Transcript Mathematical theory of democracy and its applications 1

Mathematical theory of
democracy and its applications
1. Basics
Andranik Tangian
Hans-Böckler Foundation, Düsseldorf
University of Karlsruhe
[email protected]
Fundamental distinction
Two aspects of social decisions:
Quality – How good they are
Procedure – How they are acheived
Democracy deals with the procedure
2
Plan of the course
Three blocks :
1. Basics
History, Arrow‘s paradox, indicators of
representativeness, solution
2. Representative bodies
President, parliament, government,
parties and coalitions
3. Applications
MCDM, traffic control, financies
3
Cleisthenes’ constitution 507 BC
New governance structure
New division of Attica
represented in the Council of
500
New calendar
Ostracism
4
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
5
Culmination of Athenian democracy
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
Pericles (495 – 429 BC)
6
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
7
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
8
Decline of democracy
322 BC Abolishment of Athenian democracy
Republicanism (= lot + elections + hereditary
power) in Rome and medieval Italian towns
American and French Revolutions 1776-89
promoted republicanism not democracy
Lot survived in juries and administrative rotation
(deans in German universities)
Prohibition of selection by lot of members of the
French Superior Council of Universities 1985
No democratic labeling of Soviet Republics
9
Democracy during the Cold War
Communist propaganda
German Democratic Republic
Korean People’s Democratic Republic
Federal Democratic Republic of Ethiopia
Democratic Republic of Afghanistan …
Western response
Democratic (!!) elections
Human rights
Free press
10
Democracy, elections and voting
Voting for decisions (direct democracy) ≠
voting for election of candidates (oligarchy,
now representative democracy)
Voting, regardless of the way it is used, is
considered an instrument of
democratization, that is, involving more
people into political participation.
11
1st analysis of complex voting situation
Letter by Pliny the Younger
(62–113) about a session in the
Rome Senate on selecting a
punishement for a crime
– leniency (Pliny‘s choice and
simple majority)
– execution (minority), or
– banishment (finally accepted)
subsequent binary vote?
ternary vote (simple majority)?
12
Voting in case of more than two issues
Ramon L(l)ull (1232 – 1316)
1st European novel
Blanquerna (1283 – 87):
method reinvented by
Borda in 1770
De arte eleccionis (1299):
method reinvented by
Condorcet in 1785
13
Caridnal method of Borda (1733–99)
Memoire sur les élections… (1770/84)
Example: The most undesirable wins!
Preference
A
B
C
B
C
A
C
B
A
8
7
6
Score of A = 3 · 8 + 1 · 7 + 1 · 6 =
37
Score of B = 2 · 8 + 3 · 7 + 2 · 6 =
49
Score of C = 1 · 8 + 2 · 7 + 3 · 6 =
14
Dependence on irrelevant alternatives
Preference
A
D
E
B
C
D
E
B
C
A
D
E
C
B
A
8
7
6
Score of A = 5 · 8 + 1 · 7 + 1 · 6 = 53
Score of B = 2 · 8 + 3 · 7 + 2 · 6 = 49
Score of C = 1 · 8 + 2 · 7 + 3 · 6 = 40
15
Laplace (1749 –1827): Why integer
points for the degree of preference?
Théorie…sur les probabilités (1814)
Let n independent electors evaluate m
candidates with real numbers from [0;1]. If the
evaluation marks of every elector are equally
distributed then their ordered expected ratio
µ1 : µ2 : … : µm = 1 : 2 : … : m .
Since, by the law of large numbers, a sum of n
independent random variables approaches the
sum of their expectations as n → ∞, the sum of
n real-valued points for every candidate is well
approximated by integer-valued Borda method.
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Why summation of evaluation points?
If f – function in evaluation marks xi of n electors
n
f ( x1…  xn )  f  x10 …  xn0   
f  x10 …  xn0  
xi
i 1
n
f  x10 … xn0 
i 1
xi
 f  x10 …  xn0   
Constant C


