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 opera- tions 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 in- troduces 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 convert- ing votes into seats within the elections to the Chamber of Deputies of the Czech Parliament which are designed to minimize the level of disproportion- ality. 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 pro- cedure . 28 2.4 The impact of the distinct rules on efficiency within the pro- cedure 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 pro- cedure 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 ma- jority 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 elec- tions 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 po- litical groups and constituencies (Czech national Council 1990) 105 4.18 Disproportionality of seats allocation to combinations of po- litical groups and constituencies (Czech national Council 1992) 105 4.19 Disproportionality of seats allocation to combinations of po- litical groups and constituencies (Chamber of Deputies 1996) . 106 4.20 Disproportionality of seats allocation to combinations of po- litical groups and constituencies (Chamber of Deputies 1998) . 107 4.21 Disproportionality of seats allocation to combinations of po- litical groups and constituencies (Chamber of Deputies 2002) . 107 4.22 Disproportionality of seats allocation to combinations of po- litical groups and constituencies (Chamber of Deputies 2006) . 108 4.23 Disproportionality of seats allocation to combinations of po- litical 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 sci- ence 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 opera- tions research, especially the mathematical optimization, used on the issues of political economy. The first two essays deal with the concept of the effi- ciency 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 prob- ability 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.