Extended Abstract AAMAS 2019, May 13-17, 2019, Montréal, Canada Proportional Representation in Elections: STV vs PAV Extended Abstract Piotr Faliszewski Piotr Skowron AGH University University of Warsaw Krakow, Poland Warsaw, Poland [email protected] [email protected] Stanisław Szufa Nimrod Talmon Jagiellonian University Ben-Gurion University Krakow, Poland Be’er Sheva, Israel [email protected] [email protected] ABSTRACT from all the elections. We see that STV is better at mimicking the We consider proportionality in multiwinner elections and observe distribution of the voters, whereas PAV is better at selecting more that PAV and STV are fundamentally different. We argue that the central candidates, that represent all the voters. former is proportional and the latter is degressively proportional. We conclude that there is no single notion of proportionality. We refer to the kind of proportionality represented by STV as KEYWORDS individual proportionality and to the kind represented by PAV as group proportionality. We seek to understand these notions better by multiwinner voting; committee scoring rules; proportionality; Sin- expressing STV in the same language as PAV (i.e., as an OWA-based gle Transferable Vote; Proportional Approval Voting rule). Specifically, we find that STV is degressively proportional. ACM Reference Format: Piotr Faliszewski, Piotr Skowron, Stanisław Szufa, and Nimrod Talmon. 2019. Proportional Representation in Elections: STV vs PAV. In Proc. of the 2 PRELIMINARIES 18th International Conference on Autonomous Agents and Multiagent Systems For an integer t, we denote the set f1; ...; tg by »t¼. By an election (AAMAS 2019), Montreal, Canada, May 13–17, 2019, IFAAMAS, 3 pages. E = ¹C;V º, we mean a pair that consists of a set of candidates C = fc1; ...;cm g and a set of voters V = fv1; ...;vn g, where each 1 INTRODUCTION voter vi 2 V has a linear preference order over all the candidates from C. A multiwinner voting rule R is a function that given an We consider the problem of selecting a committee that represents election E = ¹C;V º and a committee size k, outputs a family of the voters proportionally. One way of finding such a committee is size-k subsets of C, the committees that tie as election winners. to apply the Single Transferable Vote (STV) rule, which is considered to give proportional results, while another one is to use Propor- Single Transferable Vote (STV). We consider the following vari- tional Approval Voting (PAV) [8, 15]; it has recently been shown ant of the STV rule (see, e.g., [10]). Let E = ¹C;V º be the input b n c that PAV satisfies a number of appealing properties pertaining to election with n voters. We use Droop quota q = k+1 + 1 . To proportionality [1–3, 11, 12]. Yet, even though both these rules aim compute a winning committee of size k, start with an empty com- at achieving proportional representation, they might produce fun- mittee and execute the following steps until the committee has k damentally different results. For example, let us consider elections candidates: (1) If some candidate is ranked first by at least q voters, (in two dimensional Euclidean space) where the ideal points of the then find a candidate c that is ranked first by the largest number candidates are drawn uniformly at random from a disc in the centre of voters and remove him from the election, together with some (dark grey points in the picture on the left of Figure 1) while the q voters that rank him first. (2) If no candidate is ranked first by ideal points of the voters are drawn uniformly at random from the ring that surrounds that disc (light grey points in Figure 1). Each voter ranks the candidates from best to worst by sorting their ideal points with respect to the Euclidean distance from the voter’s points [4, 5]. We generated 100 elections of this form (each with 100 voters and 20 candidates) and for each of them we computed PAV and STV winning committees of size 10; we present the ideal points of the members of these winning committees in the pictures in the centre (PAV) and on the right (STV) of Figure 1, as black points; both Figure 1: 2D scatter plots for the simulation in the intro- pictures contain the points of the winning committee members duction. Light grey ring of points in each picture represent Proc. of the 18th International Conference on Autonomous Agents and Multiagent Systems the voters. Central grey points in the left picture represent (AAMAS 2019), N. Agmon, M. E. Taylor, E. Elkind, M. Veloso (eds.), May 13–17, 2019, the candidates. Black points in the middle and right pictures Montreal, Canada. © 2019 International Foundation for Autonomous Agents and represent the winners under PAV and STV rule respectively. Multiagent Systems (www.ifaamas.org). All rights reserved. 