Majority decisions when abstention is possible

Paul Larson^{a,1}, Nick Matteo^{b}, Saharon Shelah^{c,2,3}

aDepartment of Mathematics, Miami University, Oxford, OH 45056, USA bDepartment of Mathematics, Miami University, Oxford, OH 45056, USA cThe Hebrew University of Jerusalem, Einstein Institute of Mathematics Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel

Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA

Abstract

Suppose that we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential “voters.” The question is, what choice functions agree, on every pair, with the majority of some finite subfamily of the voters? For the problem as stated, a complete characterization was given in Shelah [1], but here we allow voters to abstain. Aside from the trivial case, the possible families of (partial) choice functions break into three cases in terms of the functions that can be generated by majority decision. In one of these, cycles along the lines of Condorcet’s paradox are avoided. In another, all partial choice functions can be represented.

Keywords: choice function, majority decision, Condorcet’s paradox, tournament 2000 MSC: Primary 91B14, Secondary 05C20

1. Introduction

Condorcet’s “paradox” demonstrates that given three candidates A, B, and C, majority rule may result in the society preferring A to B, B to C, and C to A [2]. McGarvey [3] proved a far-reaching extension of Condorcet’s paradox: For every asymmetric relation R (so a R b implies b R̸ a) on a set X of n candidates, there are m linear order relations on X: R_{1}, …, R_{m}, with R as their strict simple majority relation. That is, for every a, b ∈ X,

a R b ⟺ |\{i : a R_{i} b\}| > \frac{m}{2}.

Email addresses: larsonpb@muohio.edu (Paul Larson), matteo.n@neu.edu (Nick Matteo)

1The research of the first author is supported by NSF grant DMS-0801009.

2Partially supported by the United States-Israel Binational Science Foundation.

3This is publication number 965 in the third author’s publication list.

Preprint submitted to Discrete Mathematics June 24, 2011

In other words, given any set of choices from pairs from a set of n candidates, there is a population of m voters, all with simple linear-order preferences among the candidates, who will yield the given outcome for each pair in a majority-rule election between them.

McGarvey’s proof gave m = n(n − 1). Stearns [4] found a construction with m = n and noticed that a simple counting argument implies that m must be at least \frac{n}{log n}. Erdős and Moser [5] were able to give a construction with

m = O (\frac{n}{log n}). Mala [6] and Alon [7] showed that there is a positive constant c_{1} such that, given any asymmetric relation R, there is some m and linear orders R_{1}, …, R_{m} with

a R b ⟺ |\{i : a R_{i} b\}| > \frac{m}{2} + c_{1}\sqrt{m}

and that this is not true for any c_{2} greater than c_{1}.

Gil Kalai asked to what extent the assertion of McGarvey’s theorem holds if we replace the linear orders by an arbitrary isomorphism class of choice functions on pairs of elements. Namely, when can we guarantee that every asymmetric relation R on X could result from a finite population of voters, each using a given kind of asymmetric relation on X?

Informally, one may think of the situation we study here as follows. Imagine that we have n candidates running for office, and that each voter votes in a two-step process as follows. First, the voter chooses one element from a set of ballot types, which are partial tournaments (or asymmetric relations) with n vertices. Then, having chosen a ballot type, the voter labels the corresponding vertices with the names of the candidates, in such a way that an edge from candidate A to candidate B indicates a preference for A over B (of course, not all ballot types allow one to express one’s preferences, but for the purposes of this illustration we assume that each voter does the best he or she can with the available options). The outcome of the election is the partial tournament with a node for each candidate and an edge from A to B if more voters had edges from A to B on their ballots than had the reverse. The question we address here is, what sets of ballot types can generate which partial tournaments via this process, assuming any number of voters? As we shall see, the nontrivial sets of ballot types (or, kinds of symmetric relations, as above) fall into three classes (see Main Claim 3.1 and the last paragraph of this introduction). Among these are the partisan case, in which voters are offered the chance to separate out a preferred subset of the candidates (which is how elections are often run, with the preferred set having size 1). We will show that the partisan case is the only one that does not lead to paradoxical results (i.e., directed cycles). We will also characterize the sets of ballot types which allow every partial tournament to be generated.

More formally, we use the following terminology. Let \binom{X}{k} denote the family of subsets of X with k elements:

\binom{X}{k} = \{Y ⊆ X : |Y | = k\}.

Either an individual voter’s preferences among candidates, or the outcomes that would result from each two-candidate election, may be represented as any of:

• An asymmetric relation R where a R b if and only if a beats b; it is possible that a R̸ b, b R̸ a, and a ≠ b.

• A choice function defined on some subfamily of \binom{X}{2}, choosing the winner in each pair. Such a choice function is called “full” if its domain is all of \binom{X}{2}, and “partial” otherwise.

• An oriented graph, i.e. a directed graph with nodes X and edges a → b when a beats b.

We shall treat these representations as largely interchangeable throughout this paper. Total asymmetric relations, full choice functions, and tournaments (complete oriented graphs) all correspond to the case of no abstaining. Using notation from Shelah [1], we will let Tor(c) denote the oriented graph (i.e., partial tournament) associated with a choice function c. For any set X, c ↦ Tor(c) is a bijection of full choice functions onto tournaments on X, and a bijection of all choice functions onto oriented graphs on X.

For the rest of this paper, we fix a finite set X. We let n (as opposed to n, which is used as a variable) denote the size of X, and assume that n ≥ 3. We let ℭ denote the set of choice functions on pairs from X, that is, the set of functions c: Y → X where Y ⊆ \binom{X}{2} and c\{x, y\} ∈ \{x, y\} for all \{x, y\} ∈ Y. When c\{x, y\} is not defined it is interpreted as abstention or having no preference.

Definition 1.1. (a) Per(X) is the set of permutations of X.

(b) A choice function d is a permutation of a choice function c if there is σ ∈ Per(X) such that

d{σ(x), σ(y)} = σ(x) ⟺ c{x, y} = x;

we write d = c^{σ}.

(c) A set of choice functions 𝒞 ⊆ ℭ is symmetric if it is closed under permutations of X. So for each σ ∈ Per(X), if c ∈ 𝒞 then c^{σ} ∈ 𝒞.

Note that a choice function d is a permutation of a choice function c if and only if Tor(c) and Tor(d) are isomorphic directed graphs. Symmetric sets of choice functions are what was meant above by “kind of asymmetric relation.”

The main result of Shelah [1] pertained to full choice functions for the voters. It was shown that an arbitrary choice function d could result from a symmetric set 𝒞 consisting of full choice functions (i.e. for each d there is a finite set \{c_{1}, …, c_{m}\} ⊆ 𝒞 such that d\{x, y\} = x ⟺ |\{i : c_{i}\{x, y\} = x\}| > \frac{m}{2}) if and only if for some c ∈ 𝒞 and x ∈ X,

|\{y : c\{x, y\} = y\}| ≠ \frac{n − 1}{2}.

We shall call this condition “imbalance.”

Definition 1.2. For a choice function c,

(a) Let dom(c) denote the domain of c.

(b) for any pair (x, y) in X^{2}, the weight of x over y for c is

W_{y}^{x}(c) = \begin{cases}1 & if c\{x, y\} = x \\ 0 & if \{x, y\} ∉ dom c \\ −1 & if c\{x, y\} = y \end{cases}

(c) c is balanced if for all x ∈ X

∑_{y∈X} W_{y}^{x}(c) = 0.

That is, |{y : c{x, y} = x}| = |{y : c{x, y} = y}|, for every x in X.

(d) c is unbalanced if c is not balanced.

(e) c is pseudo-balanced if every edge of Tor(c) belongs to a directed cycle.

Gil Kalai further asked whether the number m can be given bounds in terms of n, and what is the result of demanding a “non-trivial majority,” e.g. 51%. We shall consider loose bounds while addressing the general case, when voters are permitted to abstain.

To determine what symmetric sets of partial choice functions could produce an arbitrary outcome, we shall characterize the set of all possible outcomes of a symmetric set of choice functions, the “majority closure” maj-cl(𝒞). The symmetric set 𝒞 satisfies this extension to McGarvey’s theorem if and only if maj-cl(𝒞) = ℭ.

Definition 1.3. For 𝒞 ⊆ ℭ, let maj-cl(𝒞) be the set of d ∈ ℭ such that, for some set of weights {r_{c} ∈ [0, 1]_{ℚ} : c ∈ 𝒞 } with ∑_{c∈𝒞} r_{c} = 1,

d{x, y} = x ⟺ ∑_{c∈𝒞} W_{y}^{x}(c)r_{c} > 0.

Here and throughout this paper, [0, 1]_{ℚ} refers to the rationals in the unit interval of the real line. Note that we cannot assume that maj-cl(maj-cl(𝒞)) = maj-cl(𝒞), and in fact we shall see this is not true. We must show that each function d in the majority closure, thus defined, can in fact be the outcome of a finite collection of voters using choice functions in 𝒞; this is Claim 2.6.

