, 3, … , m, 5, 11, 17, …).
1 3 A Society Can Always Decide How to Decide: A Proof

Proof of Lemma 1
We first prove (13).By (i) of Definition 10, every voter ranks the elements of G 1 in the same order.Let g ′ 1 be the top-ranked element.Then, voters in U 1 rank g ′ 1 the first among G while the other voters rank g higher.Therefore, inequality (13) holds.Next, we prove ( 14).First, let us focus on voters in N ( = 1, 2, … , m! ).Sup- pose, for instance, that G 2 = g 2 , g � 2 , g �� 2 and the th linear order of Y is x 1 , x 3 , x 2 , … ( x 1 is the best, x 3 is the second, x 2 is the third, and so on).While we will argue based on this particular G 2 , this should not lose generality.Rigorous proof is easily obtained by substituting G 2 with G j , mutatis By the definition of CI profiles, every voter ranks , where q, r ∈ ℤ and 0 ≤ r  < q .By (ii) of Definition 10, we can assume that (These six lines correspond with the first to the | !vot- ers defined above, we can say that the scores of g 2 , g ′ 2 , g ′′ 2 given by these voters are exactly the same.Hence, the scores of g 2 , g ′ 2 , g ′′ 2 can differ by the remaining r vot- ers.This means that the difference of the scores of g 2 , g ′ 2 , g ′′ 2 from N is, at most, r .By taking the sum of = 1, 2, … , m! , we can say that This completes the proof.■ The aim of Lemma 1 is to give a lower bound of s g 1 and an upper bound of s g 2 , s g 3 , … , s g m to our purpose (we do not intend to determine the upper/ lower "limits": indeed, the last part of the proof evaluates m or roughly).Rather, the point is that m 0 !⋅ (|G|!) does not depend on n .It is completely deter- mined when we specify the set of alternatives X (with |X| = m 0 ) and the menu G.
The first q voters in N rank G 2 as g
Let 1 ≤ i < j ≤ m 2 .As a trivial inequality, we have Therefore, it follows, Furthermore, (16) implies i + 1 ≥ (m − 1) j + 1 .So, we have Δ |G|,m ( )∶=|G| − (m − 1) , and 1 3 A Society Can Always Decide How to Decide: A Proof Using ( 17) and (18) to evaluate in H i , we have Since we assumed H i ≤ 0 , we have that Note that the above argument holds for any (i, j) with 1 ≤ i < j ≤ m 2 .Therefore, let (i, j) = (1, 2), (2, 3), (3, 4) ⋯ , m 2 − 1, m 2 in (19)-and take the sum of them.Then, we have Let A and B be the left-hand and the right-hand sides of (20), respectively.For the left-hand side, we have Δ 1 ≤ Δ 2 ≤ ⋯ ≤ Δ m 2 by definition.So, we have On the other hand, we have The inequality holds because (16) implies that Since . (20) But this can never hold.Proof of (ii): suppose that there is no j ∈ 1, 2, … , m 2 with H  j > 0 .Then, (i) implies that s Δ 1 +1 = 1 .By ( 16) and the definition of score assignment, we have So, H 1 = 0 .This proves that = 1 satisfies the required property.■ C.4 Finding a Rule-Free ı Among M 3 Candidates Lemma 2 tells us the existence of ∈ 1 , 2 , … , m 2 satisfying either (a) or (b) of statement (ii).Technically, however, such can differ by each scoring rule f (this is because H f |G|,m ( ) depends on f ).Let F be any set of M(≥ m) scoring rules.Lemma 3 proves that if we investigate M 3 candidates of ; i.e., 1 , 2 , … , M 3 ; at least one of them must satisfy either [(a) or (b)] for all f ∈ F.

Proof of Lemma 3
Let P(f , ) be a proposition that [either (a) or (b)].Also, let Then, we can confirm that Suppose to the contrary that there is no required .Then, for all ∈ D , there is f ∈ F such that P(f , ) fails to hold.This implies that | 1 3 A Society Can Always Decide How to Decide: A Proof On the other hand, fix some f ∈ F .By (ii) of Lemma 2, for every m 2 different elements among D , at least one element must satisfy P(f , ) .Therefore, the com- plement of Therefore, we have Inequalities ( 25), ( 26), and ( 27) together imply Definition 11 (Function F and Δ F ). Let F be the set of M scoring rules.We define functions as follows: as follows.Let |G|∶=2M ⋅ M M 3 and j = M M 3 −j+1 ( j = 1, 2, … , M 3 ).One can verify that such |G| and 1 , 2 , … , satisfies 0. Therefore, Lemma 3 shows that there exists for all f ∈ F .For m ∈ {2, 3, … , M} , let F (m) be one of such j .In addition, let Δ F (m)∶=|G| − (m − 1) ⋅ F (m).

