Charles University in Prague Faculty of Social Sciences Institut of Economic Studies

Three Essays on Operations Research in Political Economy

Dissertation Thesis

DoleˇzelPavel

Academic year 2011/2012 2 3

Abstract

Thesis consists of three essays dealing with the exact methods of operations research, mainly the mathematical optimization, used on the issues of political economy. The first two essays deal with the concept of the efficiency of weighted voting games (systems), the third essay is more practical and introduces three electoral methods that could be used in the real elections in order to minimize the level of disproportionality. The first essay deals with estimating the efficiency of weighted voting games using our own heuristic algorithm. We show the preciseness of our results in terms of probabilityand we apply the proposed algorithm to the efficiency of several institutions of the European Union, especially to the efficiency of the qualified majority rule used in the Council of the EU both under the Lisbon treaty and under the Treaty of Nice. The second essay provides a theoretical analysis of the efficiency of weighted voting games with focus on the maximal and minimal attainable efficiency given the quota and the number of voters. We present a proof of a theorem which enables us to find the efficiency maxima and minima in linear time and some corollaries of this theorem providing some further knowledge on the structure of the efficiency as a function of quota and number of voters. The third essay introduces three methods of converting votes into seats within the elections to the Chamber of Deputies of the Czech Parliament which are designed to minimize the level of disproportionality. All the methods can be used for a large set of general electoral systems and are based on solving integer programming problems. All the methods are designed to be able to cope with the election threshold and division of the elections into constituencies. 4 LIST OF FIGURES

2.1 Deviances of simulated efficiencies from their average (100 trials) 18 2.2 Histogram of simulated efficiencies ...... 20 2.3 The efficiency with respect to the probability of acceptance within the 2009 European Parliament absolute majority procedure ...... 28 2.4 The impact of the distinct rules on efficiency within the procedure of the qualified majority via the Treaty of Nice . . . . . 29 2.5 The efficiency with respect to probability of acceptance within the procedure of qualified majority via the Treaty of Nice and the Lisbon Treaty ...... 31 2.6 The impact of the distinct rules on efficiency within the procedure of a qualified majority via the Lisbon Treaty ...... 32

3.1 Eﬃciency symmetry ...... 51 3.2 Eﬃciency structure is driven by the Farey sequence ...... 54 6 List of Figures LIST OF TABLES

2.1 Some estimated efficiencies via the HRA algorithm ...... 19 2.2 The results of the 2006 Parliamentary elections in the Czech Republic ...... 23 2.3 Portions and numbers of mandates assigned to each country in the European Parliament via the Maastricht Treaty . . . . 25 2.4 The efficiency of the absolute majority of the 2009 European Parliament given the probabilities of acceptance and country- homogeneous voting ...... 26 2.5 The political structure of the European Parliament as of the end of 2009 ...... 26 2.6 The efficiency of the absolute majority of the 2009 European Parliament given the probabilities of acceptance and party- homogeneous voting ...... 27 2.7 The population and weights assigned to states in qualified majority voting in the Council of Ministers of the EU via the Treaty of Nice ...... 27 2.8 The efficiency of the qualified majority via the Treaty of Nice for the given probabilities of acceptance ...... 28 2.9 The impact of distinct rules on the efficiency of the qualified majority via the Treaty of Nice for the given probabilities of acceptance ...... 29 2.10 The efficiency of the qualified majority via the Treaty of Lis- bon for the given probabilities of acceptance ...... 30 2.11 The impact of distinct rules on the efficiency of the qualified majority via the Lisbon Treaty for the given probabilities of acceptance ...... 31

4.1 Number of votes cast in districts (columns) and parties (rows) given the numbers of seats ...... 63 4.2 Allocation of seats by one of our methods ...... 64 4.3 Number of votes cast in districts (columns) and parties (rows) given the numbers of seats ...... 64 4.4 Optimal solution to the problem ...... 65 8 List of Tables

4.5 Allocation rules ...... 80 4.6 Average ranks of electoral systems over all analyzed elections to PS PCRˇ (measures of disproportionality of allocation of seats to political groups) ...... 81 4.7 Average ranks of electoral systems over all analyzed elections to PS PCRˇ (measures of disproportionality of allocation of seats to constituencies) ...... 82 4.8 Average ranks of electoral systems over all the analyzed elections to PS PCRˇ (measures of disproportionality of allocation of seats to combination of political groups and constituencies) 82 4.9 List of all possibilities ...... 88 4.10 Disproportionality of seats allocation to political groups (Czech national Council 1990) ...... 101 4.11 Disproportionality of seats allocation to political groups (Czech national Council 1992) ...... 102 4.12 Disproportionality of seats allocation to political groups (Cham- ber of Deputies 1996) ...... 102 4.13 Disproportionality of seats allocation to political groups (Cham- ber of Deputies 1998) ...... 103 4.14 Disproportionality of seats allocation to political groups (Cham- ber of Deputies 2002) ...... 103 4.15 Disproportionality of seats allocation to political groups (Cham- ber of Deputies 2006) ...... 104 4.16 Disproportionality of seats allocation to political groups (Cham- ber of Deputies 2010) ...... 104 4.17 Disproportionality of seats allocation to combinations of political groups and constituencies (Czech national Council 1990) 105 4.18 Disproportionality of seats allocation to combinations of political groups and constituencies (Czech national Council 1992) 105 4.19 Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 1996) . 106 4.20 Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 1998) . 107 4.21 Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 2002) . 107 4.22 Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 2006) . 108 4.23 Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 2010) . 108 4.24 Disproportionality of seats allocation to constituencies (Czech national Council 1990) ...... 109 List of Tables 9

4.25 Disproportionality of seats allocation to constituencies (Czech national Council 1992) ...... 109 4.26 Disproportionality of seats allocation to constituencies (Cham- ber of Deputies 1996) ...... 110 4.27 Disproportionality of seats allocation to constituencies (Cham- ber of Deputies 1998) ...... 110 4.28 Disproportionality of seats allocation to constituencies (Cham- ber of Deputies 2002) ...... 111 4.29 Disproportionality of seats allocation to constituencies (Cham- ber of Deputies 2006) ...... 111 4.30 Disproportionality of seats allocation to constituencies (Cham- ber of Deputies 2010) ...... 112 10 List of Tables 1. INTRODUCTION

Political economy is quite interesting part of the economic theory recently extensively studied as politics play more and more important role in the overall decision making process. Political economy is closely related to the political science, however it focuses mainly on the quantitative aspects of decision making, coalition formation and social choice theory as opposed to political science, which is rather descriptive and focuses mainly on the theory and practice of political systems, political parties and political behaviour. Substantial difference between the political economy and political science is also in the methodology. Political economy emphasizes exact formal mathematical models (game theory, graph theory, combinatorics, statistics and operations research) and use them to describe political processes, while the political science uses rather the methods of less formal social sciences - description, historical analogy, comparative study, philosophical analysis, etc. Thesis consists of three essays dealing with the exact methods of operations research, especially the mathematical optimization, used on the issues of political economy. The first two essays deal with the concept of the efficiency of weighted voting games (systems). Weighted voting game is given by a set of players (voters) each endowed with some real weight. Voters make coalitions in order to get the total weight of the coalition over an a priori given threshold called a quota. Efficiency of such a game is given as a ratio of the number of coalitions with the sum of weights of its members higher than or equal to the quota and the number of all possible coalitions. The first essay deals with estimating the efficiency of weighted voting games using our own heuristic algorithm, which employs the Fisher-Snedecor probability distribution function to estimate the efficiency with limited probability of low preciseness and we generalize the algorithm to be able to handle the multi-rule voting systems. We show the preciseness of our results in terms of probability using the Hoeffding’s inequality. We also apply the proposed heuristic algorithm to the efficiency of several institutions of the European Union, especially to the efficiency of the qualified majority rule used in the Council of the EU both under the Lisbon treaty and under the Treaty of Nice and we also apply the algorithm on showing, how far was the efficiency 12 1. Introduction of outcome of the Chamber of Deputies of the Czech Parliament elections of 2006 from the highest attainable efficiency. The second essay provides a theoretical analysis of the efficiency of weighted voting games with focus on the maximal and minimal attainable efficiency given the quota and the number of voters. We present a proof of a theorem which enables us to find the efficiency maxima and minima in linear time and some corollaries of this theorem providing some further knowledge on the structure of the efficiency as a function of quota and number of voters. We formaly describe the function of efficiency minima as well as efficiency maxima with respect to the quota which is somewhat connected to mathematical structures such as the Euler’s totient function or Farey sequence. The third essay introduces three apportionment methods for the elections to the Chamber of Deputies of the Czech Parliament which are designed to minimize the level of disproportionality. All the methods can be used for a large set of general electoral systems and are based on solving integer programming problems. All the methods are designed to be able to cope with the election threshold and division of the elections into constituencies. We prove, that our methods always attain an optimal solution of the allocation of seats among political groups regardless of the constituencies as well as among the constituencies regardless of the political groups. We also prove that the allocation of seats to the political groups in particular constituencies is optimal as well. We compare the proposed allocation methods with a set of other methods based on common allocation rules and their combinations. We include also the comparison with the apportionment currently used in the elections to the Chamber of Deputies of the Czech Parliament. 2. ESTIMATING THE EFFICIENCY OF VOTING IN BIG SIZE COMMITTEES

Abstract In a simple voting committee with a finite number of members, in which each member has a voting weight, the voting rule is defined by the quota (a minimal number of voting weights is required to approve a proposal), and the efficiency of voting in the committee is defined as the ratio of the number of winning coalitions (subsets of the set of members with total voting weights no less than the quota) to the number of all possible coalitions. A straightforward way of calculating the efficiency is based on the full enumeration of all coalitions and testing whether or not they are winning. The enumeration of all coalitions is NP-complete problem (the time required to find the solution grows exponentially with the size of the committee) and is unusable for big size committees. In this paper we are developing three algorithms (two exact and one heuristic) to compute the efficiency for committees with high number of voters within a reasonable timeframe. Algorithms are applied for evaluating the voting efficiency in the Lower House of the Czech Parliament, in the European Parliament and in the Council of Ministers of the EU. Keywords Efficiency of voting, committee, European Parliament, EU Coun- cil of Ministers JEL classification D71, D72

2.1 Introduction

While the problem of legitimacy (allocating voting weights to voters) is rather complex and is handled using different concepts of voting power [3], the problem of efficiency is relatively simpler: What is the probability that a proposal submitted for voting is approved (i.e., the probability of changing the status quo)? The generally accepted measure of efficiency is the so called Coleman index of the power of a collective to change the status quo (the probability of the appearance of the winning coalition under the assumption of the equal probability of any coalition formation), see [2]. For empirical studies of different alternative voting rules in the EU Council of Ministers 14 2. Estimating the Efficiency of Voting in Big Size Committees from the standpoint of legitimacy and efficiency see [4] and [5]. The main aim of this article is to provide a fast algorithm (running with polynomial time complexity) for the computation of the efficiency of voting systems, as well as some basic analysis of this algorithm. In the context of the European integration process, the minimum number of voters to be investigated is 30. This number of voters is hardly reachable by standard algorithms since the total number of coalitions to be investigated is 230 = 1, 073, 741, 824. Voting is a way of transforming many individual preferences into one final preference. In this paper, I investigate the efficiency of basic voting systems, also known as a simple weighted majority game, i.e. the probability of a proposal to be approved in a voting process, while the probabilities of rejection are prescribed to each individual voter. The efficiency can be influenced not only by the preferences and probabilities of rejection by each voter, but also by the voting system itself, i.e. by the quota and weights assigned to each individual voter. In the first part I analyze the efficiency of the simple voting system based on one voting rule while the preference of each voter is given by the rejection of any proposal with 0.5 probability and the acceptance of any proposal with 0.5 probability. This simplification can be used for the selection and tuning of the newly introduced voting systems, where no additional information about the individual preferences are known. It is straightforward that by giving each voter the same weight in the voting, the probability of exactly k ∈ N out of n ∈ N, k < n voters approving the proposal is driven by the binomial probability distribution. The main aim of this paper is to support the creators of voting rules (this can be voting rules in Parliament, general meetings of stockholders as well as in any other organizations, institutions or companies where voting is employed to make one unique decision as a representation of many individual preferences) with some basic knowledge of the efficiency of voting rules. It is important to know which voting system leads to which probability of changing the status quo (approval of the proposal) under the specified preferences of the individuals. Suppose, we have a set N = {1, . . . , n} . The cardinality of N is then |N| = n ∈ N. This set will represent the set of voters (voting bodies), so that each voter is represented by just one index from N. Suppose there is a vector space Vn above the field of real numbers and the set Sn ⊂ Vn Pn of all vectors w = (w1, . . . , wn) , is such that k=1 wk = 1 and for all k = 1, . . . , n, wk ≥ 0. The set of all real numbers between 0 and 1 (including both) is denoted Λ, i.e. Λ = {λ ∈ R : 0 ≤ λ ≤ 1} . We call the ordered couples (λ, w) ∈ Λ × Sn a committee and the set of all possible committees Λ × Sn is denoted Mn. The vector w from Sn is called a vector of weights 2.2. Computing efficiency 15 and the number λ from Λ is called a quota. A coalition will be called any set of voters, which are represented by a set of indices Q ⊆ N, so that j ∈ Q ⇔ voter represented by index j accepts the proposal. In other words, a coalition is a set of all the voters who accept the proposal. In the formal definitions it is quite convenient to use the n-dimensional unit cube representation. Suppose there is an n-dimensional unit cube and denote the set n n of all its vertices C , i.e. C = {(c1, ··· , cn): ci ∈ {0, 1} , i ∈ {1, ..., n}} . The cardinality of Cn is 2n. A proposal is approved if and only if the sum of the weights of those voters who accept the proposal is equal to or greater than the quota, i.e.:

n X wici ≥ λ, (2.1) k=1 n where c = (c1, . . . , cn) ∈ C is defined as ci = 0 if the i-th voter rejected the proposal and ci = 1 if he or she approved it. n Hence a coalition can be given by j ∈ Q ⇔ cj = 1. Each c ∈ C represents just one coalition. The efficiency of a committee (λ, w) ∈ Mn is defined as the ratio of all the winning coalitions (coalitions that change the status quo) and all the possible coalitions. This can be formally expressed:

" n # 1 X X ε (λ, w) := I w c ≥ λ , (2.2) 2n i i c∈Cn i=1 where the I [A] is the identiﬁer of A, i.e. I [A] = 1 if and only if the condition A is true, otherwise it is I [A] = 0. The eﬃciency ε (λ, w) attains only values from the set:

k Q := : k = 1, ..., 2n . (2.3) 2n Since the efficiency function can only attain a finite number of values, it attains its maximum and minimum and it is not continuous. The efficiency is invariant to the order of voters, i.e.: Let Πn be a set of all permutations of the set {1, . . . , n} . Then:

n (∀π ∈ Π ) ε (λ, w1, . . . , wn) = ε λ, wπ(1), . . . , wπ(n) . (2.4)

2.2 Computing eﬃciency

Standard algorithms that compute the exact eﬃciency of voting could be used only for committees with a low number of voters, as they have to check 16 2. Estimating the Eﬃciency of Voting in Big Size Committees

all possible coalitions. Computing the efficiency of voting is an NP-complete problem, see [7] and [6], since adding one more voter approximately doubles the computation time. I have created two algorithms for computing the exact efficiency for general committees. The first one is a simple recursive algorithm (or SRA), the second one is a recursive algorithm with a branch and bound technique employed (or SRB) so as to omit some of the irrelevant committees, once they are known, from further computation. The SRA algorithm seems to be faster for quotas around one half, the SRB algorithm is faster for quotas close to 1 or 0. The reason is clear. The SRB algorithm is faster due to the branch and bound method, which can prune some branches of irrelevant solutions when the quota is close to 1 or 0. On the other hand, the branch and bound technique itself consumes time in verifying the solution relevance in each recursive step. These verifications are very time consuming and can be justified only by saving time via the pruning. However, for quotas around one half, the pruning is not sufficient to prevail over the negative impact of checks on the procedural time. We can omit computing the exact efficiency and try to estimate it. The question is whether the estimation can be done as precisely as needed in practical applications. In real life we need to be able to estimate the efficiency in percents with reliable certainty. Estimation error can be controlled in probability. I have created a heuristic algorithm (or HRA), which gives not an exact, but a sufficiently precise solution to the efficiency of a simple weighted voting system for committees with a high number of voters. The HRA algorithm is based on simulating coalitions and checking whether they are winning or not. The number of simulated coalitions will only be a negligible part of the total number of coalitions. Suppose we define a random variable Xi as Xi = 0 if and only if the i-th coalition (in the infinite sequence of randomly chosen coalitions from the set of all coalitions) is losing, and Xi = 1 if and only if it is winning. The random sequence Xi is i.i.d. as the draws from the set of all coalitions are performed independently∗ and each coalition is generated with the same probability. Then, due to the strong law of large numbers,

n 1 X lim Xi = EX1, [P ] − a.s., n→∞ n i=1

because E |X1| < ∞. We know the average converges to a unique number. In addition, it can be shown that this number equals the eﬃciency. The remain-

∗ In each step of the simulation some coalitions are generated independently of the previously generated coalitions and hence each coalition can appear more than once in the generated sequence of coalitions. 2.2. Computing efficiency 17 ing question is the speed of the convergence, or rather the deviance of the 1 Pn average Xn = n k=1 from the exact efficiency. When P (Xk ∈ [ak, bk]) = 1 for 1 ≤ k ≤ n, we can use the well-known Hoeffding’s inequality:

n ! ! X 2t2n2 P X − nε (λ, w) ≥ tn ≤ 2 exp − . (2.5) k Pn 2 k=1 k=1 (bi − ai)

In our case ak = 0 and bk = 1 for all 1 ≤ k ≤ n, and so we get:

n ! 1 X P X − ε(λ, w) ≥ t ≤ 2 exp −2nt2 n k k=1 and apply it for t = 0.01 as a margin of error. We get the probability of an error that is greater than or equal to 0.01 (the efficiency will be different from the exact one by at least 0.01), equal to or lower than 2e−0.0002n. When using the HRA algorithm we perform 50,000 iterations, and so the probability will be at most 0.0000908. As previously mentioned, the HRA algorithm is based on independent draws (with repeats) from the set of all possible coalitions and verifying whether they are winning or not. Each draw is done in two steps. In the first step the size of the simulated coalition is randomly chosen from the binomial distribution of the probability, and in the second the voters in this coalition are chosen from the set of all voters performing independent draws from the uniform discrete distribution without repeats (since we do not want to have one voter in any coalition more than once). In the first step the HRA algorithm has to randomly generate the size of the coalition from the binomial distribution. This procedure has proven to be the most time consuming part of the whole process of efficiency estimation, namely because of the very large numbers of voters, since the binomial distribution requires computing the binomial coefficients. The higher the number of voters, the higher the factorial that needs to be computed. If it were really necessary, we could have not utilized the algorithm for very big committees as it would become time consuming to compute the factorials. Fortunately, computing the factorials is not necessary as we can employ the equality: s X n n−k (s + 1)(1 − p) pk (1 − p) = F , (2.6) k 2(n−s),2(s+1) p(n − s) k=0 where F is the distribution function of the Fisher-Snedecor probability distribution, see Andˇel(2004). 18 2. Estimating the Efficiency of Voting in Big Size Committees

0.0468

0.0463

0.0458

0.0453

0.0448

0.0443

0.0438

0.0433 Efficiency estimates Real efficiency 0.0428

Fig. 2.1: Deviances of simulated eﬃciencies from their average (100 trials)

The HRA algorithm uses the equality (2.6) to simulate the size of the coalitions from binomial distribution and thus huge committees can be analyzed. I have tested the performance of the HRA algorithm and it works well in terms of processing time: It always runs 50,000 iterations, no matter the number of voters, and so it has a constant time complexity. Some simple results showing the variability of the estimations are shown in the following images: In Figure 1, there are 100 estimations of the efficiencies of a simple weighted voting system with 35 voters (with the given distribution of weights and a quota of 0.6; the exact values are not important for now). In this figure, the deviance of each observation from their average is shown. In Figure 2 the histogram of the estimated efficiencies is shown. In Table 2.2, some results of the heuristic algorithm for specified committees are shown.

2.3 Eﬃciency analysis

1 1 Lemma 1. The maximum eﬃciency for a quota higher than 2 is 2 for any committee size n ≥ 2. 2.3. Eﬃciency analysis 19

Tab. 2.1: Some estimated efficiencies via the HRA algorithm 2654) 4247) 14357) 89044) 27983) 88251) 50367) . . . 16498) 99421) . . . . 0 . . 0 0 0 0 0 0 , , 0 0 , , , , , , , 26622 42445 14357 88972 27865 88294 49862 . . . 16434 99433 . . . . 0 . 0 . 0 0 0 0 0 , , 0 0 , , , , , , , 26418 1665 42324 14161 89062 27860 88287 50274 . . . . 99414 . . . . 0 . 0 0 0 0 0 0 0 , , , 0 , , , , , , Efficiency estimates 26395 16482 42444 14113 89099 27958 88223 50055 . . . . 99457 . . . . 0 . 0 0 0 0 0 0 0 , , , 0 , , , , , , (5 independent computations) 26895 9939 16514 42403 14378 27947 88471 89129 49863 ...... , 265625 (0 279296875994140625 (0 164661884 (0 882782936 (0 (0 890625146484375 (0 (0 5423583984 (0 (0 ...... Exact efficiency 0 0 0 0 0 0 0 0 0 , , 1 2 2 36 64 64 1 , , , 5 100 100 1 2 3 , 36 64 64 , , , , 1 2 50 5 100 1 3 3 100 36 64 64 , , 1 1 , , 64 64 , , 1 50 , , 2 1 3 4 36 64 64 , 100 10 0 1 1 100 3 , 64 64 , , 20 , , 1 , , 50 , , , 1 6 1 3 4 0 36 64 64 , 3 , , 1 1 1 100 , 64 64 10 , , 20 100 1 0 6 1 50 , , , , 2 , 3 5 , , 36 64 64 , 1 1 1 1 6 1 4 , 64 64 , , 10 10 1 100 10 50 , , , 100 , 2 , 4 5 36 , 64 64 6 , 1 , 1 1 2 1 , 64 64 , , 15 , 10 3 5 50 , , 1 6 , 4 , 4 5 100 15 36 64 64 , 100 1 6 , 2 2 , 8 , 64 64 , , , 10 6 1 2 , 50 , , 4 5 5 2 5 8 36 64 64 , 100 15 3; 2 2 100 , 64 64 , , . , 55; 7; 18 50 Quota and weights , . , , 0 . 4 6 5 36 64 64 , 0 0 2 2 20 , 64 64 , , 100 15 100 50 22 , , , 5 7 5 36 , 64 64 2 2 , 64 64 9; , , 55 . 100 15 100 5 9 0 36 10 64 64 1; 75; . . 55; 35; 63; 0 . . . 0 0 0 0 =21 =6 =6 =9 =9 =9 =12 =21 =6 n Size n n n n n n n n 20 2. Estimating the Efficiency of Voting in Big Size Committees

40 35 35 32 30

Frequency 15 15 13

5 3 2 0 0.043372 0.044116 0.04486 0.045604 0.046348 0.047092 Efficiency

Fig. 2.2: Histogram of simulated eﬃciencies

Proof: Assume n ≥ 2. The eﬃciency of 1 is certainly attained for any quota  2 

1   Pn higher than 2 by assigning w = 1, 0,..., 0  . When i=1 wici > λ >  | {z }  (n-1)-times 1 Pn Pn 2 , then i=1 wi (1 − ci) < λ and i=1 wi (1 − ci) is surely not a winning coalition. So for each winning coalition there is at least one losing coalition 1 and so the efficiency can not exceed 2 . 1 As we show in Lemma 1, no efficiency for a quota above 2 can be above 1 2 , and so we would change any heuristic estimate of the efficiency that is 1 higher than the attainable maximum to 2 .

