Mathematical theory of democracy and its applications 1. Basics

1. Mathematical theory of democracy and its applications1. Basics Andranik TangianHans-Böckler Foundation, Düsseldorf University of Karlsruheandranik-tangian@boeckler.de

2. Fundamental distinction Two aspects of social decisions: Quality – How good they are Procedure – How they are acheived Democracy deals with the procedure

3. Plan of the course Three blocks : • Basics History, Arrow‘s paradox, indicators of representativeness, solution • Representative bodies President, parliament, government, parties and coalitions • Applications MCDM, traffic control, financies

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

5. 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

6. Culmination of Athenian democracy 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 Pericles (495 – 429 BC)

7. 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

8. 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

9. Decline of democracy 322 BC Abolishment of Athenian democracy Republicanism (= lot + elections + hereditary power) in Rome and medieval Italian towns American and French Revolutions 1776-89 promoted republicanism not democracy Lot survived in juries and administrative rotation (deans in German universities) Prohibition of selection by lot of members of the French Superior Council of Universities 1985 No democratic labeling of Soviet Republics

10. Democracy duringthe Cold War Communist propaganda German Democratic Republic Korean People’s Democratic Republic Federal Democratic Republic of Ethiopia Democratic Republic of Afghanistan … Western response Democratic (!!) elections Human rights Free press

11. Democracy, elections and voting Voting for decisions (direct democracy) ≠ voting for election of candidates (oligarchy, now representative democracy) Voting, regardless of the way it is used, is considered an instrument of democratization, that is, involving more people into political participation.

12. 1st analysis of complex voting situation Letter by Pliny the Younger (62–113) about a session in the Rome Senate on selecting a punishement for a crime • leniency (Pliny‘s choice and simple majority) • execution (minority), or • banishment (finally accepted) subsequent binary vote? ternary vote (simple majority)?

13. Voting in case of more than two issues Ramon L(l)ull (1232 – 1316) 1st European novel Blanquerna (1283 – 87): method reinvented by Borda in 1770 De arte eleccionis (1299): method reinvented by Condorcet in 1785

14. Caridnal method of Borda (1733–99) Memoire sur les élections… (1770/84) Example: The most undesirable wins! A B C Preference B C B C A A 8 7 6 Score of A = 3 · 8 + 1 · 7 + 1 · 6 = 37 Score of B = 2 · 8 + 3 · 7 + 2 · 6 = 49 Score of C = 1 · 8 + 2 · 7 + 3 · 6 = 40

15. Dependence on irrelevant alternatives A D D D E E Preference E B C B C B C A A 8 7 6 Score of A = 5 · 8 + 1 · 7 + 1 · 6 = 53 Score of B = 2 · 8 + 3 · 7 + 2 · 6 = 49 Score of C = 1 · 8 + 2 · 7 + 3 · 6 = 40

16. Laplace (1749 –1827): Why integer points for the degree of preference? Théorie…sur les probabilités (1814) Let n independent electors evaluate m candidates with real numbers from [0;1]. If the evaluation marks of every elector are equally distributed then their ordered expected ratio µ1 : µ2 : … : µm = 1 : 2 : … : m . Since, by the law of large numbers, a sum of n independent random variables approaches the sum of their expectations as n→ ∞, the sum of n real-valued points for every candidate is well approximated by integer-valued Borda method.

17. Why summation of evaluation points? If f – function in evaluation marks xi of n electors Constant C can be omitted Weight coefficients ai0 are equal, since voters are considered equal

18. Ordinal method of Condorcet (1743–94) Book Essai sur l‘application… (1785) Condorcet paradox: Cyclic majority A B C Preference B C A C A B 8 7 6 A > B > C > A→ A > B > C 14:7 15:6 13:8 weakest link

19. Condorcet Jury theorem A majority vote in a large electorate almost for sure selects the better candidate provided each elector recognizes rather than misrecognizes the right one. In our days this fact is perceived as an easy corollary of the law of large numbers. However in 1785 neither the central limit theorem, nor the Tchebyshev inequality were known, and Condorcet had to develop a direct proof.

20. Equivalence of Borda and Condorcet methods in a large society Theorem (2000) (1) for any pair of alternatives A,B and for every individual, the ordinal and cardinal preference constituents are independent (2) every pairwise vote has probabilities other than 0, ½, or 1 Then Borda and Condorcet methods tend to give equal results as the number of probabilistically independent individuals n→ ∞.

21. Theory of voting till mid-20th century Charles Dodgson=Lewis Carroll (1832 – 98) 3 mixed (ordinal/cardinal) methods (1873-76) Sir Francis Galton (1822 – 1911) Median solution for ordered options (1907) Duncan Black (1908 – 91) No majority cycles for single-peaked preferences (1948)

22. Arrow‘s Impossibility Theorem, or Arrow‘s paradox (1951) Theorem: Five natural requirements to collective decisions (axioms) are inconsistent Sensation: No universal formula of decision-making Informality of choice: decision rules should depend on decisions Axiomatic approach to social sciences Provable impossibility from fundamentals of mathematics came to social sciences

23. Preferences and weak orders A binary relation > (preferred to) on set X asymmetric:x > y→ not y > x negatively transitive: not x > y, not y > z → not z > x Two alternatives are indifferent if none is preferred: x ~ y iff not x > y and not y > x x≥ y(weakly or non-strictly preferred)iff not y > x Theorem. Complementarity of strict and non-strict preferences Theorem. Preference falls into classes of indifferent alternatives which constitute a linear order (antisymmetric preference: x≥ y and y≥ x → x =y) P – set of all preferences on X

