Impossibility theorems with countably many individuals
Abstract
The problem of social choice is studied on a domain with countably many individuals. In contrast to most of the existing literature which establish either nonconstructive possibilities or approximate (i.e. invisible) dictators, we show that if one adds a continuity property to the usual set of axioms, the classical impossibilities persist in countable societies. Along the way, a new proof of the Gibbard–Satterthwaite theorem in the style of Peter Fishburn’s well known proof of Arrow’s impossibility theorem is obtained.
Keywords
Arrow’s impossibility theorem The Gibbard–Satterthwaite theorem Infinite society ContinuityJEL Classification
D70 D711 Introduction
Shortly after the publication of Arrow (1963), researchers started to deal with the problem of social choice for infinite populations. Infinite society models open up many challenges; both mathematical and interpretational. They allow us to establish interesting connections between the major themes in economics, such as the well known core equivalence theorem. They are also useful, even necessary if one approaches to the problem of social choice from a statistical point of view as its analysis is based on asymptotic properties of statistical estimators, e.g. Condorcet jury theorems, Kemeny rule etc.
The early literature on social choice theory for large (or arbitrary) societies begins with Fishburn (1970), Kirman and Sondermann (1972), Brown (1974), Hansson (1976), Pazner and Wesley (1977) and Armstrong (1980). Among these, Kirman and Sondermann (1972) and Armstrong (1980) are more interested in persistence of Arrow’s impossibility theorem (see Theorem 1) for large societies, while Fishburn (1970), Brown (1974), Hansson (1976) and Pazner and Wesley (1977) focused on its failure.
A notable result is given by Kirman and Sondermann (1972) showing that when we model societies as arbitrary nonempty sets, every social welfare function which is Pareto efficient and IIA yields a unique nested collection of decisive subsocieties (i.e. an ultrafilter). They also give a more intuitive measure theoretic interpretation of this theorem and prove a stronger impossibility result when the set of alternatives is finite. A survey of this literature can be found in Lauwers (1998), and as can be seen from more recent works the exploration along this direction continues today [see e.g. Torres (2005)]. Consistent with the recurrent trend to obtain close connections between Arrow’s impossibility theorem and the Gibbard–Satterthwaite theorem (see Theorem 2), Mihara (2000) eventually showed that a result analogous to that in Kirman and Sondermann (1972) holds for social choice functions when we require coalitional strategyproofness [see also Rao et al. (2018)].
Recently, some authors focused either on countable society models or on continuity properties of social welfare and choice functions, but to our knowledge, none both at the same time as we did. Mihara (1997), Mihara (1999) and Tanaka (2007a, b) are examples of the former, while Salonen and Saukkonen (2005), Saukkonen (2007) are examples of the latter. Mihara (1997) was, to our knowledge, the first to concentrate on peculiarities of the smallest large society in order to establish a connection between computability theory and social choice theory. It shows that if Turing computability axiom is added to the usual set of axioms in Arrow’s theorem, we obtain an impossibility theorem. The other three papers in the former category deal with the possibility of social choice for countable societies. One of the key problems there is to construct concrete examples which violate classical impossibilities. Difficulty of this task has already been mentioned in the earlier literature which focused on failures of Arrow’s theorem for large societies, and the main conclusion of these recent papers is that the task is as delicate as ever.
An earlier work which focused on topological continuity of social welfare functions is Chichilnisky (1980). Recently, Salonen and Saukkonen (2005) and Saukkonen (2007) are also concerned with topological continuity of a social welfare function and social choice function, respectively. Based on the framework in Armstrong (1980), they introduce a topology on the domain and range of these functions and defined continuity in the usual sense, i.e. mapping open sets into open sets. They show that adding this continuity axiom to the usual set of axioms gives impossibility results similar to those in Sect. 3.

In Sect. 3, we show that two classical impossibility results, namely Arrow’s theorem and the Gibbard–Satterthwaite theorem can be extended to a countable society setting in an exact manner if we add a continuity property to the usual set of axioms, and

In Sect. 4, we obtain a new proof for the Gibbard–Satterthwaite theorem à la Fishburn (1970) as a byproduct.
In terms of the first objective, our paper is somewhat similar to Mihara (1997) as both papers introduce a new axiom to obtain impossibility theorems for countably infinite populations, and both axioms are peculiar to the set of natural numbers. But the newly added axioms differ from each other and the tools Mihara (1997) use are innovative while those we use are conservative. As mentioned above, some authors focused on topological continuity of social welfare and choice functions already. However, their notion of continuity depends on the topology introduced and is not directly related to ours. In particular, while continuity in our sense is guaranteed whenever society is finite, this is not true for the other (topological) continuity axioms as one can arbitrarily tear apart images of preference profiles.

Deductive proofs Proofs in the style of Arrow (1963) and Sen (1986). The common feature is to treat the impossibility result under consideration as a self contained mathematical structure, and deduce the result from its setting without referring to an external mathematical device [see A. Sen’s discussions in Maskin and Sen (2014)].