   1 1  Lemma 2. Suppose n ∈ N, n > 1 and w =  ,..., . If n is odd, then n n | {z } n-times n ( n ) 1 1 1 1 2 ε 2 , w = 2 . If n is even, then ε 2 , w = 2 1 + 2n .

n Proof. The coalitions that are winning are all 2 −element coalitions and 2.3. Eﬃciency analysis 21 all coalitions with more elements.    n b 2 c n 1  1 1  X n−k ε  ,  ,...,  = (2.7) 2 n n 2n | {z } k=0 n-times

n ( n ) 1 1 2 For odd n (2.7) equals 2 . For even n (2.7) equals 2 1 + 2n .  

1  1 1  Lemma 3. Suppose n ∈ N, n > 1, λ > and w =  ,..., . Then: 2 n n | {z } n-times n n X ε (λ, w) = n−k . (2.8) 2n k=dnλe

Proof. The coalitions that are winning are all dnλe −element coalitions as well as all coalitions with more elements.    n n   1 1  X n−k ε λ,  ,...,  = (2.9)  n n 2n | {z } k=dnλe n-times From this simple analysis, we know how to find the weights maximizing 1 the efficiency, when the quota is greater or equal to 2 . The set of rules, which have to be fulfilled in order to approve a proposal, can be larger than just one-element sets. The multi-rule voting system is a system where more than one set of weights is assigned to the voters and more than one quota is employed. The proposal is approved only if accepted by all members of a coalition which is a winning coalition under each of the single rules. These systems are closely studied in Leech et al. (2007). Up to now, we have studied only one-rule systems represented by (2.1) and here we define multi-rule systems analogously. A multi-rule weighted voting system is a system in which each proposal is approved if and only if all of the following hold at the same time: Pn k=1 w1ici ≥ λ1, . . (2.10) Pn k=1 wmici ≥ λm, 22 2. Estimating the Efficiency of Voting in Big Size Committees

n where c = (c1, . . . , cn) ∈ C is deﬁned as ci = 0 if the i-th voter rejected the proposal and ci = 1 if he approved it. Suppose m ∈ N, m > 1. The eﬃciency of a committee

1 m λ1, . . . , λm, w ,..., w ,

i Pn where w = (wi1, . . . , win), j=1 wij = 1 for all i = 1, . . . , m and wij ≥ 0 for all i = 1, . . . , m and all j = 1, . . . , n, is deﬁned as the number of all winning coalitions divided by the number of all possible coalitions:

1 m ε λ1, . . . , λm, w ,..., w = (2.11) P I [Pn w c ≥ λ ∧ ... ∧ Pn w c ≥ λ ] c∈Cn i=1 1i i 1 i=1 mi i m , 2n where the I [A] is the identifier of A, i.e. I [A] = 1 if and only if the condition A is true, I [A] = 0 otherwise. While in the case of the one-rule voting system there is only one round of testing for which coalition is winning, in the multi-rule case once we have found the set of coalitions which are winning under the first rule, we continue with this set through to the second rule, third rule and so on. In other words, it is not a problem that would be more complex in terms of time complexity than the one-rule voting system. The efficiency defined in (2.11) can only attain values from a finite set. It attains its maximum and minimum and it is not continuous. I have enabled the algorithms SRA, SRB and HRA to treat the multi-rule voting systems simply by making them verify more than just one rule when verifying whether a given coalition is winning. I applied the HRA algorithm to compute the efficiencies of the qualified majority voting process of the Council of Ministers of the EU (a good example of the multi-rule voting procedure) under the Treaty of Nice and under the Lisbon Treaty. I also briefly analyze some of the rules separately and their contribution to the total efficiency. This approach shows the level of redundancy of the particular rules.

2.4 Applications

2.4.1 Czech Lower House of the Parliament in 2006 The Czech Parliamentary voting system used for the Lower House of the Parliament is based on the law Zákon247/1995 Sb., O volbáchdo Parlamentu Ceskérepublikyˇ a o zmˇenˇea doplnˇen´ınˇekterýchdalˇs´ıchzákon˚u (1995). It is a two-stage voting system. In the first stage, mandates are allocated 2.4. Applications 23

to regional districts defined by the law using a combination of Hagenbach- Bischoff’s quota and the method of maximal reminder. Then the limit of 5% is applied to reject all the parties which do not have over 5% of all the valid votes at the national level from further allocation. In the second stage the d’Hondt method is used to assign mandates to the parties which have not been rejected. The second stage is performed in each district separately. There are 200 mandates in the Lower House of the Parliament. The regular approval process is based on the majority rule.† This corresponds to a committee with a quota equal to q = 101/200 and weights equal to the shares of mandates assigned to each party in the elections.‡ For the constitutional proposals approval process the quota is different q = 3/5. First we compute the efficiency of the absolute majority committee resulting in the 2006 Parliamentary elections and we look at how different the political division of the Lower House of the Parliament would have to be in order to increase the efficiency to its maximum. Then we compute the efficiency for the constitutional approval process.

Party (name in 2006) Votes (in %) Deputies ODS 35.38% 81 CSSDˇ 32.32% 74 KSCMˇ 12.81% 26 KDU-CSLˇ 7.22% 13 Strana zelen´ych 6.29% 6

Tab. 2.2: The results of the 2006 Parliamentary elections in the Czech Republic

The results of the 2006 Czech Parliamentary elections are shown in the Table 2.2. The efficiency of the absolute majority committee given by the results of 2006 elections equals 0.46875. This result can be obtained via the SRA or SRB algorithms since the number of parties is low. We know from the Lemma 1 that the maximum possible efficiency for quotas above 1/2 equals 1/2. Hence we search for the closest vector of weights with the efficiency 1/2. The resulting set of weights for the quota 101/200 (corresponding to the set of parties (ODS, CSSD,ˇ KSCM,ˇ KDU-CSL,ˇ SZ) is (0.40285, 0.3689, 0.1361, 0.06305, 0.0291) compared to the observed vector of weights given to the par-

† We only analyze the absolute majority rule since the simple majority can be represented by many committees with diﬀerent weights according to how many members of each party are momentarily absent. ‡ This committee is based on the assumption that each member voting is always the same as all the other members of the same party. 24 2. Estimating the Eﬃciency of Voting in Big Size Committees

ties in the 2006 elections (0.405, 0.37, 0.13, 0.065, 0.03). The result is somewhat surprising because it is clearly not that far from the actual set of weights (the distance in L2-norm equals 0.00690326, which is about 0.8% of the maximal distance in L2-norm for 5-member voting systems). The numbers of deputies assigned to each party would have to be modified very little in order to achieve the maximal efficiency from (81, 74, 26, 13, 6) to (80, 74, 27, 13, 6) . Regardless, we have to keep in mind that this analysis is purely theoretical and neglects the political positions of each of the parties. In the case of a constitutional majority with a quota equal to 3/5 the efficiency is 0.25.

2.4.2 The European Parliament in 2009 At the end of 2009 there were 736 members representing 27 EU countries in the EU Parliament, which was governed by the Maastricht Treaty. The numbers of mandates assigned to each country are shown in the Table 2.3 and follows the degressive proportionality principle. Assume we want to revise the weights assigned to each particular country in order to increase the efficiency. To do this we have to take into account the proposals for which all the countries are voting homogeneously.§ We assume they follow the procedure of absolute majority voting. The efficiency of voting in the European Parliament for an absolute majority voting procedure under the assumption of the probability of acceptance equal to 0.5 is 0.49798815. This is very close to the maximal possible efficiency, which is 0.5. Therefore, there is no practical need to change the weights to achieve a higher a priori efficiency of the committee under the assumption of a country-homogeneous proposal. The efficiencies computed for the European Parliament absolute majority voting system work under the assumption of country-homogeneous voting as shown in Table 2.4. At the end of 2009 in the European Parliament there were 6 political groups, in which 709 members were organized, and 27 independent members not belonging to any of these 6 groups. The groups are listed in Table 5, where the independent members are assigned to a group of Independents. However, each member of the Independents is treated as one single political party for the efficiency computation. When we assume the absolute majority procedure and only consider the proposals for which the voting is party-homogeneous (all members within one political group vote the same way), we end up with a different committee. I

§ For most of the real proposals in the EU Parliament voting about proposals is based on political viewpoints rather than nationality. 2.4. Applications 25

Tab. 2.3: Portions and numbers of mandates assigned to each country in the Eu- ropean Parliament via the Maastricht Treaty

Country Weight (in %) Members Country Weight (in %) Members Germany 13.5% 99 France 9.8% 72 Italy 9.8% 72 Belgium 3.0% 22 Netherlands 3.4% 25 Luxembourg 0.8% 6 Great Britain 9.8% 72 Denmark 1.8% 13 Ireland 1.6% 12 Greece 3.0% 22 Spain 6.8% 50 Portugal 3.0% 22 Sweden 2.5% 18 Austria 2.3% 17 Finland 1.8% 13 Poland 6.8% 50 Czech Republic 2.7% 22 Hungary 2.7% 22 Slovakia 1.8% 13 Slovenia 1.0% 7 Latvia 1.1% 8 Lithuania 1.6% 12 Cyprus 0.8% 6 Estonia 0.8% 6 Malta 0.7% 5 Romania 4.5% 33 Bulgaria 2.3% 17

have analyzed the eﬃciencies of this voting system for diﬀerent probabilities of acceptance. The results are shown in Table 2.6. When we look at the graph in Figure 2.3, we can conclude: (i) When the proposal is in the category of country-homogeneous proposals (each country votes as one individual), the probability it will be approved is higher than that of a proposal from the category of party- homogeneous proposals, but only for probabilities of acceptance higher than approximately one half. (ii) When the proposal is in the category of country-homogeneous proposals (each country votes as one individual), the probability it will be approved is lower than that of a proposal from the category of party- homogeneous proposals, but only for probabilities of acceptance lower than approximately one half.

2.4.3 The Council of Ministers of the EU in 2009 In this part, I study the efficiency of qualified majority voting in the Council of Ministers of the EU under the Treaty of Nice and the Treaty of Lisbon. The Council of Ministers of the EU has 27 Members States. Their weights in 26 2. Estimating the Efficiency of Voting in Big Size Committees

Tab. 2.4: The eﬃciency of the absolute majority of the 2009 European Parliament given the probabilities of acceptance and country-homogeneous voting

p 0 0.05 0.1 0.15 0.2 ε 0% 0% 0.004% 0.104% 0.532% p 0.25 0.3 0.35 0.4 0.45 ε 1.974% 5.422% 12.130% 21.822% 35.188% p 0.5 0.55 0.6 0.65 0.7 ε 49.8500% 64.564% 77.534% 88.162% 94.348% p 0.75 0.8 0.85 0.9 0.95 ε 97.888% 99.368% 99.872% 99.998% 100.000%

Tab. 2.5: The political structure of the European Parliament as of the end of 2009

EPPD (Conservative/Christian Democrat) 265 ECR (Conservatives only) 54 S&D (Social Democrats) 184 EUL/NGL (Communists/Far-left) 35 ALDE (Liberal/Centrist) 85 G/EFA (Greens/Regionalists) 55 NI (Independents) 27 EFD (Eurosceptics) 31

the qualiﬁed majority voting rule and the most recent available population estimation of each of the Member States are shown in Table 2.7. The multi-rule voting system applied to most of the proposals that are subject to voting in The Council of Ministers of the EU under the Treaty of Nice is the qualiﬁed majority voting, which is given by the following three rules:

(i) The sum of the weights of the approving states has to be at least 255 out of 345 to accept a proposal.

(ii) The number of the approving states has to be at least 14 out of 27 to accept a proposal.

(iii) The population of the approving states has to make up at least 62% of the total population of the EU to accept a proposal. 2.4. Applications 27

Tab. 2.6: The eﬃciency of the absolute majority of the 2009 European Parliament given the probabilities of acceptance and party-homogeneous voting

p 0 0.05 0.1 0.15 0.2 ε 0% 0.308% 1.524% 3.532% 6.762% p 0.25 0.3 0.35 0.4 0.45 ε 10.936% 16.824% 23.282% 31.548% 39.708% p 0.5 0.55 0.6 0.65 0.7 ε 49.800% 59.392% 68.220% 76.342% 82.952% p 0.75 0.8 0.85 0.9 0.95 ε 89.074% 93.332% 96.400% 98.536% 99.676%

The efficiency estimation (given the probability of acceptance is 0.5 for all members) of the qualified majority multi-rule voting system in The Council of Ministers of the EU under the Treaty of Nice equals approximately 1.9%. The estimated efficiencies (probabilities of approving a proposal) for the different probabilities of acceptance of a single voter are shown in the Table 2.8 and Figure 2.5.

Tab. 2.7: The population and weights assigned to states in qualiﬁed majority voting in the Council of Ministers of the EU via the Treaty of Nice

State Weight Population (mil.) State Weight Population (mil.) Germany 29 82.3 Italy 29 59.7 France 29 64.5 United Kingdom 29 61.0 Spain 27 45.2 Poland 27 38.1 Romania 14 22.3 The Netherlands 13 16.4 Belgium 12 10.6 Czech Republic 12 10.4 Greece 12 11.2 Hungary 12 10.0 Portugal 12 10.6 Austria 10 8.3 Sweden 10 9.2 Bulgaria 10 7.7 Denmark 7 5.5 Denmark 7 5.5 Ireland 7 4.4 Lithuania 7 3.4 Slovakia 7 5.4 Finland 7 5.3 Cyprus 4 0.8 Estonia 4 1.4 Latvia 4 2.3 Luxembourg 4 0.5 Slovenia 4 2.0 Malta 3 0.4 28 2. Estimating the Eﬃciency of Voting in Big Size Committees

0.9

0.8

0.7

0.6

0.5

0.4 lity of approval (efficiency) i i 030.3

Probab 0.2 Nationaly homogeneus 0.1 Politicaly homogeneous 0 0 0.2 0.4 0.6 0.8 1 Probability of acceptance

Fig. 2.3: The eﬃciency with respect to the probability of acceptance within the 2009 European Parliament absolute majority procedure

Tab. 2.8: The eﬃciency of the qualiﬁed majority via the Treaty of Nice for the given probabilities of acceptance p 0 0.05 0.1 0.15 0.2 ε 0% 0% 0% 0% 0% p 0.25 0.3 0.35 0.4 0.45 ε 0.0005% 0.0075% 0.034% 0.173% 0.600% p 0.5 0.55 0.6 0.65 0.7 ε 1.984% 5.288% 11.587% 22.520% 38.089% p 0.75 0.8 0.85 0.9 0.95 ε 56.9167% 75.4645% 89.856% 97.452% 99.828%

We know, the total number of coalitions equals 227 = 134, 217, 728. In Table 2.9 and in Figure 2.4, the efficiencies for the modified qualified majority of The Council of Ministers of the EU under the Treaty of Nice are shown. There are two modifications: The first one leaves out the rule of population (denoted ε1) and the second leaves out the rule of artificial weights (denoted ε2). As can be seen in the graph, the impact of the rule of population in the 2.4. Applications 29

Tab. 2.9: The impact of distinct rules on the eﬃciency of the qualiﬁed majority via the Treaty of Nice for the given probabilities of acceptance

p 0 0.05 0.1 0.15 0.2 ε1 0% 0% 0% 0% 0% ε2 0% 0% 0% 0% 0.006% p 0.25 0.3 0.35 0.4 0.45 ε1 0% 0.008% 0.038% 0.170% 0.696% ε2 0.038% 0.358% 1.460% 4.386% 10.272% p 0.5 0.55 0.6 0.65 0.7 ε1 1.970% 5.162% 11.700% 22.324% 37.918% ε2 18.804% 31.156% 44.764% 59.068% 71.582% p 0.75 0.8 0.85 0.9 0.95 ε1 56.736% 75.772% 89.9313% 97.456% 99.856% ε2 82.852% 91.408% 96.330% 99.120% 99.890%

0.9

0.8

0.7

0.6

0.5

0.4 lity of approval (efficiency) i i 030.3 Without population rule 0.2

Probab Without artificial rule 0.1 Without majority rule

0 0 0.2 0.4 0.6 0.8 1 Probability of acceptance

Fig. 2.4: The impact of the distinct rules on eﬃciency within the procedure of the qualiﬁed majority via the Treaty of Nice

qualified majority voting under the Treaty of Nice is almost redundant. The only reason for it to be applied is for protection against a radical change in the population distribution among member states. Interestingly, leaving out 30 2. Estimating the Efficiency of Voting in Big Size Committees the rule of artificial weights would significantly increase the probability of acceptance. On the other hand, this would mean overwhelming the small states. This result seems to be quite intuitive since the concept of fairness and the concept of efficiency are often in contradiction. Under the Lisbon Treaty the procedure of the qualified majority can be simplified to just two rules: (i) At least 55% of all the states have to accept the proposal.

(ii) The population of the approving states has to make up at least 65% of the total population of the EU to accept a proposal. The efficiency estimation (given that the probability of acceptance is 0.5 for all members) of the qualified majority multi-rule voting system in The Council of Ministers of the EU under the Lisbon Treaty equals approximately 12.7%. The estimated efficiencies (probabilities of approving a proposal) for the different probabilities of the acceptance of a single voter are shown in the Table 2.10 and Figure 2.5.

Tab. 2.10: The eﬃciency of the qualiﬁed majority via the Treaty of Lisbon for the given probabilities of acceptance

p 0 0.05 0.1 0.15 0.2 ε 0% 0% 0% 0% 0.002% p 0.25 0.3 0.35 0.4 0.45 ε 0.030% 0.128% 0.676% 2.296% 6.090% p 0.5 0.55 0.6 0.65 0.7 ε 12.664% 23.326% 36.248% 50.034% 64.262% p 0.75 0.8 0.85 0.9 0.95 ε 78.028% 87.642% 94.768% 98.534% 99.852%

In the Table 2.11 and Figure 2.6, the efficiencies for the modified qualified majority of The Council of Ministers of the EU under the Lisbon Treaty are shown. There are two modifications: The first one leaves out the rule of population (denoted ε1) and the second one leaves out the rule of the number of states (denoted ε2). As can be seen in the graph, the impact of the rule of the number of states in the qualified majority voting under the Lisbon Treaty is almost redundant. On the other hand leaving out the population rule would significantly increase the probability of acceptance. 2.4. Applications 31

0.9

0.8

0.7

0.6

0.5

0.4

lity of approval (efficiency) 030.3

0.2 Probabi Nice treaty 0.1 Lisbon Treaty 0 0 0.2 0.4 0.6 0.8 1 Probability of acceptance

Fig. 2.5: The eﬃciency with respect to probability of acceptance within the procedure of qualiﬁed majority via the Treaty of Nice and the Lisbon Treaty

Tab. 2.11: The impact of distinct rules on the eﬃciency of the qualiﬁed majority via the Lisbon Treaty for the given probabilities of acceptance p 0 0.05 0.1 0.15 0.2 ε1 0% 0% 0% 0% 0.004% ε2 0% 0% 0% 0.028% 0.096% p 0.25 0.3 0.35 0.4 0.45 ε1 0.074% 0.422% 2.324% 7.458% 18.138% ε2 0.336% 1.060% 2.514% 5.250% 10.198% p 0.5 0.55 0.6 0.65 0.7 ε1 34.982% 55.706% 75.146% 88.650% 96.442% ε2 16.876% 26.764% 38.508% 51.716% 65.286% p 0.75 0.8 0.85 0.9 0.95 ε1 99.274% 99.908% 99.998% 100.000% 100.000% ε2 78.056% 87.620% 94.786% 98.578% 99.842%

If we compare the qualified majority under the Treaty of Nice and under the Lisbon Treaty, we end up with an expected result of a significantly higher efficiency under the Lisbon Treaty since the smaller states lose part of their 32 2. Estimating the Efficiency of Voting in Big Size Committees

0.9

0.8

0.7

0.6

0.5

0.4 lity of approval (efficiency) i i 030.3 Without population rule

Probab 0.2 Without majority rule 0.1 Both rules

0 0 0.2 0.4 0.6 0.8 1 Probability of acceptance

Fig. 2.6: The impact of the distinct rules on eﬃciency within the procedure of a qualiﬁed majority via the Lisbon Treaty

ability to block proposals.

2.5 Conclusion

I have provided two exact and one heuristic algorithm for efficiency computation and some basic theoretical analysis of the efficiency function as a function of quota and weights, focusing mainly on studying the quota and de- riving some characteristics of this function with respect to the quota. Then I applied this knowledge to some practical problems. At first finding how far is the outcome of the 2006 Parliamentary elections for the Czech Lower House of the Parliament from an outcome that would represent a vector of weights for which the maximal efficiency would be attained. I have found out that the situation in 2009 in the Czech Lower House of the Parliament was very close to what would represent a maximal a priori efficiency (as defined in the article) for the approval of ordinary proposals and the outcome is relatively far from the maximum a priori efficiency for the approval of constitutional proposals. The quota for constitutional proposal approval seems to be sufficient as a constitution safety guarantee because the closest vector of weights that represents a maximal a priori efficiency is only for two parties being elected to the Lower House of the Parliament, each with the weight 0.5. REFERENCES 33

Then I computed the efficiency of the voting system of the former Euro- pean Parliament, assuming the simple majority quota for both country and political dimensions. Finally the heuristic algorithm was used for comparing the efficiency of voting under the qualified majority rule in the Council of Ministers of the EU under the Treaty of Nice and the Lisbon Treaty. Then each of these two procedures were analyzed separately to verify the distinct rules and their impact on the final efficiency. We could see the population rule in the qualified majority under the Treaty of Nice is redundant. How- ever the most influential rule of artificially assigned weights to each state was abandoned in the procedure under the Lisbon Treaty and hence the efficiency of the voting increased significantly. The developed heuristic algorithm generally runs at a constant time no matter the number of voters, but with a limited preciseness. It can be used, with slight changes, for estimations of most of the power indices. The generalized power indices [1] would require deeper changes.