Shelah [1] gave a characterization of maj-cl(𝒞) for nonempty symmetric sets of full choice functions; there are just two cases. If some c ∈ 𝒞 is unbalanced, then maj-cl(𝒞) = ℭ; if every c ∈ 𝒞 is balanced, then maj-cl(𝒞) is the set of all pseudo-balanced functions. The situation with possible abstention is more complicated; there are three nontrivial cases, see Main Claim 3.1. Unlike in Shelah [1], it is possible to have nontrivial 𝒞 for which maj-cl(𝒞) contains no directed cycles; this happens if and only if 𝒞 is partisan (see Definitions 2.3 and 2.4).

2. Basic Definitions and Facts

In addition to the representations as relations, choice functions, or oriented graphs discussed above, we shall also find it convenient to map each choice function c to a sequence in [−1, 1]_{ℚ} indexed by X^{2}, the “probability sequence”

pr(c) = ⟨W_{y}^{x}(c) : (x, y) ∈ X^{2}⟩.

Let pr(𝒞) be {pr(c) : c ∈ 𝒞 }, and pr(X) denote the set of all sequences t̄ in [−1, 1]_{ℚ}^{X^{2}} such that t_{x,y} = −t_{y,x}; pr(X) contains pr(ℭ).

Definition 2.1. For a probability sequence t̄∈ pr(X),

(a) t̄ is balanced if for each x ∈ X, ∑_{y∈X} t_{x,y} = 0.

(b) maj(t̄) is the c ∈ ℭ such that c{x, y} = x ⟺ t_{x,y} > 0.

Note that maj and pr are mutually inverse functions from ℭ to pr(ℭ); c = maj(pr(c)) and t̄ = pr(maj(t̄)) for all c ∈ ℭ and t̄ ∈ pr(ℭ). The function maj is also defined on the much larger set pr(X), which it maps onto ℭ. The reason for reusing the name “balanced” is clear:

Claim 2.2. If c ∈ ℭ is a balanced choice function, then pr(c) is a balanced probability sequence. If t̄∈ pr(ℭ) is a balanced probability sequence, then maj(t̄) is a balanced choice function.

Proof. Suppose c ∈ ℭ is balanced. For every x ∈ X, ∑_{y∈X} W_{y}^{x}(c) = 0. The (x, y)-term of pr(c) is W_{y}^{x}(c), so ∑_{y∈X} pr(c)_{x,y} = 0, i.e. pr(c) is balanced.

Now suppose t̄ ∈ pr(ℭ) is balanced. Then t̄ = pr(c) for some c ∈ ℭ. Since t_{x,y} = W_{y}^{x}(c),

∑_{y∈X} W_{y}^{x}(c) = ∑_{y∈X} t_{x,y} = 0

and c is balanced.

However, it is not the case that maj(t̄) is balanced for every balanced t̄ ∈ pr(X). For instance, choose x, z ∈ X and define a sequence t̄which has t_{z,x} = 1, t_{x,z} = −1; for all y ∉ {x, z}

t_{x,y} = t_{y,z} = \frac{1}{n − 2},
t_{y,x} = t_{z,y} = \frac{−1}{n − 2};

and for all other pairs (u, v), t_{u,v} = 0. One may check that t̄ is balanced, yet maj(t̄) has ∑_{y∈X} W_{y}^{x}(maj(t̄)) = n − 3 > 0 for any X with at least 4 elements.

Definition 2.3. For a choice function c ∈ ℭ,

(a) c is partisan if there is nonempty W ⊊ X such that

c{x, y} = x ⟺ x ∈ W and y ∉ W.

(b) c is tiered if there is a partition {X_{1}, X_{2}, …, X_{k}} of X such that c{x, y} = x if and only if x ∈ X_{i}, y ∈ X_{j}, and i > j. We say c is k-tiered where the partition has k sets. (So a partisan function is 2-tiered.)

(c) c is chaotic if it is both unbalanced and nonpartisan.

Definition 2.4. For a subset 𝒞 ⊆ ℭ,

(a) 𝒞 is trivial if 𝒞 = ∅ or 𝒞 = {c} where dom c = ∅, i.e. c makes no decisions.

(b) 𝒞 is balanced if every c ∈ 𝒞 is balanced.

(c) 𝒞 is partisan if every c ∈ 𝒞 is partisan.

(d) 𝒞 is chaotic if there is some chaotic c ∈ 𝒞.

(e) pr-cl(𝒞) is the convex hull of pr(𝒞), i.e.

pr-cl(𝒞) = {∑_{i=1}^{k} r_{i}t̄_{i} : k ∈ ℕ, r_{i} ∈ [0, 1]_{ℚ}, ∑_{i=1}^{k} r_{i} = 1, t̄_{i} ∈ pr(𝒞)} .

We can now establish some straightforward results which allow us to describe the possible outcomes due to voters chosen from a given symmetric set more explicitly.

Claim 2.5. If 𝒞 is a symmetric subset of ℭ, then d is the strict simple majority outcome of some finite set {c_{1}, …, c_{m}} chosen from 𝒞 if and only if

d{x, y} = x ⟺ ∑_{i=1}^{m} W_{y}^{x}(c_{i}) > 0.

Proof. We have that d is the strict simple majority outcome if and only if

d{x, y} = x ⟺ |{i : c_{i}{x, y} = x}| > |{i : c_{i}{x, y} = y}|
⟺ ∑_{i=1}^{m} W_{y}^{x}(c_{i}) > 0.

Claim 2.6. Suppose 𝒞 is a symmetric subset of ℭ. Then d ∈ maj-cl(𝒞) if and only if d is a strict simple majority outcome of some {c_{1}, …, c_{m}} ⊆ 𝒞.

Proof. If d ∈ ℭ and there is a set {c_{1}, …, c_{m}} ⊆ 𝒞 with d as their strict simple majority outcome, then for each c ∈ 𝒞 let r_{c} be the number of times that c appears among the c_{i}, divided by m:

r_{c} = \frac{|\{i : c_{i} = c\}|}{m}.

Clearly ∀c ∈ 𝒞 0 ≤ r_{c} ≤ 1 and ∑_{c∈𝒞} r_{c} = 1. Furthermore,

d\{x, y\} = x ⟺ ∑_{i=1}^{m} W_{y}^{x}(c_{i}) > 0 ⟺ ∑_{i=1}^{m} \frac{W_{y}^{x}(c_{i})}{m} > 0 ⟺ ∑_{c∈𝒞} W_{y}^{x}(c)r_{c} > 0.

Conversely, suppose d ∈ maj-cl(𝒞). There are r_{c} = \frac{a_{c}}{b_{c}} for each c ∈ 𝒞 with ∑ r_{c} = 1, and

d{x, y} = x ⟺ ∑_{c∈𝒞} W_{y}^{x}(c)r_{c} > 0.

Let m = lcm{b_{c} : c ∈ 𝒞 }. Now construct a finite set {c_{1}, …, c_{m}} by taking a_{c} \frac{m}{b_{c}} copies of each c ∈ 𝒞. \frac{m}{b_{c}} is an integer since b_{c} | m. Now for each c ∈ 𝒞,

∑_{c_{i}=c} W_{y}^{x}(c_{i}) = W_{y}^{x}(c) · |\{i : c_{i} = c\}| = W_{y}^{x}(c)a_{c} \frac{m}{b_{c}} = W_{y}^{x}(c)r_{c}m.

So

∑_{i=1}^{m} W_{y}^{x}(c_{i}) = ∑_{c∈𝒞} ∑_{c_{i}=c} W_{y}^{x}(c_{i}) = ∑_{c∈𝒞} W_{y}^{x}(c)r_{c}m.

∑_{c∈𝒞} W_{y}^{x}(c)r_{c}m > 0 ⟺ ∑_{c∈𝒞} W_{y}^{x}(c)r_{c} > 0,

so d is the simple majority relation of the c_{i}.

Claim 2.7. For any symmetric 𝒞 ⊆ ℭ, maj-cl(𝒞) = {maj(t̄) : t̄∈ pr-cl(𝒞)}.

Proof. Suppose d ∈ maj-cl(𝒞). Then there are r_{c} per c ∈ 𝒞 with ∑_{c∈𝒞} r_{c} = 1 and

d{x, y} = x ⟺ ∑_{c∈𝒞} r_{c}W_{y}^{x}(c) > 0.

Let t̄= ∑_{c∈𝒞} r_{c} pr(c) be a probability sequence in pr-cl(𝒞). Then

t_{x,y} = ∑_{c∈𝒞} r_{c} pr(c)_{x,y} = ∑_{c∈𝒞} r_{c}W_{y}^{x}(c).

So

maj(t̄){x, y} = x ⟺ ∑_{c∈𝒞} r_{c}W_{y}^{x}(c) > 0,

i.e. d = maj(t̄).

Suppose d = maj(t̄) for some t̄∈ pr-cl(𝒞). Then for some r_{c} with ∑_{c∈𝒞} r_{c} = 1, d = maj (∑_{c∈𝒞} r_{c} pr(c)).

d{x, y} = x ⟺ ∑_{c∈𝒞} r_{c} pr(c)_{x,y} > 0 ⟺ ∑_{c∈𝒞} r_{c}W_{y}^{x}(c) > 0.

So d ∈ maj-cl(𝒞).

