Transcript Example I: Dictatorship

Social choice:
Information, power,
indeterminacy and chaos
Gil Kalai,
Hebrew University of
Jerusalem
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HU Economics summer school 2007
Center for rationality
Outline:
Part 0: The basic model: Social welfare
functions
Part I: Irrational social preferences
(Condorcet, Arrow);
Part II: Aggregation of information and the
Shapley-Shubik power index.
Part III: Indeterminacy;
Part IV: The noise stability/noise
sensitivity dichotomy and chaos.
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PART 0:
Our basic model:
social welfare functions
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The Basic Model:
(Neutral) Social Welfare Functions
We start with a voting rule between two
alternatives
(Like the majority rule)
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Basic Model:
Social Welfare Functions (cont.)
Given a set of m alternatives, consider a
situation where every member of the
society has an order relation describing
her preference.
The society’s preference relation
between a pair of alternatives is
determined by the voting rule.
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Basic Model:
Social Welfare Functions (cont.)
A social welfare function is thus a map
which associates to every profile of
individual order relations, a social
preference relation.
Important: Individual preferences are
assumed to be rational (order relations).
Social preferences can be arbitrary.
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Remarks:
SWF are actually more general: They allow a
different voting rule to each pair of
alternatives. Our version assumes “neutrality”
We do not assume the social preferences are
order relations.
Property “IIA” (independence of Irrelevant
alternatives) is already assumed in my
description of SWF.
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PART I:
Irrationality of social
preferences
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Example I: Dictatorship
The social preferences agrees with the
preferences of a single individual (the
dictator).
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Example 2: Majority
The preferences between two
alternatives a and b are determined
according to the majority rule.
(Assume the number
of voters is odd.)
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Phenomenon I:
Cyclic Outcomes (irrationality)
Codorcet: Majority may lead to cyclic
social preferences
Marie Jean Nicolas Caritat, marquis de Condorcet (1743-1794)
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Phenomenon I:
Cyclic Outcomes (irrationality)
Codorcet: Majority may lead to cyclic social preferences
Arrow: And so is every nondictatorial social welfare function.
Kenneth Arrow
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Example 3: Junta
The outcome is determined by the
preferences of a small number of
individuals
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PART II:
Aggregation of information
Weak individual signals aggregate to
the correct outcome.
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Phenomenon II:
Aggregation of Information
Codorcet’s Jury Theorem: Let p>1/2 be a
real number. Consider an election between
Alice and Bob and suppose that every
voter votes for Alice with probability p
and for Bob with probability 1-p, and that
these probabilities are statistically
independent.
Then the probability that Alice be elected
by the majority of voters tends to 1 as
the number n of voters tends to infinity.
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Condorcet Jury Theorem
Codorcet’s Jury Theorem follows from the
weak law of large numbers.
In modern language it asserts that the
majority rule leads to an asymptotically
complete aggregation of information.
It fails for dictatorships, as well as for
juntas.
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Asymptotic paradigm
Note: Condorcet’s Jury Theorem is an
asymptotic result.
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Stochastic paradigm
In Condorcet’s theorem, voters
preference relations are random
variables.
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Q: What kind of condition on an
election rule would guarantee
asymptotically complete
aggregation of information?
A: It has to do with power!
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How to measure power?
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Power indices: pivotality
For an election with two candidates and a
profile of voter preferences a voter is
called pivotal if given the votes of the
others, her vote determines the winner!
When is a voter pivotal for dictatorship?
When is a voter pivotal for majority?
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The Shapley-Shubik power
index
The power of the kth individual is the
probability of him being pivotal,
according to the following probability
distribution:
1) We choose p between 0 and 1 uniformly
at random.
2) Every voter vote for the first
alternative with probability p
(independently).
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The Shapley-Shubik power
index
Lloyd Shapley and Martin Shubik
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Lloyd Shapley and Martin Shubik?
This is the only two-person image
in a google search under Lloyd S.
Shapley and Martin Shubik.
Is it a drawing of them? Did Google
Based the search on physical
resemblance?
Information of aggregation
and power
Theorem (Kalai, 2002): A sequence of
monotone voting rules satisfies the
conclusion of Condorcet’s jury theorem if
and only if the Shapley-Shubik power
indices of the individual voters are
diminishing.
Note: This is a definite description. A necessary and
sufficient condition for information aggregation.
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Information of aggregation,
power and symmetry
Theorem (Kalai, 2002): A sequence of voting rules leads
to asymptotically complete aggregation of information
if and only if the Shapley-Shubik power indices of the
individual voters are diminishing.
Note: 1) Dictatorships and Junta not
included;
2) Weak forms of symmetry are
already enough. (The Shapley-Shubik
power of all voters sum up to 1.)