Acknowledgment This research was supported by the Grant Agency of the Czech Republic, project No. 402/09/1066 “Political economy of voting behavior, rational voters’ theory and models of strategic voting”.

References

[1] Fuad Aleskerov. Power indices taking into account agents’ preferences. Mathematics and Democracy, 2006.

[2] J. S. Coleman. Control of collectivities and the power of a collectivity to act. Social Choice, B. Lieberman (ed.), pages 269–300, 1971.

[3] D. S. Felsenthal and M. Machover. The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes. Edward Elgar, Chel- tenham, 1998.

[4] M. Hosli. Balancing legitimacy and eﬃciency: Council decision rules and european constitutional design. AUCO, Czech Economic Review, 2(2), 2008.

[5] D. Leech and H. Aziz. The double majority voting rule of the eu reform treaty as a democratic ideal for an enlarging union: an appraisal using voting power analysis. Paper presented at the European Public Choice Society meeting in Jena, 27.-30.3., 2008. 34 2. Estimating the Eﬃciency of Voting in Big Size Committees

[6] T. Matsui and Y. Matsui. NP-completeness for calculating power indices of weighted majority games. Theoretical Computer Science, 263(1-2):305– 310, 2001.

[7] Tomomi Matsui and Yasuko Matsui. A survey of algorithms for calculating power indices of weighted majority games. J. Oper. Res. Soc. Japan, 43:71–86, 2000. 3. OPTIMIZING EFFICIENCY OF WEIGHTED VOTING SYSTEMS

Abstract Having some group of voters endowed with weights. The simple weighted voting game (or system) represents such a system of approving propositions, in which approved is only a proposition that is accepted by voters weighted to a number that is at least equal to a prescribed number called a quota. We call the system simple if there is only one set of weights and one quota, as opposed to the multi-rule systems with more weights assigned to each voter and with more quotas. This paper presents an analysis of the efficiency of simple weighted voting systems. It assumes the Impartial Anonymous Culture (the probability of a single voter voting for a propo- 1 sition is 2 and voters act independently). This culture is used for general evaluation of voting systems, when no specific information about propositions and voters’ preferences are known, or when the voters’ preferences and proposition characteristics are not willing to be reflected in the voting system itself, having in mind its non-pragmatics, fairness and generality. The efficiency of simple weighted voting system is defined as the probability of a proposition being approved. This paper focuses on efficiency maximization and minimization with respect to weights. We proove theorem which enables to compute the efficiency maximum and efficiency minimum with respect to weights given the number of voters and quota in linear time. Keywords Weighted voting game, Integer programming, Efficiency of voting, IAC models JEL classification D71, D72

3.1 Introduction

Voting systems are systems transforming many individual preferences into one. By a simple weighted voting system we mean a voting system that processes a proposition and assigns the proposition a single value: accepted or rejected. If accepted, the proposition is applied and changes the status quo. If rejected, it is not applied and does not change the status quo. The proposition is accepted within the system if, and only if, the sum of weights 36 3. Optimizing Efficiency of Weighted Voting Systems that are assigned to all the voters who accept the proposition is above a prescribed real number called quota. For simplicity we assume that all the weights are real numbers between 0 and 1 (both including) and in addition the weights sum up to 1 (each simple weighted voting system can be normalized in this way). Simple weighted voting systems are used in various institutions all around the world, including Parliamentary and legislative institutions, United Nations institutions, European institutions, general committees of shareholders, academy awards committees, the Nobel prize committee, governments on national, district or city level, executive boards of national banks, courts of law, housing associations and many other institutions. Voting play a crucial role not only in the political markets, but also in traditional economic and financial markets, as the political decisions largely impact consumption, investment, inflation, unemployment and overall economic performance through fiscal and monetary policies. For the use of weighted voting systems in international organizations, see for example [13]. We consider two types of propositions. The first type we call simple, the second we call complex. Simple propositions are all those that can be inverted to their opposite. This means they specify such a change that admits just one alternative proposition. This alternative contradicts the original one proposed. An example of a simple proposition is one concerned with entering the European Union, as the only alternative to entering is not entering. For simple propositions there are generally only two states of the nature. On the other hand, complex propositions have many alternatives. An example of a complex proposition is changing a tax law, as the proposed change usually has many alternatives that cannot be proposed individually. Assume the law to be very simple, stating that the tax for individuals is 20% of their income. Then there is no single proposition on tax law in contradiction with the original. The tax rate can be 25% or 15%. Both these tax rates are possible alternative propositions in contradiction with the 20% tax rate. Hence the proposition on the tax rate is a complex proposition. In other words, simple propositions are those dealing with change that admits only two states of nature and complex propositions are those which admit at least three. The reason for distinguishing simple and complex propositions is to avoid the objections about concerning simple weighted voting systems with quotas under one half, which we admit. Sometimes there are complaints that once a proposition is rejected, its opposite should be accepted under the same voting system. This simple behavior is clearly not assured for quotas below one half. However, these objections are relevant only for simple propositions as for a complex proposition there is no single opposite proposition and so once a complex proposition is rejected, some of its contradictory propositions 3.1. Introduction 37

might be rejected as well under the same preference setting. Additionally in some cases, only certain types of propositions can be processed under the weighted voting system with a quota lower than one half - for example in general assembly of stock holders or in the bankruptcy proceedings∗). In many codes of rules over public and private associations or political parties, the most powerful authority (usually a congress, convention or assembly of all members) can be convened by less than half of all the members. Although to convene an authority may not be considered approving a proposition, it actually is an approval process changing the status quo. An important role the quotas below one half also play in the multi-rule voting systems, where more than just one rule must be fulﬁlled for a coalition to win. For example, consider a union of 10 countries, where two of them are very large and, based on their population, they would have majority on their own. But such a union is hardly acceptable for the other 8 countries, which would have no practical power in the union under the population majority rule. For such a union it might be useful besides the population rule, to apply another rule requiring (for example) at least 4 countries accepting the proposition for it to be approved. The quota would then be 0.4 in this rule and it makes sense to apply it. This reasoning led us to admit all possible quotas between 0 and 1. Weights can represent the share of mandates in some voting body, shares of property values of stock holders or creditors, populations of countries in supranational institutions and populations of counties or districts in national institutions†. If we wish to study the voting systems themselves, not having any information about the preferences of single voters, parties or voting countries, the natural way would be to assign each voter a theoretical probability to accept any proposition. We call it ”probability of acceptance”. This probability is considered to be the same for all the voters, as we do not have any information about the future propoitions. Having no such information about the propositions and voters’ preferences, it is natural to assume this probability is 0.5 and the voters act independently. Taking these assumptions together is often called an IAC (Impartial Anonymous Culture) assumption. The probability of a proposition being approved, we call ”probability of approval”. We will study the probability of approval (which we will also call ”an eﬃciency of the voting system”) under the IAC assumption as it is one of

∗ Under the Czech bankruptcy law, a creditor assembly can be convened when supported by at least two creditors with at least 10% share of the total value of all submitted receivables, i.e. the quota is 0.1. † For example in the Lower House of the Czech Parliament, the shares of mandates assigned to districts depend upon the number of valid votes cast in each district. 38 3. Optimizing Efficiency of Weighted Voting Systems the major characteristics of a voting procedure itself. This efficiency is also known in the literature as the Coleman index or the power of a collectivity to act and has been introduced in [3] and used (for example) in [9]. A slight generalization of Coleman index is studied in [10]. The Coleman index can be considered an a priori voting power assigned to each voter. We are interested in the behavior of a simple weighted voting system efficiency as a function of weights and quota. The questions we rise and answer are:

• What are the minima and maxima of the eﬃciency function with respect to the weights and can they be expressed analytically in terms of quota and number of voters?

• Is the eﬃciency function symmetric in some sense around the quota equal to one half?

• Are there any intervals in which the quota can move freely so that the eﬃciency maximum (resp. eﬃciency minimum) with respect to weights does not change? If so, how they can be formaly described?

• How many different values the efficiency maximum, resp. efficiency minimum attain in terms of the number of voters? Can this number be expressed explicitly as a function of n?

The concept of efficiency is quite clear, but the problem of computing the exact efficiency of a simple weighted voting system is the time complexity. It was shown to be an NP-complete problem, see [12], [11] or [14]. We do not study the multi-rule voting systems (see for example [8], [7] or [9]), as it would not bring any further significant insight to our analysis. For further information on weighted voting systems and their efficiency, see [6] (balancing legitimacy and efficiency), [16] (theoretical analysis of individual voting power), [2] (survey on computationally tractable weighted voting systems) and [4]. In the article [5], the authors defend the concept of a priori voting power opposing some of the objections being made to this concept by explicitly stating: ”The main purpose of measuring a priori voting power is not descriptive but prescriptive; not empirical but normative. It is indispensable in the proper constitutional design and assessment of decision rules. Here, it is important to quantify the voting power each member is granted by the rule itself.” Some further findings of computationally favorable assumptions for Banzhaf power index computations can be found in [1]. 3.2. Base Concepts 39

3.2 Base Concepts

Suppose we have a set N = {1, . . . , n} , n ∈ N. This set will represent the set of voters (voting bodies, parties, countries, etc.), so that each voter is represented by just one index from N. We will not consider the elements of N to be further divisible. Suppose a vector space Vn above the field n n of real numbers and the set S ⊂ V of all vectors w = (w1, . . . , wn) , Pn such that k=1 wk = 1 and wk ≥ 0 for any k ∈ {1, . . . , n} . The set of all real numbers between 0 and 1 (both including) we denote Λ. The ordered couples (λ, w) ∈ Λ × Sn we call committees. The vector w from Sn we call a vector of weights and the number λ from Λ we call a quota. Sup- pose the n-dimensional unit cube and denote the set of all its vertices Cn, n n i.e. C = {(c1, ··· , cn): ci ∈ {0, 1} , i ∈ {1, ..., n}} . The cardinality of C is clearly 2n. A proposition is approved if, and only if, the sum of the weights of those voters who accept the proposition is greater than or equal to the quota, i.e.: n X wici ≥ λ, (3.1) k=1 n where c = (c1, . . . , cn) ∈ C is defined as ci = 0 if the i-th voter rejected the proposition and ci = 1 if he or she approved it. We call coalition any subset n Q of N such that j ∈ Q ⇔ cj = 1. Each c ∈ C represents just one coalition. We say, the coalition Q ⊂ N is winning, when (3.1) and j ∈ Q ⇔ cj = 1 are fulfilled. The efficiency of a committee (λ, w) ∈ Λ × Sn is defined as the number of winning coalitions divided by the number of all possible coalitions P I [Pn w c ≥ λ] ε (λ, w) = c∈Cn i=1 i i , (3.2) 2n where the I [A] is the identifier of A, i.e. I [A] = 1 if, and only if, the condition A is true, I [A] = 0 otherwise. We are interested in finding the maximum efficiency (resp. minimum efficiency) with respect to weights given quota and number of voters. So we n n are searching for mλ and τλ given by (3.3) and (3.4)

n def mλ = max ε (λ, w) (3.3) w∈Sn n def τλ = min ε (λ, w) . (3.4) w∈Sn

3.3 Basic ﬁndings

Lemmas 4 and 5 are simple observations that we provide without rigorous proof and which provide us with the eﬃciency maxima and minima for certain 40 3. Optimizing Eﬃciency of Weighted Voting Systems quotas without any computations.

1 1 Lemma 4. The maximum eﬃciency for quota higher than 2 is 2 for any size committee n ≥ 2.

1 1 Lemma 5. The minimum eﬃciency for quota lower than 2 is 2 for any size committee n ≥ 2.

Lemma 6 is used in the proof of lemma 8. For many quotas it enables to lower the number of computations needed to be performed in order to find the efficiency maximum, resp. efficiency minimum.

Lemma 6. Let n ≥ 2, w = (w1, . . . , wn) and 0 < λ ≤ 1. Then at least

1 max 0, n − λ weights from the set {w1, . . . , wn} are lower than λ for any set of weights. Proof: There are two possibilities: 1 1) n ≤ λ 1 λ λ 1 We have n ≥ 1 = 1−p , where p ≥ 0 and hence n ≥ λ. So there b λ cλ exists such a set of weights where none is lower than λ. These weights are 1 1 for example, the weights n ,..., n . This is consistent with the statement 1 as max 0, n − λ = 0. 1 2) n > λ We will make the proof by contradiction. Assume that the statement of 1 the lemma does not hold. There would be exactly n − λ − m weights lower 1 than λ, for some m such that 0 < m ≤ n − λ , m ∈ N. Then there must 1 be exactly λ + m weights higher than or equal to λ and for the sum W of all these weights that are higher than or equal to λ must hold

1 1 W ≥ λ + m ≥ λ + 1 ≥ 1 − p + λ > 1, λ λ

1 because λ λ = 1 − p, where p < λ. But this is simply in contradiction with Pn i=1 wi = 1, wi ≥ 0. Hence the statement of the lemma must hold.

Examples: 3.4. Maximizing and minimizing eﬃciency with respect to weights 41

3.4 Maximizing and minimizing eﬃciency with respect to weights

The following theorem can be used for maximum and minimum eﬃciency computations, given quotas and numbers of voters.

Theorem 1. Suppose n ∈ N, λ ∈ Λ and w ∈ Sn. Then the maximum as well as minimum of voting eﬃciency ε (λ, w) is attained in at least one of the points from the set

         1 1    1 1  λ,  ,...,  , λ, 0, ,...,  ,...,  n n   n − 1 n − 1  | {z } | {z } n−times (n−1)−times       1 1  λ,  0,..., 0 , ,  , λ,  0,..., 0 , 1 (3.5) | {z } 2 2 | {z }  (n−2)−times (n−1)−times

The cardinality of this set is n.

Proof: At ﬁrst we express the problem as a maximization problem of integer 42 3. Optimizing Eﬃciency of Weighted Voting Systems

programming:

max 1Td + 0Tw (3.6) n d∈R2 ,w∈Rn

subject to 1¯Tw = 1 (3.7) T i di w c − λ ≥ 0 (3.8i) for all i ∈ {1,..., 2n} w ≥ 0 (3.9)

w1 ≤ w2 ≤ ... ≤ wn (3.10) n d ∈ {0, 1}2 (3.11)

where 1 is an 2n × 1-type vector of ones, 1¯ is an n × 1-type vector of ones, w is an n × 1-type vector of real numbers, d is an 2n × 1-type vector of real numbers and 0 ≤ λ < 1 is a real number and ci stands for the vector representing a vertex of an n-dimensional unit cube such that ci 6= cj ⇔ i 6= n j, for any i, j ∈ {1,..., 2n} . By changing the condition (3.11) to d ∈ [0, 1]2 (which we denote (3.11’)), we enlarge the set of all feasible solutions, but the optimal solution (d∗, w∗) remains the same as for whatever i ∈ {1,..., 2n} ∗ if 0 < di < 1, then the value of the objective function would increase when ∗ shifting the value of di to 1 while all the conditions remain to hold (because ∗ T i 0 < di , we have also w c ≥ λ and so (3.8i) remains to hold). Therefore, ∗ ∗ ∗ whenever 0 < di < 1, the solution (d , w ) cannot be optimal. Let’s denote F the set of all feasible solutions (i.e. all the vectors (d, w) , for which the conditions (3.7), (3.8i) for all i ∈ {1,..., 2n} , (3.9), (3.10) and (3.11’) hold). Now we prove the set F is convex. Assume, we have two diﬀerent vectors from F, let’s denote them d(1), w(1) and d(2), w(2) . We need to show, that α d(1), w(1) + (1 − α) d(1), w(1) ∈ F.

We easily confirm this fact by checking that the set given by each single constraint‡ is convex and use the fact, that any intersection of finite number of convex sets is also a convex set. For vectors fulfilling (3.7): n The set of all vectors (d, w) ∈ R2 +n which are given by (3.7) is convex

‡ Each single constraint from the set of constraints {(3.7), {(3.8i), i ∈ {1,..., 2n}} , (3.9), (3.10), (3.11’)}. 3.4. Maximizing and minimizing eﬃciency with respect to weights 43 as having any two diﬀerent vectors d(1), w(1) and d(2), w(2) , for which

n X (1) wj = 1 j=1 and n X (2) wj = 1 j=1 we have for any real 0 ≤ α ≤ 1 αw(1) + (1 − α) w(2) = α w(1) − w(2) + w(2) and so 1¯T α w(1) − w(2) + w2 = α − α + 1 = 1. So each convex combination of any two vectors from F fulfilling (3.7) fulfills (3.7) as well. For vectors fulfilling (3.8i): n The set of all vectors (d, w) ∈ R2 +n which are given by (3.8i) is convex, because having any two different vectors d(1), w(1) and d(2), w(2) , for which n ! (1) X (1) i di wj cj − λ ≥ 0 j=1 and n ! (2) X (2) i di wj cj − λ ≥ 0 j=1 we have for any real 0 ≤ α ≤ 1

" n # (1) (2) X (1) (2) i αdi + (1 − α) di αwj + (1 − α) wj cj − λ = j=1

" n # " n # (1) X (1) i (2) X (2) i αdi αwj cj − αλ +(1 − α) di (1 − α) wj cj − (1 − α) λ + j=1 j=1 n n (1) X (2) (2) X (1) αdi (1 − α) wj + (1 − α) di αwj = j=1 j=1 " n # " n # 2 (1) X (1) i 2 (2) X (2) i α di wj cj − λ + (1 − α) di wj cj − λ + j=1 j=1 | {z } | {z } ≥0 ≥0 44 3. Optimizing Eﬃciency of Weighted Voting Systems

n n (1) X (2) (2) X (1) αdi (1 − α) wj + (1 − α) di αwj ≥ 0. j=1 j=1 | {z } | {z } ≥0 ≥0 Hence each convex combination of any two vectors from F fulfilling (3.8i) fulfills (3.8i) as well. For vectors fulfilling (3.9): n The set of all vectors (d, w) ∈ R2 +n which are given by (3.9) is convex, because any convex combination of two non-negative vectors is a non-negative vector. Hence each convex combination of any two vectors from F fulfilling (3.9) fulfills (3.9) as well. For vectors fulfilling (3.10): n The set of all vectors (d, w) ∈ R2 +n which are given by (3.10) is convex, because having any two different vectors d(1), w(1) and d(2), w(2) , for (1) (1) (1) (2) (2) (2) which w1 ≤ w2 ≤ ... ≤ wn and w1 ≤ w2 ≤ ... ≤ wn , we get (1) (2) (1) (2) (1) (2) αw1 + (1 − α) w1 ≤ αw2 + (1 − α) w2 ≤ ... ≤ αwn + (1 − α) wn . Hence each convex combination of any two vectors from F fulfilling (3.10) fulfills (3.10) as well. For vectors fulfilling (3.11’): n The set of all vectors (d, w) ∈ R2 +n which are given by (3.11’) is convex as convex combination of two real numbers in [0, 1] is in [0, 1]. Hence each convex combination of any two vectors from F fulfilling (3.11’) fulfills (3.11’) as well. So we are maximizing linear function on a convex set and hence, the solution to the problem must be on the frontier of the set of all feasible solutions F. We easily find out, that§ 1 w ≤ . i n − i + 1 Let’s have a vector (d∗, w∗) that maximizes the objective function. We show this vector can be expressed as a convex combination of vectors from (3.5). The convex combination parameters 0 ≤ ak ≤ 1, k = 1, . . . , n can be explicitly computed by solving the following system of equations¶:

n X ak = w∗, for r = 1, . . . , n. (3.12) k r k=n+1−r

§ It is due to the condition (3.10) as otherwise, the weights would sum up to a number strictly higher than 1. ¶ For example by Gauss elimination method it is very easy, as the system matrix is already triangular. 3.4. Maximizing and minimizing eﬃciency with respect to weights 45

Each ak represents the parameter in convex combination that corresponds to the vector  

 ∗ 1 1 d , 0,..., 0, ,...,  .  k k  | {z } k−times ∗ We can compute the ak, k = 1, . . . , n from the wk, k = 1, . . . , n using the Gauss elimination.

∗ ∗ a1 = wn − wn−1 · 1 ∗ ∗ a2 = wn−1 − wn−2 · 2 ...... (3.13) ∗ ∗ an−1 = (w2 − w1) · (n − 1) ∗ an = w1 · n Now we have   n ∗ X  1 1 w = ak · 0,..., 0, ,...,  ,  k k  k=1 | {z } k−times Pn which is a convex combination of the vectors from (3.5) as k=1 ak = Pn ∗ ∗ ∗ k=1 wk = 1 and ak ≥ 0 for all k = 1, . . . , n. Finally the vector (d , w ) can be expressed as a convex combination of the vectors from (3.5) because ∗ Pn ∗ of course di k=1 ak = di . When minimizing the same objective function on the set F, the optimal solution is also on the frontier and hence the minimum must be also attained in the set (3.5).

This theorem tells us, that to find the maximum efficiency given quota, is a very easy and fast exercise. We need just to check n values to find out the maximum efficiency for any quota. In fact, we don’t even have to do that, as we will see later on. In the following lemma we use the theorem 1 and express the maximum and minimum efficiency explicitly as a function of quota and the number of voters.

Lemma 7. Assume the quota 0 < λ ≤ 1 and the number of voters n ∈ N, n > 1. Then n − i max 2i · Pn−i i∈{0,...,n−1} j=d(n−i)λe j mn = (3.14) λ 2n 46 3. Optimizing Eﬃciency of Weighted Voting Systems

and

n − i min 2n − 2i · Pn−i i∈{0,...,n−1} j=b(n−i)(1−λ)c+1 j τ n = . (3.15) λ 2n Proof: First we prove the maximization part. The proof is easy, once we know the eﬃciency maximum is attained in the set (3.5). Number of winning coalitions of size 0 < k ≤ n when considering the vector    1 1    0,..., 0, ,...,  (3.16) | {z } n − i n − i i-times | {z } (n−i)-times

n − i with no zero weight is , but only if k ≥ d(n − i) λe . So in total, k there are n−i X n − i k k=d(n−i)λe winning coalitions made of voters with non-zero weights. For each winning coalition made of voters with non-zero weights, there are 2i −1 other winning coalitions, including the voters with zero weights, as we have to add all the subsets of voters with zero weights except the empty setk. There are in total

n−i X n − i 2i · k k=d(n−i)λe

winning coalitions corresponding to the vector (3.16). When we compute this value for all the i = 0, . . . , n − 1 and we take the highest of them, we must have the maximum attainable number of winning coalitions. To compute the maximum eﬃciency, we just divide it by 2n. Now we prove the minimization part. We know from theorem 1 that minimum will be attained for vector of weights from the set (3.5) and hence we will further just consider the vectors from (3.5). Let’s denote the sum of weights of one of the winning coalitions W ≥ λ. Then clearly 1 − W ≤ 1 − λ. k Because the empty set represents the coalition made just of the voters with non-zero weights and this coalition has already been accounted. 3.4. Maximizing and minimizing eﬃciency with respect to weights 47

1 − W is the sum of weights of all the voters in complementary coalition∗∗. For each winning coalition for committees with quota λ, there is just one complementary coalition with a sum of weights lower than or equal to 1 − λ. Assume we have such a vector of weights that minimizes the number of winning coalitions for the quota λ. Then we also have the minimum number of complementary coalitions with sum of weights lower than or equal to 1−λ. As in total there are always 2n coalitions, minimizing the number of coalitions with sum of weights lower than or equal to 1 − λ with respect to weights is the same as maximizing the number of coalitions with sum of weights over 1 − λ with respect to weights. In total, there are

n−i X n − i k k=b(n−i)(1−λ)c+1

winning coalitions made of voters with non-zero weights for vector of weights    1 1    0,..., 0, ,...,  . (3.17) | {z } n − i n − i i-times | {z } (n−i)-times

Analogously to the ﬁrst part of the proof, the total number of winning coalitions corresponding to the vector (3.17) from the set (3.5) is

n−i X n − i 2n − 2i · . k k=b(n−i)(1−λ)c+1

Computing this for each vector from (3.5) we obtain this value for i = 0, . . . , n − 1 and by taking the minimum of them, we obtain the minimum number of winning coalitions under the quota λ. To compute the minimum eﬃciency, we just divide it by 2n.