3. A Characterization

Main Claim 3.1. Given a symmetric 𝒞 ⊆ ℭ,

(i) 𝒞 is trivial ⟺ maj-cl(𝒞) is trivial. (ii) 𝒞 is balanced but nontrivial ⟺ maj-cl(𝒞) is the set of all pseudo-

balanced choice functions on X. (iii) 𝒞 is partisan and nontrivial ⟺ maj-cl(𝒞) is the set of all tiered choice

functions on X. (iv) 𝒞 is not balanced and not partisan ⟺ maj-cl(𝒞) = ℭ.

Proof. No nontrivial 𝒞 is both balanced and partisan, so the sets

{𝒞 ⊂ ℭ : 𝒞 is trivial},

{𝒞 ⊂ ℭ : 𝒞 is symmetric, nontrivial, and balanced}, {𝒞 ⊂ ℭ : 𝒞 is symmetric, nontrivial, and partisan}, and {𝒞 ⊆ ℭ : 𝒞 is symmetric, not balanced, and not partisan}

partition the family of all symmetric subsets of ℭ. Thus proving each forward implication in the claim will give the reverse implications.

i. If no-one in 𝒞 makes any choices, no combination can have a majority choice.

ii. This is the content of Section 4, below.

iii. It is shown in Inada [8] (using different terminology) that if 𝒞 is partisan and nontrivial then every element of maj-cl(𝒞) is a tiered choice function. We give an alternate proof. Suppose that 𝒞 is partisan and d ∈ maj-cl(𝒞). Then there is a finite set {c_{1}, …, c_{m}} ⊆ 𝒞 with d as the strict simple majority outcome. For each x ∈ X, let k_{x} be the number of c_{i} such that x is in the “winning” partite set. For any (x, y) ∈ X^{2}, let k be the number of c_{i} where x and y are both in the winning subset; the number of c_{i} with c_{i}{x, y} = x is k_{x} − k, and the number with c_{i}{x, y} = y is k_{y} − k, so

d{x, y} = x ⟺ k_{x} − k > k_{y} − k ⟺ k_{x} > k_{y}.

Partition X into subsets for each value of k_{x},

X = ⋃{{x ∈ X : k_{x} = k} : k ∈ {k_{x} : x ∈ X}}.

These subsets form the tiers, so d is a tiered function.

For the reverse inclusion, suppose d is an arbitrary tiered function with tiers {X_{1}, …, X_{k}}. Let c ∈ 𝒞. For each x ∈ X, let Γ_{x} ⊂ Per(X) be all permutations holding x fixed and c_{x} be a choice function in 𝒞 with x in its winning subset. (If y is in the winning subset of c, c^{(x,y)} will suffice for c_{x}.) Now construct {c_{1}, …, c_{m}} by taking, for 1 ≤ i ≤ k, each x ∈ X_{i}, and each σ ∈ Γ_{x}, i copies of c_{x}^{σ}. Suppose x ∈ X_{i}, y ∈ X_{j}, and i > j (so d{x, y} = x). Then for any z ∈ X_{i} \ {x}, any z ∈ X_{j} \ {y}, and any z in other tiers, x and y are chosen by equal numbers of {c_{z}^{σ} : σ ∈ Γ_{z}}. So they are chosen by an equal number of {c_{1}, …, c_{m}} except among the functions c_{x}^{σ} and c_{y}^{σ}; there are i such functions c_{x}^{σ} for each σ ∈ Γ_{x}, and j such functions c_{y}^{σ} for each σ ∈ Γ_{y}. The number of permutations fixing x and the number fixing y are the same, |Γ_{x}| = |Γ_{y}|. The number of permutations fixing x and putting y in the losing subset and the number fixing y and putting x in the losing subset is the same, l. There are il functions which choose x over y, and jl which choose y over x; il > jl so the strict simple majority outcome of the c_{i} chooses x.

If i = j, then il = jl, so x and y tie in the majority outcome. Thus d is the strict simple majority outcome of {c_{1}, …, c_{m}} and d ∈ maj-cl(𝒞).

iv. If 𝒞 is a chaotic set, the conclusion follows from Section 5, below.

Otherwise, 𝒞 is not balanced and not partisan, but not chaotic. In this case, 𝒞 contains unbalanced functions which are all partisan, and nonpartisan functions which are all balanced. By symmetry, 𝒞 contains a nontrivial balanced symmetric subset ℬ and a nontrivial partisan symmetric subset 𝒫.

Let d ∈ ℭ. Suppose b → a is any edge of d. For each z ∈ X \ {a, b}, by Claim 4.5 there is a voter population T_{z} from ℬ such that a beats z, z beats b, b beats a, and no other pairs are decided; moreover, the same number of voters, say m, choose the winner in each pair, and no dissenting votes occur. In the combination ⋃_{z} T_{z}, m voters choose a over each z, and each z over b. m(n − 2) voters choose b over a.

Let c ∈ 𝒫 be a partisan function, and l be the size of the set of winning candidates under c. Let A ⊂ 𝒫 be the set of permutations of c so that a is on the losing side, B ⊂ 𝒫 be the permutations of c so that b is on the winning side, and C = A ∩ B—the permutations with b winning and a losing.

|A| = \binom{n − 1}{l}, |B| = \binom{n − 1}{l − 1}, |C| = \binom{n − 2}{l − 1}.

Any candidates besides a or b are tied over all the voters of A, B, or C, since each is selected in the winning set an equal number of times. Furthermore, a does not beat b in any choice function in A, B, or C.

Now we seek to take appropriate numbers of copies of the sets A, B, and C so that, for some constant k, any z ∈ X \ {a, b} defeats a by k votes, and loses to b by k votes. Say we have k_{0} copies of A, k_{1} copies of B, and k_{2} copies of C in a population D. Now for each z ∈ X \ {a, b},

|\{c ∈ D : c\{z, a\} = z\}| = k_{0}\binom{n − 2}{l − 1} + k_{1}\binom{n − 3}{l − 2} + k_{2}\binom{n − 3}{l − 2},
|\{c ∈ D : c\{z, a\} = a\}| = k_{1}\binom{n − 3}{l − 2},
|\{c ∈ D : c\{b, z\} = b\}| = k_{0}\binom{n − 3}{l − 1} + k_{1}\binom{n − 2}{l − 1} + k_{2}\binom{n − 3}{l − 1},
|\{c ∈ D : c\{b, z\} = z\}| = k_{0}\binom{n − 3}{l − 1}.

So we solve k_{0}\binom{n−2}{l−1} + k_{2}\binom{n−3}{l−2} = k_{1}\binom{n−2}{l−1} + k_{2}\binom{n−3}{l−1}, with coefficients not all zero. There are solutions

k_{0} = n − 2l,  k_{1} = 0,  k_{2} = n − 2,  if l ≤ \frac{n}{2};
k_{0} = 0,  k_{1} = 2l − n,  k_{2} = n − 2,  if l > \frac{n}{2}.

Let k = k_{0}\binom{n−2}{l−1}+k_{2}\binom{n−3}{l−2}, the number of votes for b over z, or z over a, for any z ∈ X \ {a, b}. Now take the union of k copies of the population ⋃_{z} T_{z}, mk_{0} copies of A, mk_{1} copies of B, and mk_{2} copies of C for our voter population. b defeats any z mk times among the partisan functions, and z defeats b km times among the balanced functions, so they tie; similarly, a and z tie. Thus we are left with b → a. The union of such populations for each edge in d yields d as the majority outcome.

Hence ℭ ⊆ maj-cl(𝒞).

4. Balanced Choices

Definition 4.1. For a choice function c ∈ ℭ,

(a) c is triangular if for some \{x, y, z\} ∈ \binom{X}{3}, c\{x, y\} = x, c\{y, z\} = y, c\{z, x\} = z, and for any {u, v} ⊄ {x, y, z}, {u, v} ∉ dom c. We write c^{x,y,z} for such c.

(b) c is cyclic if for some \{x_{1}, …, x_{k}\} ∈ \binom{X}{k},

• c{x_{i}, x_{j}} = x_{i} if j ≡ i + 1 mod k.

• No other pair {x, y} is in dom c.

We write c^{x_{1},...,x_{k}} for such c. (Triangular functions are a specific case of cyclic functions.)

(c) Let pr^{bl} be the set of all balanced t̄∈ pr(X).

Note that if 𝒞 is symmetric and c^{x,y,z} ∈ 𝒞, then c^{u,v,w} ∈ 𝒞 for any \{u, v, w\} ∈ \binom{X}{3}; similarly for cyclic functions.

Claim 4.2. If a choice function c = maj(t̄) for some t̄∈ pr^{bl}, then c is pseudobalanced.

Proof. Assume c = maj(t̄) for some t̄ ∈ pr^{bl}. Let x → y be any edge of Tor(c); then t_{x,y} > 0. Suppose x → y is in no directed cycle. Let Y be the set of z ∈ X with a winning chain to x, i.e.

Y = ⋃ \{\{z_{1}, …, z_{k}\} ∈ \binom{X \ \{y\}}{k} : k < n, ∀i < k c\{z_{i}, z_{i+1}\} = z_{i}, z_{k} = x\} .

Let \overline{Y} = X \ Y. Note that y ∈ \overline{Y} and x ∈ Y, so Y and \overline{Y} partition X into nonempty sets. Suppose z ∈ Y and v ∈ \overline{Y}. Say z, z_{1}, …, z_{k} is a winning chain from z to x. If t_{z,v} < 0, then v, z, z_{1}, …, z_{k} forms a winning chain from v to x. If v is y, the chain forms a directed cycle with the edge x → y and we are done. If v ≠ y, this contradicts that v ∉ Y. So we assume that t_{z,v} ≥ 0.

