Mathematical theory of democracy and its applications

1. 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 courseThree blocks :

1. BasicsHistory, Arrow‘s paradox, indicators of representativeness, solution

2. Representative bodiesPresident, parliament, government, parties and coalitions

3. ApplicationsMCDM, 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

Athenian democracy in 507 BCPresident of Commitee (1 day)

Strategoi= military generals

(Elections)

Magistratesheld by board of 10

(Lot)

Courts>201 jurors

(Lot)

Boule: Council of 500 (to steer the Ekklesia)

Ekklesia: people‘s assembly (quorum 6000, >40 sessions a year)

Citizenry: Athenian males >20 years, 20000-30000

(Rotation)

Committee of 50 (to guide the Boule)

(Lot)

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’ constitution 600 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 democracy322 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 during the Cold WarCommunist 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 situationLetter 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

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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.

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Why summation of evaluation points? 0 010 0 0

1 1 1 11

0 0 0 01 10 0 0

11 1

01

( )

Constant Weighted sum

nn

n ni i

n nn n

n i ii ii i

ni ii

f x … xf x … x f x … x x x

x

f x … x f x … xf x … x x x

x x

C a x

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

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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

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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 ordersA 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

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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

( )f P

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Theorem of Fishburn (1970)

If the number of individuals is infinite then there exists a non-dictatorial Arrow Social Welfare Function σ(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)

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Lemma of Kirman and Sondermann

A nonempty coalition A of individuals is decisive if for every preference profile f

An ultrafilter U is a maximal nonempty set of nonempty coalitions which contains their supersets and finite intersections

For every Arrow Social Welfare Function σ(f) which satysfies Axioms 1 – 4, all decisive coalitions A constitute an ultrafilter U

Ultrafilters are the model invariants!

( ) ( ) ( )i A

f A f i f

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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?

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Binary representation of preferences

Preferences >i x >1 y >1 z x >2 z >2 y z >3 x ~3 y

Their matrices

x y z

x 0 1 1

y 0 0 1

z 0 0 0

x y z

x 0 1 1

y 0 0 0

z 0 1 0

x y z

x 0 0 0

y 0 0 0

z 1 1 0

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Matrix A of preference profileQuestions q Opinion of individual i = Question

weight µi1 2 3

1: x > x 0 0 0 0

2: y > x 0 0 0 1/6

3: z > x 0 0 1 1/6

4: x > y 1 1 0 1/6

5: y > y 0 0 0 0

6: z > y 0 1 1 1/6

7: x > z 1 1 0 1/6

8: y > z 1 0 0 1/6

9: z > z 0 0 0 0

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Matrix R of representativenessQuestion q,

q≠1,5,9

Representativeness rqi of individual i on question q

Question weight

µq

i = 1 i = 2 i = 3

q = 2 1 1 1 1/6

q = 3 2/3 2/3 1/3 1/6

q = 4 2/3 2/3 1/3 1/6

q = 6 1/3 2/3 2/3 1/6

q = 7 2/3 2/3 1/3 1/6

q = 8 1/3 2/3 2/3 1/6

Popularity Pi 11/18 13/18 10/18 Expected P=17/27

Universality Ui 2/3 1 1/2 Expected U=13/18

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Probability measure

Question weights constitute a probability measure:

non-negativity: µq ≥ 0 for all q

additivity: µQ′ = ∑qєQ′ µq

normality: ∑q µq = 1

µ = {µq}

34

Notation 1

0 5

probability measure on the set of individuals

{ } 0 1 matrix of opinions of profile

representativeness

P popularity

U round[ ] universality

P P expected

qj qi

qi

qi qi

qi jj a a

i q qiq

i q q qiq r q

i ii

a a

r

r

r

A

popularity

U U expected universalityi ii

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

One level (total indifference)

Single level and double level

Double level and single level

Three levels

x ~ y ~ z x > y ~ zy > x ~ zz > x ~ y

x ~ y > zx ~ z > yy ~ z > x

x > y > zx > z > y y > x > zy > z > xz > x > yz > y > x

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

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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

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Evaluation of dictators in the simplest model 3x3

Number of individuals 3

Number of alternatives 3

Number of preferences 13

Number of preference profiles 133 = 2197

Total number of questions (6 for a preference) 13182

Number of elements in the opinion matrix A 39546

Popularity of a dictator (mean % of individuals represented)

68.44%

Universality of a dictator (% of majority opinions represented by the dictator)

76.33%

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Theorem 1: Popularity of dictators

22

2

1

0

0.5 2( 0.5)0.5 2( 0.5)

0.5 2( 0.5) 0 5

0 52

( 1)

P P

where

probability that an individual

number of preferences

mi m

mn

mm

m

j m l jm l

j l m

pp

n

p

Np x y

N

N C j

y

40

Theorem 1: Universality of dictators

1

1

1

0

1 2

1 1 1 1(1 ) 2

2 2 2 2

2 2 2 2(1 ) 2

2 2 2 2

1 0 5

( 1)!( ) (1

( 1)!( 1)!