The previous lemma shows how to compute the minimum and maximum attainable efficiency given quota λ and number of voters n. However as we mentioned, we need not compute all the n values to find the maximum and n values to find the minimum efficiency. In the following lemma we show it is not necessary to compute all the n values and that it suffices to compute

∗∗ We call complementary coalition such a coalition, which consists of all the voters that are not in the original coalition. 48 3. Optimizing Eﬃciency of Weighted Voting Systems

1 just n − min n, λ + 1 values. This means we can compute less than n 1 values to find the efficiency maximum whenever λ ≤ 2 , resp. the efficiency 1 minimum whenever λ > 2 . But according to lemmas 4 and 5, these are the only situations, that are of any interest.

Lemma 8. Assume the quota 0 < λ ≤ 1 and the number of voters n ∈ N, n > 1. Then i Pn−i n − i maxi∈ 0,...,n−min n, 1 2 · k=d(n−i)λe { { b λ c}} k mn = (3.18) λ 2n and n i Pn−i n − i mini∈ 0,...,n−min n, 1 2 − 2 · k=b(n−i)(1−λ)c+1 { { b 1−λ c}} k τ n = . λ 2n (3.19)

Proof: First we prove the part about maximum efficiency. Within this proof, we will always take into account only the vectors of weights from (3.5). Two situations can occur. In the first, no weight is higher than or equal to λ. In the second at least one weight is higher than or equal to λ. Now, we are interested just in the second situation. Because we have only vectors from (3.5), at least one weight higher than λ is the same as any non-zero weight is higher than or equal to λ. We need to specify, for which vectors from (3.5) all the non-zero weights are higher than or equal to λ. From lemma 6 we know, 1 that at least max 0, n − λ must be lower than λ. This implies that no 1 1 more than n − max 0, n − λ = min n, λ weights can be higher than ˆ 1 or equal to λ. Assume, there is exactly k = min n, λ weights higher than or equal to λ (this means we assume the vector      1 1  0,..., 0, ,...,   1 1   min n, λ min n, λ   | {z } 1 min{n,b λ c}-times from (3.5)). Indeed, 1 min n, ≥ λ. λ 3.4. Maximizing and minimizing efficiency with respect to weights 49

Hence for any kˆ < min n, 1 is 1 > λ and so any non-zero weight is also λ kˆ higher than λ and the eﬃciency equals to

Pn−1 j ˆ 2 j=n−k , 2n as there is 2n−1 coalitions containing the first non-zero weight, 2n−2 coalitions containing the second non-zero weight which do not contain the first non-zero weight, 2n−3 coalitions containing the third non-zero weight which contain nor the first, neither the second non-zero weight and so on up to the kˆ-th non-zero weight. In total there are exactly

n−1 X ˆ 2j = 2n − 2n−k (3.20) j=n−kˆ winning coalitions. As the sum (3.20) is strictly increasing in k,ˆ we may compute the ˆ 1 number of winning coalitions only for k = min n, λ , . . . , n as for any ˆ 1 k < min n, λ the efficiency is surely strictly lower than the one for ˆ 1 k = min n, λ . kˆ stands for the number of non-zero weights, while i in the expression (3.14) corresponds to the number of zero weights. The number of zero weights can be expressed as n − k.ˆ So we use the equation (3.14), but we take maximum just of the efficiency for vectors of weights from (3.5) which contain up 1 to max 0, n − λ zero weights. By doing this, we obtain (3.18). Now we prove the part about minimum efficiency using the equality (3.21)

    n − i  min 2n−2i·Pn−i i∈ 0,...,n−min n, 1 k=b(n−i)(1−λ)c+1  { { b 1−λ c}} k  n = 2   (3.21)  n − i  max 2i·Pn−i i∈ 0,...,n−min n, 1 k=b(n−i)(1−λ)c+1  { { b 1−λ c}} k  1 − 2n . We denote µ := 1 − λ and use slightly modified lemma 7 for quota equal to µ. The modification is in requiring sum of weights being strictly above the quota. In other words, we maximize the number of coalitions with sum of weights strictly above 1 − λ and then subtract this maximized number from the total number of coalitions, obtaining the minimum number of coalitions with sum of weights higher than or equal to λ. 50 3. Optimizing Efficiency of Weighted Voting Systems

3.5 The symmetry of eﬃciency

Lemma 9. Assume n ≥ 2 and 0 < λ ≤ 1. Then

n n τλ = 1 − m1−λ+ε, (3.22) for any ε such that λ > ε ≥ 0 and ε is smaller than the smallest diﬀerence between any two distinct sums of weights of two distinct coalitions.

Proof: Consider all the winning coalitions for the quota β and denote their number k. Clearly, for each such a coalition there exists unique complementary coalition made of all the voters which are not in the original winning coalition. Hence the number of winning coalitions for quota β is the same as the number of all coalitions that are made of voters whose weights sum up to at most 1 − β. When k is maximal attainable number of winning coalitions, then it is also the maximal attainable number of coalitions with voters whose weights sum up to at most 1 − β. Because there is always just 2n of all coalitions, maximizing the number of all coalitions with total weight lower than or equal to 1 − β is the same as minimizing the number of all coalitions with total weight strictly higher than β. Assume now we have added very small number ε to the quota β such that the maximal number of winning coalitions have lowered just by the coalitions with total weight equal to β (if there were any). Then the maximum number of winning coalitions K for the quota β +ε equals to the maximum number of all loosing coalitions for quota 1 − β, which is the same as the minimum of 2n − K of coalitions winning n n for the quota 1 − β. So we have τ1−β = 1 − mβ+ε. Denoting λ = 1 − β as n n 0 ≤ β ≤ 1 we obtain τλ = 1 − m1−λ+ε.

1 It is clear the problem is not fully symmetric around quota equal to 2 as the coalition is winning whenever the sum of its voters’ weights is at least equal to the quota, which means higher than or equal to the quota. The slight asymmetry is caused by reaching the quota being sufficient for the coalition to be winning. 1 The symmetry of efficiency around the quota λ = 2 is schematically shown in the figure 3.5. 3.6. Efficiency structure 51

Fig. 3.1: Eﬃciency symmetry

3.6 Eﬃciency structure

In this section, we analyze the structure of the efficiency maxima and minima with respect to weights, when we take the maximum (resp. minimum) as a 3n2 function of the quota. We show, there are at most asymptotically 2π2 + o (n log (n)) different efficiency maxima (resp. efficiency minima) that can be attained when moving the quota within Λ (n stands for the number of voters). We also show in which intervals the quota can move freely without changing the efficiency maximum (resp. efficiency minimum). For k voters, k ∈ N, we divide the interval (0, 1) in the following way: We i denote dij = k−j for i ∈ N, 0 ≤ i ≤ (k − j) , j = 0, ..., k − 1 and sort all the dij’s in nondecreasing order. We obtain some finite sequence (di) of numbers within the interval [0, 1] sorted in nondecreasing order. Now we create a ˆ subsequence di of the sequence (di) by dropping the members, that are not unique except just one. In other words, whenever there are at least two equivalent members of the sequence (di), we drop all of them except just ˆ one. In the final sequence di , each value is hence unique and the sequence becomes sorted in ascending order and contains all the values, which were in ˆ the original sequence (di). The elements of the final sequence di we denote ˆ ˆ ˆ 0 = d1 < d2 < ... < dkˆ = 1 and call them milestones. The interval (0, 1) is 52 3. Optimizing Efficiency of Weighted Voting Systems

then divided as follows

ˆ ˆ h ˆ ˆ h ˆ ˆ (0, 1) = d1, d2 ∪ d2, d3 ∪ ... ∪ dkˆ−1, dkˆ . (3.23)

Lemma 10. The number of milestones (when we include 0 and 1) for n Pn voters in the interval (0, 1) is kn = 2 + dj, where dj = j − 1 − P j=1 {p:p/j∧p0} dp.

Proof: The proof can be done by mathematical induction. We say k-th step will be the process of computing the number of milestones for denominator equal to k. In each step those milestones are added, which are not equal to milestones from any of the previous steps. The milestones from k-th step are equal to milestones added in the l-th step, when l < k if and only if there is a number p ∈ N so that p divides k and l. To the total number of milestones we add 2 representing 0 and 1. Because the sequence of milestones is in literature known as the Farey sequence and the cardinality of the set of all the elements of the sequence is expressed using the Euler’s totients function, we will not present the rigorous detailed proof here.

An interesting observation is that the number of milestones for n voters 3n2 is asymptotically equal to π2 + o (n log (n)) , see for example p. 155 in [15]. As the efficiency maximum for quotas above one half is just a single value (see lemma 4) and the efficiency minimum for quotas below one half is also just a single value (see lemma 5), we see that at most just about half of the milestones are in fact relevant†† for the efficiency maximum, or 3n2 minimum to change. Hence there is asymptotically at most 2π2 +o (n log (n)) of different efficiency maxima (resp. efficiency minima) that can be attained when moving the quota within Λ. Any two milestones, such that there is no other milestone between them we will call adjacent milestones (Fareys pairs). Having two adjacent mile- ˆ ˆ ˆ stones di and di+1 we will call di the lower adjacent milestone with respect ˆ ˆ ˆ to di+1 and di+1 the higher adjacent milestone with respect to di.

Lemma 11. Eﬃciency maximum (resp. eﬃciency minimum) is the same for all the quotas between two adjacent milestones, or for quotas such that ˆ the higher quota is equal to some milestone dj and the lower quota is strictly ˆ above the adjacent lower milestone with respect to dj.

†† 1 As for each milestone, we can ﬁnd another one, which has the same distance from 2 . 3.6. Eﬃciency structure 53

Proof: Assume, we have two quotas 0 < λ1 < λ2 < 1 and k different milestones. ˆ ˆ Assume there is an index i ∈ {1, . . . , k − 1} such that di < λ1 < λ2 ≤ di+1. In other words, both the quotas are between two adjacent milestones or the higher quota is equal to a milestone while the lower quota is strictly above the adjacent lower milestone. This implies, there is no milestone strictly between λ1 and λ2. From the definition of a milestone, we know there is also no rational number with denominator equal to or lower than n strictly between the two quotas. From lemma 8 we know, the efficiency maximum (resp. efficiency minimum) does not depend directly on the quota λ, but on the expression dλje for some j ∈ {1, . . . , n} . If for every j ∈ {1, . . . , n} is dλ1je = dλ2je , then the efficiency maximum (resp. efficiency minimum) surely is the same for both the quotas. We make the proof by contradiction assuming the efficiency maximum (resp. efficiency minimum) is different for the quota λ1 and for the quota λ2. Then there must be some j ∈ {1, . . . , n} so that djλ1e < djλ2e as if such a j would not exist, the efficiency maximum (resp. efficiency minimum) could not be different for the two quotas. There must exist a positive integer p, such that djλ2e = djλ1e + p. However for us it is sufficient to know, that

djλ2e ≥ djλ1e + 1. (3.24)

We will prove the statement bjλ c λ < 2 ≤ λ . 1 j 2

The inequality bjλ2c ≤ jλ2 is easy as b•c stands for the highest lower integer. To prove the strict inequality jλ1 < bjλ2c is a little bit more challenging. From (3.24) we know that jλ1 ≤ djλ1e ≤ djλ2e − 1 ≤ bjλ2c . However, if jλ1 jλ1 = bjλ2c , then jλ1 ∈ N and hence j = λ1 must be a milestone. But this ˆ ˆ bjλ2c would be in contradiction with di < λ1 < di+1. So in total we get λ1 < j . We have proven, that bjλ c λ < 2 ≤ λ , 1 j 2

bjλ2c but if j = λ2, we are not yet in contradiction with the statement of the bjλ2c lemma as λ2 is allowed to be a milestone. If j < λ2, we are finished, as bjλ2c j is surely a milestone and it is strictly between the two quotas. To show, it is a milestone, it suffices to recall that milestone is each rational number 54 3. Optimizing Efficiency of Weighted Voting Systems

Fig. 3.2: Eﬃciency structure is driven by the Farey sequence

between 0 and 1 with denominator equal to an integer lower than or equal to n.

bjλ2c Now we ﬁnish the proof by solving the case j = λ2 (which means that λ2 is a milestone). If this holds, we have jλ2 ∈ N. In this case, there is a djλ1e number j , which is a milestone and moreover, we can easily show, that

djλ e λ ≤ 1 < λ . 1 j 2

The strict inequality is simple as bjλ2c = jλ2 implies jλ2 = djλ2e and we have (3.24) stating that djλ1e < djλ2e . The weak inequality is easy as d•e stands for the lowest higher integer. This completes the proof as whenever the efficiency maximum (resp. efficiency minimum) is different for quota λ1 from the one for quota λ2, we have just shown there must be a milestone between the two quotas or the higher quota must be a milestone and the lower quota must be below or equal to the adjacent lower milestone.

This lemma tells us that whenever the quota is between any two adjacent milestones, the efficiency maximum (resp. efficiency minimum) does not change. However it does not tell us that moving the quota ”accross” a milestone would always change the efficiency maximum (resp. efficiency minimum) as this is generally not true. In the figure 3.6 we show the efficiency structure for n = 5 (the values of 1 efficiency except for 2 , 1 and 0 are not shown in their real values in order to make the picture lucid). 3.7. Application 55

3.7 Application

The theorem 1 can be used to provide information on efficiency when setting the quota for a newly designed electoral system or voting rule, but it can also be used for other economic problems, where an optimal alocation is being searched. Assume we are in a role of a pension fund manager who has promised to make (1 + r) W at the end of the year which he started with W. Assume, he has n opportunities of investment, while the output of each investment is with probability one half equal to the original investment (there is zero gain) and with probability one half, there is a positive gain equal to the original investment plus some interest λ on that investment. If the interests are more or less the same for all the opportunities (or the manager has no relevant information on them), he faces a decision of how much to invest in which investment opportunity in order to maximze the probability of making at least (1 + r) W. Intuitively, he should divide his original wealth W into n equal parts and invest each part in exactly one of the opportunities. This seems to be optimal in order to diversify the investments. However it is generaly not. The optimal solution depends heavily on the numbers r and λ. Assume for example that 1 1 r = 7 , λ = 2 and n = 5. If the manager invests 20% of the original wealth W to each investment opportunity, the probability of getting over (1 + r) W is 26 32 . However, if he invests 25% to whatever four of the opportunities and one 31 leaves unutilized, the probability of getting over (1 + r) W equals 32 . Using the theorem 1 enables us to find this maximal possible probability of geting over what the manager has promised very fast and moreover it assures that no better allocation exists. 56 3. Optimizing Efficiency of Weighted Voting Systems

3.8 Conclusions

We were interested in the behavior of the efficiency of weighted voting system (probability of approval) as a function of the weights assigned to particular voters and the quota. Especially we wished to find the maximal and minimal attainable efficiency with respect to the vector of weights, given the number of voters and the quota. We have answered all the raised questions: • What are the minima and maxima of the efficiency function with respect to the weights and can they be expressed analytically in terms of quota and number of voters? This question has been answered by the theorem 1 and lemmas 4, 5 and 3.15. • Is the efficiency function symmetric in some sense around the quota equal to one half? This question has been answered by the lemma 9. • Are there any intervals in which the quota can move freely so that the efficiency maximum (resp. efficiency minimum) with respect to weights does not change? If so, how they can be formaly described? This question has been answered by the lemmas 10 and 11. • How many different values the efficiency maximum, resp. efficiency minimum attain in terms of the number of voters n? Can this number REFERENCES 57

be expressed explicitly as a function of n? This question has been partialy answered in the text. We have noted the number of efficiency maxima, resp. efficiency minima assymptotically 3n2 equals 2π2 + o (n log (n)) . We have presented theorem 1 allowing us to express the efficiency minimum and maximum explicitly as functions of the quota and the number of voters. Moreover we have provided a proof of this theorem. So far as we know, there is no other work where this would already have been proven. The practical use of our theorem is mainly in the control over the quota setting when designing voting rules for the weighted voting systems and it can be used more generally within the context of traditional economy when searching for an optimal allocation. Usually the quota for regular propositions to be accepted by legislative institutions equals one half plus one mandate. However the constitutional quotas are often higher, as the most important democratic rules imbedded in the constitutions are more to be secured than regular laws. Corollaries of our theorem help us to analyze the minimum (resp. maximum) efficiency of weighted voting systems as a function of quota and number of voters.

Acknowledgements This research was supported by the Grant Agency of the Czech Republic, project No. 402/09/1066 “Political economy of voting behavior, rational voters’ theory and models of strategic voting”

References

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[9] Dennis Leech. Voting power in the governance of the international monetary fund. Annals of Operations Research, 109:373–395, 2002.

[10] Ines Lindner. The power of a collectivity to act in weighted voting games with many small voters. Social Choice and Welfare, 30(4), 2008.

[11] T. Matsui and Y. Matsui. NP-completeness for calculating power indices of weighted majority games. Theoretical Computer Science, 263(1- 2):305–310, 2001.

[12] Tomomi Matsui and Yasuko Matsui. A survey of algorithms for calculating power indices of weighted majority games. J. Oper. Res. Soc. Japan, 43:71–86, 2000.

[13] Elizabeth McIntyre. Weighted voting in international organizations. In- ternational Organization, 8:484–497, 1954.

[14] K. Prasad and J. S. Kelly. NP-completeness of some problems concerning voting games. Int. J. Game Theory, 19(1):1–9, 1990.

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[16] Michael Zuckerman, Piotr Faliszewski, Yoram Bachrach, and Edith Elkind. Manipulating the quota in weighted voting games. Conference on Artiﬁcial Intelligence, 2008. 4. APPORTIONMENT IN PROPORTIONAL ELECTORAL SYSTEMS BASED ON INTEGER PROGRAMMING

Abstract In this work we introduce three apportionment methods for proportional electoral systems with constituencies based on integer programming. The reason for their introduction we see in the fact, that the commonly used apportionment methods often lead to substantial deviations from perfect proportionality, beyond the scope given by the elections threshold. There are generally three different sources of disproportionality-allocation of whole seats and not just their fractions, division of elections into constituencies in which the seats are allocated separately and elections threshold. Our apportionment methods are designed as to get rid of the first two sources keeping the elections threshold, which prevents excessive fragmentation of the representation and hence low efficiency. The proposed methods we compare with several other apportionment methods on the results of all the historically held elections to the Chamber of Deputies of the Czech Parliament (PS PCR)ˇ and the Czech National Council since 1990 in terms of achieved level of proportionality. We show their introduction has a real impact on reducing disproportionality no matter of which of the three considered disproportionality measures is used (Gallagher index, Loosemore-Hanby index, or Sainte-Laguëindex). Keywords Apportionment, Electoral systems, Proportionality, Integer Pro- gramming, Constituency JEL classification D71, D72

4.1 Introduction and motivation

Any algorithm that assigns seats to political groups or candidates based on the number of votes cast in elections we call an apportionment method. There are many different apportionment methods used in the world, see. for example, [27], but we will mainly focus on the one used in the elections to the Chamber of Deputies of the Czech Parliament (PS PCR)ˇ that is given by the Electoral Act (Act 247/1995 Coll.). One of the most important characteristics of apportionment methods is 60 4. Apportionment in Proportional Electoral Systems Based on Integer Programming the achieved level of proportionality, which indicates how much is the distribution of valid votes cast to political groups different from the final distribution of seats among these groups (or alternatively for constituencies, states, etc.). The definition of proportionality and analysis of its importance can be found for example in [27]. TomáˇsLebeda in his publication [26] states: ”The degree of proportionality is the main and most important quantifiable phe- nomenon reflecting the political consequences of electoral systems. It is an indicator of wrong representation - indicates the over-representation, under- representation or mis-representation of the political parties - and largely de- termines the party system itself.” In his study, he also describes the efforts of some other authors to sort apportionemtn methods according to the achieved level of proportionality. Partly in oposition to proportionality stands the efficiency (see for example [12]). It expresses ability of electoral system to produce changes in status quo. In the case of PS PCR,ˇ efficiency is the probability that a stable and strong government will be formed. It is obvious that the efficiency and proportionality often go against each other, as high level of proportionality usually leads to a large number of mutually incompatible and disparate political entities entering the voting body. It can be difficult for them to agree on joint participation in a government and even if they agree, such coalitions are rather unstable. In order to achieve a reasonable level of efficiency, election thresholds are usually applied. Elections threshold prevents the political groups that obtained less then some given percentage of votes (on the national level) from gaining any seats. In the case of PS PCR,ˇ the elections threshold is set to 5%. Another very important feature of the elections to PS PCRˇ is their division into electoral districts (constituencies). Under the current electoral system at first the election threshold is applied at the national level, then the allocation of seats to constituencies takes place and then the seats are allocated to political groups in each constituency separately. There are several reasons for introducing constituencies. First, they enable close connection between the particular seats and specific territorial units. This leads to fairer geographical distribution of seats. They also enable easier and less expen- sive election campaigns. Constituencies are commonly used in elections all around the world and play irreplaceable role in the supranational or federal institutions (EU, House of Representatives in the United States, IMF, UN, etc.). The aim of the proposed apportionment methods is to minimize the deviation from perfect proportionality for elections with nonzero election threshold and with division into constituencies. The advantage of these proposals is their general applicability (number of constituencies as well as elections 4.1. Introduction and motivation 61 threshold are not limited). The disadvantage is their technical complexity and a need to use a computer. All the algorithms that we use in this text vere made in VBA (Visual Basic for Applications) implemented in Microsoftr Excel and we have created them ourselves. The paper is organized as follows: In Section 4.2 we analyze studies which have delt with the problem. In Section 4.3, we introduce methodology start- ing with notation and definitions of disproportionality measures which will be used throughout the paper. Then we introduce allocation algorithms Ψ and Ψ• which are used within the proposed apportionment methods. In Section 4.4 we present the results of the proposed methods on real life data and compare the achieved levels of proportionality with some other apportionment methods applied in literature or commonly used. In Section 4.5 we conclude and in Section 4.6 there are proofs and detail results which would take the eye off the main problem if placed directly in the text. 62 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

