Mathematical theory of democracy

The mathematical theory of democracy is an interdisciplinary branch of the public choice and social choice theories conceptualized by Andranik Tangian. It operationalizes the fundamental idea to modern democracies – that of political representation, 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,^[1] who explained the difference between democracy, oligarchy and mixed constitution in terms of vote weighting.^[2] The historical mathematization of social choice principles is reviewed by Iain McLean and Arnold Urken.^[3] 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.^[4][5]

In 1960s, the notion of policy representation has been introduced.^[6] 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^[7] and monitored through the MANIFESTO data base that quantitatively characterizes parties' election programs in about 50 democratic states since 1945.^[8] 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.^[9]

The theoretical aspects of how to best satisfy a society with a composite program first considered by Andranik Tangian^[10] and Steven Brams with coauthors^[11] is now studied within the relatively new discipline of judgment aggregation.^{[12][13][14][15]} The mathematical theory of democracy focuses, in particular, on the practical aspects of the same topic.^[16][17] 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.^[18][19][20]

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.^[10] 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).^{[17][21][22][23]}

Third Vote

Main article: 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.^[17][24][25] The name "Third Vote" has been used in electoral experiments where the new method had to complement the two-vote German system.^{[26][27][28][29]} 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.^[30]

The multi-voter paradoxes of Condorcet and Kenneth Arrow 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^[31]
• Quantitative analysis and alternative interpretation of Arrow's impossibility theorem^[17][32]
• Analysis of Athenian democracy based on selection of public officers by lot^[33]
• Analysis of election outcomes with estimations of the representativeness of election winners and parliament factions^{[16][34][35][21]}
• Analysis of national political spectra^[36][22]

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 fluctuations, since some of them (e.g. in the USA) "represent in advance" some other share price fluctuations (e.g. in Germany)^[37]
• Traffic light control and coordination, since situations at certain crossroads represent in advance the situation at some other crossroads^[38]