Inductive proofs Proofs in the style of Satterthwaite (1975) and Svensson and Reffgen (2014), where mathematical induction is used explicitly.

Proofs by contradiction Proofs in the style of Fishburn (1970), Suzumura (1988) and the proof given in Sect. 4 where ‘reductio ad absurdum’ is used.

Proofs by construction Proofs in the style of Barberà (1980, 1983) and Reny (2001) where an explicit procedure for finding a dictator is given. Their treatment of impossibility theorems is similar to that of the intermediate value theorem in calculus, and the underlying procedure (i.e. pivotal voter) has been modified several times, see e.g. Fey (2014).

Indirect proofs Proofs in the style of Gibbard (1973) and Denicolò (1993), where a result is obtained through other known impossibility theorems.
The third task is more feasible than the former two which are metatheoretical, and the ‘program’ of completing it is worthwhile as in each successful attempt we can reasonably hope to discover something new about the results, as well as the methods.^{1} Accordingly, in Sect. 4 we give a new proof for the Gibbard–Satterthwaite theorem following Fishburn (1970)’s approach.
2 The preliminaries
A denotes the set of alternatives with \(3\le A<\infty \), X and \(X^{*}\) denote, respectively, the set of strict and weak rankings on A.^{2} Let \(I=\{1,2,\ldots \}\) be a countable set representing individuals in the society. Thus, I can be of two types: either \(I=\{1, 2, \ldots , n\}\) for some \(n\in \mathbb {N}\) or \(I=\mathbb {N}\). Let \(X^{I}\) denote the set of all preference profiles, i.e. the set of all functions from I into X. A function \(F:X^{I}\rightarrow X^{*}\) is called as a social welfare function (SWF) and a function \(f:X^{I}\rightarrow A\) is called as a social choice function (SCF).In practice, an argument or result is “combinatorial” if it is not overtly modeltheoretic, topological, or measuretheoretic.
James Baumgartner
A member x of \(X^{I}\) is called a profile and its \(i'\)th component, \(x_{i}\), is called individual \(i'\)s ranking. A member of \(X^{*}\) is called a social order or society’s ranking. When \(a\in A\) is ranked above \(b\in A\) according to \(x_{i}\) we write \(a\succ ^{x}_{i}b\), and for the same ranking of a vs. b according to F(x), we write \(a\succ _{F(x)}b\). Similarly, when a is ranked at least as good as b according to F(x), we write \(a\gtrsim _{F(x)}b\). For any \(x\in X^{I}\) and \(i\in I\), let \((x_{i}^{\prime },x_{i})\in X^{I}\) denote the profile that has \(x_{i}^{\prime }\in X\) in its \(i'\)th component instead of \(x_{i}\in X,\) and otherwise the same as \(x\in X^{I}\).
A group of individuals \(G\subseteq I\) is decisive over \(a, b\in A\) for F if \(a\succ ^{x} _{i}b\) for all \(i\in G\) implies \(a\succ _{F(x)}b\) for all \(x\in X^{I}\). \(G\subseteq I\) is decisive for F if it is decisive over all \(a, b\in A\). Similarly, a group of individuals \(G\subseteq I\) is decisive over \(a\in A\) for f if \(a\in A\) is ranked on top of \(x_{i}\) for all \(i\in G\) implies \(f(x)=a\) for all \(x\in X^{I}\). \(G\subseteq I\) is decisive for f if it is decisive over all \(a\in A\). We say that \(F:X^{I}\rightarrow X^{*}\) is Pareto efficient (PE) if I is decisive for F. It is independent of irrelevant alternatives (IIA) if whenever \(a,b\in A\), \(x,x'\in X^{I}\) are such that \(a\succ ^{x}_{i}b\) if and only if \(a\succ ^{x'}_{i}b\), for all \(i\in I\), we have \(a\gtrsim _{F(x)}b\) if and only if \(a\gtrsim _{F(x')}b\). Finally, it is dictatorial (D) if there is a decisive group for F consisting of a single individual. The following result is known as Arrow’s impossibility theorem:
Theorem 1
(Arrow 1963) Let I be finite. Then \(F:X^{I}\rightarrow X^{*}\) is PE and IIA if and only if it is D.
We say that \(f:X^{I}\rightarrow A\) is unanimous (UNM) if I is decisive for f. It is manipulable (MNP) at \(x\in X^{I}\) by \(i\in I\) via \(x_{i}^{\prime }\in X\) if \(f(x_{i}^{\prime },x_{i})\succ ^{x} _{i}f(x)\). It is strategy proof (STP) if it is not manipulable. Finally, it is dictatorial (DT) if there is a decisive group for f consisting of a single individual. The following result is known as the Gibbard–Satterthwaite theorem:
Theorem 2
(Gibbard 1973; Satterthwaite 1975) Let I be finite. Then, \(f:X^{I}\rightarrow A\) is UNM and STP if and only if it is DT.

