Expressive voting, graded interests and participation

Klein, Dominik Article — Published Version Expressive voting, graded interests and participation Public Choice Provided in Cooperation with: Springer Nature Suggested Citation: Klein, Dominik (2020) : Expressive voting, graded interests and participation, Public Choice, ISSN 1573-7101, Springer US, New York, NY, Vol. 188, Iss. 1-2, pp. 221-239, https://doi.org/10.1007/s11127-020-00825-2 This Version is available at: https://hdl.handle.net/10419/288793 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Public Choice (2021) 188:221–239 https://doi.org/10.1007/s11127-020-00825-2 1 3 Expressive voting, graded interests andparticipation DominikKlein1,2 Received: 11 November 2019 / Accepted: 23 May 2020 / Published online: 16 June 2020 © The Author(s) 2020 Abstract I assume that voters mark ballots exclusively to express their true preferences among par‑ ties, leaving aside any considerations about an election’s possible outcome. The paper then analyzes the resulting voting behavior.In particular, it studies how effective different voting systems such as plurality rule, approval voting, and range voting are in fostering high turn‑ out rates of such expressive voters. Keywords Expressive voting· Voting theory· Approval voting· Range voting· Issue voting· Spatial voting· Directional voting· Proximity voting 1 Introduction In reasoning about elections, much analysis builds on a simple question: why do people vote the way they do? A major received answer to that question invokes the election’s out‑ come. Briefly put, the instrumental account of voting stipulates that rational voters cast their votes in order to render the election’s expected outcome as preferable as possible. Such instrumental analysis of voting has been employed successfully to explain a number of observed patterns, for instance concerning the expected number of parties in a given political system (Duverger 1959). However, the account also has some shortcomings. In particular, as Downs (1957) observed, it cannot satisfactorily answer why voters would participate in large scale elections at all.1 A second framework fills that lacuna. Often, voting behavior is not guided by the elec‑ tion’s possible outcome. Rather, we may choose a certain option because we judge it ethi‑ cally correct, attractive, fair, in line with our general political convictions, or simply to cohere with our values and preferences. In short, in the perspective of expressive voting, utility is derived directly from the act of expressing one’s preferences, rather than from any * Dominik Klein dominik.klein@uni‑bamberg.de 1 Philosophy Institute, University ofBayreuth, Universitätsstraße 30, 95447Bayreuth, Germany 2 Institute forPolitical Science, University ofBamberg, Feldkirchenstraße 21, 96052Bamberg, Germany 1 The argument, briefly, goes as follows. The chances of making a difference in the outcome of a large scale election are miniscule. Hence, the expected benefit of casting one’s vote will be smaller than that of many other thing one could do at the same time. 222 Public Choice (2021) 188:221–239 1 3 considerations about possible outcomes (Brams and Fishburn 1978, 2007; Brennan and Lomasky 1993). In the present paper, we explore some consequences of the expressive take on voting, mainly in relation to the phenomenon of abstentions. Within expressive voting, abstentions are not explained by the cost of voting. Rather, abstentions occur if an empty ballot is the best available expression of a voter’s preferences. We compare three voting systems, plu‑ rality rule, approval voting, and range voting, with respect to their propensity for creating high voter turnout. In what follows, we will build a formal model of expressive voting, within which we assume that voters’ preferences range over an agenda of issues. Parties then will be evalu‑ ated by their attitudes towards the different agenda items. A similar, topic‑based perspec‑ tive on voting has been assumed in two recent papers by Aragones etal. (2011) and Dean and Parikh (2011). Within our