Now

∑_{z∈Y,v∈\overline{Y}} t_{z,v} > 0,

since every t_{z,v} ≥ 0, and t_{x,y} > 0 is among them. For each u ∈ \overline{Y} let

r_{u} = ∑_{z∈Y} t_{z,u},
r̄_{u} = ∑_{v∈\overline{Y}} t_{v,u}.

Since t̄ is balanced, r_{u} + r̄_{u} = 0, so

∑_{u∈\overline{Y}}(r_{u} + r̄_{u}) = 0 = ∑_{u∈\overline{Y}} r_{u} + ∑_{u∈\overline{Y}} r̄_{u}.

We have seen that the first summand is positive, so the second summand is negative. But it is zero because for each pair (u, v) ∈ \overline{Y}^{2} we have t_{u,v} +t_{v,u} = 0. This is a contradiction; so x → y must be in some directed cycle.

Claim 4.3. pr^{bl} is a convex subset of pr(X).

Proof. Suppose t̄ and s̄ are balanced probability sequences, and a ∈ [0, 1]_{ℚ}. Then ū = at̄+ (1 − a)s̄ has, for any x ∈ X,

∑_{y∈X} u_{x,y} = ∑_{y∈X} (at_{x,y} + (1 − a)s_{x,y})

= a ∑_{y∈X} t_{x,y} + (1 − a) ∑_{y∈X} s_{x,y}

= a0 + (1 − a)0 = 0.

So ū is a balanced probability sequence.

Claim 4.4. If 𝒞 ⊂ ℭ is symmetric, balanced, and nontrivial, then every d ∈ maj-cl(𝒞) is pseudo-balanced.

Proof. By Claim 2.2, every t̄∈ pr(𝒞) is balanced. So by Claim 4.3, the convex hull pr-cl(𝒞) is contained in pr^{bl}. Any d ∈ maj-cl(𝒞) is maj(t̄) for some t̄ ∈ pr-cl(𝒞), by Claim 2.7, hence is pseudo-balanced, by Claim 4.2.

For the reverse inclusion, that every pseudo-balanced function is in the majority closure, we shall consider the graph interpretation. First we shall show that every triangular function is in the majority closure of a balanced symmetric set.

Claim 4.5. If 𝒞 ⊂ ℭ is symmetric, balanced, and nontrivial, then for any \{x, y, z\} ∈ \binom{X}{3}, the triangular function c^{x,y,z} ∈ maj-cl(𝒞). Moreover, there is a voter population generating c^{x,y,z} such that the same number of votes choose x over y, y over z, or z over x, and there are no opposing votes in any of these cases.

\begin{matrix}x \\ y \\ z \end{matrix}  \begin{matrix}x \\ y \\ z \end{matrix}  \begin{matrix}x \\ y \\ z \end{matrix}

\begin{matrix}x \\ y \\ z \end{matrix}  \begin{matrix}x \\ y \\ z \end{matrix}  \begin{matrix}x \\ y \\ z \end{matrix}

Figure 1: Constructing a triangle from permutations of a cycle, as in Claim 4.5. Note that every edge is matched by an opposing one, except for xy, yz, and zx.

Proof. Let c_{0} ∈ 𝒞 and let m be the size of the smallest directed cycle in Tor(c_{0}). Index the distinct elements of this cycle, in order, as x_{1}, x_{2}, …, x_{m}. Let c = c_{0}^{σ} where σ ∈ Per(X) takes x_{1} to x, x_{2} to y, and x_{3} to z. Now consider the cycle in Tor(c), so x_{1} = x, x_{2} = y, and x_{3} = z. We define

c_{1} = c,
c_{2} = c^{ρ} where ρ takes x ↦ y, y ↦ z, z ↦ x,
c_{3} = c^{ρ} where ρ takes x ↦ z, y ↦ x, z ↦ y.

Let Γ_{x,y,z} ⊂ Per(X) be the permutations fixing x, y, and z, and take {c_{i}^{σ} : 1 ≤ i ≤ 3, σ ∈ Γ_{x,y,z}} as our finite set of voters. We claim the majority outcome of this set is c^{x,y,z}.

If u, v ∈ X \ {x, y, z}, then equal numbers of permutations σ ∈ Γ_{x,y,z} have c_{i}^{σ}{u, v} = u and c_{i}^{σ}{u, v} = v, so they are tied. If u is among {x, y, z} and v is not, then there are some out-edges from u in Tor(c). Since c is balanced, there are an equal number of in-edges to u. For each i ∈ {1, 2, 3}, v will occupy the inedges which are not on the cycle in as many permutations of c_{i} as permutations where it occupies out-edges which are not on the cycle. If m = 3, then v occupies no edges on the cycle. Otherwise, v occupies an out-edge from u along the cycle only in

• c_{1}^{σ}, for some set of σ ∈ Γ_{x,y,z}, if u = z; then v occupies an in-edge to z on the cycle in c_{3} for an equal number of permutations.

• c_{2}^{σ}, for some set of σ ∈ Γ_{x,y,z}, if u = x; then v occupies an in-edge to x on the cycle in c_{1} for an equal number of permutations.

\begin{matrix}x_{0} \\ x_{1} \\ x_{2} \\ x_{0} \\ x_{2} \\ x_{3} \\ + \\ x_{0} \\ x_{1} \\ x_{2} \\ x_{3} \\ ↦ \\ x_{0} \\ x_{1} \\ x_{m−1} \\ x_{0} \\ x_{m−1} \\ x_{m} \\ + \\ x_{0} \\ x_{1} \\ x_{m−1} \\ x_{m} \\ ↦ \end{matrix}

Figure 2: Constructing a cycle from triangles, as in Claim 4.6.

• c_{3}^{σ}, for some set of σ ∈ Γ_{x,y,z}, if u = y; then v occupies an in-edge to y on the cycle in c_{2} for an equal number of permutations.

Thus u and v are tied. Otherwise, {u, v} ⊂ {x, y, z}. For each i ∈ {1, 2, 3}, there are |Γ_{x,y,z}| many c_{i}^{σ}. We have that c_{i}^{σ}{x, y} = x if i = 1 or i = 3; this is two-thirds of the voters, so x beats y. Similarly, c_{i}^{σ}{y, z} = y if i = 1 or i = 2, and c_{i}^{σ}{z, x} = z if i = 2 or i = 3. Thus the strict simple majority outcome is c^{x,y,z}.

The collection {c_{i}^{σ} : 1 ≤ i ≤ 3, σ ∈ Γ_{x,y,z}} is the voter population in the claim. There are no opposing votes between x, y, or z because such a vote would imply an edge between two vertices of the cycle which is not itself on the cycle, contradicting our choice of the smallest cycle.

Claim 4.6. If 𝒞 ⊆ ℭ and c^{x,y,z} ∈ maj-cl(𝒞) for any \{x, y, z\} ∈ \binom{X}{3}, then for any k ≥ 3 and \{x_{1}, …, x_{k}\} ∈ \binom{X}{k}, c^{x_{1},...,x_{k}} ∈ maj-cl(𝒞).

Proof. If k = 3, we have c^{x_{1},x_{2},x_{3}} ∈ maj-cl(𝒞) by assumption.

Suppose that 3 < k ≤ n and for any k−1 distinct candidates {x_{1}, …, x_{k−1}} ⊂ X we have c^{x_{1},...,x_{k−1}} ∈ maj-cl(𝒞). Say {c_{1}, …, c_{m_{0}}} is a set of m_{0} voters with c^{x_{1},...,x_{k−1}} as their strict simple majority outcome. Let

l_{0} = ∑_{i=1}^{m_{0}} W_{x_{1}}^{x_{k−1}}(c_{i}).

Then l_{0} > 0 since c^{x_{1},...,x_{k−1}}{x_{1}, x_{k−1}} = x_{k−1}. By hypothesis c^{x_{1},x_{k−1},x_{k}} ∈ maj-cl(𝒞). Say {c′_{1}, …, c′_{m_{1}}} is a set of m_{1} voters with c^{x_{1},x_{k−1},x_{k}} as their strict simple majority outcome. Let

l_{1} = ∑_{i=1}^{m_{1}} W_{x_{k−1}}^{x_{1}}(c′_{i}).