x1  x10 
xi0 
n
f  x10 … xn0 
i 1
xi

xi
Weighted sum  ai0 xi
n
i 1
Constant C can be omitted
Weight coefficients ai0 are equal, since voters
are considered equal
17
Ordinal method of Condorcet (1743–94)
Book Essai sur l‘application… (1785)
Condorcet paradox: Cyclic majority
Preference
A
B
C
B
C
A
8
C
A
B
7
6
A > B > C > A → A > B > C
14:7
15:6
13:8
weakest link
18
Condorcet Jury theorem
A majority vote in a large electorate almost
for sure selects the better candidate
provided each elector recognizes rather
than misrecognizes the right one.
In our days this fact is perceived as an easy corollary
of the law of large numbers. However in 1785
neither the central limit theorem, nor the
Tchebyshev inequality were known, and Condorcet
had to develop a direct proof.
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Equivalence of Borda and Condorcet
methods in a large society
Theorem (2000)
(1) for any pair of alternatives A,B and for every
individual, the ordinal and cardinal preference
constituents are independent
(2) every pairwise vote has probabilities other than
0, ½, or 1
Then Borda and Condorcet methods tend to give
equal results as the number of probabilistically
independent individuals n → ∞.
20
Theory of voting till mid-20th century
Charles Dodgson=Lewis Carroll (1832 – 98)
3 mixed (ordinal/cardinal) methods (1873-76)
Sir Francis Galton (1822 – 1911)
Median solution for ordered options (1907)
Duncan Black (1908 – 91)
No majority cycles for single-peaked
preferences (1948)
21
Arrow‘s Impossibility Theorem, or
Arrow‘s paradox (1951)
Theorem: Five natural requirements to
collective decisions (axioms) are
inconsistent
Sensation:
No universal formula of decision-making
Informality of choice: decision rules should
depend on decisions
Axiomatic approach to social sciences
Provable impossibility from fundamentals of
mathematics came to social sciences
22
Preferences and weak orders
A binary relation > (preferred to) on set X
asymmetric: x > y → not y > x
negatively transitive: not x > y, not y > z → not z > x
Two alternatives are indifferent if none is preferred:
x ~ y iff not x > y and not y > x
x ≥ y (weakly or non-strictly preferred) iff not y > x
Theorem. Complementarity of strict and non-strict
preferences
Theorem. Preference falls into classes of indifferent
alternatives which constitute a linear order
(antisymmetric preference: x ≥ y and y ≥ x → x = y)
P – set of all preferences on X
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Arrow‘s axioms
Axiom 1 (Number of alternatives) m = |X| ≥ 3
Axiom 2 (Universality) For every preference profile, i.e.
a combination of individual preferences f:I→ P, there
exists a social preference  (f ) P denoted also >
Axiom 3 (Unanimity) An alternative preferred by all
individuals is also preferred by the society:
x >i y for all i → x > y
Axiom 4 (Independence of irrelevant alternatives) If
individual preferences on two alternatives remain the
same under two profiles then the social preference
on these alternatives also remains the same under
these profiles: f |xy = f ′|xy → σ(f )|xy = σ(f ′)|xy
Axiom 5 (No dictator) There is no i: x >i y → x > y
24
Theorem of Fishburn (1970)
If the number of individuals is infinite
then there exists a non-dictatorial
Arrow Social Welfare Function σ(f)
(= which satysfies Axioms 1 – 4)
25
Theorem of Kirman and Sondermann
(1972)
Even if there is no dictator in an
infinite Arrow’s model, there exists an
invisible dictator “behind the scene”.
That is, the infinite set of individuals
can be complemented with a limit
point which is the dictator
All of these make the situation even
more unclear:
Dictator is not the model invariant!
26
About paradoxes
How wonderful that we have met
with a paradox. Now we have
some hope of making progress.
Niels Bohr (1885–1962)
27
Lemma of Kirman and Sondermann
A nonempty coalition A of individuals is decisive
if for every preference profile f
f (i )   (f )
f ( A) 
iA
An ultrafilter U is a maximal nonempty set of
nonempty coalitions which contains their
supersets and finite intersections
For every Arrow Social Welfare Function σ(f)
which satysfies Axioms 1 – 4, all decisive
coalitions A constitute an ultrafilter U
Ultrafilters are the model invariants!
28
Natural next step
An ultrafilter of decisive coalitions is a kind of
decisive hierarchy, whose top is the dictator
An infinite decisive hierarchy can have no top
(no dictator), but it can be inserted by
„continuity“ (invisible dictator)
Since the dictator makes decisions with decisive
coalitions, the question emerges, how large
are they? – If they are large on the average,
then the dictator is rather a representative
So, how dictatorial are Arrow‘s dictators?
29
Binary representation of
preferences
Preferences >i
x >1 y >1 z x >2 z >2 y z >3 x ~3 y
x y z
x y z
x y z
x
Their matrices y
z
0 1 1 x
0 0 1 y
0 0 0 z
0 1 1 x
0 0 0 y
0 1 0 z
0 0 0
0 0 0
1 1 0
30
Matrix A of preference profile
Questions q
1:
2:
3:
4:
5:
6:
7:
8:
9:
x>x
y>x
z>x
x>y
y>y
z>y
x>z
y>z
z>z
Opinion of individual i =
1
0
2
0
3
0
0
0
1
0
0
1
0
1
0
0
0
1
1
0
1
1
0
0
1
0
0
0
0
0
Question
weight µi
0
1/6
1/6
1/6
0
1/6
1/6
1/6
0
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Matrix R of representativeness
Question q,
q≠1,5,9
Representativeness rqi of
individual i on question q
i=1
i=2
i=3
Question weight
µq
q=2
q=3
q=4
1
2/3
2/3
1
2/3
2/3
1
1/3
1/3
1/6
1/6
1/6
q=6
1/3
2/3
2/3
1/6
q=7
2/3
2/3
1/3
1/6
q=8
1/3
2/3
2/3
1/6
11/18
13/18
10/18
Expected P=17/27
2/3
1
1/2
Expected U=13/18
Popularity Pi
Universality Ui
32
Probability measure
Question weights constitute a probability
measure:
non-negativity: µq ≥ 0 for all q
additivity: µQ′ = ∑qєQ′ µq
normality: ∑q µq = 1
µ = {µq}
33
Notation 1