\(x^{1}\) is not a minimal element in \((X^{I}, R_{z})\), i.e. the first component of \(x^{1}\) is the same as that of z, and

for all \(m,n\in \mathbb {N}\) with \(m<n\) and \(m<I\), we have \(x^{n}P_{z}x^{m}\).
Let us give another interpretation. Recall that X can be endowed with the so called Kemeny distance which takes the number of pairwise swaps needed to change \(x_{i}\in X\) to \(z_{i}\in X\) as the distance between \(x_{i}\) and \(z_{i}\) [for details see e.g. Can and Storcken (2018)]. With this notion we can say that \((x^{n})_{n\in \mathbb {N}}\in X^{I}\) converges (pointwise) to \(z\in X^{I}\), which is denoted as \((x^{n})\rightarrow z\), if the distance between \(x^{n}_{i}\) and \(z_{i}\) converges to 0 as \(n\rightarrow \infty \), for all \(i\in I\). Clearly, \((x^{n})\uparrow z\) implies \((x^{n})\rightarrow z\).

\(a\gtrsim _{F(x^{n})}b\) for all \(n\in \mathbb {N}\) implies \(a\gtrsim _{F(z)}b\).

\(f(x^{n})=a\) for all \(n\in \mathbb {N}\) implies \(f(z)=a\), and

\(f(x^{n})\ne a\) for all \(n\in \mathbb {N}\) implies \(f(z)\ne a\).

\(F: X^{I}\rightarrow X^{*}\) is CNT if for all \(z\in X^{I}\), \(A', B' \subseteq A\), and \((x^{n})_{n\in \mathbb {N}}\in X^{I}\) such that \((x^{n})\uparrow z\), \(a\gtrsim _{F(x^{n})}b\) for all \(a\in A'\), \(b\in B'\) and for all \(n\in \mathbb {N}\) implies \(a\gtrsim _{F(z)}b\) for all \(a\in A'\) and \(b\in B'\); and

\(f: X^{I}\rightarrow A\) is CNT if for all \(z\in X^{I}\), \(A' \subseteq A\) and \((x^{n})_{n\in \mathbb {N}}\in X^{I}\) such that \((x^{n})\uparrow z\), \(f(x^{n})\in A'\) for all \(n\in \mathbb {N}\) implies \(f(z)\in A'\).

\((x^{n})\uparrow z\) if and only if for all \(n\in \mathbb {N}\) with \(n\le I\), \(\min \{i\in I: x^{n}_{i}\ne z_{i}\}>n\), i.e. at least the first n coordinates of \(x^{n}\) are the same as those of \(z\in X^{I}\).
Lemma 1
(Pareto) Let \(f:X^{I}\rightarrow A\) be UNM, STP and CNT, and \(a,b\in A\), \(x\in X^{I}\) be such that \(a\succ ^{x}_{i}b\) for all \(i\in I\). Then \(f(x)\ne b\).
Proof

\(f(x^{0})=b\), and

if \(f(x^{k})=b\), then \(f(x^{k+1})=b\) for all \(k\in \mathbb {N}\) with \(0\le k<I\). To see this, note that \(f(x^{k+1})\in \{a, b\}\), as otherwise f is MNP by \(k+1\) at \(x^{k+1}\) via \(x_{k+1}\), and \(f(x^{k+1})\ne a\), as otherwise f is MNP by \(k+1\) at \(x^{k}\) via \(x'_{k+1}\).
\(f:X^{I}\rightarrow A\) is ONTO if \(\forall a\in A\), \(\exists x\in X^{I}: f(x)=a\). Clearly, UNM implies ONTO. The following result concerns the reverse implication.
Lemma 2
If \(f:X^{I}\rightarrow A\) is ONTO, STP and CNT, then it is UNM.
Proof

