Transcript Mathematical theory of democracy and its applications 1

Mathematical theory of
democracy and its applications
2. Fundamentals
Andranik Tangian
Hans-Böckler Foundation, Düsseldorf
University of Karlsruhe
[email protected]
Plan of the course
Three blocks :
1. Basics
History, 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
2
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
3
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
4
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
5
Cleisthenes’ constitution 507 BC
New governance structure
New division of Attica
represented in the
Council of 500
New calendar
Ostracism
6
Athenian democracy in 507 BC
President of Commitee (1 day)
Committee of 50 (to guide the Boule)
Strategoi
= military generals
(Elections)
Magistrates
held by board of 10
(Lot)
Courts
>201 jurors
(Lot)
(Rotation
)
Boule: Council
of 500 (to steer the Ekklesia)
(Lot)
Ekklesia: people‘s assembly (quorum 6000, >40 sessions a year)
Citizenry: Athenian males >20 years, 20000-30000
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
Example: Athens 462 BC
Three leaders
Pericles
495–429 BC
Ephialtes
Cimon
495–461 BC
510–450 BC
democratic party democratic party aristocratic party
10
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
11
Areopagus
The Areopagus (view from the Acropolis) – a monolith where
Athenian aristocrats decided important matters of state
12
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
13
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 >40
assemblies a year
took 3-5 days every
week which
complicated
economic activity
14
Example: Question at issue 3
Help 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
15
Ancient Grece
233 km
16
Example: Evaluation of leaders
17
Questions
q
dichotomous questions (Y/N answers)
m
total number of questions
µ  { µq } m-vector of question weights probability measure:
non-negativity: µq  0 for all q
additivity: µQ '   qQ ' µq
normality:

q
µq  1
Equal weights µq  1/ m
18
Individuals
i
n
individuals (Athenian citizens)
total number of individuals
ν  { i } n-vector of individual weights - probability
Equal chances  i  1/ n
A  {aiq } (n  m)-matrix of  1 opinions of
individuals i on questions q
a  A'ν  {aq } m-vector balance of opinions predominance of protagonists over antagonists
19
Candidates
c
candidates
N
total number of candidates
ξ  {c } N -vector of candidate weights-probability
Equal chances c  1/ N
B  {bcq } (N  m)-matrix of  1 opinions of
candidates c on questions q
b  B'ξ  {bq } m-vector balance of
candidate opinions
20
Representativeness
The size of group with the same opinion:
 weight of protagonists if bcq  1
rcq  
 weight of antagonists if bcq  1
   i representativeness of c on q
i aiq bcq
Example: b11 = 1, b12 = -1;
q1 Protagonists ai1=1
q2
ai2=-1
ai2=1
r1q shown by color
Antagonists aiq=-1
ai2=-1
ai2=1
21
Indicator of popularity –
„spatial“ representativeness
Average size of the group represented:
Pc   q rcq
popularity of c
q
P   cPc expected popularity
c
of a candidate selected by lot
22
Indicator of universality –
„temporal“ representativeness
Frequency of representing a majority:
Uc 

qrcq 05
q   q round[rcq ] universality of c
q
U   cUc expected universality
c
of a candidate selected by lot
23
Indicator of goodness –
„specific“ representativeness
Average ratio "group represented-to-majority":
rcq
Gc   q
goodness of c
weight of majority for q
q
G   c Gc
expected goodness
c
of a candidate selected by lot
24
Notation
'
.
.k
vector-matrix transpose
element-by-element vector product,
(1,2) .(3,4)  (3,8)
element-by-element vector power,
(2,3) .  (4,9)
|a | vector of absolute values of coordinates
signa vector of signs of coordinates
sign(0.5,0,-3)  (1,0,-1)
μ'δa total weight of questions with tie opinion
2
25
Theorem: Computing popularity
Pc
popularity of
candidate c
P
expected
popularity of
a candidate
selected
by lot
 0.5  0.5 (μ.a)
bc
p
m -vector
μ-weighted
of opinions
social m -vector of candidate c
of balance
of opinions
 0.5  0.5 (μ.a)
b
m -vector
p
of opinions
μ-weighted
social m -vector all candidates
of balance
of opinions
 0.5 if candidates from individuals
 0.5 if also  a non-tie opinion
26
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
27
Theorem: Computing universality
Ui
universality
of
candidate c
U
expected
universality
of a
candidate
selected
by lot
 0.5  0.5 μ '  a  0.5(μ.signa)
weight of
questions
with tie
opinion
 0.5  0.5 μ '  a
weight of
questions
with tie
opinion
bc
m -vector
u
μ-weighted of opinions
of
social m -vector
candidate c
of majority
opinions
 0.5(μ.signa)
b
m -vector
u
of opinions
μ-weighted
of all
social m -vector candidates
of majority
opinions
 0.5 if candidates from the individuals
28
Theorem: Computing goodness
Gi
goodness
of
candidate c
G
expected
goodness
of a
candidate
selected
by lot
1
1


 μ'
  μ.
.a  '
1 | a |  1 | a | 
g
μ-weighted
social m -vector
of specific
opinion balance
1
1


