Mathematical theory of democracy and its applications 2. Fundamentals

1. Mathematical theory of democracy and its applications2. Fundamentals Andranik TangianHans-Böckler Foundation, Düsseldorf University of Karlsruheandranik-tangian@boeckler.de

2. Plan of the course Three blocks : • Basics History, Arrow‘s paradox, indicators of representativeness, solution • Fundamentals: Model of Athens governance (president, assembly, magistrates, courts) and German Bundestag (parties and coalitions) • Applications MCDM, traffic control, financies

3. Athens: Draco 621 BC In the 7th century BC Athens was governed by magistrates formed from Eupatridai (=well born), that is, leading clans Polarization between the rich and the poor First laws „written not in ink but in blood“ The rich lost their legislative and juridical monopoly, since the laws became obligatory for all citizens Selection by lot of minor magistrates Draconian laws had little success

4. Solon 638 BC–558 BC 594 BC: general amnesty no enslavement for debt freedom for slaves for debt general political reforms The laws remained valid with minor modifications till 322 BC

5. Solon‘s political reform 594 BC Election depend on wealth rather than birth Offices can be held by the top property class of four, in case of archons (Athens governers) of top two classes Council of 400 making agenda for the People‘s Assembly Selection by lot of all magistrates from an elected short list

6. Cleisthenes’ constitution 507 BC New governance structure New division of Attica represented in the Council of 500 New calendar Ostracism

7. Strategoi = military generals (Elections) Magistrates held by board of 10 (Lot) Courts >201 jurors (Lot) (Rotation) Athenian democracy in 507 BC President of Commitee (1 day) Committee of 50 (to guide the Boule) Boule: Council of 500 (to steer the Ekklesia) (Lot) Ekklesia: people‘s assembly (quorum 6000, >40 sessions a year) Citizenry: Athenian males >20 years, 20000-30000

8. Historic concept of democracy Plato, Aristotle, Montesquieu, Rousseau: Democracy  selection by lot (=lottery) Oligarchy  election by vote Vote is appropriate if there are common values + of selection by lot: gives equal chances - of election by vote: tend to retain at power the same persons good for professional politicians who easily change opinions to get and to hold the power

9. Athenian democracy by Aristotle 621 BC Draconic Laws selection by lot of minor magistrates 594 BC Solon’s Laws selection by lot of all magistrates from an elected short list 507/508 BC Cleisthenes’ constitution600 of 700 offices distributed by lot 487 BC selection by lot of archons from an elected short list 403 BC selection by lot of archons and other magistrates

10. Example: Athens 462 BC Three leaders Pericles 495–429 BC democratic party Cimon 510–450 BC aristocratic party Ephialtes 495–461 BC democratic party

11. Example: Question at issue 1 Remove powers from the Court of the Areopagus, an ancient aristocratic institution composed of “men of noble birth” who held office for lifeEphialtes opposed aristocrats led by Cimon. Together with Pericles he removed many powers from the Areoopagus and gave them to the People’s Court or the Assembly

12. Areopagus The Areopagus (view from the Acropolis) – a monolith where Athenian aristocrats decided important matters of state

13. Example: Question at issue 2 Pay for political participation The payment for public office and attending the Assembly had been adopted on the initiative of Pericles who promoted total participation of Athenian citizens in politics

14. ATTICA Pericles: We do not say that a man who takes no interest in politics is a man who minds his own business; we say that he has no business here at all But: Trips to >40 assemblies a year took 3-5 days every week which complicated economic activity

15. Example: Question at issue 3 Help Spartans to put down a rebellionIn 462 BC Sparta asked for help in putting down a rebellion of helots in Ithomi (Messinia). Ephialtes opposed sending help, but Athenians delegated Cimon with a military force. In his absence, Ephialtes and Pericles limited the power of the Areopagus. Spartans did not appreciate it and refused to accept the help. The army returned to Athens in rage. Cimon was ostracized for 10 years

16. Ancient Grece 233 km

17. Example: Evaluation of leaders

18. Questions

19. Individuals

20. Candidates

21. Representativeness Example: b11 = 1, b12= -1; r1qshown by color q1 q2 Protagonists ai1=1 Antagonists aiq=-1 ai2=-1 ai2=-1 ai2=1 ai2=1

22. Indicator of popularity – „spatial“ representativeness

23. Indicator of universality –„temporal“ representativeness

24. Indicator of goodness – „specific“ representativeness

25. Notation

26. Theorem: Computing popularity

27. Proof for popularity aq is the balance of opinions = predominance of protagonists over antagonists for question q bcq= ±1 opinion of candidate c on question q rcq= 0.5 + 0.5 aq bcq (think!). Hence, Pc = ∑q µqrcq= ∑q µq (0.5 + 0.5 aqbcq) = 0.5 + 0.5 ∑q µqaqbcq = 0.5 + 0.5 (µ.a)′bc P = ∑c Pc ξc= ∑c [0.5 + 0.5 ∑q µqaqbcq]ξ c = 0.5+0.5(µ.a)′b

28. Theorem: Computing universality

29. Theorem: Computing goodness

30. Back to the example of Athens

31. Geometric interpretation

32. Analogy with vectors of forces in physics The best candidate has the largest projection of his opinion vector bc on the µ-weighted social vector, defined for each indicator appropriately Variety of candidate opinions is reduced to a one-dimensional evaluation

33. Assembly, Council of 500, Committee of 50, and juries

34. Magistrate (Cabinet, Ministry)

35. Representativeness of decisive bodies

36. Indicators of decisive bodies

37. Theorem: Computing the indices

38. Theorem: Computing the indices

39. Absolute maxima of the indicators

40. Theorem: Saturation of decisive bodies “recruited” from the society

41. Theorem: Stability of decisive bodies “recruited” from the society

42. Implications Much superior performance of magistrates over parliaments of the same size k The larger the size k of decisive body, the higher the indices. Indices of large decisive bodies are close to absolute maxima Performance of a decisive body depends on its size k rather than on the size of the society n(Monaco needs as large parliament as China)

43. Implications 2 Statistical viewpoint: If candidates are “recruited” from the society, a representative body is a sample of the society and statistically tends to represent rather than not to represent the totality Moreover, the larger the sample, the better representation. A sufficiently large sample represents the society with almost 100% reliability Analogy to quality control and Gallup polls

44. Goodness as a function of majority-to-minority ratio Society is unstable if the majority-to-minority ratio is close to 50:50

45. Inefficiency of democracy in an unstable society A political power is efficient if good results are achieved by moderate means. If a president satisfies the same percentage of population as a large Assembly then his efficiency is superior In an unstable society (majority-to-minority ratio close to 50:50) the democratic institutions provide the same power quality as single representatives, implying a higher efficiency of personal power

46. Minimal expected goodness of Athenian decisive bodies

47. Election to Bundestag 2009

48. Source data: 32 Y/N-questions (like in Wahl-o-mat)

49. Representativeness

50. Reminding the indicators Popularity: % of the electorate represented, averaged on 32 questions Universality: frequency of representing a majority (% of 32 questions)