\(f(y^{0})=a\), and

if \(f(y^{k})=a\), then \(f(y^{k+1})=a\) for all \(k\in \mathbb {N}\) with \(0\le k<I\), as otherwise f is MNP at \(y^{k+1}\in X^{I}\) by \(k+1\) via \(y_{k+1}\).
Let \(I=\mathbb {N}\) and we say that a SCF or SWF is infinitarian (INF) if an infinite support always wins against a finite opposition. More formally, \(f: X^{\mathbb {N}}\rightarrow A\) is INF if for all \(x\in X^{\mathbb {N}}\) and for all \(a,b\in A\) with infinitely many individuals rank a above b, and finitely many of them have the opposite ranking at \(x\in X^{\mathbb {N}}\), we have \(f(x)\ne b\). Similarly, \(F: X^{\mathbb {N}}\rightarrow X^{*}\) is INF if for all \(x\in X^{\mathbb {N}}\) and for all \(a,b\in A\) with infinitely many individuals rank a above b, and finitely many of them have the opposite ranking at \(x\in X^{\mathbb {N}}\), we have \(a\succ _{F(x)}b\). Thus, INF implies UNM and PE. The following result shows that INF is inconsistent with CNT.
Theorem 3
Let \(I=\mathbb {N}\). Then, there exist no SCF (or SWF) which is INF and CNT.
Proof
Let \(f: X^{\mathbb {N}}\rightarrow A\) be INF, hence also UNM, and let \(x\in X^{\mathbb {N}}\) be such that \(x_{i}=(a\succ \cdots \succ b)\) for all \(i\in \mathbb {N}\). Let \((x^{n})\in X^{\mathbb {N}}\) be a sequence with \(x^{n}_{i}=x_{i}\) for all \(1\le i\le n\), and \(x^{n}_{j}=(b\succ \cdots \succ a)\) for all \(n<j\), for all \(n\in \mathbb {N}\). Then, \((x^{n})\uparrow x\), and by INF, \(f(x^{n})\ne a\) for all \(n\in \mathbb {N}\). But by UNM, \(f(x)=a\), thus f is not CNT. A very similar argument shows that every \(F: X^{\mathbb {N}}\rightarrow X^{*}\) which is PE and INF is not CNT. \(\square \)
Remark
In proving the results above it suffices to assume \(A\ge 2\), instead of \(A\ge 3\). The analysis above allows us to provide a motivation for CNT axiom within the framework of social choice. Let us interpret the infinite society as literally, or as a collection of state contingent selves as Mihara (1997), or as that of timecontingent selves. The justification for the latter is as follows. Suppose there are finitely many individuals in the society, but each of them forms an opinion (i.e. preference) at every instant of time. The social aggregation process (e.g. election) takes place once in a while and when that happens it aggregates all the formed (interim) opinions which are indexed by the time contingent selves.^{5}
Suppose the society is in a transformation from an initial state to a target state, both expressed as profiles, through a dynamic process of changing some individuals’ rankings into their target rankings in each step. Then, CNT ensures that a SWF or SCF must decide within a finite step whether or not to respond to this social change. To be more concrete, consider the setting in Theorem 3 and suppose that the society is becoming increasingly concerned about an issue so that \(b\in A\) is the social alternative in which the concern is taken care of. Then, under UNM, CNT ensures that SCF learns this growing trend (or pressure) before it is too late and responds to it, whereas any INF rule will ignore it until the end. In this sense, CNT which could also be called as ‘finite responsiveness’ is desirable in every forward looking or intelligent social design.
3 The main results
Theorem 4
\(F:X^{I}\rightarrow X^{*}\) is PE, IIA and CNT if and only if it is D.
Lemma 3
(Expansion) If \(G\subseteq I\) is decisive for F over some \(a, b \in A\), then it is decisive.
Proof
See the proof of the field expansion lemma in Sen (1986). \(\square \)
Lemma 4
(Contraction) If \(G\subseteq I\) with \(G\ge 2\) is decisive for F then it has proper subset which is decisive for F.
Proof
See the proof of the group contraction lemma in Sen (1986). \(\square \)
Let us now turn to Theorem 4. For \(k\in \mathbb {N}\), let \(G(k)=\{1, \ldots , k\}\cap I\) and further let \(x^{k}\in X^{I}\) be the profile which is constant over G(k) with the fixed ranking \((a\succ \cdots \succ b)\), and so is over \(I{\setminus } G(k)\) with the ranking \((b\succ \cdots \succ a)\), for some \(a, b \in A\). Then, we claim that \(a\succ _{F(x^{k})}b\) for some \(k\in \mathbb {N}\). Suppose by contradiction that no such k exists. Then, \(b\gtrsim _{F(x^{k})}a\) for all \(k\in \mathbb {N}\) and \((x^{k})\uparrow x\) where \(x\in X^{I}\) is the ranking which is constant over I with the same fixed ranking \((a\succ \cdots \succ b)\). CNT implies that \(b\gtrsim _{F(x)}a\), but this contradicts PE. We proved our claim.
Let \(k^{*}=\min \{k\in \mathbb {N}: a\succ _{F(x^{k})}b\}\) and take any \(c\in A{\setminus }\{a,b\}\). Let \(u\in X^{I}\) be a profile such that \(c\succ ^{i}_{u}a\succ ^{i}_{u}b\) for all \(i\in G(k^{*})\), and \(b\succ ^{j}_{u}a\), \(c\succ ^{j}_{u}a\) for all \(j\in I{\setminus } G(k^{*})\). Then, by PE \(c\succ _{F(u)}a\) and by IIA \(a\succ _{F(u)}b\); hence, \(c\succ _{F(u)}b\) by transitivity. Since the relative ranking of \(b,c\in A\) is unspecified across \(u_{j}\) for all \(j\in I{\setminus } G(k^{*})\), this together with IIA imply that \(G(k^{*})\) is decisive for F over c, b. Then, by Lemma 3 we conclude that \(G(k^{*})\) is decisive and repeated application of Lemma 4 to \(G(k^{*})\) gives the result in Theorem 4. \(\square \)
Remark
Theorem 4 implies Theorem 1 as when I is finite every SWF is CNT. Sen (1986) writes that “The Field Expansion Lemma and the Group Contraction Lemma both continue to hold for infinitely large communities and decisive sets can be endlessly curtailed, effectively disenfranchising nearly everybody.” We believe that Theorem 4 is a way to make this insight formal.
Fishburn (1970) gives a SWF which is PE, IIA, nonDT. It is easy to see that this example satisfies INF, noting that under the finitely additive probability measure he used every finite coalition has zero mass. By Theorem 3 such SWF is not CNT, hence this example assures the role of CNT in Theorem 4. However, this example is not constructive as it uses Zorn’s lemma, and ideally, we should try to construct explicitly a SWF which is PE, IIA, nonDT and not CNT. But that task is very difficult, if not impossible, because of the Mihara’s impossibility theorem, saying that every PE, IIA and Turing computable SWF is DT (see Mihara 1997).^{6}
Theorem 5
\(f:X^{I}\rightarrow A\) is ONTO, STP and CNT if and only if it is DT.
Proof
By Lemma 2 we may assume that f is UNM. Our proof is based on the proof of Theorem 2 in Reny (2001).
Step 1 Consider \(a,b \in A\) and let \(x\in X^{I}\) be a profile in which a is ranked as the top and b is ranked as the bottom in \(x_{i}\) for all \(i\in I\). By UNM, \(f(x)=a\). Consider now changing individual \(1'\)s ranking by raising \(b'\)s position one by one. By UMN the social choice remains at a as long as a is on the top. When b rises above a to the top, the social choice either changes to b or remains at a by Lemma 1. Begin the same process with individual 2, then 3, etc. Let \(x^{i}\) be the resulting profile after b got the top position in \(x_{1}\), ..., \(x_{\min \{i, I\}}\), and let \(y\in X^{I}\) be the resulting profile after b got the top position in everyone’s ranking. We shall prove that there is \(i\in I\) such that \(f(x^{i})=b\). Assume that \(f(x^{i})\ne b\) for all \(i\in I\). Then, \(f(x^{i})\ne b\) for all \(i\in \mathbb {N}\). Since \((x^{i})\uparrow y\), this implies \(f(y)\ne b\) by CNT which contradicts UNM. Let \(j=\min \{i\in I: f(x^{i})=b\}\). Since \(f(x^{i})\in \{a,b\}\) for all \(i\in \mathbb {N}\), \(f(x^{j1})=a\) and \(f(x^{j})=b\), by definition (see Fig. 1).
Step 2 Let \(z\in X^{I}\) be such that \(z_{i}=(b\succ \cdots \succ a)\) for \(1\le i\le j1\), \(z_{j}=(b\succ a\succ \cdots )\), and \(z_{i}=(\cdots \succ a\succ b)\) for \(i>j\). We shall prove that \(f(z)=b\). Transform \(x^{j}\in X^{I}\) by replacing \(x^{j}_{i}\) with \(z_{i}\) for all \(i\in I\), one at a time. Let \(z^{i}\in X^{I}\) be the profile obtained after \(x^{j}_{1}, \ldots , x^{j}_{\min \{i, I\}}\) are replaced. Then, STP ensures that \(f(z^{i})=b\) for all \(i\in \mathbb {N}\). Since \((z^{i})\uparrow z\), we then conclude that \(f(z)=b\) (see Fig. 2).