probability measure on the set of individuals
A  {aqi } aqi  0 1 matrix of opinionsof profile
rqi 

j aqj aqi
j
Pi   q rqi
representativeness
popularity
q
Ui 

qrqi 05
q   q round[rqi ] universality
P    i Pi
q
expected popularity
i
U    i Ui
i
expected universality
34
13 preferences on three
alternatives
One level
Single level
(total
and
indifference) double level
Double level Three levels
and
single level
x~y~z
x~y>z
x~z>y
y~z>x
x>y~z
y>x~z
z>x~y
x>y>z
x>z>y
y>x>z
y>z>x
z>x>y
z>y>x
Yes to „x > y?“ with probability p = 5/13
No to „x > y?“ with probability q = 1 – p = 8/13
35
Popularity in the 3x3 model
Expected weight of the group represented by ind.1:
P1= p · [1/3 + Eν{i: i ≠ 1, x >i y} ]
+ (1 – p) · [1/3 + Eν{i: i ≠ 1, not x >i y} ] identity
= p · [1/3 + 2·1/3·p] + (1 – p) · [1/3 + 2 ·1/3·(1 – p)]
=1/3 + 2/3 · [p2 + (1 – p)2]
p = 5/13
= 347/507
≈ 0.6844
36
Universality in the 3x3 model
The probability of the event that ind. 1
represents a majority = Probability that ind.
2,3 are not both opposite to ind.1:
U1= p · [1 – (1 – p)2] + (1 – p) · [1 – p2]
=1 – p ·(1 – p)
p = 5/13
= 129/169
≈ 0.7633
identity
37
Evaluation of dictators in the
simplest model 3x3
Number of individuals
Number of alternatives
Number of preferences
Number of preference profiles
Total number of questions (6 for a preference)
Number of elements in the opinion matrix A
Popularity of a dictator (mean % of individuals
represented)
Universality of a dictator (% of majority opinions
represented by the dictator)
3
3
13
133 = 2197
13182
39546
68.44%
76.33%
38
Theorem 1: Popularity of dictators
2
0.5