discussion, we make explicit how those frameworks fit within the current approach. The first of them, the paper by Aragones etal. (2011), is related to our theme of voter participation. It addresses the propensities for various voting systems to promote high turn‑ outs among expressive voters. We are sympathetic to their general approach and adopt a similar underlying framework. However, we raise various criticisms of their conceptualiza‑ tion of approval voting and the resulting comparison of voting systems. Responding to that criticism, we offer an alternative conceptualization of approval voting and then use it to compare expected voter participation within different voting systems. The remainder of this paper is structured as follows. In Sect.2, we introduce our gen‑ eral model of expressive voting over an agenda consisting of various items. In Sect.3, we then present Aragones etal. (2011)’s analysis of approval voting, before raising a criticism of that framework in Sect.4. We then proceed to introduce our own conceptualization of voters’ behavior under three voting methods, plurality, approval and range voting (Sect.5), and compare them with respect to expected voter turnouts (Sect.6). All proofs and calcula‑ tions are provided in the “Appendix”. 2 The model In this section we present our basic electoral model. The central object of concern is a finite agenda of topics or issues for the upcoming election, denoted by A={1…n} . We assume the agenda items to be propositions that the individual parties and candidates can either endorse or oppose. Additionaly, we fix a set C of parties or candidates. Each voter is represented by a vector 𝐯∈ [−1, 1]n , representing her positions on the various topics. The intended reading is that vi∈ [−1, 1] , the i‑th entry of 𝐯 , is the degree to which voter 𝐯 supports the proposition underlying issue i, where +1 stands for total sup‑ port and −1 for total opposition to the statement in question. Notably, we allow voters to have positions anywhere in [−1, 1] in order to allow for uncertainty about the right course of action, or to mirror graded degrees of interest in the different topics. The only case we exclude are universally disinterested voters. Thus, we assume that 𝐯≠𝟎 . As with voters, each party is characterized by its positions on the various issues of con‑ cern A={1, …,n} . Unlike voters, though, we assume parties to have extremal positions on each topic. That is, they are represented by vectors in {−1, 1}n . Briefly, that assumption has two different justifications. Firstly, parties are identified with the policies they would implement if elected. We assumed the individual agenda items to be propositional, thus 223 Public Choice (2021) 188:221–239 1 3 they only can be implemented or not implemented. Of course, a party could break some of its promises and act differently to what they claimed prior to Election Day. Yet, any policy choice will either implement some agenda item i or not. Thus, no space for graded judg‑ ments is available, but parties eventually will have to decide for or against implementing any particular item on the agenda. Secondly, Aragones etal. (2011) argue that political discourse moves parties to extreme positions. In an attempt to position themselves on the political scale and to stand out from their opponents, they ultimately will have to take a clear position on each topic. For notational convenience, we will use the letter 𝐩 in different forms to denote parties or candidates, whereas 𝐯 and all of its variants denotes voters. Bold letters always refer to vectors, while their entries are denoted by italics, for example 𝐯=⟨v1…vn⟩ . Finally, we define the three voting systems we study: plurality rule, approval voting and range voting. In line with the underlying intuitions of expressive voting, our main empha‑ sis lies on the possible ballots a voter can choose amongst, rather than the outcome of an election. Under plurality rule, each voter can vote for a single candidate 𝐩∈C . The candidate with the most votes then wins the election. The set of possible ballots