Like l_{0}, l_{1} is a positive integer. Let L = lcm(l_{0}, l_{1}), and take \frac{L}{l_{0}} copies of each c_{i}, and \frac{L}{l_{1}} copies of each c′_{i}, to make a finite set of voters \{d_{1}, …, d_{m}\} where m = \frac{L}{l_{0}} m_{0} + \frac{L}{l_{1}} m_{1}. Let d be their strict simple majority outcome. For any \{u, v\} ∈ \binom{X}{2}, if \{u, v\} ⊄ \{x_{1}, …, x_{k}\} then u ties v among the c_{i} and the c′_{i}, so {u, v} ∉ dom d. Any two points on the cycle (x_{1}, …, x_{k−1}) which are not adjacent are tied among the c_{i} and among the c′_{i}. Any point on the cycle besides x_{1} or x_{k−1} is also tied with x_{k} among the c_{i} and among the c′_{i}. Any pair of consecutive points (x_{i}, x_{j}) on the cycle, besides (x_{k−1}, x_{1}), are tied among the c′_{i}, but the majority of the c_{i} pick x_{i}. So d{x_{i}, x_{j}} = x_{i}. x_{k−1} and x_{k} tie among the c_{i}, but the majority of the c′_{i} pick x_{k−1}, so d{x_{k−1}, x_{k}} = x_{k−1}. Similarly d{x_{k}, x_{1}} = x_{k}. The only remaining pair to consider is {x_{1}, x_{k−1}}. d{x_{1}, x_{k−1}} is defined if and only if

∑_{i=1}^{m} W_{x_{k−1}}^{x_{1}}(d_{i}) ≠ 0.

The left hand side is equal to

\frac{L}{l_{0}} ∑_{i=1}^{m_{0}} W_{x_{k−1}}^{x_{1}}(c_{i}) + \frac{L}{l_{1}} ∑_{i=1}^{m_{1}} W_{x_{k−1}}^{x_{1}}(c′_{i})

= \frac{L}{l_{0}} (−l_{0}) + \frac{L}{l_{1}} (l_{1}) = 0.

Thus {x_{1}, x_{k−1}} ∉ dom d, and d{x_{i}, x_{j}} = x_{i} if and only if j ≡ i + 1 mod k, so d = c^{x_{1},...,x_{k}} ∈ maj-cl(𝒞).

By induction on k, any cyclic choice function on X is in maj-cl(𝒞).

Claim 4.7. If 𝒞 ⊂ ℭ is symmetric, balanced, and nontrivial, then every pseudobalanced choice function d ∈ ℭ is in maj-cl(𝒞).

Proof. Suppose d is an arbitrary pseudo-balanced choice function. Every edge of Tor(d) is on a directed cycle, so consider the decomposition of Tor(d) into cycles C_{x,y} for each edge x → y in Tor(d). Let d_{x,y} be the cyclic function corresponding to C_{x,y} for each edge x → y in Tor(d). By Claim 4.5, 𝒞 meets the requirements of Claim 4.6, so every cyclic function on X is in maj-cl(𝒞); in particular, each d_{x,y} ∈ maj-cl(𝒞). If we combine all the finite sets of voters which yielded each d_{x,y}, we get another finite set of voters from 𝒞, since there are finitely many edges in Tor(d).

For any \{u, v\} ∈ \binom{X}{2} \ dom d, the edge u → v is not in any cycle C_{x,y}. We have that ∑ W_{v}^{u}(c_{i}) = 0 over the c_{i} yielding any d_{x,y}, so ∑ W_{v}^{u}(c_{i}) = 0 over all our voters and u and v tie in their strict simple majority outcome.

For any \{u, v\} ∈ \binom{X}{2} with d\{u, v\} = u, the edge u → v is in Tor(d). Since each cycle C_{x,y} has edges only from Tor(d), each population of c_{i} yielding some d_{x,y} has ∑ W_{v}^{u}(c_{i}) ≥ 0, and in particular the population yielding d_{u,v} has ∑ W_{v}^{u}(c_{i}) > 0, so over all our voters ∑ W_{v}^{u}(c_{i}) > 0.

Thus the strict simple majority outcome is d, and d ∈ maj-cl(𝒞).

Combining Claims 4.4 and 4.7, we have that maj-cl(𝒞) is the set of all pseudo-balanced choice functions whenever 𝒞 is symmetric, balanced, and nontrivial.

5. Chaotic Choices

Definition 5.1. For a choice function c ∈ ℭ,

(a) For x ∈ X, the valence of x with c is

val_{c}(x) = ∑_{y∈X} W_{y}^{x}(c).

(b) For ℓ ∈ {−1, 0, 1}, let

V_{ℓ}(c) = \{(val_{c}(x) − ℓ, val_{c}(y) + ℓ)) : \{x, y\} ∈ \binom{X}{2}, W_{y}^{x}(c) = ℓ\}.

These are the valence pairs, with winners on the right in V_{−1}, ties in V_{0}, winners on the left in V_{1}, and with 1 subtracted from the valence of the winner. In fact,

V_{ℓ}(c) = \{ ∑_{z∈X\\{x,y\}} (W_{z}^{x}(c), W_{z}^{y}(c)) : \{x, y\} ∈ \binom{X}{2}, W_{y}^{x}(c) = ℓ\} .

(c) For a subset A of ℚ × ℚ, let conv(A) be the convex hull of A in ℚ × ℚ.

(d) Let

V^{∗}(c) = {av̄_{1} + (1 − a)v̄_{0} : v̄_{1} ∈ conv (V_{1}(c)) , v̄_{0} ∈ conv (V_{0}(c)) , a ∈ (0, 1]_{ℚ}}.

These are the convex hulls of the “winning” valence pairs, together with ties, requiring some contribution from a non-tied pair.

Claim 5.2. For each ℓ ∈ {−1, 0, 1}, (k_{0}, k_{1}) ∈ V_{ℓ}(c) ⟺ (k_{1}, k_{0}) ∈ V_{−ℓ}(c).

Proof. A pair (val_{c}(u) + 1, val_{c}(v) − 1) is in V_{−1}(c) if and only if W_{v}^{u}(c) = −1 ⟺ W_{u}^{v}(c) = 1, so (val_{c}(v) − 1, val_{c}(u) + 1) ∈ V_{1}(c).

A pair (val_{c}(u), val_{c}(v)) is in V_{0}(c) if and only if {u, v} = {v, u} ∉ dom c, so (val_{c}(v), val_{c}(u)) ∈ V_{0}(c).

Claim 5.3. An unbalanced c ∈ ℭ is partisan if and only if

• V_{1}(c) lies on a line parallel to the line y = x, and

• V_{0}(c) is contained in the line y = x.

Proof. Suppose that V_{0}(c) is all on y = x. Thus, candidates are only ever tied with others of the same valence. Suppose further that V_{1}(c) is on a line y = x−b. Then whenever c{w, v} = w, val_{c}(w)−val_{c}(v) = b+2 is constant; the valence of any winner is always b + 2 more than the defeated. Since c is unbalanced, there is such a pair {w, v}. If any candidate z has a valence different from that of w, it cannot be tied with w, so {w, z} ∈ dom c. Therefore the valences differ by b + 2, but if val_{c}(z) ≠ val_{c}(v), then z can neither tie nor be comparable to v, a contradiction. So every candidate has the valence of w or the valence of v. If two candidates with the same valence do not tie, then b = −2 and val_{c}(v) = val_{c}(w). In this case, every candidate has the same valence, which would mean that c is balanced; a contradiction. Furthermore, every candidate with the high valence must defeat everyone of the low valence, since they cannot be tied. This is the definition of a partisan function.

Conversely, suppose that c is partisan. Then two elements are tied only if they are both in the winning subset, or both in the losing subset; in either case, they have the same valence, so V_{0}(c) is contained in the line y = x. V_{1}(c) is the single point (val_{c}(w) − 1, val_{c}(v) + 1) for any w in the winning subset and v in the losing subset. Naturally, this point is contained in a line parallel to y = x.

Claim 5.4. If c ∈ ℭ is unbalanced, then a point of V^{∗}(c) lies above the line y = −x, and a point of V^{∗}(c) lies below it.

Proof. Let v_{1} and v_{2} be two candidates in X having the highest valences with c. Since c is unbalanced, val_{c}(v_{1}) > 0. The sum of the corresponding pair in any V_{ℓ}(c) is val_{c}(v_{1}) + val_{c}(v_{2}). If this is less than or equal to 0, then val_{c}(v_{2}) ≤ −1. Thus all other valences are at most −1, so the average valence is strictly smaller than the average of val_{c}(v_{1}) and val_{c}(v_{2}), at most 0. This is a contradiction, since the average valence is always 0. Therefore, the sum of the valence pair for v_{1} and v_{2} in any V_{ℓ}(c) satisfies y + x > 0. If the pair is in V_{1}(c), it is in V^{∗}(c). If it is in V_{−1}(c), then its reflection in V_{1}(c) (hence in V^{∗}(c)) still satisfies x + y > 0. If it is in V_{0}(c), then points arbitrarily close to the pair, along any line to a point of V_{1}(c), are in V^{∗}(c). Some of these points are above y = −x.

Let u_{1} and u_{2} be two candidates in X having the smallest valences with c. By imbalance, val_{c}(u_{1}) < 0. Suppose val_{c}(u_{1}) + val_{c}(u_{2}) ≥ 0. Then the average of these two valences is at least 0, and the average over all candidates is larger; it is strictly larger, since val_{c}(v_{1}) > 0 figures in the average. This is a contradiction, so the sum of the valence pair for u_{1} and u_{2} in any V_{ℓ}(c) satisfies y + x < 0. If this pair is in V_{−1}(c), then its reflection in V_{1}(c) still satisfies x + y < 0. Otherwise it is in V^{∗}(c), or arbitrarily close points are in V^{∗}(c).