C.5 Every Rule Chooses a Nonempty Subset of G 1
Lemma 4 (Every rule chooses a nonempty subset of G 1 ).
Proof of (i): Since L 0 is assumed to be SC and NTP, Lemma 1 implies that For N ′ ⊆ N and g ∈ G , let s g;N ′ be the score of g assigned by the voters in N ′ .Then, it follows that With this and Lemma 1, we have By the definition of the CI profile, we can estimate the positions of elements in G 2 at L 1 as follows: • voters in U 1 must rank each element of G 2 at the Δ F (m) + 1 th or lower, • voters in U 2 must rank each element of G 2 between the first to the F (m) th , and • voters in U j ( 3 ≤ j ≤ m ) must rank the elements of G 2 at the F (m) + 1 th or lower.

Proof of (ii):
By exist who rank g 2 as the worst of the whole G .Let i be such a voter.Then, we have s g 2 , {i} = 0 .So, s g 2 = s g 2 , N − {i} + s g 2 , {i} ≤ n − 1 .Now, we have con- firmed that s g 1 = n > s g 2 .Suppose (b).Then, by (i) and

C.6 Definition of G F
Definition 12 (menu G F ).Given a set F of M(≥ 2) scoring rules, we define G F ⊇ F to be a set of 2M ⋅ M M 3 SCCs such that for any nonempty set A and L 0 ∈ L(A) n , (i) if L 0 satisfies SC with respect to F and NTP-let us follow basic symbols with respect to F, L 0 .Then, exactly Δ F (m) elements of G select x 1 and for each j = 2, 3, … , m , exactly F (m) elements of G select x j .
(ii) Otherwise, for each g ∈ F ⧵ F, By definition of G F ⧵ F , we have the following: Lemma 5 If L 0 satisfies SC with respect to F and NTP, then it is also SC with respect to G F .Otherwise; i.e., under case (ii); suppose that g j (L) the greatest element among {f (L)| f ∈ F} for individual j ∈ N (such a definition is possible when |N| is suffi- ciently large).

C.7 Convergence
Theorem 3 is a straightforward result from the following Lemma 6 (as well as Assumption 1).
1 3 A Society Can Always Decide How to Decide: A Proof Lemma 6 If L 0 satisfies SC, NTP, and DVR with respect to F-and that n is suffi- ciently large-then L 0 weakly converges under G F .
Proof of Lemma 6 Suppose L 0 satisfies SC with respect to F , NTP, and DVR.By Lemma 5, it must also be SC with respect to G F .Therefore, we can follow basic symbols with respect to G F , L 0 .By Definition 12, it must be that = F (m) for all j = 2, 3, … , m .So, we can apply Lemma 4 to obtain s g 1 > s g 2 s g 3 , … , s g m at sufficiently large n .This implies that g L 1 ⊆ G 1 for all level-2 rules g ∈ F .By (ii) of Definition 12, g L 1 ⊆ G for all g ∈ G F ⧵ F , too.Therefore, L 0 converges to x 1 .■

Fig. 2
Fig. 2 A simple example where convergence fails n on B as: for all a, b ∈ B and i ∈ N , aL i b ⟺ (a)L i (b).An SCC is called neutral if; for any sets A and B with 2 ≤ |A| = |B| < +∞ , b ∈ B , bijection ∶ B → A , and profile L ∈ L(A) n ; we have that (a) ∈ f (L) ⟺ a ∈ f (L ).
For a finite nonempty set A , L ∈ L(A) n , B ⊆ A , and b ∈ B-we define the B -plurality score of b at L as By definition, plscore(b, B, L) is the score of b among B at L evaluated by plu- rality (i.e., the number of voters who rank b the first among B at L).

a
if a wins b in majority rule, b if b wins a in majority rule, and a Otherwise(tie between a and b)

Remark 4 (
Example of G F ) The above definition only specifies the conditions on G F .However, such a G F surely exists.Let us construct one example.Letc∶= | | G F | | = 2M ⋅ M M 3 , F = f 1 , f 2 , … , f M , and G F ⧵ F = g 1 , g 2 , … , g c−M .When a profile L ∈ M A is given such that f (L) is a singleton for each f ∈ F , let By Definition 11, each F (m), Δ F (m) is at least M .So, we have d 1 , d 2 , … , d m ≥ 0 and d 1 + d 2 + ⋯ + d m = c − M .Therefore, we can define the outcomes as follows:

ı Among m 2 Candidates For
positive integers |G|, m, ∈ ℕ with 2 ≤ m and 2m ≤ |G| and a scoring rule f , let where s 1 , s 2 , … , s |G| is the score assignment of f for |G| options.Lemma 2 (Finding a rule-specific among m 2 candidates).Let |G|, m ∈ ℕ with 2 ≤ m and f be a scoring rule.Let 1 , 2 , … , m 2