4.2 Literature survey

The literature on proportional apportionment is comprehensive. There are several directions, that treat the apportionment differently. One direction uses an axiomatic approach aiming to preserve some general properties and avoid voting paradoxes and undesirable anomalies (Alabama paradox, population paradox, new state paradox, etc.), see for example [2]. We can find some axioms defining various properties of apportionment in the book [25] (axiom of homogeneity, axiom of house monotonicity, axiom of population monotonicity, axiom of uniformity, quota satisfaction property, etc.). Second direction focuses on searching fast, proportional and simple apportionment algorithms. In [25], apportionment methods are divided in three main groups: divisor methods, parametric methods and rank-index methods. There is also shown, that some of the methods have some properties, such as apportionment fulfills the axiom of population monotonicity if and only if it is a divisor method apportionment. Widely used are mainly Hamilton’s method (the method of largest reminder, the Vinton’s method), Jefferson’s method (the method of greatest divisors, the d’Hondt method or the Hagenbach- Bischoff method), Adams’ method (the method of smallest divisors), Web- ster’s method (the Webster-Willcox method, the method of major fractions) and Huntington-Hill method (the method of equal proportions). The third direction deals with empirical studies on apportionment and its practical applications, see for example [32] or [3]. The most influential work can be found in [1], [2], [4], [5], [6], [7], [15], [18], [17], [21], [29], [33], [14] and [31]. Most of them deal with the bi-proportional apportionment and the iterative proportional fitting method, which is convenient under certain assumptions for the measure of disproportionality ξ. As the topic is still actual, there are several other recent articles dealing with the uniproportional∗ and biproportional † apportionment methods. In the article [28] two algorithms for biproportional apportionment are presented: the alternating scaling algorithm‡ and the tie-and-transfer algorithm proposed in [9], which is based on transforming the problem into a bipartite graph. As the first one does not work generally and the second might be slow, in [28] a hybrid algorithm combining these two is proposed. However these algorithms are generally not leading to an optimal solution regarding disproportionality measures which we define in subsection 4.3.2. In [8] the author characterizes divisor methods for vector and matrix apportion prob-

∗ Allocation to parties, or allocation to constituencies. † Allocation to parties and constituencies simultaneously. ‡ Well known from statistics as a Stephan-Deming algorithm or RAS algorithm, see for example [13], or [19] for a proof of convergence for the measure ξ. 4.2. Literature survey 63

lems, which are re-scaleable, but again not minimizing disproportionality as we define it. In [16] an apportionment algorithm of minimum total deviation is presented combining two other simple algorithms developed for vector apportionment: Adams and Jefferson apportionment methods. An interesting empirical article [32] deals with the flawed procedure for biproportional apportionment in the Italian electroal law, which is based on two common allocation rules: Hare’s quota and the Method of the largest reminder. Very interesting and closely related to our work is the work presented in [35] dealing with the current issues of apportionment methods, especially the biproportional ”apportionment methods for systems with multiple constituencies”. It is based on [34], where a Java programm BAZI for proportional apportionment is introduced. In [3] is analyzed the new Zürich’s electoral law based on bi-dimensional proportional apportionment. The apportionment methods introduced in this article do not directly follow reasoning of any of the ones presented in the literature. We are interested just in finding the optimal apportionment in terms of disproportionality§ and do not impose any additional properties to be fullfiled, such as homogeneity, monotonicity, uniformity or quota satisfaction. Hence the proposed methods do not generaly prevent the Alabama and population paradoxes or obtaining seats for no votes. It is a price paid for their generality. To illustrate this, we can use the example from [8] where allocation of seats is performed in seven different constituencies for 4 parties, see the table 4.1. Using the divi-

Votes 1st 2nd 3rd 4th 5th 6th 7th Seats Party 1 100 100 100 100 100 100 100 2 Party 2 100 100 100 100 100 100 100 5 Party 3 100 100 100 0 0 0 0 4 Party 4 100 100 100 0 0 0 0 6 Seats 4 2 3 1 1 2 4 17

Tab. 4.1: Number of votes cast in districts (columns) and parties (rows) given the numbers of seats sor apportionment method described in [8], the setting described in table 4.1 can not be solved. However using one of our methods would give the result shown in table 4.2. Note that for some of the seven constituencies where parties 3 and 4 obtained no vote, they are being allocated some seats. This can be considered strange, but it is a valid solution. The reader can check also [10] where authors describe methods to meet fair representation and in [23] where the author presents some solvability check for biproportional multiplier methods. § Which we measure via the measures deﬁned in subsection 4.3.2. 64 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

Mandates 1st 2nd 3rd 4th 5th 6th 7th Seats Party 1 1 0 0 0 0 0 1 2 Party 2 1 0 1 0 1 1 1 5 Party 3 1 1 1 0 0 0 1 4 Party 4 1 1 1 1 0 1 1 6 Seats 4 2 3 1 1 2 4 17

Tab. 4.2: Allocation of seats by one of our methods

Let’s summarize our approach. The proposed methods always give the best possible results regarding the disproportionality measures deﬁned in subsection 4.3.2 assuming the order of importance (in terms of lexicographic ordering):

1. uni-proportional allocation of seats to parties,

2. uni-proportional allocation of seats to constituencies,

3. bi-proportional allocation of seats to constituencies and parties jointly,

except for the situation, when the optimal solution (without the non- negativity constraint) to the bi-proportional apportionment is not non- negative¶. If the optimal solution (without the non-negativity constraint) is not non-negative, our methods generaly ﬁnd only a suboptimal non-negative solution. To see, that optimal solution (without the non-negativity constraint) to the bi-proportional apportionment does not have to be non-negative, see the tables 4.3 and 4.4. Optimal solution to this problem is shown in the table 4.4

1st 2nd 3rd Seats Party 1 0 501 499 9 Party 2 302 399 199 9 Party 3 102 0 498 6 Seats 3 9 12 24

Tab. 4.3: Number of votes cast in districts (columns) and parties (rows) given the numbers of seats

and it is not non-negative as to the Party 1 in 1st district there is assigned negative number of seats, which is of course impossible.

¶ By non-negative solution we mean a matrix with all entries non-negative. 4.3. Methodology 65

1st 2nd 3rd Seats Party 1 -1 5 5 9 Party 2 3 4 2 9 Party 3 1 0 5 6 Seats 3 9 12 24

Tab. 4.4: Optimal solution to the problem

4.3 Methodology

4.3.1 Notation

Let’s introduce some notation. Number of all valid cast votes we denote v0 and the number of seats m (in the case of elections to the PS PCRˇ m = 200). We assume m < v0. Let’s denote P 0 the number of all candidating political groups and R the number of all constituencies (in case of PS PCRˇ since 2002 there are 14 constituencies, in case of elections to the Czech national Council and elections to PS PCRˇ up to 1998 there were 8 constituencies). The set of all candidating political groups we denote P0 and this set we arrange in descending order according to the number of obtained votes so that index i ∈ {1,...,P 0} represents just one of the candidating political groups. We denote pi the number of valid votes cast to political group i for all i = 1,...,P 0. The set of all constituencies we arrange in descending order according to the number of votes cast in each of them, so that each index j ∈ {1,...,R} represents just one constituency. The number of valid 0 votes cast to political group i in constituency j we denote vij, for all i = 1,...,P 0 and all j = 1,...,R. The election threshold we denote 0 < u < 1. Moreover we denote v the total number of votes cast to all the political pi parties passing through the elections threshold (i.e. v0 > u). We denote P the set of all the political groups that have passed through the elections threshold and for each party we remain its original index i ∈ {1,...,P } , where P stands for the number of all the political groups that have passed through the elections threshold. Number of votes cast to political group i ∈ {1,...,P } in constituency j we denote vij and the number of votes cast to all the political groups that have passed through the elections threshold in constituency j we denote rj. 66 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

4.3.2 Disproportionality measures We will measure the deviation from the theoretical perfect proportionality using three diﬀerent measures (4.1), (4.2) a (4.3).

P R X X m 2 ρ = m − v , (4.1) ij v ij i=1 j=1

P R X X m φ = m − v , (4.2) ij v ij i=1 j=1

P R m 2 X X mij − vij ξ = v , (4.3) m v i=1 j=1 v ij

where mij stands for the number of seats allocated to the political group i in constituency j. Note, that we measure the level of disproportionality only for the political groups that have passed through the election threshold. The reason is that in any two apportionments with the same election threshold the political groups that have not passed through the threshold have the same contribution to the overall level of disproportionality for any of the three measures. Hence to compare the apportionment methods in terms of proportionality it is not necessary to account for these small (not passed through the threshold) political groups. When we need to measure the level of disproportionality of the vector of seats allocated to political groups on the national level, resp. to constituencies across all the political groups, which have passed through the election threshold, we use the same three measures, but we put R = 1 and use pi instead of vi1 and mi instead of mi1, resp. put P = 1 and use rj instead of v1j and mj instead of m1j. The three given measures are common disproportionality measures, but not the only ones. The index ρ is an increasing transformation of the Gallagher index, see [20], index ψ is an increasing transformation of the Loosemore-Hanby index, see [22], and ξ is an increasing transformation of the Sainte-Laguë index, see [24]. Many different measures of proportionality are defined in literaturek. As mentioned in [20], every method of converting votes to seats minimizes disproportionality with respect to a measure, k Rae index, Monroe index, Gini index, Farina index, Borooah index, Grofman index, Li- jphart index, Gatev index, Ryabtsev index, Szalai index, Aleskerov-Platonov index, Atkin- son index, index of generalized entropy and d’Hondt index are described in [24], where their basic characteristics are shortly analyzed. Measuring proportionality is researched in detail in the book [14]. 4.3. Methodology 67

which itself arbitrarily chooses. Therefore it is not possible to proclaim one of the methods to by the most favorable in terms of proportionality generaly. General evaluation of the achieved level of proportionality is always subject to normative selection of its measure. In spite of this fact, there are some articles dealing with comparisons of the levels of proportionality achieved by particular methods, see for example [11], [30], [27].

4.3.3 General conditions of aplicability For an apportionment method to be applicable in real elections, we consider the following three conditions to be necessary to be met: 1. Determinism - the method has to be deterministic, i.e. it should not contain any randomness and when applied to the same data again, it should give the same results. 2. Non-negative solution - the method has to always lead to assigning non-negative number of seats for all political groups in all the constituencies. 3. Speed - the method has to find the solution in reasonable time, i.e. in polynomial time. So by designing the methods our work is not finished. We must also verify that each of the proposed methods fulfills all these three conditions∗∗. If not, they can not be used in practice.

4.3.4 Allocation algorithm Ψ In all the methods of converting votes into seats that we propose, we will utilize an allocation algorithm, which leads to allocating seats to constituencies, resp. political groups, that have passed through the election threshold. This algorithm we have created to ﬁnd the solution of the following problem of integer programming in case of measuring disproportionality over the political parties using the measure ρ:

min PP m 2 i=1 mi − v pi mi, i=1,...,P

PP (4.4) subject to i=1 mi = m

mi ∈ N0 i = 1,...,P. ∗∗ Note that we do not require uniqueness of the solution. 68 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

Similarly, the algorithm Ψ can be used to ﬁnd the solution of the following problem of integer programming in case of measuring disproportionality over the constituencies using the measure ρ:

min PR m 2 j=1 mj − v rj

mj , j=1,...,R

PR (4.5) subject to j=1 mj = m

mj ∈ N0 j = 1,...,R.

Both the problems (4.4) and (4.5) have always an optimal solution, which is non-negative even when condition mi ∈ Z replaces the condition mi ∈ N0. In case of measuring disproportionality via the measure ψ we have the problems (4.4) and (4.5), only with slightly diﬀerent objective function

P X m m − p , i v i i=1 resp. R X m m − r j v j j=1 and in case of measuring disproportionality via the index ξ with objective function P m 2 X mi − pi v , m p i=1 v i resp. R m 2 X mj − rj v . m r j=1 v j Let Q ∈ N. Allocation algorithm Ψ proceeds in the following steps:

1. Assign mi := m for all i = 1, . . . , Q.

2. For all i = 1,...,Q compute

m 2 mi − v vi , when minimizing ρ m mi − v vi , when minimizing ψ m 2 (mi− v vi) m , when minimizing ξ. v vi 4.3. Methodology 69

vi 3. From all the values obtained in step 2, for which mi > m v ﬁnd the argument of maximum, i.e. ﬁnd the lowest index i∗, for which

2 2 vi∗ vi m ∗ − m ≥ m − m i v i v

vi holds for all i = 1,...,Q for which mi > m v when minimizing ρ,

vi∗ vi m ∗ − m ≥ m − m i v i v

vi holds for all i = 1,...,Q for which mi > m v when minimizing ψ and

1 1 m ∗ − m − i 2 ≥ i 2 vi∗ vi

vi holds for all i = 1,...,Q for which mi > m v when minimizing ξ.

4. Assign mi∗ := mi∗ − 1.

PQ 5. If i=1 mi = m, then the algorithm ends with the allocation of seats: PQ (m1, . . . , mQ) . If i=1 mi > m, we go back to step 2.

When allocating seats to constituencies we put Q := R and to vj we assign the value rj for all j = 1,...,R and when allocating seats to political groups we put Q := P and to vi we assign the value pi for all i = 1,...,P.

4.3.5 Allocation algorithm Ψ• We create an algorithm, that will assign number of seats to political groups in each constituency. We will use it to ﬁnd the optimal solution to the following problem of integer programming:

min PP PR vij 2 i=1 j=1 mij − m v mij , i=1,...,P , j=1,...,R

subject to: PP m = L j = 1,...,R, i=1 ij j (4.6) PR j=1 mij = Mi i = 1,...,P,

mij ∈ N0 i = 1,...,P , j = 1,...,R, 70 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

where Lj, j = 1,...,R and Mi, i = 1,...,P are positive integers which PP PR †† are subject to the condition of balanace m = i=1 Mi = j=1 Lj . The constraints correspond to a balanced transportation problem. If we had a linear utility function, then we would get a balanced transportation problem, which would automatically have an integer optimal solution. The linear transportation problem could be solved in polynomial time using for example the ellipsoid method. But our objective function is not linear and so we can not expect the optimal solution to be integer. We will divide algorithm Ψ• in three steps. First of them is the allocation • • algorithm which we denote Ψ1, second is the iterative algorithm denoted Ψ2 • • and third is the additional iterative algorithm denoted Ψ3. Algorithm Ψ is then a successive execution of these three ”sub-algorithms” in the order of their lower index. To make the formal description of the algorithm simpler, we denote F = {1,...,P } × {1,...,R} . • The allocation algorithm Ψ1 has the following steps:

1. Assign mij := m for all i = 1,...,P, and all j = 1,...,R.

2. For every ordered couple of indices (i, j) ∈ F we compute vij 2 mij − m v .

∗ ∗ 3. We ﬁnd such an ordered couple of indices (i , j ) for which holds mi∗j∗ > vi∗j∗ m v and 2 2 vi∗j∗ vij m ∗ ∗ − m ≥ m − m , i j v ij v vij for all (i, j) ∈ F such that mij > m v in case of minimizing ρ,

vi∗j∗ vij m ∗ ∗ − m ≥ m − m , i j v ij v vij for all (i, j) ∈ F such that mij > m v in case of minimizing φ and

1 1 ∗ ∗ mi j − 2 mij − 2 vi∗j∗ ≥ vij , m v m v

vij for all (i, j) ∈ F such that mij > m v in case of minimizing ξ. If there are more couples that could be chosen, we choose the one with the lowest ﬁrst index and if there are even more couples with the same ﬁrst index we choose the couple with the lowest second index.

†† In our particular usage, the Lj will be the number of seats assigned to constituency j regardless of political groups by algorithm Ψ and Mi will be the number of seats assigned to political group i on national level (regardless of constituencies) by algorithm Ψ. 4.3. Methodology 71

4. Assign mi∗j∗ := mi∗j∗ − 1.

PP PP 5. If i=1 j=1 mij = m, then we go to step 6, otherwise we go back to step 2.

n PR o 6. We deﬁne a set Ω := i ∈ {1,...,P } : j=1 mij > Mi , a set Φ := n PP o j ∈ {1,...,R} : i=1 mij > Lj and a set ∆ := Ω × Φ. If Ω = ∅, then we go to step 8.

7. We put

2 2 0 0 vij vij (i , j ) := arg min mij − 1 − m − mij − m (i,j)∈∆ v v

when minimizing ρ,

0 0 n vij vij o (i , j ) := arg min mij − 1 − m − mij − m (i,j)∈∆ v v when minimizing φ and

vm (i0, j0) := arg max ij (i,j)∈∆ mvij

when minimizing ξ. We put mi0j0 := mi0j0 − 1 and get back to step 6.

n PR o 8. We deﬁne a set Λ := i ∈ {1,...,P } : j=1 mij < Mi , a set Ξ := n PP o j ∈ {1,...,R} : i=1 mij < Lj and a set Θ := Λ×Ξ. If Λ = ∅, then the allocation algorithm ends with allocation of seats mij for a political group i in constituency j for all i ∈ {1,...,P } and all j ∈ {1,...,R} .

9. We put

2 2 0 0 vij vij (i , j ) := arg min mij + 1 − m − mij − m (i,j)∈Θ v v

when minimizing ρ

0 0 n vij vij o (i , j ) := arg min mij + 1 − m − mij − m (i,j)∈Θ v v when minimizing φ 72 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

vm (i0, j0) := arg min ij (i,j)∈Θ mvij

when minimizing ξ. Further we put mi0j0 := mi0j0 + 1 and get back to step 8.

• The allocation algorithm Ψ1 generaly finds only a suboptimal solution to (4.6) and moreover not necessarily non-negative. Hence we have to add an • iterative algorithm Ψ2, which finds the optimal solution (as proved in 4.6.2) in case the optimal solution is non-negative. This iterrative algorithm we • apply to the solution found by the allocation algorithm Ψ1 and it is based on the idea of ε-change. Integer ε(i,j),(k,l)-change will be called a mapping from the set of integer matrices into the set of integer matrices of the same type given by the prescription:  . . . . .   . . . . .  ......      . . . ai,j . . . ai,l ...   . . . ai,j + ε . . . ai,l − ε . . .   . . . . .   . . . . .   ......  7−→  ......  ,      . . . ak,j . . . ak,l ...   . . . ak,j − ε . . . ak,l + ε . . .   . . . . .   . . . . .  ...... where ε ∈ Z. When we want to include a general ε-change without specifi- cation of the row and column indices, we will denote it simply a ε-change. Whenever we write about a ε-change, we will always mean an integer ε- change as we have defined it, however sometimes we stress, that we have in mind just an integer changes and sometimes we even reduce it to just 1-changes or −1-changes. The most important feature of ε-change is the fact, that when applied to a matrix fulfilling all the constraints of the problem (4.6) (except for the constraint on non-negativity of variables), then the outcome fulfils the • same constraints as well. The iterative algorithm Ψ2 detects the existence of such 1-change or −1-change, which would reduce the value of the objective • function when applied to the outcome of the allocation algorithm Ψ1. If at least one such 1-change or −1-change exists, then the iterative algorithm would apply the one that reduces the objective function the most. Algorithm • Ψ2 then continues in the iterations until there is no 1-change or −1-change which would reduce the value of the objective function. If the resulting solution is non-negative, then we have the optimal solution to the problem (4.6) as we prove in subsection 4.6.2. If it is not non-negative, we denote ∗ ∗ it A = aij and apply an additional iterative algorithm using 1-changes 4.3. Methodology 73 and −1-changes on the objective function prescribed by 4.7. This additional • iterative algorithm we have denoted Ψ3 and it proceeds in the following steps.

1. Put cij := 0 for all i = 1,...,P and j = 1, . . . .R 2. Find the minimum P R " # 2 2 X X (m + 1) ∗ vij ∗ min PR 1 − sgn cij + aij + m − cij − aij cij ∈Z 2 v i=1 j=1 (4.7) subject to PP i=1 cij = 0, for all j = 1,...,R, PR j=1 cij = 0, for all i = 1,...,P. by computing all the possible 1-changes and −1-changes and apply the one, that decreases the value of the objective function the most. 3. Repeat the step 2 as long as there is nor 1-change neither −1-change that would decrease the value of the objective function.

∗ ∗ The solution C = cij found by this iterative algorithm we add to the ∗ ∗ • solution A = aij that has been found by the algorithm Ψ2 and we get ∗ ∗ ∗ ∗ ∗ ∗ a feasible solution P := C + A = cij + aij . Matrix P is an optimal solution to (4.6) with the non-negativity constraint mij ∈ N0 as we prove in the subsection 4.6.2.

4.3.6 Common basis for the proposed methods First three steps are common to all three proposed methods and so we describe them just once. 1. The ﬁrst step of all the proposed methods is aplication of the election threshold. This step is not necessary and it has no impact on aplicability of the methods. In order to make it reasonable to compare the achieved rates of proportionality with other electoral systems, we always apply the same election threshold for all electoral systems which we compare to each other. 2. The second step of all the proposed methods is the application of allocation algorithm Ψ to the allocation of seats to political groups independently of the constituencies. For each of the three methods we insert the appropriate objective function into the algorithm Ψ based on the disproportionality measure which we minimize. 74 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

3. The third step of all the proposed methods is the application of allocation algorithm Ψ to the allocation of seats to constituencies independently of the political groups. For each of the methods we insert the appropriate objective function into the algorithm Ψ based on the disproporcionality measure which we minimize.

Lets describe these three steps formally.

1. The ﬁrst step means that for all i = 1,...,P 0 we verify the inequality pi v0 > u. If for a given i this inequality holds, the political group i pass through the election threshold. If it does not, political group i is excluded from the set P0 and hence no seats can be allocated to it. After all the candidating political groups that have not passed through the election threshold are excluded from the set P0, we obtain the set P. The political groups that are not in the set P are not taken into account any more.

2. In the second step we apply the allocation algorithm Ψ to the allocation of seats to particular political groups (independently of the consituen- cies). We denote Mi the number of seats assigned to political group i for all i = 1,...,P.

3. In the third step we apply the allocation algorithm Ψ to the allocation of seats to particular constituencies (regardless of the political groups). We denote Lj the number of seats assigned to constituency j for all j = 1,...,R.

4.3.7 Method of seats allocation based on the measure ρ The ﬁrst three steps are described in the section: Common basis for the proposed methods.

4. Now we create a table with numbers of valid votes, where columns represent particular constituencies and rows candidating political groups, that have passed through the election threshold. This table contains in i−th row and j−th column the value vij. Perfectly proportional outcome of elections is given by the matrix of type P × R with elements vij m˜ ij := m v , where i = 1,...,P and j = 1,...,R.