hence are Under approval voting, each voter selects any subset of candidates of which they approve. Again, the candidate receiving the most approval wins the election. Thus, the set of ballots are Range voting, finally, refers to a family of related procedures, sometimes also going by the name of score voting or majority judgment (Fishkin 1997; Balinski and Laraki 2010). In range voting, a fixed set of grades g1,…,gk∈ℝ is provided that the voters use to assess candidates.2 Depending on the exact formulation, the candidate with the highest average or median grade then wins the election. The set of admissible ballots thus is Clearly, the set of ballots available in approval voting is a superset of that available under plurality rule. Hence, within expressive voting, abstentions under approval voting should be less frequent that under plurality rule. To determine voting behavior, we need to specify how an expressive voter 𝐯 chooses among her available ballots. That is, we need to explicate the utility 𝐯 gains from the differ‑ ent ballots, which depends on her own standpoint as well as the various parties’ positions. However, we will not provide any specific utility‑function u∶F∗ → ℝ for ∗∈ {M,A,R} here. Rather, we state a condition that every reasonable payoff function should satisfy and that is sufficient to determine the voter’s choice. We present the condition used by Aragones etal. (2011), before introducing our own framework. FM∶= {{𝐩}∣𝐩∈C} ∪ {�}. FA∶= {J∣J ⊆ C}. FR ∶= { f ∣ f ∶ C →{ g1 ,…, gk }} . 2 Unlike ranked voting methods such as Borda Count, no structural requirements limit how often each vote can be assigned. 224 Public Choice (2021) 188:221–239 1 3 3 Voting decisions inAragones etal. Aragones etal. (2011) offer an analysis of the first two voting rules introduced above, plurality rule and approval voting. The core principle within their approach is that a voter will prefer a party that is closer to her own position to a party that is further away. To make this precise, we measure distances in [−1, 1]n in the Euclidean distance: dist (x,y)= √∑ (x i −y i ) 2 . Aragones, Gilboa and Weiss then define the following choice rules: Rule (Aragones etal.-plurality) Under plurality rule, voter 𝐯 votes for the candidate that is closest to her and abstains if the empty ballot 𝟎 is closer than any of the candidates. In other words, 𝐯 chooses the ballot 𝐱m∈FM that is closest to her own standpoint in the Euclidean distance. Formally speaking, For approval voting, the above rule must be extended to the set of approval ballots. To that end, Aragones etal. represent every approval set 𝐱J with its arithmetic mean3 1 �J�∑ j∈J𝐩 j , leading to the following decision rule: Rule (Aragones etal.-approval) Under approval voting, voter 𝐯 chooses the ballot 𝐱a ∈ FA that is closest to her own standpoint in the Euclidean distance. Formally, Building on that framework, Aragones etal. show two results, both related to the question of when voters participate in an election. Note that within approval voting, a voter abstains only if the corresponding position vector 𝟎 is closer to her than any other possible ballot. That is, abstentions are not caused by an external cost, as in Downs’ analysis, but by the fact that the voter fails to find any alternative that is more appealing. The central results of Aragones etal. compare the two voting systems with respect to their potential for generat‑ ing a high level of electoral involvement, measured by the number of abstentions. The first result studies a best‑case scenario, while the second analyzes a case wherein parties’ posi‑ tions are absolutely uncorrelated. Both results confine themselves to situations when voters are fully opinionated, i.e., 𝐯∈ {−1, 1}n . Theorem1 (Theorem1 of Aragones etal. 2011) (i) Under approval voting, four strategically positioned parties are sufficient to ensure that no fully opinionated voter abstains (ii) Under plurality vote, the number of parties necessary to ensure that no fully opinionated voter abstains is exponential in the number n of agenda items. Theorem2 (Theorem2 of Aragones etal. 2011) Assume that the agenda consists of n topics and we place n parties randomly on that agenda (i.e., for every party we have a fair lottery over the 2n possible positions). As 𝐱m=argmin𝐱∈F M dist(𝐯,𝐱). 