Mathematical%20theory%20of%20democracy%20and%20its%20applications%202.%20Fundamentals

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Mathematical theory of democracy and its
applications2. Fundamentals

• Andranik TangianHans-Böckler Foundation,
  Düsseldorf University of Karlsruheandranik-tangia
  n_at_boeckler.de

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Plan of the course

• Three blocks
• Basics
• History, Arrows 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

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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

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Solon 638 BC558 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

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Solons 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 Peoples
  Assembly
• Selection by lot of all magistrates from an
  elected short list

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Cleisthenes constitution 507 BC

• New governance structure
• New division of Attica represented in the Council
  of 500
• New calendar
• Ostracism

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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 peoples assembly (quorum 6000, gt40
sessions a year)
Citizenry Athenian males gt20 years, 20000-30000
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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

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Athenian democracy by Aristotle

• 621 BC Draconic Laws selection by lot of minor
  magistrates
• 594 BC Solons 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

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Example Athens 462 BC

• Three leaders

Pericles 495429 BC democratic party
Cimon 510450 BC aristocratic party
Ephialtes 495461 BC democratic party
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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 Peoples
  Court or the Assembly

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Areopagus

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

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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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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 gt40 assemblies a year took 3-5
  days every week which complicated economic
  activity

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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

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Ancient Grece
233 km
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Example Evaluation of leaders
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Questions
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Individuals
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Candidates
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Representativeness
Example b11 1, b12 -1 r1q shown by
color
q1 q2
Protagonists ai11
Antagonists aiq-1
ai21
ai2-1
ai2-1
ai21
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Indicator of popularity spatial
representativeness
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Indicator of universality temporal
representativeness
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Indicator of goodness specific
representativeness
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Notation
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Theorem Computing popularity
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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 ?c ?c 0.5 0.5 ?q µqaqbcq? c
• 0.50.5 (µ.a)' b

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Theorem Computing universality
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Theorem Computing goodness
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Back to the example of Athens
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Geometric interpretation
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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
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Magistrate (Cabinet, Ministry)
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Representativeness of decisive bodies
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Indicators of decisive bodies
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Theorem Computing the indices
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Theorem Computing the indices
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Absolute maxima of the indicators
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Theorem Saturation of decisive bodies
recruited from the society
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Theorem Stability of decisive bodies
recruited from the society
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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

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Goodness as a function of majority-to-minority
ratio

• Society is unstable if the majority-to-minority
  ratio is close to 5050

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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 5050) 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 Opinions of parties and unions Opinions of parties and unions Opinions of parties and unions Opinions of parties and unions Opinions of parties and unions Question weights 1-5 Question weights 1-5 Survey results, 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
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Reminding the indicators

• Popularity of the electorate represented,
  averaged on 32 questions
• Universality frequency of representing a
  majority ( of 32 questions)

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National indices of the parties
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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

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Opinion of a coalition on question q

• 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

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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

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Normalizing the weights for the coalitions
considered
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Theorem computing the coalition indicators
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Indices of coalitions
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Indices of coalitions
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Principal components for 3 indicators
For all coalitions For all coalitions For all coalitions For coalitions with gt50 seats For coalitions with gt50 seats For coalitions with gt50 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 all coalitions For coalitions with gt50 seats For coalitions with gt50 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 gt50 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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Conclusions

• 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)

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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

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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

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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)

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Seats proportional to universality
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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

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Conclusions

• Instruments Indicators 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 elections Finding best
  representatives and representative bodies with
  indicators
• Bridge between direct democracy and
  representative democracy

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Sources

• 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, 294307.
• Tangian A. (2010c) Decision making in politics
  and economics 4 Bundestag elections 2009 nd
  direct democracy. Karlsruhe, Karlsruhe Institute
  of Technology, Working paper 8