5. As allocating non-integer numbers of seats is not possible, we have to 4.3. Methodology 75

solve the following problem of integer nonlinear programming: PP PR 2 min i=1 j=1 (m ˜ ij − dij)

dij , i=1,...,P , j=1,...,R

subject to: PP d = L j = 1,...,R, i=1 ij j (4.8) PR j=1 dij = Mi i = 1,...,P,

dij ∈ N0 i = 1,...,P , j = 1,...,R.

We search for a matrix D∗ = d∗ j=1,...,R , which is an optimal solution ij i=1,...,P to (4.8). If the optimal solution to (4.8) with constraint dij ∈ Z instead of constraint dij ∈ N0 is non-negative, we ﬁnd it using the algorithm Ψ• for objective function given by ρ. If it is not, the algorithm Ψ• ﬁnds just some estimation of the optimal solution.

∗ The seats allocated to political group i in constituency j is given by dij.

4.3.8 Method for seats allocation based on the measure φ The ﬁrst three steps are described in the section: Common basis for the proposed methods.

PP PR min i=1 j=1 |m˜ ij − dij|

dij , i=1,...,P , j=1,...,R

subject to PP d = L j = 1,...,R, i=1 ij j (4.9) PR j=1 dij = Mi i = 1,...,P,

+ dij ∈ N0 i = 1,...,P , j = 1,...,R, 76 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

We search for a matrix D∗ = d∗ j=1,...,R , which is an optimal solution ij i=1,...,P to (4.4). If the optimal solution to (4.4) with constraint dij ∈ Z instead of constraint dij ∈ N0 is non-negative, we ﬁnd it using the algorithm Ψ• for objective function given by ψ. If it is not, the algorithm Ψ• ﬁnds just some estimation of the optimal solution.

∗ The seats allocated to political group i in constituency j is given by dij.

4.3.9 Method for seats allocation based on the measure ξ

The first three steps are described in the section: Common basis for the proposed methods, but this method has in contrast with the previous two some additional assumptions. These assumptions arise from the construction of the measure ξ, which is defined as a ratio with denominator equal to the number of valid votes vij cast to political group i (which have passed through the election threshold) in constituency j. This number should never be zero, otherwise the index is not well defined. Moreover none of the values Mi, i = 1,...,P and none of the values Lj, j = 1,...,R should be zero, but this will not happen for any Mi as each political group had to passed through the election threshold (in extreme case, there can be P = 0 which would end up with no allocation). If Lj = 0, for some j ∈ {1,...,R} , we could reject the constituency j from any consideration and make the whole allocation process start without this constituency‡‡.

5. We search for a matrix D∗ = d∗ j=1,...,R , which is an optimal soultion ij i=1,...,P

‡‡ Except for the ﬁrst step of applying the election threshold, because there might be some votes cast to small political groups that do not pass through the election threshold! 4.3. Methodology 77

to (4.10).

2 min PP PR (dij −m˜ ) i=1 j=1 m˜ dij , i=1,...,P

subject to PP d = L j = 1,...,R, i=1 ij j (4.10) PR j=1 dij = Mi i = 1,...,P,

dij ∈ N0 i = 1,...,P , j = 1,...,R.

We search for a matrix D∗ = d∗ j=1,...,R , which is an optimal solution ij i=1,...,P to (4.10). If the optimal solution to (4.10) with constraint dij ∈ Z instead of constraint dij ∈ N0 is non-negative, we ﬁnd it using the algorithm Ψ• for objective function given by ξ. If it is not, the algorithm Ψ• ﬁnds just some estimation of the optimal solution.

∗ The seats allocated to political group i in constituency j is given by dij.

4.3.10 Verifying aplicability of the proposed methods Let us now return to the aforementioned three properties which must be fulﬁled in order to be able to apply the methods in real elections. Certainly all the three introduced methods are deterministic whenever there are no two parties with the same number of obtained votes which passed through the election threshold. So under this assumption each application to the same inputs leads to the same output. This is because none of the algorithms would use any random selection. Algorithms are aplicable on real data and according to the constraints on non-negativity of variables it never allocates a negative number of seats. Algorithm Ψ• can technically end up with a solution that is not non-negative after its allocation and iterative part, but this possibility is then treated via the additional algorithm (north-west corner rule followed by the heuristic iterrative algorithm). Hence any solution produced by the whole algorithm Ψ• will always be non-negative. The time needed to ﬁnd the optimal solution via algorithm Ψ is O(mQ2) and via algorithm Ψ• is O ((m + 1) (P 2R2)) . In other words, the solution is always found in polynomial time. In the practical application on data from the historical elections to PS PCRˇ the algorithm Ψ have always found the optimal solution faster than in 1 second and the algorithm Ψ• have always 78 4. Apportionment in Proportional Electoral Systems Based on Integer Programming found the optimal solution at most in minutes, but have never exceeded three-minute limit.

All the three required properties are met by all the three proposed methods. Anyway, we should discuss one important property-uniqueness of the optimal solution. It is quite clear the optimal solution does not have to be unique.

Lets consider a situation with just two political groups in which one political group obtains exactly 5% of all the valid votes and the second exactly the remaining 95%. Assume we have to allocate 10 seats and the threshold is lower than 5%. Then the output of the algorithm Ψ depends on the order of the parties. The lower party can be allocated one or zero seats and the bigger nine, or ten seats. The reason for this strange behavior is the selection of the maximum in step 3. of the algorithm Ψ (and for algorithm Ψ• it holds as well), which would not be given unambiguously if the ordering of the parties entering the algorithm is set arbitrarily. The algorithm Ψ is designed so it chooses the maximum corresponding to the political group with the lowest index. This fenomenon can not be fully eliminated and it is not eliminated by all the other currently used electoral systems, including the d’Hondt method.

Another problem that arises from this ambiguity is the problem of allocating diﬀerent numbers of seats to parties with exactly the same numbers of obtained votes. Assume a situation with three candidating political groups, two of them obtaining exactly 5% of all the cast valid votes and the third obtaining the remaining 90%. Assume again we have just 10 seats to allocate. The biggest party obtains 9 seats and the two smaller have to share one seat, which of course will be allocated to just one of them depending on the ordering of the parties entering the algorithm.

Because the ordering of the parties entering the algorithm is given by the number of obtained votes and so it is not a matter of chance or arbitrariness of any subject, we face only to the problem of ordering the parties with the same number of obtained votes, resp. the ordering of constituencies with the same number of cast valid votes. Now we have to use a random selection between the parties with the same number of obtained votes. A common method is to use a lot.

Let us add at the end of this chapter that the described, not very desirable behavior of the proposed algorithms can not be completely removed and it is treated by the other electoral systems (including the currently used electoral system to PS PCR)ˇ analogously (including the use of the lot). 4.4. Comparison of commonly used and proposed apportionment methods 79

4.4 Comparison of commonly used and proposed apportionment methods

All the three proposed electoral methods (systems) are reasonable to be introduced only if the constituencies are large enough as to have at least five seats to be allocated in each of them. According to the classification of constituencies (see http : //cs.wikipedia.org/wiki/V olebn%C3%ADobvod) the constituencies should be at least medium-sized. If there are many one- mandate constituencies (as in the case of election to the Senate), then it is not very reasonable to implement the proposed electoral allocation methods. The results of the proposed allocations of seats under the proposed methods we have compared with the results of some other methods on the data from all the elections to the Czech National Council (which can be considered a predecessor of the Chamber of Deputies) and from all the elections to PS PCRˇ which took place after 1989. These are the elections in 1990, 1992, 1996, 1998, 2002, 2006 and 2010. We compared all the three proposed methods with the currently used electoral system and other eighteen electoral systems that we have chosen randomly as combinations of various commonly used methods of converting votes into seats. We always used 5% election threshold. These eighteen electoral systems represent a sample from the set of all possible combinations of methods for converting votes into seats, which contains thousands of different electoral systems. Random selection from this set was made as comparing the results of thousands of electoral systems (which moreover often give the same results) is out of the scope of this study. For each of the compared electoral systems, we calculate all three measures of disproportionality (4.1) (4.2) and (4.3). Whenever the calculation of disproportionality (4.3) was not meaningful due to division by zero, the resulting value of disproportionality were marked as NA (not applicable). All three measures of disproportionality were used for each electoral system three times:

1. on allocation of seats to political groups,

2. on allocation of seats to constituencies,

3. on allocation of seats to combination of political groups and constituencies.

The compared electoral systems are made of commonly used allocation rules. Each of the allocation rules which is used by at least one of the eighteen chosen electoral systems, are denoted by abbreviations listed in the 80 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

table 4.5§§.

Allocation rule Abbreviation Allocation rule Abbreviation Haare quota HA Sainte-Laguë divisor SL Hagenbach-Bischoff quota HB d’Hondt divisor DH Droop quota DK Modified Sainte-Laguë divisor MSL Quota Imperiali KI Enhanced quota Imperiali PKI Divisor Imperiali DI Danish divisor DD Huntington divisor HU Method of maximal reminder Z

Tab. 4.5: Allocation rules

The proposed algorithms we denote N1 (minimizing the measure ρ), N2 (minimizing the measure ψ) and N3 (minimizing the measure ξ). Assume there are two stages of conversion of votes into seats similar to the one, which is currently used. In the first stage an allocation of seats to constituencies is performed and in the second stage an allocation of seats to political groups is performed in each constituency separately. The allocation rules HA, HB, DK, KI and PKI does not necessarily lead to an allocation of all the seats and therefore they must be followed by one of the other rules, which allocate the remained seats. On the other hand all the rules DH, DI, DD, SL, MSL, Z and HU always allocate all the seats. That is why it is always necessary to amend any rule from the first group by a rule from the second group. This fact we denote using the summation symbol ”+”. For example, if the Haare quota is amended by the method of the largest remainder, we denote the resulting rule HA + Z. Because at first are seats allocated to constituencies, and then to the political groups, while the rules used for these two separate allocations does not have to be identical, we separate the first part (allocation of seats to constituencies) from the second part (allocation of seats to political gtoups) with the symbol ”-”. The electoral system used to allocate seats to constituencies using the Hagenbach-Bischoff quota amended by the largest remainder method and then in each constituency allocating seats to political groups using the d’Hondt divisor, we denote HB+Z-DH. However, because just this particular election system is currently used in elections to PS PCRˇ we will exceptionally denote it ACT. All the other electoral systems we denote using the described scheme. The detail results of disproportionality comparisons are presented in the Appendix. Because the measure of disproportionality depends also on the number of constituencies and the number of political groups that have passed through the election threshold, we can not directly compare its values attained in different elections. Instead of this, we present the average rank (in

§§ The allocation rules are described in detail for example in [27]. 4.4. Comparison of commonly used and proposed apportionment methods 81 ascending ordering according to the measure of disproportionality) of each electoral system over all the elections. In the table 4.6 there are shown the average ranks of electoral systems when measuring the disproportionality of allocation of seats to political groups regardless of constituencies, in the table 4.7 average ranks of electoral systems when measuring the disproportionality of allocation of seats to constituencies regardless of the political groups and in the table 4.8 average ranks of electoral systems when measuring the disproportionality of allocation of seats to political groups in particular constituencies.

Method ρ φ ξ N1 1,00 1,00 1,29 N2 1,00 1,00 1,00 N3 1,00 1,00 1,00 ACT 11,29 11,14 11,14 DH-DH 10,71 10,43 10,57 SL-SL 5,29 5,00 5,29 HA+Z-DH 11,29 11,14 11,14 HU-HU 17,57 17,00 17,57 HA+Z-HU 17,43 17,14 17,43 DD-DD 6,86 6,86 7,29 DD-HU 17,57 17,57 17,57 DD-DI 20,71 20,43 20,71 DD-DH 11,57 11,57 11,43 DH-DI 20,43 20,29 20,43 DI-DI 21,29 21,00 21,29 DK+DH-DK-DH 8,43 9,14 8,86 DK+HU-SL 5,14 4,86 5,00 KI+Z-HA+Z 5,57 5,43 5,86 PKI+DI-PKI-DI 15,57 15,14 15,29 SL-KI+Z 10,43 10,71 10,43 SL-DH 11,71 11,57 11,57 HB+DI-MSL 6,57 5,86 5,86

Tab. 4.6: Average ranks of electoral systems over all analyzed elections to PS PCRˇ (measures of disproportionality of allocation of seats to political groups) 82 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

Method ρ φ ξ N1 1,00 1,00 1,29 N2 1,00 1,00 1,00 N3 1,00 1,00 1,00 ACT 5,86 5,86 5,86 DH-DH 4,00 4,00 4,00 SL-SL 7,57 7,57 7,57 HA+Z-DH 5,86 5,86 5,86 HU-HU 11,14 11,29 12,14 HA+Z-HU 5,86 5,86 5,86 DD-DD 12,29 12,43 12,14 DD-HU 12,29 12,43 12,14 DD-DI 12,29 12,43 12,14 DD-DH 12,29 12,43 12,14 DH-DI 4,00 4,00 4,00 DI-DI 14,57 14,00 14,86 DK+DH-DK-DH 5,86 5,86 5,86 DK+HU-SL 8,29 7,71 8,00 KI+Z-HA+Z 4,57 4,57 4,57 PKI+DI-PKI-DI 4,00 4,00 4,00 SL-KI+Z 7,57 7,57 7,57 SL-DH 7,57 7,57 7,57 HB+DI-MSL 12,86 12,14 12,43

Tab. 4.7: Average ranks of electoral systems over all analyzed elections to PS PCRˇ (measures of disproportionality of allocation of seats to constituencies)

Method ρ φ ξ N1 3,14 3,14 4,67 N2 3,14 3,14 4,67 N3 4,17 4,17 3,83 ACT 10,71 10,71 11,00 DH-DH 10,14 10,14 11,00 SL-SL 4,14 4,14 2,17 HA+Z-DH 10,71 10,71 11,17 HU-HU 17,14 17,14 17,17 HA+Z-HU 17,43 17,43 17,83 DD-DD 7,14 7,14 6,83 DD-HU 18,00 18,14 18,33 DD-DI 20,71 20,43 21,00 DD-DH 13,00 13,00 13,83 DH-DI 20,57 20,43 20,83 DI-DI 21,29 21,00 21,17 DK+DH-DK-DH 7,43 7,43 7,67 DK+HU-SL 3,86 3,86 2,67 KI+Z-HA+Z 2,43 2,43 3,50 PKI+DI-PKI-DI 15,43 15,43 15,50 SL-KI+Z 10,43 10,43 10,50 SL-DH 11,43 11,43 11,83 HB+DI-MSL 7,29 7,29 7,00

Tab. 4.8: Average ranks of electoral systems over all the analyzed elections to PS PCRˇ (measures of disproportionality of allocation of seats to combination of political groups and constituencies) 4.5. Conclusion 83

4.5 Conclusion

In the text we proposed three methods of allocation of seats to political groups in constituencies. We have compared the proposed methods with randomly chosen electoral systems and the electoral system, which is currently used, on the data of all historically held elections to the Chamber of Deputies and the Czech National Council between 1989 and 2010. Our results show that the proposed methods using mathematical programming, generally lead to a lower degree of disproportionality than all the other considered electoral systems. The proposed algorithms lead to a lower level of disproportionality of allocation of seats to political parties, as well as lower levels of disproportionality of allocation of seats to individual constituencies, than all the examined alternative electoral systems. It does not realy depend on whether the degree of disproportionality is measured using the Gallagher index, Loosemore-Hanby index, or Sainte-Laguë index. In addition, we keep the electoral threshold and so the lower level of disproportionality is not achieved at the expense of a larger number of parties entering the PS PCR.ˇ Besides the allocation of seats to political groups and to constituencies separately, we also examine the degree of disproportionality of allocation of seats to their combinations, i.e. of allocation of seats to political groups in particular constituencies. Even in this case the proposed methods attain very low levels of disproportionality, but not necessarily the lowest. The reason for this behavior is in the fact, that sometimes it is possible to attain lower level of disproportionality of allocation of seats to political groups in particular constituencies for the expense of higher level of disproportionality of the allocation of seats to political groups regardless of the constituencies, or for the expense of higher level of disproportionality of the allocation of seats to constituencies regardless of the political groups. The presented proofs show, that each of the three proposed methods leads to an optimal allocation of seats to political groups as well as optimal allocation of seats to constituencies. We moreover prove, that the disproportionality of allocation of seats to combinations of political groups and constituencies is minimized by all the three proposed methods unless the optimal solution is not non-negative. If the optimal solution is not non-negative, our methods find generally only suboptimal solution. Two of the three proposed allocation methods can be used generally, are fast enough and lead to a unique solution (unless there are parties with exactly the same number of votes). They can be used, by our opinion, as electoral systems to PS PCRˇ with no substantial difficulties and immediately. The third method can be used only if there is no political group that have 84 4. Apportionment in Proportional Electoral Systems Based on Integer Programming passed through the electoral threshold, but obtained zero votes in at least one constituency. Anyway, this constraint is not given by the method itself, but by the disproportionality measure (Sainte-Laguë index) which it minimizes.

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4.6 Appendix

4.6.1 Prooving optimality of solution found by algorithm Ψ Here we prove that the algorithm Ψ produce optimal solution to (4.4), resp. (4.5) when minimizing ρ. The proof will be made by contradiction. Lets denote x∗ the solution found by the alagorithm Ψ. Assume, this solution is not optimal to (4.4), resp. to (4.5). Hence there must be a way of decreasing the value of the objective function. x∗ is surely a feasible solution, because algorithm Ψ ends only when m seats are allocated. Hence there must be some couple of indices i and j so that 1 ≤ i 6= j ≤ P, for which holds the following

v 2 v 2 v 2 v 2 x∗ + 1 − m i + x∗ − 1 − m j < x∗ − m i + x∗ − m j (4.11) i v j v i v j v v v ⇔ x∗ + 1 − m i < x∗ − m j , i v j v Now we treat two cases separately.

∗ vi ∗ vj 1. If xi + 1 − m v > 0, then by (4.11) is also xj − m v > 0 and in the step 3 of algorithm Ψ there must have been a situation in which the values v 2 v 2 x∗ + 1 − m i and x∗ − m j i v j v had to be compared and as a maximum must had been chosen the value

v 2 x∗ + 1 − m i , i v

∗ because otherwise the value xi + 1 would never been lowered by one to ∗ ∗ ∗ xi , unless the value xj would previously been lowered by one to xj − 1. Hence the following inequality must hold

v 2 v 2 x∗ + 1 − m i ≥ x∗ − m j i v j v and that is in contradiction with (4.11).

2. The inequality v x∗ + 1 − m i ≤ 0 i v can never hold as it would imply v x∗ − m i < 0 i v 88 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

∗ ∗ and lowering xi + 1 by one to xi could never happen as in the step 3 of algorithm Ψ the argument of maximum is chosen only over such members, for which v x − m i > 0. i v Hence we have found out, that couple of indices i and j such that (4.11) does not exist and so there is no way how to lower the objective function. Therefore the solution x∗ must be an optimal solution to (4.4), resp. (4.5) when minimizing ρ. Now we prove, that algorithm Ψ always ﬁnds an optimal solution to (4.4), resp. (4.5) also when minimizing φ. We will again make the proof by contradiction, assuming x∗ is a solution found by algorithm Ψ when minimiznig φ, but is not an optimal solution. Building on the analogy to (4.11).

v v v v x∗ + 1 − m i + x∗ − 1 − m j < x∗ − m i + x∗ − m j (4.12) i v j v i v j v

∗ vi ∗ vj For simplicity we denote A := xi − m v and B := xj − m v . There are 9 possibilities:

A B |A + 1| + |B − 1| < |A| + |B| when it holds? 1. A < −1 B < 0 0 < 0 never 2. A < −1 0 ≤ B < 1 −B < B when 0 < B 3. A < −1 B ≥ 1 −2 < 0 always 4. −1 ≤ A < 0 B < 0 A < −1 never 5. −1 ≤ A < 0 0 ≤ B < 1 A + 1 < B when A + 1 < B 6. −1 ≤ A < 0 B ≥ 1 A < 0 always 7. A ≥ 0 B < 0 2 < 0 never 8. A ≥ 0 0 ≤ B < 1 2 < 0 never 9. A ≥ 0 B ≥ 1 0 < 0 never

Tab. 4.9: List of all possibilities

The condition (4.12) holds if and only if one of the situations from rows 2., 3., 5., or 6. of table 4.9 comes true. We know, that A < −1 can never ∗ vi hold as xi − m v > −1. This follows from step 3 of algorithm Ψ, in which vi we choose maximum only from those values xi − m v , which are strictly positive and only from the corresponding xi we subtract one in step 4. Hence ∗ vi we can never get to a situation, where xi − m v ≤ −1. Therefore situations from rows 2. and 3. will never come true and so it is sufficient to show, that conditions from rows 5. and 6. also lead to contradictions. Row 5. means (see the first three columns of row 5. of table 4.9) that it ∗ vi ∗ vj ∗ vi ∗ vj holds −1 ≤ xi − m v < 0, 0 ≤ xj − m v < 1 and xi + 1 − m v < xj − m v at 4.6. Appendix 89 the same time. Algorithm Ψ had to face a situation when it had to compare ∗ vi ∗ vj the value xi + 1 − m v with the value xj − m v and as a maximum of ∗ vi them it had chosen the value xi + 1 − m v , as otherwise it would had to ∗ ∗ ∗ before lowering the value xi + 1 by one to the value xi lower the value xj by ∗ vi ∗ vj one and this did not happen. Hence xi + 1 − m v ≥ xj − m v . According ∗ vi ∗ vj to non-negativity of xi + 1 − m v and non-negativity of xj − m v we obtain ∗ vi ∗ vi the inequality xi + 1 − m v ≥ xj − m v , which is in contradiction with ∗ vi ∗ vj xi + 1 − m v < xj − m v . Hence the situation described in row 5. can never come true. Row 6. (see the first two columns of row 6. of table 4.9) implies 0 ≤ ∗ vi ∗ vj xi + 1 − m v < 1 and xj − m v ≥ 1. Using the same reasoning as for row 5. we get v v x∗ + 1 − m i ≥ x∗ − m j . i v j v Because the absolute values are in this expression always applied to non- negative values, we can abandon them and we get v v 1 > x∗ + 1 − m i ≥ x∗ − m j ≥ 1, i v j v which is equivalent to 1 > 1. Therefore the situation from row 6. of table (4.9) can also never happen. Hence we’ve got the desired contradiction with the assumption, that x∗ is not an optimal solution to (4.4), resp. (4.4) when minimizing φ. It remains to prove that the algorithm Ψ finds the optimal solution of the problem (4.4), resp. (4.5) even when minimizing ξ. We will proceed again by contradiction, assuming x∗ is the solution found by the algorithm Ψ when minimizing ξ, but is not an optimal solution. Building on the analogy to (4.11).

