Anonymous Voting Rules

We consider finite setsN={1, . . . ,n}ofnplayers or voters such that each voteri∈N has strict preferencesPiover a setA={a1, . . . ,am}ofm≥2 alternatives. We will write abcin abbreviation of aPibPic. The set of all m! strict preference orderings on A is denoted byP(A). A(resolute) voting rule r: P(A)ⁿ→ Amaps each preference profile P=(P1, . . . ,Pn) to a single winning alternativea^∗=r(P).

Throughout our analysisrwill be defined by truthful voting.³ We will consider the following five anonymous rules that treat all votersi∈Nsymmetrically:⁴

3Our assumption of truthful voting might, e.g., be justified by players being political agents to principals (constituents) that value truth telling. There is also experimental evidence that players often lack or ignore the information required to make manipulation profitable (cf. Kube and Puppe 2009). Even if they are aware of the other voters’ preferences, they might face a hard (NP-complete) problem in identifying profitable deviations. See Nurmi (2016).

4Formally, constructP⁰by applying a permutationπ:N →NtoP, so thatP⁰ =(P_π(1), . . . ,P_π(n)).

Thenrisanonymousif for all suchP,P⁰the winning alternativea^∗=r(P)=r(P⁰) is the same.

5.2. Preliminaries 99 Plurality

The most simple voting method is plurality rule r^P: each voter endorses his or her top-ranked alternative; then the alternative which is ranked first by the most voters is chosen. That is,a^∗=r^P(P) implies

Similarly, plurality runoff rule r^PR asks each voter to name his or her top ranked alternative. If an alternative is ranked first by more than half of the voters, it is chosen. Otherwise a runoff is held between the alternatives a(1) and a(2) which are lexicographically minimal among the alternatives that received the (second-)most votes. Formally, conceive ofa(1),a(2), . . . ,a(m)as a permutation of the alternatives inA such that they are ordered according to decreasing numbers of votes and, in case of ties, in decreasing lexicographic order. Pi. Then it selects the alternative with the highest total number of points (known as Borda score). Formally, let

be the number of alternatives ranked below a according to i’s preferences. Then a^∗=r^B(P) implies

Copeland

The fourth benchmark isCopeland rule r^C. Pairwise majority votes are held between all alternatives; the alternative that beats the most others is selected. Formally, let the majority relation^P

Ruler^Cis a Condorcet method: if some alternativeais a Condorcet winner, i.e., beats all others in pairwise majority comparisons, thenr^C(P)=a.

Schulze

A fairly recent voting method isSchulze rule r^S, as introduced by Schulze (2011). It is also a Condorcet method and based on indirect comparisons of alternativesai andaj

by defining apatha_i^→_j:=a(1), . . . ,a(k) fromai toaj with the following properties:

1. a(1)=ai

2. a(k)=aj

3. For alli=1, . . . ,(k−1) : a(i)^P

M a(i+1)

Every alternativea(i) in the sequencea(1), . . . ,a(k) wins the binary comparison against its successora(i+1). Thestrength s(ai→j) of a patha_i→j =a(1), . . . ,a(k) is defined as the

. Thestrength of the strongest pathfrom aitoaj is then

p[ai,aj] :=maxn

s(ai→j)|a_i→jis a path fromai toaj

o.

That is, one calculates the strength of every possible path fromaitoajand then selects the path with the highest strength. If there is no path fromai toaj,p[ai,aj] :=0.

5.2. Preliminaries 101 Thus,a^∗=r^S(P) implies

a^∗ ∈n

a∈A|p[a,a⁰]≥p[a⁰,a]∀a⁰ ,a∈Ao

. (5.5)

Several private organizations adopted the Schulze rule for internal elections in recent years. Among them are the Pirate Parties of Austria, Germany and Sweden, Debian, Ubunto and the Wikimedia Foundation. We remark that the Schulze method is closely related to theSimpson-Kramer rule, which is also known asmaximin rule.⁵

We assume that whenever there is a non-singleton setA^∗={a^∗_i1, . . . ,a^∗_i

Real committees are more likely than not to involve a non-anonymous voting rule.

This can be because designated members like a chairperson or agenda-setter have veto rights or procedural privileges. Or an anonymous decision rule rapplies not at the voter level but at the level of their respective votes, shareholdings, etc. We could also think of the relevant playersi ∈ N in a committee to be differently sized but well-disciplined parties or interest groups with homogeneous preferences. It is then obvious that the introduction of voting weights implies non-anonymity at the level of voter blocks – even if the utilized voting rule is anonymous at the level of individual voters.

Single voters who differ in their numbers of votes and homogeneous groups of voters are formally equivalent: the committee’s voting rule amounts to the combina-tion of an anonymous baseline voting rule r with voting weights w1, . . . ,wn associated to the relevant players.

Following Chapter 4, we conceive of a ruleras representing the entire associated family of mappings from n-tuples of strict preferences over A = {a1, . . . ,am} to a winnera^∗∈Afor arbitraryn. Then, the indicated combination is simply areplication operation. It defines the weighted voting ruler|w: P(A)^wΣ →Aby

5Form=3, the Schulze winner and the Simpson-Kramer winner are identical; for largermthey coincide in about 99 % of all instances. These small differences notwithstanding, the Schulze rule fixes the most common flaws of the Simpson-Kramer rule. See Schulze (2011) for an elaborate discussion.

for a given anonymous rule r and a non-negative, non-degenerate weight vector w=(w1, . . . ,wn)∈Nⁿ₀ withw_Σ :=Pn

i=1wi >0. Moreover, we assume thatr|0(P) := a1

in the degenerate casew=(0, . . . ,0).

The combination (N,A,r|w) of a set of voters, a set of alternatives and a particular weighted voting rule is called aweighted committee game. Such games were introduced and their structure was extensively studied in Chapter 4.

When the underlying anonymous rule is plurality rule r^P, we call (N,A,r^P|w) a(weighted) plurality committee. Similarly, (N,A,r^PR|w), (N,A,r^B|w), (N,A,r^C|w) and (N,A,r^S|w) are respectively referred to as aplurality runoffcommittee,Borda committee, Copeland committee, andSchulze committee.

Weighted representations of given committee games are far from unique. For instance consider the weighted plurality committee (N,A,r^P|w). Since having more than half of the total voting weight renders playerja dictator, all weighted committee games (N,A,r^P|w) with w ∈ E = n P∈ P(A)ⁿ. Chapter 4 investigates situations where r =r⁰ and tries to capture struc-tural equivalencein the general sense that (N,A,r|w) and (N⁰,A⁰,r|w⁰)reflect the same decision environmenteven though weights and labels of players or alternatives may differ.⁶ For instance, it turns out that whenn=3 voter groups decide onm=3 alter-natives with Borda rule, one can distinguish 51 equivalence classes of committees, i.e., families of weights or group sizes that induce the same mapping from preference profiles to outcomes; in contrast, there are only 4 classes for binary voting. There are also just 4 with Copeland rule form = 3: either the largest group has dictatorial power, the largest two share power equally, or all three share power either equally or with an advantage to the largest.

We here takeN, Aand the distribution of voting weightswas given and seek to identify the implications of choosingr ∈ {r^P,r^PR,r^B,r^C,r^S}on players’ possibilities to influence the outcome. To do so, we first try to quantify influence on the outcome in a suitable way.

6In particular this means that, e.g., the three games (N,A,r|w) withw∈ n

(6,5,3),(5,3,6),(3,6,5)o are treated as equivalent. Although they obviously differ from the perspectives of players 1, 2 and 3, they do not from the perspective of the large, medium and small player, i.e., the decision environment stays the same.