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PART III:
Indeterminacy
Everything can happen
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Phenomanon III:
Indeterminacy
Suppose the number of alternatives is fixed.
McGarvey (1954): For many voters, Majority
leads to indeterminacy - every asymmetric
relation can occur as the social outcome.
(“Everything can happened:” “nothing can be concluded”
Erdos-Moser, Gilboa-Vieille, Alon, Sonnenschein...)
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Phenomanon III:
Indeterminacy
McGarvey (1954): Given a fixed number of alternatives,
for many voters, Majority leads to indeterminacy every asymmetric relation can occur as the social
outcome.
Theorem (Kalai 2002): And so is
every monotone voting rule provided
the individual power according to
Shapley-Shubik is sufficiently small.
Note:
Junta not included
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Example 4:
Apex game
The dictator decides unless ALL other
individuals have opposite
preferences.
Note: No dummies. (A dummy is a
voter who is never pivotal.)
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Phenomenon III’: Weak (bipartite) indeterminacy
Weak determinacy asserts that for every
partition of the alternatives into two parts A
and B, all preference relations between
elements in A to those in B can be prescribed.
Theorem (Beigman, 2004). Weak
indeterminacy holds for every monotone
voting rule provided there are no dummies and
many players.
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Asymptotic paradigm
(again)
Note: The results on indeterminacy are
asymptotic. We study the case where
the number of voters is very large.
(This is implied by vanishing power of
voters.) So far, the results we
mentioned are not stochastic.
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Random Uniform Voter profile
assumption
Next we consider the situation where voters
preferences are random, uniformly
distributed, and statistically independent.
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Phenomenon III’’:
Stochastic indeterminacy
As the number of voters grows, the probability
for every social preference relation is bounded
away from zero!
It Follows from results (or rather from the
proofs) by Erdos and Moser, Alon, and others
that:
The majority voting rule leads to
Stochastic indeterminacy
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Stochastic indeterminacy
Theorem: Kalai 2004
Every monotone voting rule leads to
stochastic indeterminacy provided the
individual power according to Banzhaf is
diminishing.
(Based on a result by Mossel, O’Donnell and
Oleszkiewicz. )
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Banzhaf power index
The power of the kth individual is the
probability of him being pivotal,
according to the following probability
distribution: Every voter vote for
the first alternative with probability
1/2 (independently).
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Banzhaf power index (cont.)
John F. Banzhaf III
(“legal terrorist” “Mr anti
smoking” “the man who is
taking fat to court”
“radical feminist”.)
The Banzhaf power index
is related also to the
“Penrose method” that was
introduced by Lionel Penrose.
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Example 5: Majority and a
Judge
There are n voters
The rule is majority
But if the gap between
two candidates is smaller
than n2/3 the results are
determined by a judge.
Here we have asymptotically complete aggregation of
information and indeterminacy but not stochastic
indeterminacy. The Shapley-Shubik power indices of the
judge is diminishing but not the Banzhaf value.
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Part IV :The
Noise Stability/Noise sensitivity
Dichotomy
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PARTIALLY CHAOTIC
STOCHASTIC
STABILTY
COMPLETE CHAOS
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Random Uniform Voter profile
assumption
For the rest of this talk we consider the
situation where voters preferences are
random, uniformly distributed, and
statistically independent.
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Majority is noise stable
Sheppard Theorem: (’99)
Suppose that there is a probability t for a
mistake in counting each vote.
The probability that the outcome of the
election are reversed is: arccos(1-t)/π
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Gulibaud Theorem
For three alternatives, the probability
for cyclic outcomes for the majority
rule is:
1/4 - 3/(2 π) arcsin (1/3) = 0.08744
(Sheppard  Gulibaud )
Note: ’99 stands for 1899.
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Phenomenon IV:
Noise stability
(informally)
As the number of voters tends to infinity:
If the amount of noise becomes small, the
probability for reversing the election
outcomes also becomes small.
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Interpretation: Some “stochastic
rationality” for the outcomes of the
majority voting method .
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Example 6: Hierarchy
Recursive majority rules:
The country is divided to three states,
Every state is divided to three regions
Every region is divided to three counties
Every county is divided to three sub-counties
In every sub-county there are three areas
In each area there are three cities
In every city there are three neighborhoods
In every neighborhood there are three streets
In every street there are three houses
In every house there are three families
In every family there are three people
Election rule: majority of majority of majority of majority of majority,...
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The Soviet Tier System
Party members in a local organization
for example, the Department of Mathematics in Budapest, elected
representatives to the Science Faculty party committee who in
turn elected representatives to the University council. The next
levels were the council of the 5th District of Budapest the
Budapest council, the Party Congress, the Central Committee and
finally the Politburo.