for

U U for odd

for even

where

m m

m m

i m p m p

m p m p

mn

p ap

n

n n n np I p I n

n n n np I p I n

p

a bI a b t

a b1) [0;1] 0

is the incomplete beta function

bt dt p a b

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of dictators, in %

Number of alternatives m

Number of individuals n=

3 4 5 6 … ∞

3 …

4 …

5 …

6 …

68 4

76.3

67.7

75.8

58 9

81.3

59.1

81.3

61.2

69.9

62 1

70.7

50.7

55.8

67.3

75.5

59.6

81.3

60 6

81.3

67.1

75.3

51.0

56.9

51.5

58.7

52.7

61.5

63.2

87.5

63.0

87.5

60 5

69.2

60 8

69.5

63.6

87.5

64 5

87.5

Popularity P

Universality Ui

i

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Notation 2 { } 0 1 matrix of opinions of profile

2( 0.5) opinion matrix converted to 1

2( 0.5) opinions of converted to 1

probability measure on the set of individuals

balance of opinions in the

qi qi

i i

a a

i

A

B A A

b a

b B society

probability measure on the set of questions

total weight of questions with tie opinion

. element-by-element vector product,

e.g. (1,2) . (3,4) = (3,8)

b

μ

μ'δ

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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

1 1P ( . )

2 21 1 1

P ( . ) 2 2 21 1 1

U ( sign )2 2 21 1 1 1

U ( sign )2 2 2 2

i i

i i'

'

μ b b

μ b b

μ μ b bb

μ μ b bb

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Proof for popularitybq 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

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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

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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) is never empty

Arrow‘s dictators

Dictators in a proper sense(should be prohibited)

Pi<0.5 and Ui<0.5

Dictators-representatives(should not be prohibited)

Pi>0.5 or Ui>0.5

Representatives selected by lotexpected to be representativerather than non-representative

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

Aristotle 384 – 322 BC

Systematic book Logic

Parmenides of Elea (Velia)540/535–483/475 BC ?

Logial arguments to statements

Zeno of Elea 490 – 430 BC ?

Reduction ad absurdum

50

Zeno‘s paradoxes

as disproof of Pythagoras’ “atomic” time

Achilles and the tortoise: Achilles cannot overtake the tortoise who is always ahead

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Explanation 1

Arrow’s “impossibility” is relevant to the first meaning only; two other meanings require no prohibition of dictators

52

Example: War and Peace (L.Tolstoy)When an event is taking place people express their opinions and wishes about it, and as the event results from the collective activity of many people, some one of the opinions or wishes expressed is sure to be fulfilled if but approximately. When one of the opinions expressed is fulfilled, that opinion gets connected with the event as a command preceding it.

Men are hauling a log. Each of them expresses his opinion as to how and where to haul it. They haul the log away, and it happens that this is done as one of them said. He ordered it. There we have command and power in their primary form.

53

Explanation 2

Arrow’s definition of dictator assumes causality (first dictatorial, then social preference) which is misleading

– Logic ≠ causality (logic is static; causality is dynamic)– Causal equations are not formal equations

54

Example: Lenin’s equation

Communism = Soviet power + Total electrification

?

Total electrification = Communism – Soviet power

55

Explanation 3

Arrow’s impossibility arises from the emotional metaphor “dictator” which prompts its prohibition

Aristotle’s warning (Logic):

“Obscurity may arise from the use of equivocal expressions, of metaphorical phrases, or of eccentric words”

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Conclusions

Voting as manifestation of democracy

Putting voting in question by Arrow’s paradox

Resolution of the paradox: valid Arrow’s theorem, but its interpretation refined

Calculus instead of rigid Yes/No axiomatic logic: Finding compromises instead of sorting out all but unobjectable solutions

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SourcesArrow K. (1951) Social Choice and Individual Values. New York, Wiley

Black D. (1958) The Theory of Committees and Elections. Cambridge, Cambridge University Press

Fishburn, P.C. (1970) Arrow's impossibility theorem: Concise proof and infinite voters. Journal of Economic Theory, 2(1), 103–106

Kirman A. and Sondermann D. (1972) Arrow's theorem, many agents, and invisible dictators. Journal of Economic Theory, 5(2), 267–277

Tangian A. (1991) Aggregation and Representation of Preferences: Introduction to the Mathematical Theory of Democracy. Berlin, Springer

Tangian A. (2000) Unlikelihood of Condorcet's paradox in a large society. Social Choice and Welfare, 17 (2), 337–365.

Tangian A. (2003) Historical Background of the Mathematical Theory of Democracy. Diskussionspapier 332, FernUniversität Hagen

Tangian A. (2003) Combinatorial and Probabilistic Investigation of Arrow's dictators. Diskussionspapier 336, FernUniversität Hagen (Forthcoming in Social Choice and Welfare 2010)

Tangian A. (2008) A mathematical model of Athenian democracy. Social Choice and Welfare, 31, 537 – 572