𝐱 a=argmin𝐱J∈FAdist ( 𝐯,1 | J |∑ 𝐩∈J 𝐩 ). 3 To facilitate our presentation, we set 1 ���∑ j∈� pj∶= 𝟎 . Thus, the empty approval set is represented by the zero‑vector. 225 Public Choice (2021) 188:221–239 1 3 n→∞ , the probability that a fully opinionated voter abstains in such a setting goes to 1 under plurality rule and to 0 under approval voting. Hence, approval voting is judged categorically better than plurality voting in prevent‑ ing voter abstentions. Later, we analyze the same cases within our framework, arriving at partially conflicting findings. 4 A shortcoming ofAragones etal.’s take onapproval voting While we are sympathetic to Aragones etal.’s general approach and their treatment of plurality rule, we identify a conceptual shortcoming within their treatment of approval voting. That shortcoming is then taken to motivate our alternative account, which is pre‑ sented in the following section. By its underlying assumptions, expressive voting is blind to any possible outcome of an election, as voters obtain their utility straight from the act of submitting their ballots. However, we maintain that some mild consistency requirement between expressed con‑ sent and electoral outcomes is in order. We take the following to be an uncontroversial desideratum. Unanimity desideratum In elections wherein all voters share exactly the same prefer‑ ences, and thus submit the same ballots, any single voter should approve of the resulting outcome. As we will show, Aragones etal.’s approach can violate the desideratum maximally, at least if the election is assumed to have a single winner. In short, we will show that a voter may end up approving exactly of those parties she prefers least. Accordingly, if everybody voted the same way, the winner would be a party the voter maximally dislikes, thus violat‑ ing the unanimity desideratum. Example 1 (below) constructs a situation wherein that conclusion is true. The gist of the example is that a moderate voter 𝐯 ’s position may happen to be exactly the average of two extremist parties—even though every moderate party is closer to her than each of the extremists; see Fig.1 for an illustration. Under the above semantics of approval voting, 𝐯 ’s approval set would consist of precisely the two extremist parties. Now if every voter had the same preference as 𝐯 , all votes would go to the two extremist parties and, thus, assum‑ ing a single winner election, one of the two would be voted into office, clearly producing the outcome 𝐯 dislikes most. Since we represent parties by their fully opinionated positions on a vector of topics, rather than by degrees of extremism, we cannot straightforwardly translate Fig.1 into a formal counterexample. The following example, though, shares the relevant characteristics with our informal story. v p1 p2 p 3p4 p8 p5 p 6 p7 Fig. 1 Voter 𝐯 ’s position is (exactly) the arithmetic mean of the two most extreme parties 226 Public Choice (2021) 188:221–239 1 3 Example 1 Assume that the agenda consists of nine issues t1…t9 . The first four items con‑ cern the economy, taxes, environmental issues, and the social system, four issues about which 𝐯 has some mild opinions. She assigns positions 1 3 ,− 1 3 , 1 3 ,− 1 3 , respectively, to t1…t4 . The other five topics concern difficult decisions in foreign policy on which 𝐯 finds it hard to choose sides, so she assigns them weights of zero. The two extremist parties are 𝐞+ , assigning 1 to every topic, and 𝐞 − , assigning −1 to every topic. Every other party 𝐩i assigns weights 1, −1, 1, −1 , respectively, to the first four topics and 1 to all remaining items. Then the setup is as claimed above, i.e., all moderate parties 𝐩i are closer to 𝐯 than both 𝐞+ and 𝐞 − , but {𝐞+,𝐞−} is the approval set chosen by 𝐯 . We detail the relevant calculations in the “Appendix”. 