\(f(u^{0})=b\),

if \(f(u^{i})=b\), but \(f(u^{i+1})\ne b\) for some \(i\in \{0, \ldots , j2\}\), then f is MNP at \(u^{i+1}\in X^{I}\) by individual \(i+1\) via \(z_{i+1}\),

if \(f(u^{j1})=b\) but \(f(u^{j})\notin \{a,b\}\) then f is MNP at \(u^{j}\in X^{I}\) by individual j via \(z_{j}\),

if \(f(u^{i})\in \{a,b\}\), but \(f(u^{i+1})\notin \{a,b\}\) for some \(i\in \{j, j+1, \ldots \}\), then f is MNP at \(u^{i}\in X^{I}\) by individual \(i+1\) via \(u_{i+1}\).

\(f(v^{0})=b\),

if \(f(v^{i})=b\), but \(f(v^{i+1})\ne b\) for some \(i\in \{0, \ldots , j2\}\), then f is MNP at \(v^{i+1}\in X^{I}\) by individual \(i+1\) via \(u_{i+1}\),

if \(f(v^{j1})=b\) but \(f(v^{j})\notin \{a,b\}\) then f is MNP at \(v^{j}\in X^{I}\) by individual j via \(u_{j}\). Thus, \(f(v^{j})\in \{a,b\}\). But if \(f(v^{j})=a\) then f is MNP at \(v^{j1}\in X^{I}\) by individual j via \(x^{j1}_{j}\). Thus, \(f(v^{j})=b\),

if \(f(v^{i})=b\), but \(f(v^{i+1})\ne b\) for some \(i\in \{j, j+1, \ldots \}\), then f is MNP at \(v^{i}\in X^{I}\) by individual \(i+1\) via \(x^{j1}_{i+1}\).
Step 3 Take \(c\in A{\setminus }\{a,b\}\) and let \(w\in X^{I}\) be the profile in Fig. 3 (left).