2 v 2 2 v 2 x∗ + 1 − m vi x∗ − 1 − m j x∗ − m vi x∗ − m j i v + j v < i v + j v (4.13) vi vi vj vj m v m v m v m v x∗ + 1 x∗ − 1 ⇔ i 2 < j 2 vi vj

x∗+ 1 At some point the algorithm Ψ faced comparison of the values i 2 and vi x∗− 1 x∗+ 1 j 2 and as the higher one had been chosen the value i 2 . Hence vj vi

x∗ + 1 x∗ − 1 i 2 ≥ j 2 . vi vj 90 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

But this is again in contradiction with (4.13). In other words, there is no couple of indices i and j so that (4.13) holds and so there is no way, how to lower the value of the objective function. Therefore the solution x∗ is not just the solution found by the algorithm Ψ when minimizing ξ, but also an optimal solution to (4.4), resp. (4.5) when minimizing ξ. Lemma 12. For all i = 1,...,P it holds p p m i − 1 < M ≤ m i + 1 v i v and for all j = 1,...,R r r m j − 1 < L ≤ m j + 1. v j v

pi Proof: First we prove Mi ≤ m v + 1 for all i = 1,...,P. We do it by contradiction. Assume there is some ¯i ∈ {1,...,P } , such that p¯i m + 1 < M¯ (4.14) v i PP The algorithm Ψ ends only if it has allocated all the m seats, so i=1 Mi = m P pi and clearly i m v = m. The algorithm Ψ always chooses in the step 3 the pk 2 pk maximum of all the diﬀerences mk − m v for which mk − m v > 0. Be- cause the ﬁnal outcome of the algorithm is the optimal solution (M1,...,MP ) the following inequality must hold for all the k ∈ {1,...,P }

2 2 pk p¯i M + 1 − m ≥ M¯ − m > 1. (4.15) k v i v The ﬁrst inequality in (4.15) comes from the fact, that the algorithm Ψ had pk 2 pi 2 in one iteration faced the comparison of Mk + 1 − m v and M¯i − m v pk 2 and as the higher or equal one it has chosen the value Mk + 1 − m v , because otherwise it would in step 4 decrease the value M¯i by one prior to decreasing the value Mk + 1 by one to Mk. But this clearly didn’t happen as the ﬁnal values are Mk and M¯i. The second inequality in (4.15) comes from (4.14) as for whatever x > 1 is x2 > 1. From the fact, that maximum in step 3 of algorithm Ψ is always chosen pk only from those mk for which mk − m v > 0 we get p M + 1 − m k > 0, (4.16) k v which must hold for any k = 1,...,P. Now from (4.15) and (4.16) we get p p M + 1 − m k > 1 ⇒ M − m i > 0. (4.17) k v k v 4.6. Appendix 91

Because (4.17) must hold for all the k ∈ {1,...,P } we get

P P X X pk M > m = m k v k=1 k=1 PP and this contradicts k=1 Mk = m. pi The inequality m v − 1 ≤ Mi for all i = 1,...,P which is part of the lemma statement comes directly from the fact, that maximum is in step 3 pi of algorithm Ψ always chosen only from the mi, for which mi − m v > 0. This completes the proof. The proof for Lj is analogous.

4.6.2 Prooving optimality of solution found by algorithm Ψ• Here we prove that the algorithm Ψ• ﬁnds an optimal solution to the problem (4.6). We need to prove several statements: 1. Algorithm Ψ• always ends in a ﬁnite time. 2. The solution found by algorithm Ψ• is always feasible solution to (4.6). 3. Optimal solution to (4.6) always exists. 4. The solution found by algorithm Ψ• is always optimal.

• The fact that the allocation algorithm Ψ1 always ends in a ﬁnite time we show quite easily. The only step, where it can loop, is the step 3., in which there may not exist the searched ordered couple of indices. The existence of such an ordered couple (i, j) ∈ F so that v m > m ij ij v is given by the fact, that into the step we can get only when

P R X X mij > m. i=1 j=1 We know that P R X X vij m = m, v i=1 j=1 and hence also P R P R X X X X vij m > m . ij v i=1 j=1 i=1 j=1 92 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

Therefore it can not be v m ≤ m ij ij v for all (i, j) ∈ F. So there must exist at least one ordered couple of indices (i0, j0) ∈ F such that vi0j0 m 0 0 > m . i j v This assures no loop in step 3 as it always has something to choose for maximum. The maximum in step 3 is always chosen for such an ordered couple of indices (i0, j0) ∈ F, for which

vi0j0 m < m 0 0 ∈ v i j N and therefore 1 is always subtracted from a positive integer in the step 4 (it vi0j0 holds m v ≥ 0). This implies that in step 6, the actual matrix M = (mij) is non-negative. In the step 6 we just test, whether there is some row of the actual matrix M = (mij), which sum up to a value grater than the corresponding row constraint using the set formulation Ω = ∅, as Ω is empty if and only if Φ = ∅.¶¶ • The allocation algorithm Ψ1 in step 7 lowers the values mij for which PP PR k=1 mkj > Lj and k=1 mik > Mi by one as long as there are any. It stops in the moment, when no row sum and no column sum is higher than the coresponding row or column constraint. In this moment we have

P R X X mij ≤ m i=1 j=1 and the algorithm goes to the ﬁnal phase to step 8. In step 8 we test whether there is any row which sums to a lower value, than the value of the corresponding row constraint using the set formulation Λ = ∅. It is analogous to the situation in step 6. The allocation algorithm now PP PR increases all the values mij for which k=1 mkj < Lj and k=1 mik < Mi as long as they exist. It stops when no row and no column sum is lower than the corresponding row or column constraint. When it stops, it must hold

P R X X mij = m i=1 j=1

¶¶Sum of all the values of the matrix is greater than the sum of all row constraints, if and only if, it is higher than the sum of all column constraints as the problem is balanced. 4.6. Appendix 93

and all the row and column constraints are fulfilled. • The allocation algorithm Ψ1 has found a feasible solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0. Feasible solution could be obviously found using much less sophisticated method (north-west corner rule, Vogel method, MODI method, etc.), however our allocation algorithm finds a feasible solution that is close to the optimal one. However we have to keep in mind that the solution found by the allocation algorithm is not necessarily feasible to the problem (4.6) as it does not necessarily meet the constraint mij ∈ N0. It remains to prove that a finite sequence of applications of 1-changes and • −1-changes used in the iterative algorithm Ψ2 always leads to an optimal solution of the problem (4.6) with constraint mij ∈ Z instead of mij ∈ N0. First we prove the following lemma.

Lemma 13. Having two integer matrices that meet the same row and column constraints, we can always ﬁnd a ﬁnite sequence of 1-changes and −1-changes such that by their application to one of these matrices, we get the second.

Proof: Assume we have two integer matrices F = (fij) and O = (oij) of the same type with exactly the same all columns and all rows sums. We will construct a sequence of 1-changes and −1-changes which, when applied to F, give O. Assume for now both the matrices are of type 3 × 3. We will see, that the type of the matrices does not realy matter∗∗∗.

f11 f12 f13 f21 f22 f23 + (o11 − f11) -change = f31 f32 f33 (1,1),(2,2)

o11 f12 − (o11 − f11) f13 f21 − (o11 − f11) f22 + (o11 − f11) f23 f31 f32 f33

+ (o12 − f12 + o11 − f11)(1,2),(2,3) -change =

o11 o12 o13 f21 − o11 + f11 f22 − o12 + f12 f23 + o12 − f12 + o11 − f11 f31 f32 f33

+ (o21 − f21 + o11 − f11)(2,1),(3,2) -change =

o11 o12 o13 o21 f22 − o12 + f12 − o21 + f21 − o11 + f11 f23 + o12 − f12 + o11 − f11 o31 f32 + o21 − f21 + o11 − f11 f33

∗∗∗We only assume to have type m × n, where m ≥ 2 and n ≥ 2. Otherwise we can make no ε-changes and there is only one feasible solution. 94 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

+ (o22 − f22 + o12 − f12 + o21 − f21 + o11 − f11)(2,2),(3,3) -change =

o11 o12 o13 o21 o22 o23 . o31 o32 o33 This algorithm can be performed on matrices of any type with the same rows and columns sums. Assume we will now use only ε-changes of the type ε(i,j),(i+1,j+1), where i = 1, . . . , m − 1, and j = 1, . . . , n − 1. We just always start in the left upper corner and apply such an ε-change so that we get the desired value in the cell (1, 1). Then we move on to the right and apply such an ε-change so that we get the desired value in the cell (1, 2) and so on up to the cell (1, n − 1). In this point we must have already the desired value in the cell (1, n) as the sums of the ﬁrst rows of both the matrices are the same and so we move on to the other row, to cell (2, 1) and continue the same way. Because we never change the cells that we have already put equal to the desired values, we will allways end up with the desired matrix. Because each of the cells of both the matrices are integer, their sums or diﬀerences are also integer and instead of k-change, where k ∈ Z we can apply |k|-times 1-change (if k is positive) or −1-change (if k is negative). So we have shown the statement of the lemma holds.

Let’s prove now that (4.6) with constraint mij ∈ Z instead of mij ∈ N0 always has an optimal solution.

Lemma 14. Assume vij > 0 for all i = 1,...,P and all j = 1,...,R. Then the problem (4.6) with constraint mij ∈ Z instead of mij ∈ N0 always has an optimal solution for any of the three measures of disproportionality deﬁned in subsection 4.3.2. For measures ρ and φ the optimal solution exists even without any assumption on the matrix of votes V = (vij) .

Proof: Assume, we have (4.6) with constraint mij ∈ Z instead of mij ∈ N0. As- ∗ ∗ sume we have found some feasible solution A = aij to this problem using • alocation algorithm Ψ1. We already showed, this algorithm always ends up with a feasible solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0. Let’s deﬁne the set of all integer points that are closer to the point of perfect proportionality in the distance measured by each of the three measures of disproportionality: 4.6. Appendix 95

( P R P R ) 2 2 X X vij X X vij S : = X ∈ P ×R : m − x ≤ m − a∗ ρ Z v ij v ij i=1 j=1 i=1 j=1 ( P R P R ) X X vij X X vij S : = X ∈ P ×R : m − x ≤ m − a∗ φ Z v ij v ij i=1 j=1 i=1 j=1

( P R vij 2 P R vij ∗ 2 ) X X m − xij X X m − aij S : = X ∈ P ×R : v ≤ v ξ Z m vij m vij i=1 j=1 v i=1 j=1 v

Now we prove, each of the sets Sρ, Sφ and Sξ are finite. If so, the minimum of finite number of values always exists and we are finished. The fact that the optimal solution must be in each of these sets is clear. Let’s denote

P R 2 X X vij K : = m − a∗ < ∞, ρ v ij i=1 j=1 P R X X vij K : = m − a∗ < ∞ and φ v ij i=1 j=1

P R vij ∗ 2 X X m − aij K : = v < ∞, whenever v > 0, ξ m vij ij i=1 j=1 v for all i = 1,...,P and j = 1,...,R.

Now we see that in the case of Sρ, v v m ij − pK ≤ x ≤ m ij + pK (4.18) v ρ ij v ρ for each i = 1,...,P and each j = 1,...,R. As xij ∈ Z, for any i = 1,...,P and j = 1,...,R, there is only finitely many of those, fulfilling the constraint (4.18). In the case of Sφ, v v m ij − K ≤ x ≤ m ij + K (4.19) v φ ij v φ for each i = 1,...,P and each j = 1,...,R. As xij ∈ Z, for any i = 1,...,P and j = 1,...,R, there is at most 2Kφ + 1 fulfilling the constraint (4.19). In the case of Sφ,

v r v v r v m ij − m ij K ≤ x ≤ m ij + m ij K (4.20) v v ξ ij v v ξ 96 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

for each i = 1,...,P and each j = 1,...,R. As xij ∈ Z, for any i = 1,...,P q vij and j = 1,...,R, there is at most 2 m v Kξ + 1 fulﬁlling the constraint (4.20). We can conclude, that there is only ﬁnitely many integer matrices, that are closer to the point of perfect proportionality than the feasible matrix • found by algorithm Ψ1.

Let’s go back to proving optimality of the solution found by algorithm • Ψ . We know, the optimal solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0 always exists and whenever it exists, there exists a sequence of 1-changes and −1-changes such that by their successive application to the • matrix, which is an output of the allocation algorithm Ψ1, we get this optimal solution. Each ε-change can be viewed as a matrix and the application of ε-change as an addition of a matrix. Because summing integer matrices representing the ﬁnite sums of ε-changes is commutative and asociative, these matrices are closed on summing, there exists a zero matrix and for each sum of integer ε-changes, there exists an opposite sum of integer ε-changes, we know that all the matrices that are a ﬁnite sums of integer ε-changes together with the binary operation of matrix summation form a commutative (Abel) group. Adding any element of this commutative group to any feasible solution of the problem (4.6) with constraint mij ∈ Z instead of mij ∈ N0 always leads to a feasible solution of (4.6) with constraint mij ∈ Z instead of mij ∈ N0.

Lemma 15. Let’s denote F a feasible solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0. Then there always exists a sequence of 1-changes and −1-changes, such that 1. by their application to F we obtain an optimal solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0 2. none of the 1-changes or −1-changes in this sequence increases on its own the value of the objective function. Proof: Let’s have a feasible solution of the problem (4.6) with constraint mij ∈ Z instead of mij ∈ N0. We know there must always exist a sequence of integer 1-changes and −1-changes so that by their application we can get to the optimal solution of the same problem (let’s denote this sequence of ε-changes B1,..., BK ). It is an easy observation, that none of these ε-changes can itself increase the value of the objective function when applied. If there was some, we would apply the opposite ε-change and this would decrease the value of 4.6. Appendix 97 the objective function and that is in a contradiction with the assumption that the sequence B1,..., BK leads to the optimal solution. So we can conclude that from any feasible solution we can get to the optimal one by application of ﬁnite number of 1-changes and −1-changes and none of them would increase the value of the objective function.

Lemma 16. Let’s denote F a feasible solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0. If no 1-change or −1-change which would decrease the value of the objective function exists, then F is an optimal solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0. Proof: Assume, we have a feasible solution F to (4.6) with constraint mij ∈ Z instead of mij ∈ N0 and there is no 1-change or −1-change which would decrese the value of the objective function. Then the actual matrix must already represent an optimal solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0 as if it F was not optimal, we know from lemma 15 there always exists a sequence of 1-changes and −1-changes which leads from F to the optimal solution of the same problem so that none of the 1-changes or −1- changes increases the value of the objective function. But this means, the sequence of 1-changes and −1-changes leading to the optimal solution is such, that all the included 1-changes and −1-changes keep the value of the objective function the same and that is in contradiction with the non-optimality of F. Hence F must be an optimal solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0.

Note, that the optimality check is independent of the objective function. It holds for whatever objective function in (4.6) with constraint mij ∈ Z instead of mij ∈ N0. However, these arguments can not be used when we have the non-negativity constraint mij ∈ N0, because then the set of all feasible (i.e. those that lead to a non-negative solution) integer ε-changes and their sums is not an Abel group any more as for some of the members of the sequence B1,..., BK there does not have to exist a feasible opposite ε-change. In other words, the path from a feasible solution to the optimal one does not necessarily have to consist of feasible 1-changes and −1-changes such that each of them decreases the value of the objective function. That • is why we apply the additional iterative algorithm Ψ3. • Now we prove, that the additional iterative algorithm Ψ3 always ﬁnds an optimal solution to (4.6). 98 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

Lemma 17. Assume we have an optimal solution C∗ to (4.7). Then C∗ +A∗ ∗ (A denotes the optimal solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0) is an optimal solution to (4.6) no matter of which disproportionality measure (out of the three deﬁned in subsection 4.3.2) we use. Proof: We make the proof just for the measure ρ as the proof for the other two measures is analogous. The number of seats mij for political group i allocated in constituency j ∗ ∗ we write as cij + aij, where aij denotes the solution found by the iterative • algorithm Ψ2, which is the solution to (4.6) with constraint mij ∈ Z instead of mij ∈ N0. We know that

P R 2 X X (m + 1) PR 1 − sgn c + a∗ ≥ 0 (4.21) 2 ij ij i=1 j=1 for any C ∈ ZP ×R. We distinguish two cases: A∗ being non-negative and A∗ being not non- negative. Assume ﬁrst, A∗ is non-negative. Then clearly the optimal solution to ∗ (4.6) equals A , as the solution fulﬁlls the constraint mij ∈ N0. So for any P ×R PR PP C ∈ Z such that j=1 cij = 0 for all i = 1,...,P and i=1 cij = 0 for all j = 1,...,R, the following inequality holds

P R P R 2 2 X X vij X X vij m − a∗ ≤ m − c − a∗ . (4.22) v ij v ij ij i=1 j=1 i=1 j=1

Let’s denote C∗ the optimal solution to (4.7). Then for any C ∈ ZP ×R PR PP such that j=1 cij = 0 for all i = 1,...,P and i=1 cij = 0 for all j = 1,...,R, the following inequality holds

P R ! P R 2 2 X X (m + 1) X X vij PR 1 − sgn c∗ + a∗ + m − c∗ − a∗ ≤ 2 ij ij v ij ij i=1 j=1 i=1 j=1 P R ! P R 2 2 X X (m + 1) X X vij PR 1 − sgn c + a∗ + m − c − a∗ . 2 ij ij v ij ij i=1 j=1 i=1 j=1

∗ It is easy to realize, that cij = 0 for all i = 1,...,P and j = 1,...,R, because for this solution both the functions P R 2 X X vij m − c∗ − a∗ v ij ij i=1 j=1 4.6. Appendix 99 and P R 2 ! X X (m + 1) PR 1 − sgn c∗ + a∗ 2 ij ij i=1 j=1 are minimized as we know from (4.22) and (4.21). Because they are both minimized for the same matrix, this matrix has to be also the optimal solution to their sum. In other words the optimal solution C∗ to (4.7) whenever A∗ is non-negative equals to zero matrix and the optimal solution to (4.6) equals A∗. So the statement of the lemma holds. ∗ ∗ ∗ Assume, A is not non-negative. Then for the optimal solution cij + aij to (4.6) it must hold ∗ ∗ cij ≥ −aij, (4.23) for all i = 1,...,P and j = 1,...,R. We prove (4.23) by contradiction assuming it does not hold. So if it didn’t hold there would have to be some indices i0 ∈ {1,...,P } and j0 ∈ {1,...,R} such that we would have

∗ ∗ ci0j0 < −ai0j0 (4.24) and hence 2 ! (m + 1) ∗ ∗ 2 PR 1 − sgn c 0 0 + a 0 0 = (m + 1) PR ≥ (4.25) 2 i j i j

P R 2 X X vij m − (m + 1) . v i=1 j=1

vij The inequality in (4.25) is given by m v ≥ 0 for all i = 1,...,P and all j = 1,...,R. Let’s prove, that for any feasible solution M to (4.6) holds (4.26) for all i = 1,...,P and all j = 1,...,R. Note, that there is always some feasible solution to (4.6) as it is a balanced transportation problem and so it always has some integer non-negative feasible solution. v 2 v 2 m ij − m < m ij − (m + 1) . (4.26) v ij v We know, that for every i ∈ {1,...,P } and every j ∈ {1,...,R} is v 0 ≤ m ij < min {M + 1,L + 1} = min {M ,L } + 1 < m + 1, (4.27) v i j i j

PR vij pi as from lemma 12 we know that j=1 m v = m v < Mi + 1 and also PP vij rj i=1 m v = m v < Lj + 1. Therefore clearly v 2 v 2 v 2 m ij − m ≤ m ij − min {M + 1,L + 1} < m ij − (m + 1) v ij v i j v 100 4. Apportionment in Proportional Electoral Systems Based on Integer Programming for every i = 1,...,P and every j = 1,...,R. So we have proved, that there is always some feasible integer solution ∗ ∗ M = (mij) to (4.6) for which holds (4.26) and (4.25) and so cij +aij for which holds (4.24) for some indices (i0, j0) , can never be an optimal solution to (4.6). ∗ ∗ Therefore it must be cij ≥ −aij for all i = 1,...,P and all j = 1,...,R. ∗ ∗ In the statement of the lemma we assume that cij + aij is an optimal solution to (4.7). It means the following inequality holds

P R 2 2 X X (m + 1) vij PR 1 − sgn c∗ + a∗ + m − c∗ − a∗ ≤ (4.28) 2 ij ij v ij ij i=1 j=1 P R 2 2 X X (m + 1) vij PR 1 − sgn c + a∗ + m − c − a∗ 2 ij ij v ij ij i=1 j=1

PP PR for all cij ∈ Z, i=1 cij = 0, j=1 cij = 0 for all i = 1,...,P and j = 1,...,R. Because of (4.23) it holds

P R 2 X X (m + 1) PR 1 − sgn c∗ + a∗ = 0 2 ij ij i=1 j=1 and so (4.28) changes to

P R 2 2 vij X X (m + 1) m − c∗ − a∗ ≤ PR 1 − sgn c + a∗ + v ij ij 2 ij ij i=1 j=1 P R 2 X X vij m − c − a∗ . v ij ij i=1 j=1

∗ ∗ PP ∗ Now for all cij ≥ −aij ⇔ cij + aij ∈ N0 such that i=1 cij + aij = Lj, PR ∗ j=1 cij + aij = Mi for all i = 1,...,P and j = 1,...,R, is clearly

P R 2 X X (m + 1) PR 1 − sgn c + a∗ = 0 2 ij ij i=1 j=1 and hence (4.29) changes for all such C to v 2 v 2 m ij − c∗ − a∗ ≤ m ij − c − a∗ . v ij ij v ij ij ∗ ∗ So we can conclude cij + aij is an optimal solution to (4.6). 4.6. Appendix 101

4.6.3 Detail results In this section we present detail results comparing rates of disproportionality between diﬀerent electoral systems on the data from each election to the Chamber of Deputies of the Czech Parliament (PS PCR).ˇ

1990 ρ φ ξ N1 0,528 1,226 0,018 N2 0,528 1,226 0,018 N3 0,528 1,226 0,018 ACT 86,899 16,145 1,405 DH-DH 17,278 6,665 0,391 SL-SL 7,695 4,665 0,170 HA+Z-DH 86,899 16,145 1,405 HU-HU 169,776 22,145 3,103 HA+Z-HU 169,776 22,145 3,103 DD-DD 7,326 5,293 0,262 DD-HU 169,776 22,145 3,103 DD-DI 282,187 28,145 5,285 DD-DH 86,899 16,145 1,405 DH-DI 246,816 26,145 4,810 DI-DI 246,816 26,145 4,810 DK+DH-DK-DH 40,021 10,665 0,723 DK+HU-SL 7,695 4,665 0,170 KI+Z-HA+Z 13,653 6,665 0,422 PKI+DI-PKI-DI 169,776 22,145 3,103 SL-KI+Z 86,899 16,145 1,405 SL-DH 86,899 16,145 1,405 HB+DI-MSL 17,278 6,665 0,391

Tab. 4.10: Disproportionality of seats allocation to political groups (Czech national Council 1990) 102 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

