The mathematical theory of democracy is an interdisciplinary branch of the

public choice Public choice, or public choice theory, is "the use of economic tools to deal with traditional problems of political science". Gordon Tullock, 9872008, "public choice," ''The New Palgrave Dictionary of Economics''. . Its content includes the s ...

and

social choice Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Soci ...

theories conceptualized by

Andranik Tangian Andranik Semovich Tangian (Melik-Tangyan) (Russian: Андраник Семович Тангян (Мелик-Тангян)); born March 29, 1952) is a Soviet Armenian-German mathematician, political economist and music theorist. Tangian is known ...

. It operationalizes the fundamental idea to modern democracies – that of

political representation Political representation is the activity of making citizens "present" in public policy-making processes when political actors act in the best interest of citizens. This definition of political representation is consistent with a wide variety of vie ...

, in particular focusing on policy representation, i.e. 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.

History

The mathematical approach to politics goes back to

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...

, who explained the difference between democracy, oligarchy and mixed constitution in terms of vote weighting. The historical mathematization of social choice principles is reviewed by

Iain McLean Iain Donald Smith McLean (born 8 October 1983) is a Scottish international Bowls, lawn bowler. Bowls career McLean won a bronze medal in the 2016 World Outdoor Bowls Championship - Men's Fours, fours at the 2016 World Outdoor Bowls Championsh ...

and Arnold Urken. Modern mathematical studies in democracy are due to the game,

and

theories, which emerged after the

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing ...

; for reviews see. In 1960s, the notion of policy representation has been introduced. 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 and monitored through the MANIFESTO data base that quantitatively characterizes parties' election programs in about 50 democratic states since 1945. 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. The theoretical aspects of how to best satisfy a society with a composite program first considered by

and Steven Brams with coauthors is now studied within the relatively new discipline of ''judgment aggregation''. The mathematical theory of democracy focuses, in particular, on the practical aspects of the same topic. 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.

Content of the theory

Like the social choice theory, the mathematical theory of democracy analyzes the collective choice from a given list of candidates. However, these theories differ in both the methodology and the data used. The social choice theory operates on the voters’ preference orders of the candidates and applies an axiomatic approach to find impeccable solutions. The mathematical theory of democracy is based on the candidates’ and the electorate's positions on topical political questions and finds the representatives (deputes, president) and representative bodies (parliament, committee, cabinet) that best represent the public opinion. For this purpose, several quantitative indices to assess and compare the representative capability are introduced. It has been proven that compromise candidates and representative bodies can always be found, even if there is no perfect solution in terms of social choice theory. Among other things, it is proven that even among the axiomatically prohibited Arrow's dictators there always exist good representatives of the society (e.g. to be elected as presidents), which implies a principal possibility of democracy in every society – contrary to the common interpretation of

Arrow's impossibility theorem Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral syst ...

. The further results deal with the characteristics and special features of individual representatives (such as members of parliament, chairmen, presidents) and the committees (such as parliaments, commissions, cabinets, coalitions and juries).

Third Vote

The Third Vote is an election method developed within the framework of the mathematical theory of democracy to expand the concept of political representation. The name "Third Vote" has been used in electoral experiments where the new method had to complement the two-vote German system. Its aim is to draw voters' attention from individual politicians with their charisma and communication skills to specific policy issues. The question "''Who'' should be elected?'" is replaced by the question "''What'' do we choose?" (Party platform). Instead of candidate names, the Third Vote ballot asks for Yes/No answers to the questions raised in the candidates’ manifestos. The same is demanded by voting advice applications (VAA), but the answers are processed in a different way. In contrast to VAAs, the voter receives no advice which party best represents the voter's position. Instead, the Third Vote procedure determines the policy profile of the entire electorate with the balances of public opinion on each issue (pro and cons percentages on individual topics). The election winner is the candidate whose policy profile best matches with the policy profile of the entire electorate. If the candidates are political parties competing for parliamentary seats, the latter are allocated to the parties in proportion to the closeness of their policy profiles to that of the electorate. When considering decision options instead of candidates, the questions focus on their specific characteristics. The multi-voter paradoxes of

Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher and mathematician. His ideas, including support for a liberal economy, free and equal pu ...

and

Kenneth Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972. In economics ...

are circumvented because the entire electorate with its opinion profile is viewed as a single agent, or a single voter.

Applications

Societal applications

* Inefficiency of democracy in an unstable society * Quantitative analysis and alternative interpretation of

* Analysis of Athenian democracy based on selection of public officers by lot * Analysis of election outcomes with estimations of the representativeness of election winners and parliament factions * Analysis of national political spectra

Non-societal applications

Since some interrelated objects or processes "represent" one another with certain time delays, revealing the best "representatives" or "anticipators" can be used for predictions. This technique is implemented in the following applications: * Predicting

share price A share price is the price of a single share of a number of saleable equity shares of a company. In layman's terms, the stock price is the highest amount someone is willing to pay for the stock, or the lowest amount that it can be bought for. B ...

fluctuations, since some of them (e.g. in the USA) "represent in advance" some other share price fluctuations (e.g. in Germany) *

Traffic light control and coordination The normal function of traffic lights requires more than sight control and coordination to ensure that traffic and pedestrians move as smoothly, and safely as possible. A variety of different control systems are used to accomplish this, ranging fr ...

, since situations at certain crossroads represent in advance the situation at some other crossroads

References

{{Reflist Social choice theory Public choice theory