 μ'
  μ.
.a  '
1 | a |  1 | a | 
g
μ-weighted
social m -vector
of specific
opinion balance
bc
m -vector
of opinions
of candidate c
b
m -vector
of opinions
of all
candidates
29
Back to the example of Athens
30
Geometric interpretation
31
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
32
Assembly, Council of 500,
Committee of 50, and juries
P  (c1, ,ck ) Parliament with k (odd) votes -
bPq
decisive body operating on majority vote.
Multiple instances of c: multiple vote holder
opinion of parliament P on q
 sign bcq  1 since k is odd
cP
33
Magistrate (Cabinet, Ministry)
M  (c1, ,ck ) Magistrate with board of k decisive body controlled by the Assembly
bMq
opinion of magistrate M on question q
opinion of minority of the society on q

if all c  M share this opinion


opinion of majority of the society on q

if c  M who shares this opinion
34
Representativeness
of decisive bodies
D  parliament P, or magistrate M
k  size of D (k  1 corresponds to president)
rDq 

i :aiq bDq
  
k
D
 i representativeness D on q
  probability to select D by lot
with replacement
35
Indicators of decisive bodies
PD   q rDq
popularity of D
q
UD 

qrDq 05
GD   q
q
q   q round[rDq ] universality of D
q
rDq
weight of majority for q
goodness of D
Ind   DkIndD expected indices of D of size k
c
selected by lot
36
Theorem: Computing the indices
P
Index of
popularity
U
Index of
universality
G
Index of
goodness
 0.5  0.5 (μ.a)
d
m -vector
p
μ-weighted of opinions
of D ,
social m -vector
or of the
of balance
society
of opinions
 0.5  0.5 μ '  a  0.5(μ.signa)
weight of
questions
with tie
opinion
d
m -vector
u
of opinions
μ-weighted
of D ,
social m -vector or of the
of majority
society
opinions
1
1


 μ'
  μ.
.a  '
1 | a |  1 | a | 
g
μ-weighted
social m -vector
of specific
opinion balance
d
m -vector
of opinions
of D ,
or of the
society
37
Theorem: Computing the indices
sign bcq
if D  P

cP

1 k  1
signbq  Ib2  ,
if D  P selected by lot

q
2 2 


dq  signaq
if D  M with a majority representative

if D  M with no majority representative
 signaq
k



signa  1  2  1  signaq bq   if D  M selected by lot


q

2

 

The incomplete beta function:
( x  y  1)! p x 1
y 1
I p ( x y ) 
t
(1

t
)
dt  p  [0;1] x y  0

( x  1)!( y  1)! 0
38
Absolute maxima of the indicators
Absolute maxima of the indicators, if a majority
could be represented on all the questions
P   q q (0.5  0.5 | aq |)  0.5  0.5μ' | a |
size of majority
U1
G1
39
Theorem: Saturation of decisive
bodies “recruited” from the society
2

 9(k  2) min | a | for parliament selected by lot
q
PP  
q:aq 0

 ( k  2) 1
2
k
for magistrate selected by lot

2

 9(k  2) min a 2 for parliament selected by lot
q
UU  
q:aq 0

k
2
for magistrate selected by lot

4

 9(k  2) min | a | for parliament selected by lot
q
GG 
q:aq 0

 ( k  2)
2
for magistrate selected by lot

40
Theorem: Stability of decisive
bodies “recruited” from the society
VP  2(P  P) 
 0 double deficit of popularity
k 
VU  2(U  U) 
 0 double deficit of universality
k 
VG  2(G  G) 
 0 double deficit of goodness
k 
41
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)
42
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
43
Goodness as a function of
majority-to-minority ratio
Society is
unstable if the
majority-tominority ratio is
close to 50:50
44
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
45
Minimal expected goodness of
Athenian decisive bodies
46
Election to Bundestag 2009
CDU/CSU (conservators)
SPD (social democrats)
FDP (neoliberals)
Left-Party (left social democrats & communists)
Green (ecologists)
22 minor parties
Votes,
%
33.8
23.0
14.6
11.9
10.7
6.0
47
Source data: 32 Y/N-questions
(like in Wahl-o-mat)
Opinions of parties and unions
Question
weights 1-5
Survey results,
%
CDU
33.8
SPD
23.0
FDP
14.6
Linke
11.9
Grünen
10.7
DGB
1st
expert
2nd
expert
Protagonists
Antagonists
Minimal wage
No
Yes
No
Yes
Yes
Yes
5
5
52
43
Relax protection 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
48
Representativeness
49
Reminding the indicators
Popularity: % of the electorate
represented, averaged on 32 questions
Universality: frequency of representing a
majority (% of 32 questions)
50
National indices of the parties
51
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
52
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
53
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
54
Normalizing the weights for
the coalitions considered
C
coalition (subset of candidates)
 C
ξ   c 

C

 c  C  member weights
cC c

c
C
B  {bcq  c  C } matrix of member opinions
C

C
C C
b  bq  B ' ξ
balance of coalition opinions
55
Theorem: computing the
coalition indicators
C
Unanimity of C  1  μ ' s
C C
1


PC  PC  (1  p)(μ  a)  s .b 
2


1
C C
UC  UC  (1  p)(μ  sign a)'  s  b 
2


where
C
 C

 q







s  s  sign n 

 

cq  

 
b
cC

n is the number of members in C
C
C
PC   c Pc , UC   c Pc weighted member indicators
cC
cC
56
Indices of coalitions
57
Indices of coalitions
58
Principal components for 3
indicators
For all coalitions
For coalitions with
>50% seats
1st
2nd
3rd
1st
2nd
3rd
comp. comp. comp. comp. comp. 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
59
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
60
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
61
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)
62
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
63
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 voteaid 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
64
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)
65
Seats proportional to universality
66
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
67
Conclusions
1. Instruments
Indicators of popularity, universality, and
goodness
2. Evaluation of Athenian democracy
Selection 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
68
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, 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
69