5 Our model In this section, we offer an alternative framework for approval voting that squares naturally with Aragones etal.’s decision rule for plurality voting, while also satisfying the unanimity desideratum identified in the previous section. Crucially, plurality and approval voting invoke different choice strategies. The former requires the voter to optimize, that is, identify the best among the parties and either vote for that party or else abstain. The latter, in contrast, is built around the notion of satisficing. The central task a voter faces under approval voting is to identify some threshold quality requirement to impose on candidates. She will then approve of every party that meets or exceeds her threshold. In this section, we identify two different ways in which voters could formulate their threshold requirements, one in terms of expected utility, the other as geo‑ metrical proximity, and show that both lead to the same choices. We also show that a simi‑ lar formalism applies to range voting. To introduce the framework, recall that a voter’s position on some topic i is given by a number vi∈ [−1, 1] , where −1 stands for maximal opposition and 1 for consent with maxi‑ mal possible weight. We can decompose that attitude into: where sign(vi) 4 indicates whether 𝐯 is inclined in favor of or against ti , while the absolute value |vi| measures the degree of commitment5 𝐯 attaches to topic i. We assume commit‑ ment |vi| to be related to the payoff 𝐯 can obtain on the agenda item. More specifically, we assume that, by voting for some party 𝐩 , a voter 𝐯 gets a payoff |vi| on item i if 𝐯 and 𝐩 agree on whether or not i is a good thing to do, i.e., about the sign of i. Otherwise, 𝐯 receives a payoff of −|vi| . Since we have assumed that pi∈ {−1;1} , that payoff can be expressed as Thus, the total payoff u(𝐯,𝐩) a voter 𝐯 receives by voting for party 𝐩 , i.e., the sum of his or her individual payoffs is: vi=sign(vi) ⋅ |vi| vi⋅pi. 4 sign(x) is 1 if x≥0 and −1 else. 5 Here, commitment may again reflect the importance 𝐯 attaches to that topic as well as her uncertainty about the right course of action. 227 Public Choice (2021) 188:221–239 1 3 Thus, exploiting again that p∈ {−1;1} , the maximal payoff a voter can get is: In order to state our decision rule for approval voting, we also need the voter’s approval threshold, stating how much deviation from her optimal position she is prepared to accept. The threshold is given by an approval coefficient k∈ [−1, 1] , where a smaller coefficient stands for greater tolerance. As such, we can formulate our decision rule for approval vot‑ ing . Rule (approval voting) Let 𝐯 be a voter with approval coefficient k∈ [−1, 1] . Then 𝐯 approves of all parties 𝐩 that satisfy 𝐩 ⋅𝐯 =∑ p i v i ≥k⋅ ∑� v i� , or, equivalently: Note the subtle dependency on the approval coefficient k. For the extreme value of k=1 , the voter will approve only of an optimal party coinciding with her on the inclination of every topic. If no such party exists, the voter will submit an empty approval set, i.e., abstain. Conversely, a voter with an approval coefficient of −1 will approve indiscrimi‑ nately of every party, no matter what that party claims, wants, or does. Finally, a middle value of k=0 corresponds to a fairly tolerant voter, approving of every party that agrees with her more often than it disagrees. For most of the following applications we will thus assume that k≥0 . Next, we examine two alternative intuitions relating to how a voter could choose parties of which to approve. As it turns out, both of the alternatives are equivalent to our choice rule. We take that equivalence as an argument for the naturalness of our definition. The first alternative choice rule is given in terms of percental agreement. An agent chooses a percental threshold t∈[0, 100] and approves of every party that agrees with her on at least t percent of the topics. Since the voter has different degrees of commitment to the various agenda items, the percental agreement needs to be weighted by the agent’s commitments |vi| . Thus, the corresponding rule is: Rule (approval voting: 1st