References

1. Miroiu, Adrian; Partenie, Catalin (2019). "Collective choice in Aristotle". Constitutional Political Economy. 30 (3): 261–281. doi:10.1007/s10602-019-09279-1. S2CID 254421072.
2. Aristotle (340 BC). Politics, Book 3. Cambridge MA: Harward University Press; 1944. pp. 1280a.7–25.{{cite book}}: CS1 maint: numeric names: authors list (link)
3. McLean, Iain; Urken, Arnold Bernard, eds. (1995). Classics of social choice. Ann Arbor MI: University of Michigan Press.
4. Simeone, Bruno; Pukelsheim, Friedrich, eds. (2006). Mathematics and Democracy. Berlin-Heidelberg: Springer.
5. Brams, Steven (2008). Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures. Princeton, NJ: Princeton University Press.
6. Miller, Warren Edward; Stokes, Donald Elkinton (1963). "Constituency influence in Congress". American Political Science Review. 57 (1): 45–56. doi:10.2307/1952717. JSTOR 1952717. S2CID 144730217.
7. Budge, Ian; McDonald, Michael D (2007). "Election and party system effects on policy representation: Bringing time into a comparative perspective". Electoral Studies. 26 (1): 168–179. doi:10.1016/j.electstud.2006.02.001.
8. Volkens, Andrea; Bara, Judith; Budge, Ian; McDonald, Michael D; Klingemann, Hans-Dieter, eds. (2013). Mapping policy preferences from texts: Statistical solutions for manifesto analysts. Oxford: Oxford University Press.
9. Garzia, Diego; Marschall, Stefan (eds.) (2014). Matching voters with parties and candidates: voting advice applications in a comparative perspective. Colchester UK: ECPR Press. {{cite book}}: |first2= has generic name (help)
10. Tanguiane (Tangian), Andranick (1994). "Arrow's paradox and mathematical theory of democracy". Social Choice and Welfare. 11 (1): 1–82. doi:10.1007/BF00182898. S2CID 154076212.
11. Brams, Steven J; Kilgour, D Marc; Zwicker, William S (1998). "The paradox of multiple elections". Social Choice and Welfare. 15 (2): 211–236. doi:10.1007/s003550050101. S2CID 154193592.
12. List, Christian; Puppe, Clemens (2009). "Judgment aggregation: a survey". In Anand, Paul; Puppe, Clemens; Pattranaik, Prasanta (eds.). Oxford handbook of rational and social choice. Oxford: Oxford University Press. pp. 457–482.
13. List, Christian (2012). "The theory of judgment aggregation: an introductory review" (PDF). Synthese. 187 (1): 179–207. doi:10.1007/s11229-011-0025-3. S2CID 6430197.
14. Grossi, Davide; Pigozzi, Gabriella (2014). Judgment aggregation: a primer. San Rafael CA: Morgan and Claypool Publishers.
15. Lang, Jérôme; Pigozzi, Gabriella; Slavkovik, Marija; van der Torre, Leendert (Leon); Vesic, Srdjan S (2017). "A partial taxonomy of judgment aggregation rules and their properties". Social Choice and Welfare. 48 (2): 327–356. arXiv:1502.05888. doi:10.1007/s00355-016-1006-8. S2CID 12154890.
16. Tangian, Andranik (2014). Mathematical theory of democracy. Studies in Choice and Welfare. Berlin-Heidelberg: Springer. doi:10.1007/978-3-642-38724-1. ISBN 978-3-642-38723-4.
17. Tangian, Andranik (2020). Analytical theory of democracy. Vols. 1 and 2. Studies in Choice and Welfare. Cham, Switzerland: Springer. doi:10.1007/978-3-030-39691-6. ISBN 978-3-030-39690-9. S2CID 216190330.
18. Andranik Tangian (1989). "Interpretation of dictator in Arrow's model as a collective representative". Matematicheskoe Modelirovanie (in Russian). 1 (7): 51–92.
19. Andranik Tangian (1989). "A model of collective representation under democracy". Matematicheskoe Modelirovanie (in Russian). 1 (10): 80–125.
20. Tanguiane (Andranik Tangian), Andranick (1991). Aggregation and representation of preferences: introduction to mathematical theory of democracy. Berlin-Heidelberg: Springer. doi:10.1007/978-3-642-76516-2. ISBN 978-3-642-76516-2.
21. Tangian, Andranik (2022). Analysis of the 2021 Bundestag Elections 1/4. Representativeness of the Parties and the Bundestag. ECON Working Papers. Vol. 151. Karlsruhe: Karlsruhe Institute of Technology. doi:10.5445/IR/1000143156. ISSN 2190-9806. Retrieved 8 August 2022.
22. Tangian, Andranik (2022). Analysis of the 2021 Bundestag Elections 2/4. Political Spectrum. ECON Working Papers. Vol. 152. Karlsruhe: Karlsruhe Institute of Technology. doi:10.5445/IR/1000143157. ISSN 2190-9806. Retrieved 8 August 2022.
23. Tangian, Andranik (2022). Analysis of the 2021 Bundestag Elections 3/4. Tackling the Bundestag Growth. ECON Working Papers. Vol. 153. Karlsruhe: Karlsruhe Institute of Technology. doi:10.5445/IR/1000143158. ISSN 2190-9806. Retrieved 8 August 2022.
24. Tangian, Andranik (2017). "An election method to improve policy representation of a parliament". Group Decision and Negotiation. 26 (1): 181–196. doi:10.1007/S10726-016-9508-4. S2CID 157553362.
25. Tangian, Andranik (2022). Analysis of the 2021 Bundestag Elections 4/4. The Third Vote Application. ECON Working Papers. Vol. 154. Karlsruhe: Karlsruhe Institute of Technology. doi:10.5445/IR/1000143159. ISSN 2190-9806. Retrieved 8 August 2022.
26. Tangian, Andranik (2017). "The Third Vote experiment: Enhancing policy representation of a student parliament". Group Decision and Negotiation. 26 (4): 1091–1124. doi:10.1007/S10726-017-9540-Z. S2CID 158833198.
27. Amrhein, Marius; Diemer, Antonia; Eßwein, Bastian; Waldeck, Maximilian; Schäfer, Sebastian. "The Third Vote (web page)". Karlsruhe: Karlsruhe Institute of Technology, Institute ECON. Retrieved 15 December 2020.
28. "Turning a political education instrument (voting advice application) in a new election method", World Forum for Democracy 2016, Lab 7: Reloading Elections, Strasbourg: Council of Europe, 7–9 November 2016, retrieved 15 December 2020
29. "Well Informed Vote", World Forum for Democracy 2019, Lab 5: Voting under the Influence, Strasbourg: Council of Europe, 6–8 November 2019, retrieved 15 December 2020
30. Tangian, Andranik (2021). "MCDM application of the Third Vote" (PDF). Group Decision and Negotiation. 30 (4): 775–787. doi:10.1007/s10726-021-09733-2. ISSN 0926-2644. S2CID 235571433.
31. Tanguiane (Tangian), Andranick (1993). "Inefficiency of democratic decision making in an unstable society". Social Choice and Welfare. 10 (3): 249–300. doi:10.1007/BF00182508. S2CID 154339432.
32. Tangian, Andranik (2010). "Application of the mathematical theory of democracy to Arrow's Impossibility Theorem (How dictatorial are Arrow's dictators?)". Social Choice and Welfare. 35 (1): 135–167. doi:10.1007/s00355-009-0433-1. S2CID 206958453.
33. Tangian, Andranik (2008). "A mathematical model of Athenian democracy". Social Choice and Welfare. 31 (4): 537–572. doi:10.1007/s00355-008-0295-y. S2CID 7112590.
34. Tangian, Andranik (2010). "German parliamentary elections 2009 from the viewpoint of direct democracy". Social Choice and Welfare. 40 (3): 833–869. doi:10.1007/s00355-011-0646-y. S2CID 39079121.
35. Tangian, Andranik (2017). "Policy representation of a parliament: the case of the German Bundestag 2013 election". Group Decision and Negotiation. 26 (1): 151–179. doi:10.1007/S10726-016-9507-5. S2CID 157256280.
36. Tangian, Andranik (2019). "Visualizing the political spectrum of Germany by contiguously ordering the party policy profiles". In Skiadis, Christos H.; Bozeman, James R. (eds.). Data Analysis and Applications 2. London: ISTE-Wiley. pp. 193–208. doi:10.1002/9781119579465.ch14. ISBN 9781119579465. S2CID 159314655.
37. Tangian, Andranik (2008). "Predicting DAX trends from Dow Jones data by methods of the mathematical theory of democracy". European Journal of Operational Research. 185 (3): 1632–1662. doi:10.1016/j.ejor.2006.08.011.
38. Tangian, Andranik (2007). "Selecting predictors for traffic control by methods of the mathematical theory of democracy". European Journal of Operational Research. 181 (2): 986–1003. doi:10.1016/j.ejor.2006.06.036. S2CID 46111084.