Mathematical theory of democracy and its applications

2. Fundamentals

Andranik TangianHans-Böckler Foundation, Düsseldorf

University of Karlsruheandranik-tangian@boeckler.de

2

Plan of the courseThree blocks :

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

2. Fundamentals:

Model of Athens governance (president, assembly, magistrates, courts) and German Bundestag (parties and coalitions)

3. 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

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)

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’ 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

10

Example: Athens 462 BC Three leaders

Pericles

495–429 BC

democratic party

Ephialtes

495–461 BC

democratic party

Cimon

510–450 BC

aristocratic 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 life

Ephialtes 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

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ATTICAPericles: 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

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Example: Question at issue 3Help Spartans to put down a rebellion

In 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

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Example: Evaluation of leaders

18

Questions

' '

dichotomous questions (Y/N answers)

total number of questions

{ } -vector of question weights -

probability measure:

non-negativity: 0 for all

additivity:

normality: 1

Equ

q

q

Q qq Q

qq

q

m

µ m

µ q

µ µ

µ

µ

al weights 1/ qµ m

19

Individuals

individuals (Athenian citizens)

total number of individuals

{ } -vector of individual weights - probability

Equal chances 1/

{ } ( )-matrix of 1 opinions of

individuals on questions

i

i

iq

i

n

n

n

a n m

i q

ν

A

a A { } -vector balance of opinions -

predominance of protagonists over antagonists

qa m'ν

20

Candidates

candidates

total number of candidates

{ } -vector of candidate weights-probability

Equal chances 1/

{ } ( )-matrix of 1 opinions of

candidates on questions

{ } -vector balance

c

c

cq

q

c

N

N

N

b N m

c q

b m

ξ

B

b B'ξ of

candidate opinions

21

RepresentativenessThe size of group with the same opinion:

weight of protagonists if 1

weight of antagonists if 1

representativeness of on iq cq

cq

cqcq

ii a b

br

b

c q

Protagonists ai1=1

Example: b11 = 1, b12 = -1; r1q shown by color

Antagonists aiq=-1

ai2=1 ai2=-1ai2=-1 ai2=1

q1

q2

22

Indicator of popularity – „spatial“ representativeness

Average size of the group represented:

P popularity of

P P expected popularity

of a candidate selected by lot

c q cqq

c cc

r c

23

Indicator of universality –„temporal“ representativeness

0 5

Frequency of representing a majority:

U round[ ] universality of

U U expected universality

of a candidate selected by lot

cq

c q q cqq r q

c cc

r c

24

Indicator of goodness – „specific“ representativeness

c

Average ratio "group represented-to-majority":

G goodness of weight of majority for

G expected goodness

of a candidate selected by lot

cqq

q

c cc

rc

q

G

25

Notation

2

' vector-matrix transpose

element-by-element vector product,

(1,2) (3,4) (3,8)

element-by-element vector power,

(2,3) (4,9)

| | vector of absolute values of coordinates

sign vector of signs of co

k

.

.

.

.

a

a ordinates

sign(0.5,0,-3) (1,0,-1)

total weight of questions with tie opinion

aμ'δ

26

Theorem: Computing popularity

popularity of -vector-weightedcandidate of opinions

social -vector of candidate of balance of opinions

expectedpopularity of -weighteda candidate soci

selected by lot

P 0.5 0.5 ( )

P 0.5 0.5 ( )

c c

mc

m c

pμ

pμ

μ.a b

μ.a -vector

of opinionsall candidatesal -vector

of balance of opinions

0.5 if candidates from individuals

0.5 if also a non-tie opinion

m

m

b

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

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Theorem: Computing universality

weight ofuniversality -vectorquestionsof of opinions-weighted

with tiecandidate of social -vectoropinion candidate of majority

opinions

expecteduniversality

of a

U 0.5 0.5 ' 0.5( sign )

U

i c

m

c mc

a

uμ

μ μ. a b

-vectorweight of

of opinionsquestions -weighted of all with tie social -vectorcandidate candidatesopinion of majority selected opinionsby lot

0.5 0.5 ' 0.5( sign )

0.5 if candidates from the

m

m

a

uμ

μ μ. a b

individuals

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Theorem: Computing goodness

goodness -vectorof of opinions

candidate of candidate -weightedsocial -vector

of specific opinion balance

expectedgoodness

of a candidateselected

by lot

1 1G ' '

1 | | 1 | |

1G '

1 |

i c

m

c c

m

gμ

μ μ. .a ba a

μa

-vector

of opinionsof all

-weighted candidatessocial -vector

of specificopinion balance

1'

| 1 | | m

m

g

μ

μ. .a ba

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

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

1( , , ) Parliament with (odd) votes -

decisive body operating on majority vote.