24. Arrow‘s axioms Axiom 1(Number of alternatives) m = |X| ≥ 3 Axiom 2(Universality) For every preference profile, i.e. a combination of individual preferences f:I→P, there exists a social preference denoted also > Axiom 3(Unanimity) An alternative preferred by all individuals is also preferred by the society: x >i y for all i→x > y Axiom 4(Independence of irrelevant alternatives) If individual preferences on two alternatives remain the same under two profiles then the social preference on these alternatives also remains the same under these profiles: f |xy = f ′|xy→σ(f )|xy = σ(f ′)|xy Axiom 5 (No dictator)There is no i: x >i y→x > y

25. Theorem of Fishburn (1970) If the number of individuals is infinite then there exists a non-dictatorial Arrow Social Welfare Functionσ(f) (= which satysfies Axioms 1 – 4)

26. Theorem of Kirman and Sondermann (1972) Even if there is no dictator in an infinite Arrow’s model, there exists an invisible dictator “behind the scene”. That is, the infinite set of individuals can be complemented with a limit point which is the dictator All of these make the situation even more unclear: Dictator is not the model invariant!

27. About paradoxes How wonderful that we have met with a paradox. Now we have some hope of making progress. Niels Bohr (1885–1962)

28. Lemma of Kirman and Sondermann A nonempty coalition A of individuals is decisive if for every preference profile f An ultrafilterU is a maximal nonempty set of nonempty coalitions which contains their supersets and finite intersections For every Arrow Social Welfare Functionσ(f) which satysfies Axioms 1 – 4, all decisive coalitions A constitute an ultrafilter U Ultrafilters are the model invariants!

29. Natural next step An ultrafilter of decisive coalitions is a kind of decisive hierarchy, whose top is the dictator An infinite decisive hierarchy can have no top (no dictator), but it can be inserted by „continuity“ (invisible dictator) Since the dictator makes decisions with decisive coalitions, the question emerges, how large are they? – If they are large on the average, then the dictator is rather a representative So, how dictatorial are Arrow‘s dictators?

30. Binary representation of preferences

31. Matrix A of preference profile

32. Matrix R of representativeness

33. Probability measure Question weights constitute a probability measure: non-negativity: µq ≥ 0 for all q additivity: µQ′ = ∑qєQ′µq normality: ∑qµq =1 µ = {µq}

34. Notation 1

35. 13 preferences on three alternatives Yes to „x > y?“ with probability p = 5/13 No to „x > y?“ with probability q = 1 – p = 8/13

36. Popularity in the 3x3 model Expected weight of the group represented by ind.1: P1= p · [1/3 + Eν{i: i ≠ 1, x >i y} ] + (1 – p) · [1/3 + Eν{i: i ≠ 1, not x >i y} ] identity = p · [1/3 + 2·1/3·p] + (1 – p) · [1/3 + 2 ·1/3·(1 – p)] =1/3 + 2/3 · [p2 + (1 – p)2] p = 5/13 = 347/507 ≈ 0.6844

37. Universality in the 3x3 model The probability of the event that ind. 1 represents a majority = Probability that ind. 2,3 are not both opposite to ind.1: U1= p · [1 – (1 – p)2] + (1 – p)· [1 – p2] identity =1 – p ·(1 – p) p = 5/13 = 129/169 ≈ 0.7633

38. Evaluation of dictators in the simplest model 3x3

39. Theorem 1: Popularity of dictators

40. Theorem 1: Universality of dictators

41. of dictators, in %

42. Notation 2

43. Theorem 2: Revision of the paradox Analogy with force vectors in physics: The best candidate has the largest projection of his opinion vector ai on the µ-weighted social vector

44. Proof for popularity bq is the predominance of protagonists over antagonists for question q bqi= ±1 rqi= 0.5 + 0.5 bq bqi (think!) Hence, Pi = ∑q µqrqi= ∑q µq (0.5 + 0.5 bqbqi) = 0.5 + 0.5 ∑q µqbqbqi = 0.5 + 0.5 (µ.b)′bi P = ∑i Pi νi= ∑i [0.5 + 0.5 ∑q µqbqbqi]νi = 0.5+0.5(µ.b)′b

45. Corollaries Existence of good dictators: Whatever the measures on individuals, questions, and profiles are, there always exist dictator-representatives i, j with Pi>0.5 and Uj>0.5 Consistency of Arrow’s axioms: Restricting the notion of dictator to a dictator in a proper sense whose popularity and universality <0.5, we obtain the consistency of Arrow’s axioms Selecting dictators-representatives by lot: The expected popularity and expected universality of a dictator selected by lot are >0.5

46. Bridge to Arrow‘s model Arrow’s model is defined with no measures Since representatives exist for all measures, they exist in all particular realizations of Arrow’s model It means the potential existence of representatives under all circumstances, even if there are no sufficient data to reveal them (cf. with the existence of a solution to an equation and its analytical solvability)

47. Three types of Arrow‘s dictators The left-hand branch (Arrow‘s paradox) can be empty The right-hand branch (no paradox) isnever empty

48. Bridge to the traditional understanding of democracy Statistical viewpoint: Each individual (dictator) is a sample of the society and statistically tends to represent rather than not to represent the totality. This property is somewhat masked by the complex structure of preferences Analogy to quality control

49. Inventers of logic • Parmenides of Elea (Velia) • 540/535–483/475 BC ? • Logial arguments • to statements • Zeno of Elea • 490 – 430 BC ? • Reduction • ad absurdum Aristotle 384 – 322 BC Systematic book Logic

50. Zeno‘s paradoxes Achilles and the tortoise: Achilles cannot overtake the tortoise who is always ahead as disproof of Pythagoras’ “atomic” time