Mathematical theory of democracy

The mathematical theory of democracy is an interdisciplinary branch of the public choice and social choice theories. It operationalizes the fundamental idea to modern democracies — that of political representation, in particular focusing on policy representation - how well the electorate's policy preferences are represented by the party system and the government. The representative capability is measured by means of dedicated indices that are used both for analytical purposes and practical applications.

The mathematical approach to politics goes back to Aristotle, who explained the difference between democracy, oligarchy and mixed constitution in terms of vote weighting.[1] The historical mathematization of social choice principles is reviewed by Iain McLean and Arnold Urken.[2] Modern mathematical studies in democracy are due to the game, public choice and social choice theories, which emerged after the World War II; for reviews see.[3][4]

In 1960s, the notion of policy representation has been introduced.[5] It deals with how well the party system and the government represent the electorate's policy preferences on numerous policy issues. Policy representation is currently intensively studied[6] and monitored through the MANIFESTO data base that quantitatively characterizes parties' election programs in about 50 democratic states since 1945.[7] In 1989, it was operationalized in the Dutch voting advice application (VAA) StemWijzer (= ‘VoteMatch’), which helps to find the party that best represents the user's policy preferences. Since then it has been launched on the internet and adapted by about 20 countries as well as by the European Union.[8]

The theoretical aspects of how to best satisfy a society with a composite program first considered by Steven Brams with coauthors[9] is now studied within the relatively new discipline of judgment aggregation.[10][11][12][13] The mathematical theory of democracy focuses, in particular, on the practical aspects of the same topic.For this purpose, the quality of policy representation is measured by special indices. These indices are based on the same data structures as VAAs and they are used for a wide range of problems:

• for evaluating the representative capability of individual representatives (deputee, president) and representative bodies (council, committee, cabinet, magistrate, coalition, jury)[14]
• for proving the inefficiency of democracy in an unstable society[15]
• for an alternative interpretation of Arrow's impossibility theorem[16][17]
• for analysis of Athenian democracy based on selection of public officers by lot[18]
• for evaluation of election outcomes[14][19][20]
• for analysis of national political spectra[21]
• for collective multiple-criteria decision making, since alternatives can be considered representatives of individual wishes[14]
• for predicting share price fluctuations, since some of them (e.g. in the USA) `represent in advance' some other share price fluctuations (e.g. in Germany)[22]
• for traffic light control and coordination, since situations at certain crossroads represent in advance the situation at some other crossroads[23]
• in the so-called "Third Vote" election method that enhances policy representation.[24][25][26][27][28]

The name mathematical theory of democracy is due to the game theorist Nikolai Vorobyov who commented on the first findings of this kind in the late 1980s.[14] A comprehensive presentation of the theory is provided by Andranik Tangian.[29]