To establish that every choice function is in the majority closure of a chaotic symmetric set, we shall first establish that some c therein satisfies a rather abstruse condition which we’ll call “valence-imbalance.”

y = x ; p̄_{0} ; p̄_{1} ; q̄_{0} ; q̄_{1} ; ū ; v̄ ; w̄ ; ū_{N} ; v̄_{N}

Figure 3: The situation of Claim 5.6. The points ū_{N} and v̄_{N} cannot be equidistant from w̄, since one lies within the dashed parallel lines, and one lies without.

Definition 5.5. A choice function c ∈ ℭ is valence-unbalanced if (0, 0) can be represented as r_{−1}v̄_{−1} + r_{0}v̄_{0} + r_{1}v̄_{1} where

(i) Each v̄_{ℓ} is a pair in conv (V_{ℓ}(c)), (ii) r_{−1}, r_{0}, r_{1} ∈ [0, 1]_{ℚ}, (iii) r_{−1} + r_{0} + r_{1} = 1, (iv) r_{−1} ≠ r_{1}.

Note that (0, 0) ∈ V^{∗}(c) implies that c is valence-unbalanced.

Claim 5.6. If ū is strictly between two points of conv (V_{1}(c)) on a line segment not parallel to y = x, then the nearest point on y = x to ū is of the form r_{−1}v̄_{−1} + r_{1}v̄_{1}, where v̄_{ℓ} ∈ conv (V_{ℓ}(c)), r_{ℓ} ∈ [0, 1]_{ℚ}, r_{−1} + r_{1} = 1, but r_{−1} ≠ r_{1}.

Proof. Say ū = (a, a + b). If b = 0, the assertion holds. Otherwise, let w̄ = (a + \frac{b}{2}, a + \frac{b}{2}); w̄ is the nearest point to ū on y = x.

There are p̄_{0} = (a_{0}, b_{0}) and p̄_{1} = (a_{1}, b_{1}) in conv (V_{1}(c)) so that ū is strictly between p̄_{0} and p̄_{1}. If the line segment Λ between p̄_{0} and p̄_{1} is perpendicular to y = x, then either w̄ is in Λ, and the assertion holds, or w̄ is between p̄_{1} and the reflection of p̄_{0}, q̄_{0} = (b_{0}, a_{0}). So for some r_{−1} and r_{1}, r_{−1}q̄_{0} + r_{1}p̄_{1} = w̄ = \frac{1}{2} q̄_{0} + \frac{1}{2} p̄_{0}. If r_{−1} = r_{1} = \frac{1}{2}, then p̄_{1} = p̄_{0}, contradicting that ū is strictly between them. So r_{−1} ≠ r_{1}.

If Λ is not perpendicular to the line y = x, then we may choose the points p̄_{0} and p̄_{1} so that a_{0} + b_{0} < 2a + b < a_{1} + b_{1}, and on the same side of y = x. The reflection of Λ over y = x lies on a line; call it L. Then L contains v̄ = (a+b, a), q̄_{0} = (b_{0}, a_{0}), and q̄_{1} = (b_{1}, a_{1}), all in conv (V_{−1}(c)); so v̄ lies strictly between q̄_{0} and q̄_{1} within conv (V_{−1}(c)).

Consider the sequence

ū_{n} = \frac{1}{n}p̄_{1} + (1 − \frac{1}{n}) ū,

for n ≥ 1, approaching ū as n → ∞. For each n, let v̄_{n} be the intersection of the line through ū_{n} and w̄ with the line L. (If necessary, define the v̄_{n} only for n > j, if the line from ū_{j} through w̄ is parallel to L; at most one such j can exist.) As n → ∞, v̄_{n} approaches v̄. So for some N ∈ ℕ and all n ≥ N, v̄_{n} is in the interval (q̄_{0}, q̄_{1}), an open neighborhood of v̄ in L. So

w̄ = r_{−1}ū_{N} + r_{1}v̄_{N},

for some r_{−1} and r_{1} in [0, 1]_{ℚ} with r_{−1} + r_{1} = 1.

If r_{−1} = r_{1} = \frac{1}{2}, then v̄_{N} = w̄ + (w̄ − ū_{N}) is just as far from the line y = x as ū_{N} = w̄ − (w̄ − ū_{N}). But notice, either

• both the interval (ū, p̄_{1}) on Λ and the interval (v̄, q̄_{1}) on L are farther from y = x than ū is, while the intervals (p̄_{0}, ū) and (q̄_{0}, v̄) are closer, or

• the intervals (ū, p̄_{1}) and (v̄, q̄_{1}) are closer to y = x than ū is, while the intervals (p̄_{0}, ū) and (q̄_{0}, v̄) are farther.

Then ū_{N} is in the half-plane y + x > 2a + b, and w̄ is the sole intersection of the line {ū_{N} + t(w̄ − ū_{N}) : t ∈ ℚ} with the line y + x = 2a + b. Thus v̄_{N} is in the half-plane y + x < 2a + b, hence in the interval (q̄_{0}, v̄) of L. But then the distance from y = x to ū is strictly between the distances from y = x to v̄_{N} and to ū_{N}, contradicting that these distances are equal.

So r_{−1} ≠ r_{1}.

Claim 5.7. If c is chaotic, then c is valence-unbalanced.

Proof. We consider five cases. Case 1. V_{0}(c) \ {(0, 0)} is not contained in y = x, nor in one open half-plane of y = −x.

By Claim 5.4, there is a point v̄ of V^{∗}(c) on the line y = −x. There is a point (k_{0}, −k_{0}) ∈ conv (V_{0}(c)), k_{0} ≠ 0. Either (0, 0) is between v̄ and (k_{0}, −k_{0}) or it is between v̄ and (−k_{0}, k_{0}); either way, it is in V^{∗}(c), so c is valence-unbalanced. Case 2. V_{0}(c) \ {(0, 0)} is not contained in y = x, but is contained in one open half-plane of y = −x.

There is (k_{0}, k_{1}) ∈ V_{0}(c) such that k_{0} ≠ k_{1} and k_{0} ≠ −k_{1}. By Claim 5.4, there is a point v̄ ∈ V_{1}(c) strictly on the other side of y = −x. If v̄ is on y = x, then (0, 0) is on the line segment between v̄ and w̄ = (\frac{k_{0}+k_{1}}{2} , \frac{k_{0}+k_{1}}{2} ), hence in V^{∗}(c), so c is valence-unbalanced. If v̄ is not on y = x, then it has a reflection ū ∈ V_{−1}(c). The set conv{v̄, ū, w̄} contains an open disc about (0, 0). Since there are points in conv (V_{0}(c)) arbitrarily close to w̄ on the line between (k_{0}, k_{1}) and (k_{1}, k_{0}), we may choose (k′_{0}, k′_{1}) so k′_{0} ≠ k′_{1} and (0, 0) ∈ conv{v̄, ū, (k′_{0}, k′_{1})}.

Suppose c is valence-balanced. Say v̄ is (v_{0}, v_{1}). Then

(0, 0) = rv̄ + (1 − 2r)(k′_{0}, k′_{1}) + rū = r(v_{0} + v_{1}, v_{1} + v_{0}) + (1 − 2r)(k′_{0}, k′_{1}),

so k′_{0} = \frac{−r(v_{0}+v_{1})}{1−2r} = k′_{1}, a contradiction. Therefore c is valence-unbalanced.

Case 3. V_{0}(c) is contained in y = x, and V_{1}(c) is contained in a line parallel to y = x.

Then c is partisan by Claim 5.3, contradicting that c is chaotic. Case 4. V_{0}(c) is contained in y = x, conv (V_{1}(c)) contains a line segment not parallel to y = x, and V_{1}(c) has points on either side of y = −x.

Then there is a point of y = −x strictly between two points of conv (V_{1}(c)) such that the line segment between them is not parallel to y = x. By Claim 5.6, (0, 0) = r_{−1}v̄_{−1} + r_{1}v̄_{1} where v̄_{ℓ} ∈ conv (V_{ℓ}(c)), r_{ℓ} ∈ [0, 1]_{ℚ}, r_{−1} + r_{1} = 1, but r_{−1} ≠ r_{1}, i.e. c is valence-unbalanced. Case 5. V_{0}(c) is contained in y = x, conv (V_{1}(c)) contains a line segment not parallel to y = x, and V_{1}(c) is entirely on one side of y = −x.

Let v̄ be strictly between two points of V_{1}(c) on a line segment not parallel to y = x. By Claim 5.6, the nearest point w̄ to v̄ on the line y = x is r_{−1}v̄_{−1} +r_{1}v̄_{1} where v̄_{ℓ} ∈ conv (V_{ℓ}(c)), r_{ℓ} ∈ [0, 1]_{ℚ}, r_{−1} + r_{1} = 1, but r_{−1} ≠ r_{1}. By Claim 5.4, V_{0}(c) has a point v̄_{0} = (k, k) strictly on the other side of y = −x from v̄, and hence from w̄. So for some r ∈ [0, 1]_{ℚ},

(0, 0) = rv̄_{0} + (1 − r)(r_{−1}v̄_{−1} + r_{1}v̄_{1}) = (1 − r)r_{−1}v̄_{−1} + rv̄_{0} + (1 − r)r_{1}v̄_{1},

and (1 − r)r_{−1} ≠ (1 − r)r_{1}, so c is valence-unbalanced.