\(f(w^{0})=a\),

if \(f(w^{i})=a\), but \(f(w^{i+1})\ne a\) for some \(i\in \{0, \ldots , j2\}\), then f is MNP at \(w^{i}\in X^{I}\) by individual \(i+1\) via \(w_{i+1}\),

if \(f(w^{j1})=a\) but \(f(w^{j})\ne a\) then f is MNP at \(w^{j}\in X^{I}\) by individual j via \(u_{j}\),

if \(f(w^{i})=a\), but \(f(w^{i+1})\ne a\) for some \(i\in \{j, j+1, \ldots \}\), then f is MNP at \(w^{i}\in X^{I}\) by individual \(i+1\) via \(w_{i+1}\) (Note that \(f(w^{i})\ne b\) for \(i\ge j\) by Lemma 1 since everyone ranks c above b).

\(f(t^{0})=a\),

if \(f(t^{i})=a\), but \(f(t^{i+1})\ne a\) for some \(i\in \{0, \ldots , j2\}\), then f is MNP at \(t^{i}\in X^{I}\) by individual \(i+1\) via \(t_{i+1}\),

if \(f(t^{j1})=a\) but \(f(t^{j})\ne a\) then f is MNP at \(t^{j}\in X^{I}\) by individual j via \(w_{j}\),

if \(f(t^{i})\in \{a,b\}\), but \(f(t^{i+1})\notin \{a,b\}\) for some \(i\in \{j, j+1, \ldots \}\), then f is MNP at \(t^{i}\in X^{I}\) by individual \(i+1\) via \(t_{i+1}\).
Step 5 Let \(s\in X^{I}\) be any profile where \(a\in A\) is ranked as the top in \(s_{j}\). By transforming \(t\in X^{I}\) into \(s\in X^{I}\) and using a similar argument as above we can conclude that \(f(s)=a\). So, we may say that \(j\in I\) is a dictator for alternative a. Because a was arbitrary, we have shown that for each alternative \(a\in A\), there is a dictator for it. But clearly there cannot be distinct dictators for distinct alternatives, hence there is a single dictator. \(\square \)
Remark
Theorem 5 implies Theorem 2, as when I is finite every SCF is CNT. Cato (2011) shows that when I is arbitrary, STP is implied by any of the following axioms: Maskin monotonicity, independent weak monotonicity, independent personbyperson monotonicity and coalitional strategy proofness. Thus, one can replace STP in Theorem 5 by any of them; in particular, the MullerSatterthwaite theorem holds for the countable society under CNT. It also reveals that when I is countable, under UNM and CNT all of these axioms are equivalent as each implies DT.^{7}
4 Another missing proof of the Gibbard–Satterthwaite theorem
This section gives a proof of Theorem 2 à la Fishburn (1970). We first prove a useful lemma. For \(x\in X^{I}\) and \(a\in A\), let \(G(a,x)=\{i\in I: x_{i} \text { ranks }a\text { as the top}\}\).
Lemma 5
(Tops only) If \(x\in X^{I}\) and \(a,b\in A\) are such that \(G(a,x)\cup G(b,x)=I\) and at least one of G(a, x) and G(b, x) is finite, then \(f(x)\in \{a, b\}\).
Proof

\(f(x^{0})\notin \{a,b\}\), and

if \(f(x^{i})\notin \{a,b\}\), but \(f(x^{i+1})\in \{a,b\}\) for some \(i\in \{0, \ldots , k1\}\), then f is MNP at \(x^{i}\in X^{I}\) by individual \(i+1\) via \(x'_{i+1}\).

\(f(y^{0})\in \{a,b\}\), and

if \(f(y^{i})\in \{a,b\}\) then \(f(y^{i+1})\in \{a,b\}\) for all \(i\in \{0, \ldots , k1\}\). To see this, suppose for some \(i\in \{0, \ldots , k1\}\), \(f(y^{i})\in \{a,b\}\) but \(f(y^{i+1})\notin \{a,b\}\). Start with \(y^{i+1}\in X^{I}\) and transform everyone’s preferences across \(\{k+1, \ldots \}\) by bringing \(a\in A\) to the top. Then STP ensures that social choice is never in \(\{a,b\}\), and by CNT, neither the social choice at the profile obtained after making all transformations, which then contradicts UNM.

\(f(z^{k})\in \{a,b\}\), and

if \(f(z^{j})\in \{a,b\}\) then \(f(z^{j+1})\in \{a,b\}\) for all \(j\in \{k, k+1, \ldots \}\). To see this suppose for some \(j\in \{k, k+1,\ldots \}\), \(f(z^{j})\in \{a,b\}\) but \(f(z^{j+1})\notin \{a,b\}\). Start with \(z^{j+1}\in X^{I}\) and transform everyone’s preferences across \(\{1, \ldots , k\}\) by bringing \(b\in A\) to the top. Then STP ensures that social choice is never in \(\{a,b\}\), which then eventually contradicts UNM after transforming the individual \(k'\)s preferences.

