Mathematical theory of democracy and its applications

3. Applications

Andranik TangianHans-Böckler Foundation, Düsseldorf

University of Karlsruheandranik-tangian@boeckler.de

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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

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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

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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

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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

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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

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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

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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

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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

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MCDM: FU-Hagen outing Candidate

destinations 1. Casino of

Hohensyburg

2. Hagen open air museum

3. Wiblingwerde

4. Schmallenberg

5. Soest

6. Münster

7. Cologne opera

5

6

4

32

1

7

11

Team wishes (individual opinions)

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Indicators of candidate destinations

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2-year plans to satisfy a majority once in each respect (cabinets)

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3- and 5-year plans to satisfy a majority in each respect half

the times (parliaments)

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Traffic control

10

11

1 13

12

14

15

1917

18

23

5

9

876

4

20

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Surveillance cameras

1-5 Hagen Ring

6-20 Important inter-sections

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Prevailing traffic flow at the Hagen ring during the lunch time

+ clockwise traffic flow prevails

- counterclockwise traffic flow prevails

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Traffic flows all over Hagen+/- main traffic flow outside the Ring is towards/outside the city

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Intersection representativeness

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Best collective predictors of the Ring flow and statistical tests

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Predicting DAX from DJ

NYSE Frankfurter Börse

15:30

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Week changes of DAX stock prices

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Week changes of DJ stock prices

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Anticipative representativeness of DJ stocks

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Best DAX collective predictors and statistical tests

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General summaryInstruments: Indicators of representativeness:

popularity, universality, goodness, …

Theoretical implications: Arrow’s paradox, selection of representatives by lot, estimation of size of representative bodies, analogy with forces in physics, inefficiency of democracy in unstable society

Societal applications: finding best candidates, parties, and coalitions

Non-societal applications: MCDM, traffic control, finances (finding compromises, finding predictors, …)

Calculus instead of logic check: finding optimal compromises, whereas ‘yes’–‘no’ logic retains only unobjectionable solutions

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SourcesTangian A. (2003) MCDM-Applications of the Mathematical Theory of

Democracy: Choosing Travel Destginations, Preventing Traffic Jams, and Predicting Stock Exchange Trends. FernUniversität Hagen, Discussion Paper 333

Tangian A.S. (2007) Selecting predictors for traffic control by methods of the mathematical theory of democracy. European Journal of Operational Research, 181, 986–1003

Tangian A.S. (2008) Predicting DAX trends from Dow Jones data by methods of the mathematical theory of democracy. European Journal of Operational Research, 185, 1632--1662

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

Tangian A. (2005) Decision making in politics and economics: 1. Mathematical model of classical democracy. University of Karlsruhe. Discussion Paper. http://www.wior.uni-karlsruhe.de/LS_Puppe/Personal/Papers-Tangian/modelclassdem.pdf