2(
p

0.5)
m
Pi  P  0.5  2( pm  0.5)2 
n
2


0.5

2(
p

0.5)
 05
m
n 
where
N m 1
pm  05 
probability that an individual x yy
2Nm
Nm 

0 j l  m
Cl j j m ( 1)l  j number of preferences
39
Theorem 1: Universality of dictators


1
for
n2


 n 1 n 1
 n 1 n 1
Ui  U   pm I pm 


for odd n  2
  (1  pm ) I1 pm 

2 
2 
 2
 2


n2 n2
n2 n2


 pm I pm 
  (1  pm ) I1 pm 
 for even n  2
2 
2 
 2
 2


1  pm  05
n 
where
( a  b  1)! p a 1
b 1
I p ( a b) 
t
(1

t
)
dt p  [0;1] a b  0

0
( a  1)!(b  1)!
is the incomplete beta function
40
Popularity Pi
Universality Ui
Number of
alternatives m
3
4
5
6
of dictators, in %
Number of individuals n=
3
4
5
6
…
∞
684
76.3
67.7
75.8
645
87.5
63.6
87.5
621
70.7
61.2
69.9
606
81.3
59.6
81.3
…
52.7
61.5
51.5
58.7
67.3
75.5
67.1
75.3
63.2
87.5
63.0
87.5
608
69.5
605
69.2
59.1
81.3
589
81.3
…
…
…
51.0
56.9
50.7
55.8
41
Notation 2
A  {aqi }
aqi  0 1 matrix of opinionsof profile
B  2( A  0.5)
opinion matrix A converted to  1
bi  2(ai  0.5)
opinions of i converted to  1

probability measure on the set of individuals
b  B balance of opinions in the society
μ probability measure on the set of questions
μ'δb total weight of questions with tie opinion
. element-by-element vector product,
e.g. (1,2) . (3,4) = (3,8)
42
Theorem 2: Revision of the paradox
1 1
Pi   (μ.b)bi
2 2
1 1
1
P   (μ.b)b