alternative) Let 𝐯 be a voter with percental threshold t∈[0, 100] . Then, 𝐯 approves of all parties 𝐩 that satisfy That decision rule is equivalent to our original rule, as expressed by the following lemma. Lemma 1 A voter 𝐯 approves of some party 𝐩 with approval coefficient k∈ [−1, 1] if and only if she approves of 𝐩 in the alternative definition with percental threshold t =100 ⋅ 1+k 2 . The second alternative rule is of a geometric nature. Recall that we represent voters and parties by their positions on the agenda items, that is, as a vector in ℝn . So why not define the voter’s approval decision by geometric proximity? Arguably, an adequate measure of proximity is the angle between two position vectors, showing how far the two diverge in u (𝐯,𝐩)=𝐯⋅𝐩= ∑ v i p i. | 𝐯 | ∶= ∑ i| vi |. (1) 𝐩⋅𝐯 |𝐯| ≥k . 1 ∑� vi �� {i∶p i v i >0} � vi � ≥ t 100 . 228 Public Choice (2021) 188:221–239 1 3 their political opinions.6 The maximal angle of 180◦ between a voter 𝐯 and some party 𝐩 implies that pi ⋅ vi≤0 for every i, that is, 𝐯 and 𝐩 disagree about every single topic. Con‑ versely, a relatively small angle between a party and a voter corresponds to a high degree of agreement between the voter’s inclination and the party’s position; see Fig.2. Again, we need to fix a threshold angle 𝛼 for formulating the corresponding decision rule. For some given threshold angle 𝛼 , let C(𝐯,𝛼) be the cone of all vectors 𝐲 in ℝn⧵{0} such that the angle between 𝐯 and 𝐲 is at most 𝛼 . Rule (approval voting: 2nd alternative) Let 𝐯 be a voter with threshold angle 𝛼∈[0, 180] . Then 𝐯 approves of all parties 𝐩 that satisfy: Again, the alternative is related closely to the original decision rule. This time, though, the exact relationship between the approval coefficient k and threshold angle 𝛼 depends upon the exact position of voter 𝐯 . The correspondence is: Lemma 2 Let 𝐯 be a voter with approval coefficient k. Then some angle 𝛼 depending upon n, k and 𝐯 exists such that 𝐯 approves of some party 𝐩 exactly if 𝐩∈C(𝐯,𝛼) . Furthermore, the angle 𝛼 satisfies arccos (k)≤𝛼≤arccos( k √ n ) . Thus, for any possible voter 𝐯 , the three different possible interpretations of approval thresholds are equivalent to one another. Before proceeding to some general results, we will return quickly to the unanimity desideratum that was at the heart of our argument against Aragones etal.’s (2011) approach. Briefly, the desideratum demanded that, assum‑ ing a single‑winner election, if all voters submitted the same ballot, each should approve of the electoral result. That is indeed the case. The approval set of an agent contains only those parties of which the voter approves individually. If every voter happened to submit the same approval set as 𝐯 , the winner would be some member of that approval set and, thus, a party of which 𝐯 approves. 𝐩∈C(𝐯,𝛼). Fig. 2 The approval cone of voter 𝐯 (shaded) v α α 6 To elaborate a bit further on why we take the angle between two vectors and not, for instance, their length, recall that a change in the length of some vector 𝐯 , that is, replacing 𝐯 by 𝜆𝐯 for some 𝜆>0 , simply denotes a change in political commitment while leaving the general position intact. Conversely, a non‑zero angle between two voters 𝐯 ad 𝐯′ implies that the two disagree about the relative importance attributed to the various topics or even about the right course of action about some agenda item i. 235 Public Choice (2021) 188:221–239 1 3 Thus U𝐩 is the set of indices where the signs of 𝐯 and 𝐩 disagree. Now we have Only the last term depends on 𝐩 . Thus for any 𝐩 , 𝐩�∈C : On the other hand we have: where, again, the first term is independent of 𝐩 . Thus also Before we can prove Theorems 3 and 4 we need the following lemma: Lemma 5 Let m∈ ℕ ⧵{0} . Then we have for any natural number n Proof We make a case distinction between n even and odd. We show the formula for n even. For n odd, the proof is similar. First we show that for any natural number i∈[ 0, n m] we have that To this end observe that dist (𝐯,𝐩)= √∑ i (vi−pi)2= √ n+ ∑ i v2 i−2 ∑ i vipi = √ n+ ∑ i v2 i−2 ∑ i | vi | +4 ∑ i∈U𝐩 | vi | . dist (𝐩,𝐯)≤dist (𝐩 � ,𝐯)⇔ ∑ i∈U 𝐩| vi | ≤ ∑ i∈U 𝐩 � | vi |. ∑ i vipi= ∑ i | vi | −2 ∑ i∈U 𝐩| vi |, ∑ vipi ∑� vi � ≥ ∑ vip � i ∑� v� i � ⇔ � i∈U 𝐩� vi � ≤ � i∈U� 𝐩� vi �. (3) ∑ n k=⌈n(1 2+1 m)⌉ � n k � 2n≤1 (1+1 m ) ⌈ n m ⌉ −1 . (4) � n n 2 +i � ≥ � 1+ 1 m �⌈n m ⌉� n n 2 + ⌈ n m⌉ +i �. � n n 2+i � �n n 2+⌈n m⌉+i�= (n 2+⌈n m⌉+i)!