Ted Friedgut's book Political Participation in the USSR
is a good source on the early writings
of Marx, Lenin, and others, and for an analysis of the Soviet
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election systems that were prevalent in the 70's.
Phenomenon V:
Noise sensitivity
No matter how small the noise is,
as the number of voters tends to
infinity the probability for reversing
the outcome of the elections is ½.
Benjamini, Kalai, and Schramm (1999)
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Phenomenon VI: Social chaos
We have at least 3 alternatives.
As the number of voters tends to
infinity the probability for every
social preference relation is the
same.
Interpretation: Even under probabilistic
assumptions we cannot learn anything new from
observing society’s preferences.
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Phenomenon VI: Social chaos
As the number of voters tends to infinity
the probability for every social
preference relation is the same.
When there are 3 alternatives all eight
preference relations occur with the same
probability.
If you know that society prefers Alice to Bob
and Bob to Carol still the probability of
society prefering Alice to Carol tend to ½.
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Multi level majority is noise
sensitive
Theorem (Kalai, 2004): (i) The multi levels
majority rule is noise sensitive when the
number of levels grows to infinity with
the number of voters
(ii) Under the same condition it leads to
social chaos.
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Equivalence Theorem
Theorem (Kalai, 2004): Noise
sensitivity is equivalent to social
chaos for any (fixed) number of
alternatives.
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Example 7: Two-levels
Supermajority
The country is divided to a very large
number of counties.
The rule: Bob wins if he there are more
counties where he gets more than 2/3 of
the votes than counties where Alice gets
2/3 of the votes.
This rule is Noise sensitive!
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Example 8: HEX
The state is divided into m
by m Hexagonal regions
Bob win if the regions where
he has majority form a
continuous path from the
east cost to the west cost.
Alice wins if her regions
form a continuous path
from north to south.
It is Noise sensitive!
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A moment of reflection: why
uniform distribution on voters
preferences
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Noise
Sensitivity
Complete social chaos
Diminishing correlation with
weighted majority functions
Uniformly chaotic SWFs
Uniformly noise stable SWFs
Three levels majority
Two levels majority
Diminishing individual
Simple
Majority
Diminishing individual
Banzhaf Power
Shapley Shubik Power
Asymptotically complete aggregation of information
Noise
Stability
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Juntas
dictatorshop
Figure 3: Asymptotic picture of social wefare functions
Powerful individual exist
banzhaf
Powerful individual exist
Shapley-Shubik
DICTATORSHIPS
AND JUNTAS
Diminishing
Individual
Power (Banzhaf)
MAJORITY
MAJORITY OF MAJORITY
MAJORITY OF MAJORITY OF MAJORITY
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Figure 2: Stochastically stable SWF’s
The Fourier Tool
Fourier analysis of Boolean functions is a
useful (and rather elementary) tool here.
Gives: Formula for the probability of cyclic
outcomes for 3 alternatives (K. 2002)
Gives: Formula for probability of a
Condorcet’s winner for four alternatives
( Friedgut, K. Nisan, 2007)
Gives: Almost the most difficult proof of
Arrow’s
theorem (K. 2002).
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The Fourier Tool
Gives: Almost the most difficult proof of
Arrow’s theorem (K. 2002).
The “almost” does not indicate that there is
a more difficult proof but that the proof
“almost” gives the full theorem but not
quite. (But it allows “stability” results.)
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Other phenomena and
more general models
Voting rules like Borda and Plurality. The
IRA (independent of rejected
alternatives) condition.
Indeterminacy of Plurality (Saari)
The superiority of the majority rule;
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The mysterious drawing
No, this is not
a drawing of Shapley
and Shubik. A
serach of other
prominent
economists
will lead to the
same drawing.
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Summary and future research
Models: Social welfare functions (voting rules);
Different and more general voting rules, rules for
aggregation of utilities, exchange economies,
rational expectations, auctions and combinatorial
auctions, matrix games.
Phenomena: Social irrationality (cyclic social
preferences),aggregation of information,
indeterminacy, noise sensitivity and chaos,
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superiority of the majority rule, find more!
Summary and future
research (cont.)
Paradigms for research: Asymptotic approach,
stochastic voter behavior, random uniform
profiles; Strategic voter’s behavior, realistic
stochastic assumptions, empirics, find more...
Tools: Combinatorics, probability, Fourier analysis
... Find more
Interpretations
Applications
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Thank you!
!‫תודה רבה‬
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Finer understanding of the stable
regime
Phenomanon VII: Superiority of majority.
The Majority is the stablest theorem. (MOO)
How to classify noise-stable SWF?
What are the consequences of stability
(It allows statistical learning, but how far can
we go?)
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Finer understanding of the
noise-sensitive regime
Tribes and
bounded depth Boolean circuits;
Power-low sensitivity.
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