2 2
2
1 1
1
Ui   μ'   (μ  signb)bi
2 2 b 2
1 1
1
1
U   μ'   (μ  signb)b 
2 2 b 2
2
Analogy with force vectors in physics:
The best candidate has the largest projection of his
opinion vector ai on the µ-weighted social vector
43
Proof for popularity
bq is the predominance of protagonists over
antagonists for question q
bqi = ±1
rqi = 0.5 + 0.5 bq bqi (think!)
Hence,
Pi = ∑q µqrqi = ∑q µq (0.5 + 0.5 bqbqi)
= 0.5 + 0.5 ∑q µqbqbqi
= 0.5 + 0.5 (µ.b)′ bi
P = ∑i Pi νi = ∑i [0.5 + 0.5 ∑q µqbqbqi]νi
= 0.5+0.5 (µ.b)′ b
44
Corollaries
Existence of good dictators: Whatever the
measures on individuals, questions, and profiles
are, there always exist dictator-representatives i, j
with Pi>0.5 and Uj>0.5
Consistency of Arrow’s axioms: Restricting the
notion of dictator to a dictator in a proper sense
whose popularity and universality <0.5, we obtain
the consistency of Arrow’s axioms
Selecting dictators-representatives by lot: The
expected popularity and expected universality of a
dictator selected by lot are >0.5
45
Bridge to Arrow‘s model
Arrow’s model is defined with no measures
Since representatives exist for all measures, they
exist in all particular realizations of Arrow’s model
It means the potential existence of representatives
under all circumstances, even if there are no
sufficient data to reveal them (cf. with the
existence of a solution to an equation and its
analytical solvability)
46
Three types of Arrow‘s dictators
Arrow‘s dictators
Dictators in a proper sense
(should be prohibited)
Pi<0.5 and Ui<0.5
Dictators-representatives
(should not be prohibited)
Pi>0.5 or Ui>0.5
Representatives selected by lot
expected to be representative
rather than non-representative
The left-hand branch (Arrow‘s paradox) can be empty
The right-hand branch (no paradox) is never empty
47
Bridge to the traditional
understanding of democracy
Statistical viewpoint: Each individual
(dictator) is a sample of the society and
statistically tends to represent rather than
not to represent the totality. This property is
somewhat masked by the complex structure
of preferences
Analogy to quality control
48
Inventers of logic
Parmenides of Elea (Velia) Zeno of Elea
540/535–483/475 BC ? 490 – 430 BC ?
Logial arguments
Reduction
to statements
ad absurdum
Aristotle
384 – 322 BC
Systematic book
Logic
49
Zeno‘s paradoxes
Achilles and the tortoise: Achilles cannot
overtake the tortoise who is always ahead
as disproof of Pythagoras’ “atomic” time
50
Explanation 1
Arrow’s “impossibility” is relevant to the first
meaning only; two other meanings require no
prohibition of dictators
51
Example: War and Peace (L.Tolstoy)
When an event is taking place people express
their opinions and wishes about it, and as the
event results from the collective activity of many
people, some one of the opinions or wishes
expressed is sure to be fulfilled if but
approximately. When one of the opinions
expressed is fulfilled, that opinion gets connected
with the event as a command preceding it.
Men are hauling a log. Each of them expresses
his opinion as to how and where to haul it. They
haul the log away, and it happens that this is
done as one of them said. He ordered it. There
we have command and power in their primary
form.
52
Explanation 2
Arrow’s definition of dictator assumes causality
(first dictatorial, then social preference) which is
misleading
– Logic ≠ causality (logic is static; causality is dynamic)
– Causal equations are not formal equations
53
Example: Lenin’s equation
Communism = Soviet power + Total electrification
?
Total electrification = Communism – Soviet power
54
Explanation 3
Arrow’s impossibility arises from the
emotional metaphor “dictator” which
prompts its prohibition
Aristotle’s warning (Logic):
“Obscurity may arise from the use of
equivocal expressions, of
metaphorical phrases, or of eccentric
words”
55
Conclusions
Voting as manifestation of democracy
Putting voting in question by Arrow’s paradox
Resolution of the paradox: valid Arrow’s theorem,
but its interpretation refined
Calculus instead of rigid Yes/No axiomatic logic:
Finding compromises instead of sorting out all
but unobjectable solutions
56
Sources
Arrow K. (1951) Social Choice and Individual Values. New York, Wiley
Black D. (1958) The Theory of Committees and Elections. Cambridge,
Cambridge University Press
Fishburn, P.C. (1970) Arrow's impossibility theorem: Concise proof and
infinite voters. Journal of Economic Theory, 2(1), 103–106
Kirman A. and Sondermann D. (1972) Arrow's theorem, many agents,
and invisible dictators. Journal of Economic Theory, 5(2), 267–277
Tangian A. (1991) Aggregation and Representation of Preferences:
Introduction to the Mathematical Theory of Democracy. Berlin,
Springer
Tangian A. (2000) Unlikelihood of Condorcet's paradox in a large society.
Social Choice and Welfare, 17 (2), 337–365.
Tangian A. (2003) Historical Background of the Mathematical Theory of
Democracy. Diskussionspapier 332, FernUniversität Hagen
Tangian A. (2003) Combinatorial and Probabilistic Investigation of Arrow's
dictators. Diskussionspapier 336, FernUniversität Hagen
(Forthcoming in Social Choice and Welfare 2010)
Tangian A. (2008) A mathematical model of Athenian democracy. Social
Choice and Welfare, 31, 537 – 572
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