\(f(y^{k})=b\), or

\(f(z^{I})=a\).
To complete the proof of Lemma 5, assume \(f(y^{k})=b\) and transform \(y^{k}\in X^{I}\) into \(x\in X^{I}\) by changing preferences of the individuals in G(b, x) into their preferences in \(x\in X^{I}\), one at a time. Then, STP ensures that social choice remains at \(b\in A\) throughout this transformation, and (when I is infinite by CNT) we conclude \(f(x)=b\). If instead we had \(f(z^{I})=a\), then we can show that \(f(x)=a\) with a similar argument. Thus, in either case, \(f(x)\in \{a,b\}\). \(\square \)
We say that \(G\subseteq I\) is undecisive over \(a\in A\) if \(f(x)\ne a\) for all \(x\in X^{I}\) such that a is ranked as the top in \(x_{i}\) for all \(i\in G\), and some \(b\in A{\setminus }\{a\}\) is ranked as the top in \(x_{j}\) for all \(j\in I{\setminus } G\). \(G\subseteq I\) is undecisive if it is undecisive over all \(a\in A\). From now on we assume that f is UNM, STP, CNT and nonDT. This implies \(I>1\), as otherwise UNM and nonDT contradict.
Lemma 6
(Group expansion) For all \(n\in \mathbb {N}\), every group \(G\subseteq I\) with \(G=n\) is undecisive and \(I>n\).
Proof
We use (strong) induction on \(n\in \mathbb {N}\) in two steps.
Step 1 We prove that Lemma 6 is holds for \(n=1\).

\(f(x^{1})=b\), and

if \(f(x^{i})=b\), but \(f(x^{i+1})\ne b\) for some \(1\le i<I\), then f is MNP by individual \(i+1\) at \(x^{i+1}\in X^{I}\) via \(x_{i+1}\).

\(f(y^{1})=b\), and

if \(f(y^{i})=b\), but \(f(y^{i+1})\ne b\) for some \(1\le i<I\), then f is MNP by individual \(i+1\) at \(y^{i+1}\in X^{I}\) via \(x'_{i+1}\).

\(f(z^{1})\ne a\) as otherwise f is MNP by 1 at \(z^{0}\in X^{I}\) via \(x^{*}_{1}\), and

if \(f(z^{i})\ne a\), but \(f(z^{i+1})=a\) for some \(i\in I\) with \(1\le i<I\), then f is MNP by individual \(i+1\) at \(z^{i+1}\in X^{I}\) via \(x^{'}_{i+1}\).
Let now \(b,c\in A{\setminus }\{a\}\) be two arbitrary (distinct) alternatives and let \(y^{*}\in X^{I}\) be the profile with \(y^{*}_{1}=(a\succ b\succ \cdots )\) and for all \(i\in I{\setminus }\{1\}\), \(y^{*}_{i}=(c\succ \cdots \succ b)\). Since \(\{1\}\) is undecisive over \(a\in A\), we must have \(f(y^{*})\ne a\), and then by Lemma 5, \(f(y^{*})=c\). Let \(y^{**}\in X^{I}\) be the profile same as \(y^{*}\) except the relative ranking of \(a, b\in A\) in \(y^{*}_{1}\) is reversed in \(y^{**}_{1}\). Then \(f(y^{**})\ne b\) as otherwise f is MNP at \(y^{*}\in X^{I}\) by 1 via \(y^{**}_{1}\). Then, by repeating the above argument we can show that \(\{1\}\) is undecisive over \(b\in A{\setminus } \{a\}\). Thus, \(\{1\}\) is undecisive. This completes Step 1.
Step 2 We prove that if Lemma 6 holds for \(k<n\), then it holds for n.
Let \(G\subseteq I\) be such that \(G=n\). We may assume that \(G=\{1, 2, \ldots , n\}\) and split G into two subgroups \(G_{1}=\{1\}\), \(G_{2}=\{2, \ldots , n\}\). We know that \(G_{1}\) and \(G_{2}\) are undecisive. Let \(x\in X^{I}\) be the (Condorcet) profile with \(x_{1}=(a\succ b\succ c\succ \cdots )\), \(x_{i}=(c\succ a\succ b\succ \ldots )\) for all \(i\in G_{2}\), and \(x_{j}=(b\succ c\succ a\succ \cdots )\) for all \(j\in I{\setminus }{G}\). Then, \(f(x)\in \{a,b,c\}\) by Lemma 1 and we claim that \(f(x)=b\). Notice that \(f(x)\in \{a,b\}\) as otherwise f is MNP by \(\{1\}\) at \(x\in X^{I}\) via some \(x'_{1}=(b\succ a\succ c \succ \cdots )\). That is because for \(x'=(x'_{1}, x_{1})\), \(f(x')\ne c\) since \(G_{2}\) is undecisive, and hence \(f(x')=b\) by Lemma 5.

\(f(x^{n})=c\in \{b,c\}\), and

if \(f(x^{i})\in \{b,c\}\), but \(f(x^{i+1})\notin \{b,c\}\) for some \(n\le i<I\), then f is MNP by individual \(i+1\) at \(x^{i+1}\in X^{I}\) via \(x^{*}_{i+1}\).