(n 2−⌈n m⌉−i)! (n 2−i)!(n 2+i)! = (n 2+i+1)⋅(n 2+i+2)⋅…⋅(n 2+⌈n m⌉+i) (n 2−⌈n m⌉−i+1)⋅(n 2−⌈n m⌉−i+2)⋅…⋅(n 2−i ) = n 2+1+i (n 2 − ⌈ n m⌉ −i+1) ⋅…⋅ n 2+ ⌈ n m ⌉ +i n 2 −i . 236 Public Choice (2021) 188:221–239 1 3 Now, since k+i l−i ≥ k l for any k,l,i>0 , every quotient in the last formula is at least as large as the right‑most quotient n 2+ ⌈n m ⌉ + i n 2 −i . Moreover, this quotient satisfies as m,n,i≥0 . As the product n 2 + 1 + i ( n 2 − ⌈ n m⌉ −i+1) ⋅…⋅ n 2 +⌈n m ⌉+i n 2 −i contains ⌈n m⌉ factors, it is thus larger than ( 1+1 m ) ⌈n m ⌉ and (4) holds. In the following, let 𝛼 ∶= � (1+1 m )−1 �⌈n m ⌉ . Repeatedly applying (4) gives us for all natural numbers j with 0 ≤j< ⌈n m⌉ Using that ( n k ) = 0 whenever k>n , this implies Resubstituting 𝛼 =(1+1 m )− ⌈n m ⌉ and exploiting ∑ n j=0 � n j � =2 n gives us Proof ofTheorem3 For (i) observe that candidates 𝐩1∶= (1, 1, …1) and 𝐩2∶= −𝐩1 have the property that for any voter 𝐯 at least one of the two statements 𝐩1 ⋅ 𝐯 ≥ 0 and 𝐩2 ⋅ 𝐯 ≥ 0 holds. Thus each voter approves of at least one of these two parties. (ii) Assume k>0 and let 𝕍∶= {−1;1}n be the set of voters who have extreme positions on every single topic. We will show that the number of parties needed to ensure that all members of 𝕍 vote is exponential in n. Fix some natural number m such that 1 m ≤k . Since the number of parties some voter 𝐯 approves of is decreasing in k it suffices to show the theorem with k = 1 m . Observe that for any party 𝐩 and any voter 𝐯∈𝕍 holds: Since for any party 𝐩 and any l∈ℕ n 2 +⌈n m ⌉+ i n 2 −i =1+ ⌈n m ⌉+ 2i n 2 −i ≥1+ ⌈n m ⌉ n 2 ≥1+ n m n 2 >1+1 m , m � i=1� n n 2 +j+i ⌈ n m⌉� ≤ m � i=1 𝛼i � n n 2 +j � ≤𝛼 1−𝛼 � n n 2 +j � n � k=⌈n(1 2+1 m)⌉ � n k � = m � i=1 ⌈n m ⌉−1 � j=0 � n n 2+j+i⌈n m⌉ � ≤ 𝛼 1−𝛼⌈n m⌉−1 � j=0� n n 2+j � < 𝛼 1−𝛼 n � j=0� n j � . ∑ n k=⌈n(1 2+1 m)⌉ � n k � 2n≤ (1+1 m)−⌈n m⌉ 1−(1+1 m )− ⌈ n m ⌉ =1 (1+1 m ) ⌈ n m ⌉ −1 . 𝐯 ⋅ 𝐩 |𝐯| ≥ 1 m ⇔ | {i | vi=pi} | ≥ n 2+ n 2 m . |||||{ 𝐯∈𝕍 |||| {i∶vi=pi} | =l }||||| = ( n l ), 237 Public Choice (2021) 188:221–239 1 3 each party 𝐩 can be be approved by at most ∑ n k= ⌈ n(1 2+1 2m) ⌉� n k � many members of 𝕍 . Since |𝕍|=2n this implies that the number of parties needed to make sure that no member of 𝕍 abstains is at least By Lemma 5 this quotient is at least ( 1+1 m ) ⌈n m ⌉ − 1 and thus also at least ( 1+1 m ) n m− 1 . In particular it is at least exponential in n. Since 2n parties are enough to ensure that every‑ body votes, the number of parties needed cannot be worse than exponential. The proof of iii) consists of two parts. First, we show that at least n+1 parties are needed in order to ensure that every voter finds a party she approves of. Assume to the con‑ trary that 𝐩1…𝐩n are enough to attract every possible voter. Recall that, by the voting rule used for iii), a voter 𝐯 approves of a party 𝐩 iff 𝐯 ⋅ 𝐩>0 . For i<n define Xi to be the n−1 dimensional hypersurface defined by Thus X∶= X1∩…∩Xn−1 is a vector space of dimension at least 1 and thus Y=X∩{𝐱∈ [−1, 1]n|𝐱 ⋅ 𝐩n ≤ 0} ≠ {𝟎} . Pick some non‑zero 𝐯∈Y . Then 𝐯 ⋅ 𝐩𝐢=0 for i<n and 𝐯 ⋅ 𝐩n≤0 , thus the voter 𝐯 would abstain in an election with candidates 𝐩1…𝐩n , contradicting our assumption. Next, we show that n+1 parties are sufficient to attract all voters if there are at least n≥3 topics. To this end, let 𝟏 be the vector (1, …,1) and for i≤n let 𝐞i be the vector with 1 at the i‑th position and −1 on all others. Moreover, let P be the set {𝟏,𝐞1,…,𝐞n} . We will show that every voter 𝐯 approves of at least one party in P. We do so by case distinction. The first case is that 𝐯 ⋅ 𝐞i>0 for some i. In this case, 𝐯 ⋅ 𝐞 i |𝐯| > 0 and 𝐯 approves of party 𝐞i . The second case is that 𝐯 ⋅ 𝐞i≤0 for all i≤n . First, note that 𝐞1,…,𝐞n form a basis of ℝn . Thus, 𝐯 ⋅ 𝐞i=0 for all i would imply that 𝐯=𝟎 . Since we have excluded voters from assuming position 𝟎 , we can infer that there is some j with 𝐯 ⋅ 𝐞j<0 . Now, note that 𝟏 =− 1 n−2∑i≤n 𝐞 i . We hence get Since 𝐯 ⋅ 𝐞i ≤ 0 for all i and 𝐯 ⋅ 𝐞j < 0 , we obtain that − 1 n−2∑ i≤n𝐯⋅𝐞i> 0 . Hence 𝐯 ⋅ 𝟏 |𝐯| > 0 , showing that 𝐯 approves of party 𝟏 . ◻ Proof ofTheorem4 Fix a voter 𝐯 . Observe that for k=0 and any party 𝐩 at least one of the following two holds: 𝐯⋅𝐩 |𝐯| ≤ 0 or 𝐯⋅𝐩 |𝐯| ≥ 0 . Let P= {−1;1}n be the set of all possible parties. Since for any 𝐩∈P also −𝐩∈P and 𝐯⋅𝐩 |𝐯| ≥ 0 iff 𝐯⋅−𝐩 |𝐯| ≤ 0 , we get that Since picking a random party is the same as randomly drawing a party from P , the chance that a random party 𝐩 satisfies 𝐩 ⋅ 𝐱≥0 is, thus, at least one half. Thus the chance that 2n ∑ n k= ⌈ n(1 2+1 2m) ⌉� n k �. Xi={𝐱∈ [−1, 1]n|𝐱 ⋅ 𝐩i=0}. 𝐯 ⋅𝟏=𝐯⋅ ( − 1 n−2 ∑ i≤n 𝐞i ) =− 1 n−2 ∑ i≤n 𝐯⋅𝐞 i |{𝐩∈P|𝐩 ⋅ 𝐯≥0}| | P | ≥1 2 . 238 Public Choice (2021) 188:221–239 1 3 𝐯 approves of none of n random parties is at most 1 2 n , thus P(n,n,0) → 1 . As approval is monotonous in k, this implies P(n,k) → 1 for any k≤0 . Since P(n, k) is monotonous in k, and for every k>0 there is some m∈ℕ with 1 m ≤ k , it suffices to show that P (n,n, 1 m )→ 0 for any natural number m. Let 𝐯=(1, 1, …) be a voter who fully approves of all topics and let m∈ℕ . Observe that for any party 𝐩 holds: Thus for the uniform distribution ℙ over P we have As above Lemma 5 yields that Thus P (n,n,1 m)≤1− � 1−1 (1+1 m ) ⌈ n m ⌉ −1 �n . Note that for n large enough, It is a general fact that (1−xn)n → 1 for any x∈(0, 1) . As both the left‑ and rightmost member of the above inequality are of this form, we obtain P (n,n, 1 m )→ 0 as claimed. Proof ofTheorem5 Fix a voter 𝐯 and let i such that ti=0 . The third part of Theorem3 applied to voter −𝐯 shows that n+1 parties are enough to guarantee that some party gets graded at most gi−1 . Equally, the same theorem applied to 𝐯 herself shows that the same n+1 parties also guarantee that some candidate gets grade gi or higher. Finally assume that there is no i with ti=0 and let j be maximal such that tj−1 < 0 . Then the second part of Theorem3 applied to 𝐯 and −𝐯 shows that exponentially many parties are needed in order to ensure that some party gets a grade unequal to gj−1 . Proof ofTheorem6 First assume that there is some i with ti=0 . Then, by Theorem4, the probability that at least one out of n random parties gets grade at least gi goes to 1. Apply‑ ing Theorem4 to −𝐯 we see that also the probability that a party gets grade at most gi−1 goes to 1. In particular, the probability for two parties receiving different grade assign‑ ments goes to 1, thus proving the first part. For the second part assume that there is no such i. Let i0 be such that ti < 0 for all i≤i0 and ti > 0 for all i>i0 . Then applying Theorem4 with k=ti0+1 (if defined) yields that the probability that some party gets grade larger than gi0 goes towards 0. Applying 4 to −𝐯 yields that the probability for parties getting a grade below gi0 also goes to zero. Hence the probability of all parties getting the same grade gi0 goes towards 1. 𝐯 ⋅𝐩≥ 1 m| 𝐯 | 1⇔ | {i | pi=1} | ≥ n 2 + n 2m . ℙ� 𝐯⋅𝐩≥1 m� 𝐯 � 1 � = ∑ n k= ⌈ n(1 2+1 2m) ⌉� n k � 2 n . ℙ� 𝐯⋅𝐩≥ 1 m � 𝐯 � 1 � ≤ 1 (1+1 m ) ⌈ n m ⌉ −1 . ⎛⎜⎜⎝ 1−1 (1+1 m) n m+1 ⎞⎟⎟⎠ n ≤ ⎛⎜⎜⎝ 1−1 (1+1 m) ⌈ n m ⌉ −1 ⎞⎟⎟⎠ n ≤ � 1−1 (1+1 m)n �n 239 Public Choice (2021) 188:221–239 1 3 References Aragones, E., Gilboa, I., & Weiss, A. (2011). Making statements and approval voting. Theory and Decision, 71, 461–472. Balinski, M., & Laraki, R. (2010). Majority judgment: Measuring, ranking, and electing. Cambridge: MIT Press. Brams, S., & Fishburn, P. C. (2007). Approval voting. New York: Springer. Brams, S. J., & Fishburn, P. C. (1978). Approval voting. American Political Science Review, 72(3), 831–847. Brennan, G., & Lomasky, L. (1993). Democracy and decision—The pure theory of electoral choice. Cam‑ bridge: Cambridge University Press. Dean, W., & Parikh, R. (2011). The logic of campaigning. 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