Claim 5.8. If 𝒞 ⊆ ℭ is symmetric and c ∈ 𝒞 is valence-unbalanced, then for each \{x, y\} ∈ \binom{X}{2} there is d ∈ maj-cl(𝒞) such that dom d = \{\{x, y\}\}.

Proof. Suppose \{x, y\} ∈ \binom{X}{2}. There are v̄_{ℓ} ∈ conv (V_{ℓ}(c)) and r_{ℓ} ∈ [0, 1]_{ℚ}, r_{−1} + r_{0} + r_{1} = 1, r_{−1} ≠ r_{1}, so r_{−1}v̄_{−1} + r_{0}v̄_{0} + r_{1}v̄_{1} = (0, 0). Now,

v̄_{−1} = ∑_{v̄∈V_{−1}(c)} s−1 ; v̄v̄, v̄_{0} = ∑_{v̄∈V_{0}(c)} s0 ; v̄v̄, v̄_{1} = ∑_{v̄∈V_{1}(c)} s1 ; v̄v̄,

where ∑_{v̄∈V_{ℓ}(c)} sℓ ; v̄ = 1 for each ℓ. Each v̄ ∈ V_{ℓ}(c) is (val_{c}(uℓ ; v̄) − ℓ, val_{c}(wℓ ; v̄) + ℓ) for some \{uℓ ; v̄, wℓ ; v̄\} ∈ \binom{X}{2}. Let σℓ ; v̄ be a permutation taking uℓ ; v̄ to x and wℓ ; v̄ to y, i.e.

σℓ ; v̄ = \begin{cases}(x, y) & if uℓ ; v̄ = y, wℓ ; v̄ = x, \\ (x, wℓ ; v̄, y) & if uℓ ; v̄ = y, wℓ ; v̄ ≠ x, \\ (uℓ ; v̄, x)(wℓ ; v̄, y) & otherwise. \end{cases}

Let Γ_{x,y} be all the permutations of X which fix x and y, and let

t̄= \frac{1}{|Γ_{x,y}|} ∑_{ℓ∈\{−1,0,1\}} r_{ℓ} ∑_{v̄∈V_{ℓ}(c)} sℓ ; v̄ ∑_{τ∈Γ_{x,y}} pr (c^{τσℓ ; v̄}) .

Of course, the sum of the coefficients is

∑_{\begin{matrix}ℓ∈\{−1,0,1\} \\ v̄∈V_{ℓ}(c) \\ τ∈Γ_{x,y} \end{matrix}} \frac{r_{ℓ}sℓ ; v̄}{|Γ_{x,y}|} = \frac{|Γ_{x,y}|}{|Γ_{x,y}|} ∑_{ℓ∈\{−1,0,1\}} r_{ℓ} = 1.

Since r_{ℓ}, sℓ ; v̄, and \frac{1}{|Γ_{x,y}|} are in [0, 1]_{ℚ}, \frac{r_{ℓ}sℓ ; v̄}{|Γ_{x,y}|} ∈ [0, 1]_{ℚ}. So t̄∈ pr-cl(𝒞). Consider t_{x,y}. Each τ fixes x and y, so

c^{τσℓ ; v̄}{x, y} = x ⟺
c^{σℓ ; v̄}{x, y} = x ⟺
c{uℓ ; v̄, wℓ ; v̄} = uℓ ; v̄ ⟺ ℓ = 1.

So W_{y}^{x}(c^{τσℓ ; v̄}) = ℓ. Thus

t_{x,y} = \frac{1}{|Γ_{x,y}|} ∑_{ℓ∈\{−1,0,1\}} r_{ℓ} ∑_{v̄∈V_{ℓ}(c)} sℓ ; v̄ ∑_{τ∈Γ_{x,y}} ℓ

= \frac{|Γ_{x,y}|}{|Γ_{x,y}|} ∑_{ℓ∈\{−1,0,1\}} ℓ r_{ℓ} = r_{1} − r_{−1}.

Since r_{−1} ≠ r_{1}, t_{x,y} ≠ 0.

Consider t_{x,u} where u ∉ {x, y}.

c^{τσℓ ; v̄}{x, u} = x ⟺
c^{σℓ ; v̄}{x, τ^{−1}(u)} = x ⟺
c{uℓ ; v̄, u^{∗}} = uℓ ; v̄,

where

u^{∗} = (σℓ ; v̄)^{−1}(τ^{−1}(u)) = \begin{cases}τ^{−1}(u) & if τ^{−1}(u) ∉ \{uℓ ; v̄, wℓ ; v̄\}, \\ x & if τ^{−1}(u) = uℓ ; v̄, \\ y & if τ^{−1}(u) = wℓ ; v̄. \end{cases}

u^{∗} is never uℓ ; v̄ or wℓ ; v̄. For a given ℓ and v̄, u^{∗} varies equally over all elements of

X besides {uℓ ; v̄, wℓ ; v̄}, so

∑_{τ∈Γ_{x,y}} W_{u}^{x}(c^{τσℓ ; v̄}) = ∑_{τ∈Γ_{x,y}} W_{u^{∗}}^{uℓ ; v̄}(c) = ∑_{τ∈Γ_{x,y,u}} ∑_{u^{∗}∈X\{uℓ ; v̄,wℓ ; v̄}} W_{u^{∗}}^{uℓ ; v̄}(c)

= |Γ_{x,y,u}| (val_{c}(uℓ ; v̄) − W_{uℓ ; v̄}^{uℓ ; v̄}(c) − W_{wℓ ; v̄}^{uℓ ; v̄}(c)) = |Γ_{x,y,u}| (val_{c}(uℓ ; v̄) − ℓ) .

This follows from the fact that, for each u^{∗} ∈ X \ {uℓ ; v̄wℓ ; v̄}, there are |Γ_{x,y,u}| permutations τ which take u to u^{∗}. For any u, W_{u}^{u}(c) = 0, and by choice of uℓ ; v̄

and wℓ ; v̄, W_{wℓ ; v̄}^{uℓ ; v̄}(c) = ℓ.

Recall that the uℓ ; v̄ were chosen so that val_{c}(uℓ ; v̄) − ℓ is the first coordinate of v̄. Recall also that the r_{ℓ}, sℓ ; v̄, and valence pairs v̄ were chosen so

∑_{ℓ∈{−1,0,1}} r_{ℓ} ∑_{v̄∈V_{ℓ}(c)} sℓ ; v̄v̄ = (0, 0).

t_{x,u} = \frac{1}{|Γ_{x,y}|} ∑_{ℓ∈\{−1,0,1\}} r_{ℓ} ∑_{v̄∈V_{ℓ}(c)} sℓ ; v̄ ∑_{τ∈Γ_{x,y}} W_{u}^{x} (c^{τσℓ ; v̄})

= \frac{1}{|Γ_{x,y}|} ∑_{ℓ∈\{−1,0,1\}} r_{ℓ} ∑_{v̄∈V_{ℓ}(c)} sℓ ; v̄ |Γ_{x,y,u}| (val_{c}(uℓ ; v̄) − ℓ)

= 0.

Similarly, t_{y,u} = 0 for all u ∉ {x, y}.

Consider t_{u,w} with neither u nor w in {x, y}. Then

c^{τσℓ ; v̄}{u, w} = u ⟺
c^{σℓ ; v̄}{τ^{−1}(u), τ^{−1}(w)} = τ^{−1}(u) ⟺
c{u^{∗}, w^{∗}} = u^{∗},

where u^{∗} = (σℓ ; v̄)^{−1}(τ^{−1}(u)) and w^{∗} = (σℓ ; v̄)^{−1}(τ^{−1}(w)). But τ^{−1}(u) and τ^{−1}(w) are never x or y, so u^{∗} is never uℓ ; v̄ and w^{∗} is never wℓ ; v̄. As τ varies, u^{∗} and w^{∗} vary equally over all members of X except uℓ ; v̄ and wℓ ; v̄, so c{u^{∗}, w^{∗}} = u^{∗} just as often as c{u^{∗}, w^{∗}} = w^{∗}. Thus

∑_{τ∈Γ_{x,y}} W_{w}^{u}(c^{τσℓ ; v̄}) = 0.

Hence

t_{u,v} = \frac{1}{|Γ_{x,y}|} ∑_{ℓ∈\{−1,0,1\}} r_{ℓ} ∑_{v̄∈V_{ℓ}(c)} sℓ ; v̄ ∑_{τ∈Γ_{x,y}} W_{w}^{u}(c^{τσℓ ; v̄}) = 0.

So d = maj(t̄) ∈ maj-cl(𝒞) has dom d = {{x, y}}; all other pairs have t_{u,v} = 0.

Claim 5.9. If 𝒞 ⊆ ℭ is symmetric and chaotic, then maj-cl(𝒞) = ℭ.

Proof. Suppose f ∈ ℭ.

By Claim 5.7, there is some c ∈ 𝒞 which is valence-unbalanced. So by Claim 5.8, for any \{x, y\} ∈ \binom{X}{2} there is d ∈ maj-cl(𝒞) with dom d = \{\{x, y\}\}. If d{x, y} = y, by symmetry there is d′ ∈ maj-cl(𝒞) with domain {{x, y}} and d′{x, y} = x. So for every {x, y} ∈ dom f, let d_{x,y} ∈ maj-cl(𝒞) such that dom d_{x,y} = {{x, y}} and d_{x,y}{x, y} = f{x, y}. Now combine the voter populations which yielded each d_{x,y}.