\(f(x^{1})=b\), and

if \(f(x^{i})=b\), but \(f(x^{i+1})\ne b\) for some \(0\le i<n\), then by Lemma 1 \(f(x^{i+1})\in \{a,c\}\) and hence f is MNP by individual \(i+1\) at \(x^{i}\in X^{I}\) via \(x'_{i+1}\).

\(f(y^{n})=b\), and

if \(f(y^{i})=b\), but \(f(y^{i+1})\ne b\) for some \(n\le i<I\), then f is MNP by individual \(i+1\) at \(y^{i+1}\in X^{I}\) via \(x^{n}_{i+1}\).

\(f(z^{0})=b\ne a\), and

if \(f(z^{i})\ne a\), but \(f(z^{i+1})=a\) for some \(i\in \{0, \ldots , n1\}\), then f is MNP by individual \(i+1\) at \(z^{i}\in X^{I}\) via \(y_{i+1}\).
Let us now prove Theorem 2. Suppose by contradiction that when I is finite, there is \(f:X^{I}\rightarrow A\) which is UNM, STP and nonDT. Then, f is CNT, and by Lemma 6 we conclude that I is infinite, which is a contradiction. \(\square \)
Remark
The above proof is adopted from Fishburn (1970)’s proof of Theorem 1 with the following difference. Fishburn (1970)’s proof does not need any of CNT axiom, Lemmas 1 and 5 whereas these are building blocks of our proof. However, this difference is due to differences in the mathematical settings in which Theorems 1 and 2 are formulated, and also in the set of axioms constituting the premises of these results [see discussions in Sect. 4 in Ninjbat (2018)].
5 Final remarks
It is well known that infinity is mindboggling, paradoxical yet helpful. One can argue that a satisfactory mathematical theory should reduce the set of presumptions as much as possible. As discussed in Chap. 1 in AlósFerrer and Ritzberger (2016), among such assumptions often made in economics are cardinality restrictions. In social choice theory, researchers noticed early on that finite population axiom is critical in establishing Arrow’s impossibility theorem, and since then they got interested in knowing just how critical it is.

It adds no restriction to the classical set up with finitely many individuals, as it is embedded in this setting,

It allows us to use tools and results effective in the classical setting to a larger domain,

When an infinite society choice problem is confronted it captures, to some extend, the idea of dynamic consistency. In such a situation, it is likely that the social decision takes into account the tradeoff between the current and the future, such as the social planner’s problem in an OLG economy. Since the society is infinite, the current infinite support tends to win over a growing trend in the society which is to become the main position in the future (i.e. the future infinite support). Continuity axiom puts some restriction on the outcome of the social decision making procedure over such dynamics by asking it to be consistent.

Can we weaken conditions needed in two main theorems in Sect. 3, e.g. transitivity into quasitransitivity etc? Can we modify and/or unify these theorems (see Ninjbat 2015)?

How these results formally relate with some of the existing literature (e.g. Turing computability, topological continuity of SCW and SCF, etc.)? In fact, can we interconnect all the existing impossibility results on large societies?

Can we further extend these results to the case where societies are modelled as wellordered sets?
Footnotes
 1.
 2.
Weak ranking is a complete and transitive binary relation, while strict ranking is a weak ranking which is also antisymmetric.
 3.
Recall that in Kolmogorov’s axiomatization of a probability measure, continuity together with finite additivity imply countable additivity [see e.g. Chap. 2.1 in Borovkov (2013)]. The hidden axiom of full domain makes CNT a global property; for all \(z\in X^{I}\) there is \((x^{n})\in X^{I}\) with \((x^{n})\uparrow z\). Note that in the definition of CNT replacing \((x^{n})\uparrow z\) by \((x^{n})\rightarrow z\) makes it stronger.
 4.
Thus, CNT is implicit in the finite setting and we often employ this hidden property in our treatments of impossibility theorems; e.g. in using the pivotal voter approach.
 5.
Litak (2017) criticises infinite population models traditionally used in the context of social choice by their loose connections with reality based on the so called Hildenbrand criterion. The timecontingentselves argument is an attempt to resolve this matter.
 6.
An investigation on formal connections between Turing computability and CNT for a SWF is highly desirable. A common ground for both concepts is formal logic and in particular, model theory. CNT is basically a compactness type property, as in model theory, compactness means a formula is satisfiable if and only if any finite subsection is satisfiable. Also the notion of Scott continuity in domain theory might be relevant; see e.g. Vassilakis (1992).
 7.
It is probably true that this equivalence holds even UNM is dropped.
Notes
Acknowledgements
I am thankful to the editor and anonymous referees for helpful comments, and to seminar participants in the department of mathematics at the National University of Mongolia for discussions.
Compliance with ethical standards
Ethical statement
I testify that this article has not been published in whole or in part elsewhere; it is not currently being considered for publication in another journal; and its content is my sole creation. The author have no conflict of interest. There are no funding nor informed consents from an external source.
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