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Mathematical theory of democracy and its applications 2. Fundamentals. Andranik Tangian Hans-Böckler Foundation, Düsseldorf University of Karlsruhe Plan of the course. Three blocks : Basics History, Arrow‘s paradox, indicators of representativeness, solution

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Mathematical theory of democracy and its applications 2 fundamentals

Mathematical theory of democracy and its applications2. Fundamentals

Andranik TangianHans-Böckler Foundation, Düsseldorf University of

Plan of the course
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

Athens draco 621 bc
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

Solon 638 bc 558 bc
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

Solon s political reform 594 bc
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

Cleisthenes constitution 507 bc
Cleisthenes’ constitution 507 BC

New governance structure

New division of Attica represented in the Council of 500

New calendar


Athenian democracy in 507 bc


= military generals



held by board of 10



>201 jurors



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)


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

Citizenry: Athenian males >20 years, 20000-30000

Historic concept of democracy
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

Athenian democracy by aristotle
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

Example athens 462 bc
Example: Athens 462 BC

Three leaders


495–429 BC

democratic party


510–450 BC

aristocratic party


495–461 BC

democratic party

Example question at issue 1
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


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

Example question at issue 2
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


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

Example question at issue 3
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


Example: b11 = 1, b12= -1; r1qshown by color



Protagonists ai1=1

Antagonists aiq=-1





Indicator of popularity spatial representativeness
Indicator of popularity – „spatial“ representativeness

Indicator of universality temporal representativeness
Indicator of universality –„temporal“ representativeness

Indicator of goodness specific representativeness
Indicator of goodness – „specific“ representativeness

Proof for popularity
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

Analogy with vectors of forces in physics
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

Representativeness of decisive bodies
Representativeness of decisive bodies

Implications the society

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)

Implications 2
Implications 2 the society

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

Goodness as a function of majority to minority ratio
Goodness as a function of the societymajority-to-minority ratio

Society is unstable if the majority-to-minority ratio is close to 50:50

Inefficiency of democracy in an unstable society
Inefficiency of democracy the society 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

Source data 32 y n questions like in wahl o mat
Source data: 32 Y/N-questions the society(like in Wahl-o-mat)

Representativeness the society

Reminding the indicators
Reminding the indicators the society

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

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

Implications for paries
Implications for paries the society

Die Linkeis 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

Opinion of a coalition on question q
Opinion of a coalition on question the societyq

Opinion of a coalition on question q is influenced by two extremities

on 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

Indices of coalitions
Indices of coalitions the society

Popularityof coalition is its expected representativeness

Universality of a coalition is ist expected rounded representativeness

Unanimity of a coalition is the weight of questions with unanimousopinions of coalition members

Implications for coalitions with 50 of parliament seats
Implications for coalitions with >50% of parliament seats the society

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

Conclusions the society

German 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)

How to improve elections
How to improve elections? the society

(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

1st method processing each single ballot individually
1st method: Processing each single ballot individually the society

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

2nd method processing the totality of ballots
2nd method: Processing the totality of ballots the society

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)

Third vote for party manifestos drittstimme
Third vote for party manifestos (Drittstimme) the society

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

Conclusions the society

  • InstrumentsIndicators of popularity, universality, and goodness

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

  • Application to electionsFinding best representatives and representative bodies with indicators

    Bridge between direct democracy and representative democracy

Sources the society

Tangian 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