1992 ρ φ ξ N1 1,028 2,594 0,052 N2 1,028 2,594 0,000 N3 1,028 2,594 0,000 ACT 84,362 19,495 2,451 DH-DH 9,769 7,257 0,438 SL-SL 5,932 5,943 0,324 HA+Z-DH 84,362 19,495 2,451 HU-HU 229,283 29,495 5,478 HA+Z-HU 225,340 29,495 5,233 DD-DD 3,018 4,585 0,160 DD-HU 225,340 29,495 5,233 DD-DI 452,527 44,458 11,428 DD-DH 84,362 19,495 2,451 DH-DI 499,071 46,458 12,524 DI-DI 499,071 46,458 12,524 DK+DH-DK-DH 11,933 8,214 0,460 DK+HU-SL 5,932 5,943 0,324 KI+Z-HA+Z 2,103 3,670 0,121 PKI+DI-PKI-DI 43,295 13,495 1,229 SL-KI+Z 39,636 15,257 1,382 SL-DH 84,362 19,495 2,451 HB+DI-MSL 13,783 9,257 2,451

Tab. 4.11: Disproportionality of seats allocation to political groups (Czech national Council 1992)

1996 ρ φ ξ N1 0,527 1,577 0,017 N2 0,527 1,577 0,017 N3 0,527 1,577 0,017 ACT 32,084 11,577 1,015 DH-DH 39,065 13,577 1,250 SL-SL 3,081 3,768 0,143 HA+Z-DH 32,084 11,577 1,015 HU-HU 107,851 23,577 3,408 HA+Z-HU 127,024 25,577 3,851 DD-DD 6,502 4,833 0,270 DD-HU 127,024 25,577 3,851 DD-DI 451,176 47,577 13,159 DD-DH 32,084 11,577 1,015 DH-DI 407,895 45,577 11,762 DI-DI 436,876 47,577 12,676 DK+DH-DK-DH 5,934 5,048 0,259 DK+HU-SL 3,081 3,768 0,143 KI+Z-HA+Z 3,081 3,768 0,143 PKI+DI-PKI-DI 41,383 13,577 1,376 SL-KI+Z 14,387 7,577 0,572 SL-DH 32,084 11,577 1,015 HB+DI-MSL 4,074 3,768 0,103

Tab. 4.12: Disproportionality of seats allocation to political groups (Chamber of Deputies 1996) 4.6. Appendix 103

1998 ρ φ ξ N1 0,469 1,384 0,016 N2 0,469 1,384 0,016 N3 0,469 1,384 0,016 ACT 27,554 11,143 0,817 DH-DH 27,554 11,143 0,817 SL-SL 8,708 5,128 0,199 HA+Z-DH 27,554 11,143 0,817 HU-HU 158,381 27,143 4,397 HA+Z-HU 158,381 27,143 4,397 DD-DD 5,310 4,857 0,143 DD-HU 154,982 27,143 4,377 DD-DI 202,462 31,143 5,633 DD-DH 27,554 11,143 0,817 DH-DI 202,462 31,143 5,633 DI-DI 230,141 33,143 6,495 DK+DH-DK-DH 8,395 5,384 0,250 DK+HU-SL 8,708 5,128 0,199 KI+Z-HA+Z 8,708 5,128 0,199 PKI+DI-PKI-DI 52,081 15,143 1,585 SL-KI+Z 22,922 9,384 0,724 SL-DH 27,554 11,143 0,817 HB+DI-MSL 8,708 5,128 0,199

Tab. 4.13: Disproportionality of seats allocation to political groups (Chamber of Deputies 1998)

2002 ρ φ ξ N1 0,242 0,792 0,007 N2 0,242 0,792 0,007 N3 0,242 0,792 0,007 ACT 9,477 5,938 0,211 DH-DH 9,477 5,938 0,211 SL-SL 1,511 2,062 0,031 HA+Z-DH 9,477 5,938 0,211 HU-HU 36,500 11,938 0,824 HA+Z-HU 64,377 15,938 1,437 DD-DD 6,153 4,062 0,136 DD-HU 63,749 15,938 1,364 DD-DI 156,333 23,938 3,434 DD-DH 15,995 7,938 0,339 DH-DI 164,961 23,938 3,752 DI-DI 177,589 23,938 4,178 DK+DH-DK-DH 4,119 3,938 0,086 DK+HU-SL 1,724 2,137 0,031 KI+Z-HA+Z 3,096 2,792 0,076 PKI+DI-PKI-DI 41,128 11,938 0,991 SL-KI+Z 15,995 7,938 0,339 SL-DH 15,995 7,938 0,339 HB+DI-MSL 0,870 1,420 0,023

Tab. 4.14: Disproportionality of seats allocation to political groups (Chamber of Deputies 2002) 104 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

2006 ρ φ ξ N1 0,718 1,755 0,031 N2 0,718 1,755 0,031 N3 0,718 1,755 0,031 ACT 122,368 22,014 5,336 DH-DH 122,368 22,014 5,336 SL-SL 12,202 5,755 0,710 HA+Z-DH 122,368 22,014 5,336 HU-HU 318,689 38,014 11,646 HA+Z-HU 315,703 38,014 11,628 DD-DD 22,646 8,488 1,137 DD-HU 345,704 40,014 12,249 DD-DI 675,257 56,014 22,556 DD-DH 122,368 22,014 5,336 DH-DI 714,272 58,014 23,466 DI-DI 714,272 58,014 23,466 DK+DH-DK-DH 22,791 10,014 0,799 DK+HU-SL 12,202 5,755 0,710 KI+Z-HA+Z 4,460 3,755 0,265 PKI+DI-PKI-DI 150,627 24,014 6,680 SL-KI+Z 84,842 18,014 3,786 SL-DH 122,368 22,014 5,336 HB+DI-MSL 14,531 6,514 0,423

Tab. 4.15: Disproportionality of seats allocation to political groups (Chamber of Deputies 2006)

2010 ρ φ ξ N1 0,316 1,164 0,008 N2 0,316 1,164 0,008 N3 0,316 1,164 0,008 ACT 23,650 9,505 0,657 DH-DH 23,650 9,505 0,657 SL-SL 3,857 3,543 0,084 HA+Z-DH 23,650 9,505 0,657 HU-HU 86,259 19,180 2,475 HA+Z-HU 61,115 15,505 1,727 DD-DD 8,392 4,820 0,252 DD-HU 70,601 17,505 1,909 DD-DI 144,120 23,505 3,958 DD-DH 22,467 9,505 0,626 DH-DI 142,064 23,505 3,868 DI-DI 155,302 25,180 4,336 DK+DH-DK-DH 11,929 6,836 0,297 DK+HU-SL 1,040 1,887 0,031 KI+Z-HA+Z 5,171 3,981 0,129 PKI+DI-PKI-DI 50,392 13,505 1,449 SL-KI+Z 7,781 5,505 0,218 SL-DH 23,650 9,505 0,657 HB+DI-MSL 1,040 1,887 0,031

Tab. 4.16: Disproportionality of seats allocation to political groups (Chamber of Deputies 2010) 4.6. Appendix 105

1990 ρ φ ξ N1 2,800 7,375 NA N2 2,800 7,375 NA N3 NA NA NA ACT 23,001 19,867 NA DH-DH 17,930 16,817 NA SL-SL 16,155 15,816 NA HA+Z-DH 23,001 19,867 NA HU-HU 38,273 24,893 NA HA+Z-HU 38,273 24,893 NA DD-DD 20,601 17,818 NA DD-HU 43,239 26,893 NA DD-DI 58,191 30,473 NA DD-DH 27,967 21,867 NA DH-DI 50,155 28,473 NA DI-DI 47,003 26,588 NA DK+DH-DK-DH 18,882 17,009 NA DK+HU-SL 16,155 15,816 NA KI+Z-HA+Z 16,764 16,175 NA PKI+DI-PKI-DI 38,273 24,893 NA SL-KI+Z 23,001 19,867 NA SL-DH 23,001 19,867 NA HB+DI-MSL 17,930 16,817 NA

Tab. 4.17: Disproportionality of seats allocation to combinations of political groups and constituencies (Czech national Council 1990)

1992 ρ φ ξ N1 5,445 15,375 3,871 N2 5,445 15,375 3,871 N3 6,159 16,088 3,817 ACT 18,976 24,357 7,551 DH-DH 9,441 19,093 4,313 SL-SL 8,199 18,100 3,842 HA+Z-DH 18,976 24,357 7,657 HU-HU 41,402 34,737 14,044 HA+Z-HU 41,879 35,213 14,102 DD-DD 8,041 17,942 4,031 DD-HU 41,879 35,213 14,102 DD-DI 70,327 47,398 22,267 DD-DH 18,976 24,357 7,657 DH-DI 75,199 49,398 23,694 DI-DI 73,808 49,398 22,124 DK+DH-DK-DH 8,621 18,522 4,127 DK+HU-SL 8,199 18,100 3,948 KI+Z-HA+Z 7,589 17,490 4,041 PKI+DI-PKI-DI 12,778 20,508 5,695 SL-KI+Z 9,996 19,676 4,868 SL-DH 18,976 24,357 7,657 HB+DI-MSL 9,852 19,504 4,490

Tab. 4.18: Disproportionality of seats allocation to combinations of political groups and constituencies (Czech national Council 1992) 106 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

1996 ρ φ ξ N1 5,015 12,620 1,773 N2 5,015 12,620 1,773 N3 5,015 12,620 1,773 ACT 7,264 14,869 2,769 DH-DH 8,202 15,807 3,147 SL-SL 4,127 11,732 1,452 HA+Z-DH 7,264 14,869 2,769 HU-HU 19,851 24,650 7,274 HA+Z-HU 22,204 26,650 8,065 DD-DD 4,466 12,071 1,642 DD-HU 22,204 26,650 8,065 DD-DI 68,689 47,662 21,085 DD-DH 7,264 14,869 2,769 DH-DI 62,475 45,662 18,984 DI-DI 65,944 47,662 20,338 DK+DH-DK-DH 4,444 12,049 1,488 DK+HU-SL 4,127 11,732 1,452 KI+Z-HA+Z 4,105 11,710 1,458 PKI+DI-PKI-DI 8,436 16,040 3,729 SL-KI+Z 5,342 12,946 1,835 SL-DH 7,264 14,869 2,769 HB+DI-MSL 4,412 12,017 1,614

Tab. 4.19: Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 1996) 4.6. Appendix 107

1998 ρ φ ξ N1 5,027 11,965 1,459 N2 5,027 11,965 1,459 N3 5,280 12,218 1,447 ACT 8,516 15,454 2,246 DH-DH 8,516 15,454 2,246 SL-SL 4,607 11,545 1,265 HA+Z-DH 8,516 15,454 2,246 HU-HU 27,223 28,375 6,773 HA+Z-HU 27,223 28,375 6,773 DD-DD 5,383 12,321 1,411 DD-HU 26,000 28,109 6,781 DD-DI 33,099 31,928 8,994 DD-DH 9,238 16,176 2,388 DH-DI 32,172 31,928 8,652 DI-DI 35,852 33,928 10,094 DK+DH-DK-DH 6,618 13,556 1,733 DK+HU-SL 4,607 11,545 1,265 KI+Z-HA+Z 4,607 11,545 1,265 PKI+DI-PKI-DI 11,372 18,310 3,856 SL-KI+Z 7,805 14,743 2,141 SL-DH 8,516 15,454 2,246 HB+DI-MSL 4,607 11,545 1,265

Tab. 4.20: Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 1998)

2002 ρ φ ξ N1 6,018 15,295 2,529 N2 6,018 15,295 2,529 N3 6,018 15,295 2,471 ACT 6,384 15,661 2,753 DH-DH 6,384 15,661 2,753 SL-SL 6,532 15,809 2,545 HA+Z-DH 6,384 15,661 2,753 HU-HU 9,048 18,325 3,964 HA+Z-HU 10,614 19,704 4,410 DD-DD 6,644 15,921 2,579 DD-HU 10,238 19,328 4,040 DD-DI 20,523 27,510 7,508 DD-DH 6,858 16,135 2,850 DH-DI 20,900 27,886 7,878 DI-DI 24,077 29,671 8,996 DK+DH-DK-DH 6,287 15,563 2,729 DK+HU-SL 5,851 15,128 2,457 KI+Z-HA+Z 5,805 15,082 2,458 PKI+DI-PKI-DI 9,081 18,358 4,027 SL-KI+Z 6,858 16,135 2,850 SL-DH 6,858 16,135 2,850 HB+DI-MSL 6,922 16,199 2,758

Tab. 4.21: Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 2002) 108 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

2006 ρ φ ξ N1 6,741 18,779 5,227 N2 6,741 18,779 5,227 N3 6,741 18,779 5,003 ACT 16,598 27,381 10,921 DH-DH 16,598 27,381 10,921 SL-SL 7,711 19,748 5,139 HA+Z-DH 16,598 27,381 10,921 HU-HU 31,569 39,076 16,667 HA+Z-HU 30,717 38,898 16,695 DD-DD 8,348 20,385 5,950 DD-HU 32,802 40,854 17,479 DD-DI 59,811 56,354 29,503 DD-DH 15,781 27,337 11,018 DH-DI 61,546 58,089 30,101 DI-DI 68,160 58,089 29,192 DK+DH-DK-DH 8,680 20,718 7,338 DK+HU-SL 7,711 19,748 5,139 KI+Z-HA+Z 7,292 19,329 5,213 PKI+DI-PKI-DI 18,029 28,813 11,931 SL-KI+Z 13,045 24,310 9,534 SL-DH 16,598 27,381 10,921 HB+DI-MSL 8,246 20,283 7,100

Tab. 4.22: Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 2006)

2010 ρ φ ξ N1 7,097 17,955 2,933 N2 7,097 17,955 2,933 N3 7,097 17,955 2,978 ACT 9,068 19,926 3,722 DH-DH 9,068 19,926 3,722 SL-SL 6,566 17,423 2,918 HA+Z-DH 9,068 19,926 3,722 HU-HU 14,284 25,141 6,402 HA+Z-HU 13,049 23,907 5,530 DD-DD 7,013 17,871 3,351 DD-HU 13,446 24,284 5,795 DD-DI 20,490 30,087 7,917 DD-DH 9,974 20,812 4,047 DH-DI 19,721 29,511 7,780 DI-DI 22,373 30,375 8,387 DK+DH-DK-DH 7,111 17,969 3,100 DK+HU-SL 7,283 18,141 3,059 KI+Z-HA+Z 6,432 17,290 2,955 PKI+DI-PKI-DI 10,029 20,886 4,680 SL-KI+Z 7,556 18,413 3,167 SL-DH 9,068 19,926 3,722 HB+DI-MSL 7,283 18,141 3,059

Tab. 4.23: Disproportionality of seats allocation to combinations of political groups and constituencies (Chamber of Deputies 2010) 4.6. Appendix 109

1990 ρ φ ξ N1 0,888 2,422 0,044 N2 0,888 2,422 0,037 N3 0,888 2,422 0,037 ACT 31,283 13,067 1,016 DH-DH 31,283 13,067 1,016 SL-SL 31,283 13,067 1,016 HA+Z-DH 31,283 13,067 1,016 HU-HU 31,283 13,067 1,016 HA+Z-HU 31,283 13,067 1,016 DD-DD 40,480 15,067 1,351 DD-HU 40,480 15,067 1,351 DD-DI 40,480 15,067 1,351 DD-DH 40,480 15,067 1,351 DH-DI 31,283 13,067 1,016 DI-DI 24,823 12,024 0,866 DK+DH-DK-DH 31,283 13,067 1,016 DK+HU-SL 31,283 13,067 1,016 KI+Z-HA+Z 31,283 13,067 1,016 PKI+DI-PKI-DI 31,283 13,067 1,016 SL-KI+Z 31,283 13,067 1,016 SL-DH 31,283 13,067 1,016 HB+DI-MSL 31,283 13,067 1,016

Tab. 4.24: Disproportionality of seats allocation to constituencies (Czech national Council 1990)

1992 ρ φ ξ N1 0,367 1,297 0,016 N2 0,367 1,297 0,016 N3 0,367 1,297 0,016 ACT 8,957 6,831 0,294 DH-DH 6,357 4,831 0,185 SL-SL 8,957 6,831 0,294 HA+Z-DH 8,957 6,831 0,294 HU-HU 6,357 4,831 0,185 HA+Z-HU 8,957 6,831 0,294 DD-DD 8,957 6,831 0,294 DD-HU 8,957 6,831 0,294 DD-DI 8,957 6,831 0,294 DD-DH 8,957 6,831 0,294 DH-DI 6,357 4,831 0,185 DI-DI 3,930 4,632 0,167 DK+DH-DK-DH 8,957 6,831 0,294 DK+HU-SL 8,957 6,831 0,294 KI+Z-HA+Z 8,957 6,831 0,294 PKI+DI-PKI-DI 6,357 4,831 0,185 SL-KI+Z 8,957 6,831 0,294 SL-DH 8,957 6,831 0,294 HB+DI-MSL 8,957 6,831 0,294

Tab. 4.25: Disproportionality of seats allocation to constituencies (Czech national Council 1992) 110 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

1996 ρ φ ξ N1 0,410 1,721 0,019 N2 0,410 1,721 0,019 N3 0,410 1,721 0,019 ACT 2,446 3,271 0,092 DH-DH 0,410 1,721 0,019 SL-SL 2,446 3,271 0,092 HA+Z-DH 2,446 3,271 0,092 HU-HU 2,424 3,112 0,087 HA+Z-HU 2,446 3,271 0,092 DD-DD 2,446 3,271 0,092 DD-HU 2,446 3,271 0,092 DD-DI 2,446 3,271 0,092 DD-DH 2,446 3,271 0,092 DH-DI 0,410 1,721 0,019 DI-DI 2,424 3,112 0,087 DK+DH-DK-DH 2,446 3,271 0,092 DK+HU-SL 2,446 3,271 0,092 KI+Z-HA+Z 0,410 1,721 0,019 PKI+DI-PKI-DI 0,410 1,721 0,019 SL-KI+Z 2,446 3,271 0,092 SL-DH 2,446 3,271 0,092 HB+DI-MSL 3,506 3,721 0,175

Tab. 4.26: Disproportionality of seats allocation to constituencies (Chamber of Deputies 1996)

1998 ρ φ ξ N1 0,652 2,030 0,027 N2 0,652 2,030 0,027 N3 0,652 2,030 0,027 ACT 2,510 3,488 0,095 DH-DH 2,510 3,488 0,095 SL-SL 2,510 3,488 0,095 HA+Z-DH 2,510 3,488 0,095 HU-HU 2,510 3,488 0,095 HA+Z-HU 2,510 3,488 0,095 DD-DD 3,558 4,536 0,146 DD-HU 3,558 4,536 0,146 DD-DI 3,558 4,536 0,146 DD-DH 3,558 4,536 0,146 DH-DI 2,510 3,488 0,095 DI-DI 5,034 5,402 0,200 DK+DH-DK-DH 2,510 3,488 0,095 DK+HU-SL 2,510 3,488 0,095 KI+Z-HA+Z 2,510 3,488 0,095 PKI+DI-PKI-DI 2,510 3,488 0,095 SL-KI+Z 2,510 3,488 0,095 SL-DH 2,510 3,488 0,095 HB+DI-MSL 2,510 3,488 0,095

Tab. 4.27: Disproportionality of seats allocation to constituencies (Chamber of Deputies 1998) 4.6. Appendix 111

2002 ρ φ ξ N1 0,713 2,779 0,059 N2 0,713 2,779 0,059 N3 0,713 2,779 0,059 ACT 0,713 2,779 0,059 DH-DH 0,713 2,779 0,059 SL-SL 1,545 3,611 0,095 HA+Z-DH 0,713 2,779 0,059 HU-HU 3,059 5,125 0,321 HA+Z-HU 0,713 2,779 0,059 DD-DD 1,545 3,611 0,095 DD-HU 1,545 3,611 0,095 DD-DI 1,545 3,611 0,095 DD-DH 1,545 3,611 0,095 DH-DI 0,713 2,779 0,059 DI-DI 4,273 6,339 0,407 DK+DH-DK-DH 0,713 2,779 0,059 DK+HU-SL 0,713 2,779 0,059 KI+Z-HA+Z 0,713 2,779 0,059 PKI+DI-PKI-DI 0,713 2,779 0,059 SL-KI+Z 1,545 3,611 0,095 SL-DH 1,545 3,611 0,095 HB+DI-MSL 3,119 4,609 0,299

Tab. 4.28: Disproportionality of seats allocation to constituencies (Chamber of Deputies 2002)

2006 ρ φ ξ N1 0,834 2,666 0,063 N2 0,834 2,666 0,063 N3 0,834 2,666 0,063 ACT 0,993 2,825 0,070 DH-DH 0,993 2,825 0,070 SL-SL 0,993 2,825 0,070 HA+Z-DH 0,993 2,825 0,070 HU-HU 2,179 4,011 0,135 HA+Z-HU 0,993 2,825 0,070 DD-DD 0,834 2,666 0,063 DD-HU 0,834 2,666 0,063 DD-DI 0,834 2,666 0,063 DD-DH 0,834 2,666 0,063 DH-DI 0,993 2,825 0,070 DI-DI 9,547 9,563 0,745 DK+DH-DK-DH 0,993 2,825 0,070 DK+HU-SL 0,993 2,825 0,070 KI+Z-HA+Z 0,993 2,825 0,070 PKI+DI-PKI-DI 0,993 2,825 0,070 SL-KI+Z 0,993 2,825 0,070 SL-DH 0,993 2,825 0,070 HB+DI-MSL 0,993 2,825 0,070

Tab. 4.29: Disproportionality of seats allocation to constituencies (Chamber of Deputies 2006) 112 4. Apportionment in Proportional Electoral Systems Based on Integer Programming

2010 ρ φ ξ N1 0,731 2,247 0,047 N2 0,731 2,247 0,047 N3 0,731 2,247 0,047 ACT 2,641 4,157 0,161 DH-DH 2,641 4,157 0,161 SL-SL 2,641 4,157 0,161 HA+Z-DH 2,641 4,157 0,161 HU-HU 4,248 5,695 0,437 HA+Z-HU 2,641 4,157 0,161 DD-DD 5,141 6,157 0,321 DD-HU 5,141 6,157 0,321 DD-DI 5,141 6,157 0,321 DD-DH 5,141 6,157 0,321 DH-DI 2,641 4,157 0,161 DI-DI 6,283 5,921 0,494 DK+DH-DK-DH 2,641 4,157 0,161 DK+HU-SL 6,943 6,157 0,348 KI+Z-HA+Z 2,641 4,157 0,161 PKI+DI-PKI-DI 2,641 4,157 0,161 SL-KI+Z 2,641 4,157 0,161 SL-DH 2,641 4,157 0,161 HB+DI-MSL 6,943 6,157 0,348

Tab. 4.30: Disproportionality of seats allocation to constituencies (Chamber of Deputies 2010)