Since f was arbitrary, we have shown ℭ ⊆ maj-cl(𝒞). Hence maj-cl(𝒞) is all of ℭ.

6. Bounds

We shall consider the loose upper bounds, implied by the preceding proofs, on the necessary number of voters from a symmetric set to yield an arbitrary function in its majority closure, in terms of n. There is no doubt that much tighter bounds could be obtained.

Trivial 0 functions suffice to obtain no outcome.

Balanced

In order to create a triangular function from a chosen function c, (n − 3)! permutations of the rest of the set were used. These were repeated 3 times, to establish each pair of edges in the triangle.

An arbitrary pseudo-balanced function has each edge in a directed cycle, so one cycle per edge suffices; so we have at most \binom{n}{2} = \frac{n(n−1)}{2} cycles. Each is constructed with two fewer triangles than the number of nodes it contains. This is at most n − 2, if some cycle contains every node. So all together we have

\frac{1}{2}n(n − 1)(n − 2)(n − 3)! 3 = \frac{3n!}{2}.

Partisan

There are at most n tiers in an arbitrary tiered function, each with at most n elements (of course, no function meets both conditions.) For each such element x, we take functions for each permutation holding x fixed; there are (n − 1)! such. So we have n n(n − 1)! = n n! as an upper bound.

Not balanced, not partisan, but not chaotic

The proof of Claim 4.5 gives the size of the population T_{z} yielding c^{a,b,z} as |T_{z}| = 3(n − 3)!, with m = 2(n − 3)! votes for the winner of each pair. We chose a partisan function c with l candidates in the winning tier. For each of at most \binom{n}{2} edges a → b and each of the (n − 2) z ∈ X \ \{a, b\}, we used k copies of T_{z}, where

k = \begin{cases}(n − 2)\binom{n−3}{l−1} & if l ≤ \frac{n}{2}, \\ (n − 2)\binom{n−3}{l−2} & if l > \frac{n}{2}, \end{cases}

and m copies of a population of partisan functions D, where

|D| = \begin{cases}(n − 2l)\binom{n−1}{l} + (n − 2)\binom{n−2}{l−1} & if l ≤ \frac{n}{2}, \\ (2l − n)\binom{n−1}{l−1} + (n − 2)\binom{n−2}{l−1} & if l > \frac{n}{2}. \end{cases}

In fact, k′ = \frac{k}{gcd(m,k)} copies of T_{z}, and m′ = \frac{m}{gcd(m,k)} copies of D, would suffice.

Certainly \binom{n−3}{l−1} = \frac{(n−3)!}{(l−1)!(n−l−2)!} | (n − 3)!, and \binom{n−3}{l−2} = \frac{(n−3)!}{(l−2)!(n−l−1)!} | (n − 3)!, so we may take k′ = (n − 2).

If l ≤ \frac{n}{2}, then we take m′ = 2(l − 1)!(n − l − 2)!, and

m′ |D| = 2(l − 1)!(n − l − 2)! ((n − 2l)\binom{n − 1}{l} + (n − 2)\binom{n − 2}{l − 1})

= 2(l − 1)!(n − l − 2)! (\frac{(n − 2l)(n − 1)!}{l!(n − l − 1)!} + \frac{(n − 2)(n − 2)!}{(l − 1)!(n − l − 1)!})

= 2(n − 2)! (\frac{(n − 2l)(n − 1)}{l(n − l − 1)} + \frac{n − 2}{n − l − 1}) .

The last multiplicand is

\frac{n(n − l − 1)}{l(n − l − 1)} = \frac{n}{l} ≤ n.

If l > \frac{n}{2}, then we take m′ = 2(l − 2)!(n − l − 1)!, and

m′ |D| = 2(l − 2)!(n − l − 1)! ((2l − n)\binom{n − 1}{l − 1} + (n − 2)\binom{n − 2}{l − 1})

= 2(l − 2)!(n − l − 1)! (\frac{(2l − n)(n − 1)!}{(l − 1)!(n − l)!} + \frac{(n − 2)(n − 2)!}{(l − 1)!(n − l − 1)!})

= 2(n − 2)! (\frac{(2l − n)(n − 1)}{(l − 1)(n − l)} + \frac{n − 2}{l − 1}) .

The last multiplicand is

\frac{n(l − 1)}{(l − 1)(n − l)} = \frac{n}{n − l} ≤ n.

So the total number of voters is

\binom{n}{2} ((n − 2) |T_{z}| k′ + m′ |D|)

≤ \frac{n(n − 1)}{2} ((n − 2)3(n − 3)!(n − 2) + 2(n − 2)!n)

= n(n − 1)(n − 2)! (\frac{3}{2}(n − 2) + n)

= n! (\frac{3}{2}n − 3 + n) < \frac{5}{2}n n!.

Chaotic

Given an arbitrary function, we wish to create a “single-edged” d_{x,y}, as developed in Section 5, for each edge in the function; there are at most \binom{n}{2} of these. In creating such a single-edged d_{x,y}, we use a collection of permutations of a choice function, under permutations which take x and y to elements having certain valence combinations. Reading the proof of Claim 5.7 carefully, we see that at most 4 such valence pairs are used. For each of these, all permutations fixing the relevant pair are used, giving 4(n − 2)! choice functions used in the creation of the single-edged function. However, these are being combined by some set of rational coefficients, so in fact we must expand each such function taking as many copies as the numerator of its associated coefficient when put over the least common denominator L. The coefficients sum to 1, so the sum of the numbers of copies is L; so L(n − 2)! functions are used for each edge.

Six possible linear combinations of valence pairs yielding (0, 0) were considered in Claim 5.7. Using the fact that the original valence pairs are pairs of integers between −n and n, we can solve for the coefficients by matrix inversion and determine that the least common denominator is at most the determinant of the associated matrix. For instance, in Case 4, suppose there is a point of y = −x strictly between two points of V_{1}(c) on a line segment not parallel to y = x. (This is true in this Case, unless V_{1}(c) is contained in a line parallel to y = x except for a point on the line y = −x; such a situation actually has a smaller common denominator.)

We use Claim 5.6, with (0, 0) for w̄, and note that either the line from p̄_{0} through w̄ meets the line L, or the line from p̄_{1} through w̄ does. Say p̄ is the one which works. Then (0, 0) is an unbalanced linear combination of the three valence pairs p̄, q̄_{0}, and q̄_{1}. Say p̄ = (a, b); then one of the q̄ is (b, a) and the other is (c, d). Solving

r + s + t = 1
ar + bs + ct = 0
br + as + dt = 0

amounts to

[ \begin{matrix}r \\ s \\ t \end{matrix} ] = [ \begin{matrix}1 & 1 & 1 \\ a & b & c \\ b & a & d \end{matrix} ]^{−1} [ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] .

Since everything on the right hand side is an integer, the only denominator introduced is the determinant of the 3 by 3 matrix, bd − ac − ad + bc + a^{2} − b^{2}. Since each of a, b, c, d is an integer between −n and n, this is at most 6n^{2}.

In the cases involving four points, such as Case 1 or Case 5, an additional constraint is needed, which is not a consequence of the other three. In Case 1, use r_{1}+r_{2}+r_{3}+r_{4} = 1, a_{1}r_{1}+a_{2}r_{2}+a_{3}r_{3}+a_{4}r_{4} = 0, b_{1}r_{1}+b_{2}r_{2}+b_{3}r_{3}+b_{4}r_{4} = 0, and add b_{3}r_{3} +b_{4}r_{4} = −a_{3}r_{3} −a_{4}r_{4}, since the points v̄ and (k_{0}, −k_{0}) were linear combinations of two valence pairs on the line y = −x. The lowest common denominator is at most the determinant of

[ \begin{matrix}1 & 1 & 1 & 1 \\ a_{1} & a_{2} & a_{3} & a_{4} \\ b_{1} & b_{2} & b_{3} & b_{4} \\ 0 & 0 & a_{3} + b_{3} & a_{4} + b_{4} \end{matrix} ] ,

which expands to a 16-term sum of three a_{i} or b_{i} each, hence at most 16n^{3}. In Case 5, note that the point v̄ between two points of V_{1}(c) is arbitrary; we can choose any point between the two. Between any two pairs of integers on which do not lie on a line perpendicular to y = x, we can choose a point whose projection on y = x is of the form (\frac{l}{4}, \frac{l}{4}) for some integer l. Solving

[ \begin{matrix}1 & 1 & 1 & 1 \\ k & a & b & c \\ k & b & a & d \\ 0 & a & b & c \end{matrix} ]^{−1} [ 1 ; 0 ; 0 ; \frac{l}{4} ]

gives a determinant k(2a^{2} − b^{2} − ab − ac + bd + bc − ad) at most 8n^{3}; we must multiply by 4 because of the fraction in the column vector, giving 32n^{3}. This is the largest denominator of the six possible linear combinations in Claim 5.7.

Thus our upper bound is

L(n − 2)!\frac{n^{2} − n}{2} < 16 n^{3}n(n − 1)(n − 2)! = 16 n^{3}n!.

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