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Intro -- Preface -- Acknowledgements -- Contents -- Part I History -- Chapter 1 -- Athenian Democracy -- 1.1 Introduction: Democracy for Stability and Social Consent -- 1.2 Path to Democracy -- 1.2.1 Draco's Laws -- 1.2.2 Laws of Solon -- 1.2.3 Tyranny of Peisistratos -- 1.2.4 Rule of Peisistratos' Sons and Cleisthenes' Rise -- 1.3 Establishing Democracy -- 1.3.1 Cleisthenes' Constitution -- 1.4 Culmination of Democracy -- 1.4.1 Pericles' Leadership -- 1.5 Philosophy of Democracy -- 1.5.1 Controversial Opinions on Democracy -- 1.5.2 Rotation and Sortition Principles -- 1.5.3 Plato's Skepticism About Democracy -- 1.5.4 Plato's Ideal Governance -- 1.5.5 Plato's Ideas and Modern Politics -- 1.6 Aristotle's Mathematical Model of Three Types of Governance -- 1.7 Summary: Antique Understanding of Democracy -- References -- Chapter 2 -- Echoes of Democracy in Ancient Rome -- 2.1 Introduction: Greek Influence on Romans -- 2.2 Typology of States and the Theory of Mixed Government -- 2.3 Rotation of Government Types -- 2.4 The Roman Republic -- 2.5 Pliny's Logical Analysis of a Senate Hearing -- 2.6 Summary: Equilibrium of Power Forms in Rome -- References -- Chapter 3 -- Revival of Democracy in Italian Medieval City-Republics -- 3.1 Introduction: Propagation of Elections -- 3.2 The Venetian Republic -- 3.3 The Florentine Republic -- 3.3.1 Professional Guilds as Miniature Republics -- 3.3.2 Establishment and Abolition of the First Democracy -- 3.3.3 Second Democracy and Ordinances of Justice -- 3.3.4 Republican Constitution -- 3.3.5 Sunset of Democracy -- 3.4 Theory of Mixed Government Revitalized -- 3.4.1 Treatise by St. Thomas Aquinas -- 3.4.2 Breakthrough Ideas of Marsilius of Padua and William of Ockham -- 3.4.3 Leonardo Bruni and Francesco Guicciardini on Elections and Sortition -- 3.4.4 Nicc`olo Machiavelli's Republican Thought.

The mathematical theory of democracy deals with selection of representatives who make decisions on behalf of the whole society. In this book, the notion of representativeness is operationalized with the index of popularity (the average percentage of the population whose opinion is represented on a number of issues) and the index of universality (the frequency of cases when the opinion of a majority is represented). These indices are applied to evaluate and study the properties of single representatives (e.g. president) and representative bodies (e.g. parliament, magistrate, cabinet, jury, coalition). To bridge representative and direct democracy, an election method is proposed that is based not on voting but on indexing candidates with respect to the electorate's political profile. In addition, societal and non-societal applications are considered

AbstractWhen choosing among alternatives, group members may have various preferences regarding the properties of a solution being sought. Since the properties partially do and partially do not meet their collective wishes, the alternatives are in fact better or worse representatives of the collective will. This idea is implemented in the so-called Third Vote election method aimed at enhancing policy representation, and we show how to use it for collective multi-criteria decision making. To be specific, we consider an example of a committee charged with naming a campus library when neither plurality vote nor Condorcet method nor Borda count gives a unique solution. The committee members have differing opinions, such as whether the library should reflect the national affiliation, be named after a great man, relate to sciences, and so forth. Balancing opinion on these issues, the proposed library names are evaluated and the optimal compromise is found.

The advocates of modern western democracy promote the viewpoint that the class division of the society is becoming outdated. We attempt to disprove this statement with an example of 28 German parties who participated in the 2013 federal election. The official party positions on 38 policy issues are considered and the parties are identified with vectors of this 38-dimensional policy space. The statement in question, that there is no predominant political axis, would imply that the party vectors are scattered homogeneously, making a ball-shaped cloud of "observations". However, the Prime Component Analysis (PCA) shows that the party vectors constitute a thin ellipsoid whose two longest diameters cover 83.4% of the total variance. The consequent party ordering is the left-right axis rolled in a circumference, making the far-left and far-right ends meet. Basing on this empirical evidence, we conclude that neither the left-right characterization of parties nor the class opposition is outdated. Next, it is shown that the electoral success is highly correlated with the number of party members, but not with the party's capacity to represent public opinion. For this purpose, a representativeness index is defined which measures how well the party policy profiles match with the results of 36 public opinion polls on 36 out of the 38 policy issues mentioned. To reveal representativeness trends, the parties are ordered contiguously, with neighboring parties having close policy profiles. This contiguous ordering is found with four optimization methods: (1) dimensionality reduction by means of PCA, (2) traveling salesman problem to construct the shortest chain of proximate parties, (3) least squares to minimize the distances between parties with close profiles, and (4) largest squares to maximize the distances between parties with opposite profiles. The most salient trend is observed for the circular left–right party ordering found with the PCA. The best representatives of public opinion are the moderate left, next come the ...