Multiple instances of : multiple vote holder

opinion of parliament on

sign 1 since is odd

k

Pq

cqc P

P c c k

c

b P q

b k

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Magistrate (Cabinet, Ministry)

1( , , ) Magistrate with board of -

decisive body controlled by the Assembly

opinion of magistrate on question

opinion of minority of the society on

if all share this opinion

opinion of

k

Mq

M c c k

b M q

q

c M

majority of the society on

if who shares this opinion

q

c M

35

Representativeness of decisive bodies

:

parliament , or magistrate

size of ( 1 corresponds to president)

representativeness on

probability to select by lot

with replacement

iq Dq

Dq ii a b

kD

D P M

k D k

r D q

D

36

Indicators of decisive bodies

0 5

P popularity of

U round[ ] universality of

G goodness of weight of majority for

Ind Ind expected indices of of size

selected by lot

Dq

D q Dqq

D q q Dqq r q

DqD q

q

kD D

c

r D

r D

rD

q

D k

37

Theorem: Computing the indices

Index of -vectorpopularity of opinions-weighted

of ,social -vectoror of the of balance societyof opinions

Index of weight ofuniversality questions

with tieopinion

P 0.5 0.5 ( )

U 0.5 0.5 ' 0.5( sign

m

Dm

pμ

a

μ.a d

μ μ.

-vectorof opinions-weighted of ,social -vector or of theof majority societyopinions

Index of goodness

-weightedsocial -vector

of specific opinion balance

)

1 1G ' '

1 | | 1 | |

m

Dm

m

uμ

gμ

a d

μ μ. .aa a

-vectorof opinions

of , or of thesociety

m

D

d

38

Theorem: Computing the indices

2

sign if

1 1sign , if selected by lot

2 2sign if with a majority representative

sign if with n

q

cqc P

q b

q q

q

b D P

kb I D P

d a D M

a D M

1 1

0

o majority representative

1 signsign 1 2 if selected by lot

2

The incomplete beta function:

( 1)!( ) (1 ) [0;1] 0

( 1)!( 1)!

k

q qq

p x yp

a ba D M

x yI x y t t dt p x y

x y

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Absolute maxima of the indicators

size of majority

Absolute maxima of the indicators, if a majority

could be represented on all the questions

P (0.5 0.5 | |) 0.5 0.5

U 1

G 1

q qqa

μ' | a |

40

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

: 0

( 2) 1

2

: 0

2 for parliament selected by lot

9( 2) min | |P P

2 for magistrate selected by lot

2 for parliament selected by lot

9( 2) minU U

2 for

q

q

qq a

k

qq a

k

k a

k

k a

: 0

( 2)

magistrate selected by lot

4 for parliament selected by lot

9( 2) min | |G G

2 for magistrate selected by lotq

qq a

k

k a

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

VP 2(P P) 0 double deficit of popularity

VU 2(U U) 0 double deficit of universality

VG 2(G G) 0 double deficit of goodness

k

k

k

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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)

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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

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Minimal expected goodness of Athenian decisive bodies

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Election to Bundestag 2009

Votes,%

CDU/CSU (conservators) 33.8

SPD (social democrats) 23.0

FDP (neoliberals) 14.6

Left-Party (left social democrats & communists) 11.9

Green (ecologists) 10.7

22 minor parties 6.0

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

Opinions of parties and unions Question weights 1-5

Survey results, %

CDU33.8

SPD 23.0

FDP 14.6

Linke 11.9

Grünen 10.7

DGB 1st expert

2nd expert

Prota-gonists

Anta-gonists

Minimal wage No Yes No Yes Yes Yes 5 5 52 43

Relax protec-tion against dismissals

No No Yes No No No 5 5 17 82

Nationalisation of railways

No Yes No Yes Yes Yes 5 3 70 28

Equity holding by government in private banks

Yes Yes Yes No Yes Yes 3 3 28 67

No state control over salaries of top managers

Yes Yes Yes No No No 4 4 30 67

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Representativeness

50

Reminding the indicators

Popularity: % of the electorate represented, averaged on 32 questions

Universality: frequency of representing a majority (% of 32 questions)

51

National indices of the parties

52

Implications for paries

Die Linke is the most popular and universal party – in spite of shortage of votes

High representativeness of trade unions – no interrogation of public opinion

Weighting plays a negligible role – henceforth, only unweighted indicators are

considered

53

Opinion of a coalition on question q

Opinion of a coalition on question q is influenced by two extremitieson non-unanimous questions, the impact of

coalition fractions (probability that the opinion is decisive) is proportional to their size

total uncertainty (equal chances of alternative opinions)

Both factors are considered with weights

p and (1 - p), 0 ≤ p ≤ 1

54

Indices of coalitions

Popularity of coalition is its expected representativeness

Universality of a coalition is ist expected rounded representativeness

Unanimity of a coalition is the weight of questions with unanimous opinions of coalition members

55

Normalizing the weights for the coalitions considered

coalition (subset of candidates)

member weights

{ } matrix of member opinions

' balance of coalition opinions

C Cc

ccc C

C

cq

C C C C

q

C

c C

b c C

b

ξ

B

b B ξ

56

Theorem: computing the coalition indicators

Unanimity of 1 '

1P P (1 )( ) .