1946 Extended Abstract AAMAS 2019, May 13-17, 2019, Montréal, Canada at least q voters, then find a candidate d that is ranked first by the STV as an OWA-Based Rule. Let γ be one of our three single- fewest voters and remove d from the election. winner scoring functions and let E be a model of generating elec- OWA-Based Rules. A single-winner scoring function for the case tions, E 2 fUniform, Uniform/Gaussian, Asymmetric Gauss- of m candidates is a non-increasing function γ : »m¼ ! R that iang. We generated t = 50 elections E1; ...; Et from E and, using associates each position in a preference order with a non-negative our methodology, we computed the OWA vector Λ = ¹λ1; ...; λk º score. We focus on the Borda scoring function: βm¹iº = m − i; the that minimized the distance dist¹Rγ;Λ;W1; ...;Wt º (see Figure 2). To k-Approval scoring function: αk ¹iº = »i ≤ k¼; and the k-Truncated better understand the computed OWA vectors, we sought further k Borda scoring function: βm¹iº = ¹k − iº · »i ≤ k¼ (for a logical vectors that would be as close to the computed ones as possible, expression F, by »F¼ we mean 1 if F is true and 0 otherwise). but that would be expressed using a simple formula. We found that 1 x 1 x 1 x An OWA-based multiwinner rule R , for m candidates and com- /2 +y /3 +y /n +y f vectors of the form h¹x;yº = 1; 1+y ; 1+y ; ...; 1+y suffice. mittees of size k, is defined through a single-winner scoring func- tion γ and an OWA vector Λ = ¹λ1; ...; λk º (OWA stands for ordered weighted average [16]). If some voter v ranks members of a given Uniform Uniform 1 Uniform/Gaussian 1 Uniform/Gaussian size-k committee W on positions i ≤ ... ≤ i , then v assigns W a 1 k 0:8 Asymmetric 0:8 Asymmetric score of fm;k ¹i1; ...;ik º = λ1γ ¹i1º + ... + λkγ ¹ik º: The Rf -winning h(1.18, -0.05) h(1.79, -0.002) 0:6 0:6 committees are exactly those for which the sum of the scores given by the voters is the highest. PAV uses the k-Approval scoring func- 0:4 0:4 tion and OWA vectors of the form ¹1; 1/2; ...; 1/kº. OWA-based rules 0:2 0:2 are due to Skowron et al. [13]; see also [6, 14]. 0 0 Euclidean Elections. In 1D Euclidean elections, each voter and 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 candidate has an ideal point p 2 R. Each voter forms his prefer- Figure 2: OWA vectors for approximating STV using 10- ence order by sorting the candidates with respect to the increasing Approval (left) and 10-Tr.-Borda (right) scoring functions. distance of their ideal points from his. We focus on 1D Euclidean elections where the ideal points are drawn from the »0; 1¼ interval. In the Uniform model, the ideal points of the candidates and voters Degressive Proportionality. In the related model of apportion- are drawn uniformly at random; in the Uniform/Gaussian model, ment, where the task is to divide parliamentary seats among politi- the ideal points of the candidates are drawn uniformly at random, cal parties, proportionality has a natural interpretation—the num- whereas the ideal points of the voters are drawn from the Gaussian ber of seats obtained by each party should be proportional to the distribution with mean 0:5 and std. deviation 0:15; in the Asymmet- number of votes the party received. In contrast, a voting rule is ric Gaussian, model the ideal points of the candidates and voters proportional in a degressive way [9] if it assigns disproportionately are drawn with probability 30% from a Gaussian distribution with many seats to parties with smaller support. Interestingly, there is a mean 0:25, and with probability 70% from a Gaussian distribution strong correlation between the slope of the OWA vector and the with mean 0:75 (both Gaussians have std. deviation 0:1). type of proportionality the rule guarantees [11]. Intuitively, this relation can be explained as follows. Consider ¹1/ 1/ 1/ 1/ º 3 RESULTS an OWA vector Λ = z1; z2; z3; ...; zn , which is normalized so that z1 = 1. We define the vector cost = ¹z1; ...; znº. Then, the Our goal is to find an OWA-based rule that approximates STV as i-th element of this vector can be interpreted as the relative cost well as possible. We present our methodology and then the results. of obtaining the i-th seat. Indeed, when R decides which of the Methodology.