2

1U U (1 )( sign )

2

where

sign is the number of members in

P P , U P weighted

C

C C

C C

C C

C C

C C

q cqc C

C C

C c c C c cc C c C

C

p

p '

s n b n C

μ s

μ a s b

μ a s b

s

member indicators

57

Indices of coalitions

58

Indices of coalitions

59

Principal components for 3 indicators

For all coalitions For coalitions with >50% seats

1st comp.

2nd comp.

3rd comp.

1st comp.

2nd comp.

3rd comp.

Popularity 0.01 0.33 0.94 -0.05 0.22 0.97

Universality 0.05 0.94 -0.33 -0.12 0.97 -0.23

Unanimity 1.00 -0.05 0.01 0.99 0.13 0.02

Std deviation w.r.t. new axes

33.05 5.85 0.81 17.31 2.62 0.61

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Principal components for 2 indicators

For all coalitions For coalitions with >50% seats

1st comp. 2nd comp. 1st comp. 2nd comp.

Popularity 0.32 0.95 0.30 0.95

Universality 0.95 -0.32 0.95 -0.30

Std deviation w.r.t. new axes

6.06 0.83 3.43 0.68

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Implications for coalitions with >50% of parliament seats

Coalition CDU/FDP (took power) has the highest unanimity but lowest popularity and universality

Coalition CDU/SPD/Linke has low unanimity but highest popularity and universality

According to the principle component analysis, universality is a „more important“ indicator than popularity in the given consideration

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ConclusionsGerman Bundestag elections 2009 show that voters are little

consistent with their own political profiles, disregard party manifestos, and are likely driven by political traditions, even if outdated, or by personal images of politicians

Possible explanation: the spectrum of political landscape has shifted to the right, whereas voters still believe that the parties represent the same values as a few decades ago

Result of ‘voting errors’: the two governing parties are the least representative among the five leading ones, and the governing coalition CDU/CSU/FDP is the least representative among all imaginable coalitions

Effect: discrepancy between the electorate and the government elected (Stuttgart 21, Castor Transport)

63

How to improve elections? (a) redirect the voters' attention from candidates as

persons to manifestos (political profiles)

(b) base the election of candidates on matching their profiles to the majority will. Ballots can contain Yes/No questions on voter positions regarding selected issues. Since answers are determined by background ideologies, a few questions are sufficient to match political profiles of voters and candidates. Parties themselves can formulate the important questions and specify their positions

64

1st method: Processing each single ballot individually

Finding the best-matching candidate who then receives the given vote.

It does not change the election procedure itself (votes are given for candidates), but only a vote-aid is provided to surmount irrational behavior of voters. This method follows the advisory option of the Wahl-O-Mat.

Not possible to model results, since individual data are unavailable

65

2nd method: Processing the totality of ballots

After the balance of electorate opinions on the issues (majority will) has been revealed, the candidates are matched to the profile of the whole of electorate, e.g. with indices of universality

This method is equivalent to performing ‘sample referenda’. It bridges direct democracy with representative democracy (with elections)

No candidate undesired by a majority can be elected, and no cyclic orders can emerge (indices are numbers)

66

Seats proportional to universality

67

Third vote for party manifestos (Drittstimme)

Actual trend in job recruitment: anonymized applications and the focus on job-relevant merits rather than on personal information

Similarly, the third vote in the form of 'sample referenda' with voters‘ Y/N opinions on several important issues from party manifestos. It meets the existing logic of the German two-vote system: the first vote for a person, the second vote for a party, and the third vote for party profiles, so that the considerations are getting to be more conceptual and less personified

68

Conclusions1. Instruments

Indicators of popularity, universality, and goodness

2. Evaluation of Athenian democracySelection of representatives by lot provides social consent; random representatives are also used in quality control and Gallup polls

3. Application to elections Finding best representatives and representative bodies with indicators

Bridge between direct democracy and representative democracy

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SourcesTangian A. (2003) Historical Background of the

Mathematical Theory of Democracy. Diskussionspapier 332, FernUniversität Hagen

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

Tangian A. (2010) Evaluation of German parties and coalitions by methods of the mathematical theory of democracy. European Journal of Operational Research, 202, 294–307.

Tangian A. (2010c) Decision making in politics and economics 4: Bundestag elections 2009 nd direct democracy. Karlsruhe, Karlsruhe Institute of Technology, Working paper 8