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Elections and Manipulations:

Ehud Friedgut, Gil Kalai, and Noam Nisan

Hebrew University of Jerusalem and EF: U. of Toronto, GK: Yale

University, NN: GoogleFOCS 2008

EhudNoam

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I also have a blog!

Combinatorics and Morehttp://gilkalai.wordpress.com/

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Basic Model: Social Choice Functions

Given a set M of m alternatives, consider a situation where every member of the society has an order relation describing her preference .

A social choice function is a function from the profile of individual order

relations to the set of alternatives .

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The basic Model (cont.): Social Choice Functions

Given a set M of m alternatives, and a society N of n individuals, we denote by L the set of m! order relations on the alternatives. A social choice function is a map

f: Ln -> M

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Examples

The Plurality voting rule The Borda count rule

Single transferable vote method Methods based on approval voting or on ranking

The US method Dictatorship

…Many others

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Manipulation

Suppose that x1 x2 ,…, xn are the preference relations for the individuals.

A manipulation by voter k is an order relation x’ such that voter k prefers

f(x1 , x2 , … xk-1, x’k , xk+1, …,xn) over

f(x1 , x2 , … xk-1, xk , xk+1, …,xn).

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Dictatorship

The social choice agrees with the choice of a single individual (the dictator).

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The Gibbard-Satterthwaite theorem

The only social choice function (voting method) whose image is of size at least 3 (there are three possible outcomes or more) that cannot be manipulated is the dictatorship.

Remark: for two alternatives a voting method cannot be manipulated iff it is monotone. (BTW, the German election method is not monotone.)

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The Gibbard-Satterthwaite theorem

The only social choice function (voting method) whose image is of size at least 3 (there are three possible outcomes or more) that cannot be manipulated is the dictatorship.

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The questions we asked For what fraction of profiles does such a manipulation exist? Can it be tiny? How hard it is to find a manipulative

strategy ?

These questions are related to much recent effort in computational game theory/mechanism design.

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The “Big Program”

1 )To understanding quantitative, computational, and other new conceptual aspects of basic economics models and economics’ theorems .

2 )To study, and if needed to introduce, stochastic assumptions.

3 )To ask new questions and describe new phenomena for these basic models.

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Manipulation powerThe manipulation power of an individual kFor a social choice function f, denoted by Mk(f) is the probability that x’k is a profitable manipulation for voter k when the profile of preferences x1 x2 ,…, xn and x’k are chosen uniformly at random.

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NeutralityA social choice function is neutral if it is invariant under a permutation of the alternatives.

(The voting rules does not depend on the names of the candidates ).

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The distance from dictatorship

The distance of a social choice function from dictatorship is the minimal fraction of values that needed to be changed to turn f into a dictatorship.

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The main result

Theorem: There exist a constant C >0 such that for every t>0 if f is a neutral social choice function on 3 alternatives which is t-far from dictatorship then

M1 (f) + M2 (f) + …+ Mn (f) > C t2

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Manipulative-potent voter always exists!

Immediate Corollary :

For fixed t>0, if f is a neutral social choice function on 3 alternatives which is t-far from dictatorship then some voter has a non-negligible manipulation power .

for some k: Mk (f) > C t2 /n

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Open problems

1 )Better bounds for the maximum manipulation power

for some k: Mk (f) > C t2 /n1/2

2 )A theorem without neutrality

3( )Most important )many alternatives:

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Many Alternativs:

Conjecture: Let t>0 be fixed. There are real numbers a,b a>0, b>0, such that: For a neutral choice function on m alternatives and n voters, which is t-far from majority there is a voter with manipulation power at least m-a n-b.

This would imply that a random attempted manipulation has a non-negligible probability of being profitable and therefore the computational hardness in the average case of finding a profitable manipulation follows.

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The three steps of the proofs

Step I: The probability of “cyclic outcomes” for general social welfare functions; relies on (K. 2002)Step II: A reduction to SCF for multi-voter manipulationStep III: (most beautiful) Relating multi-voter manipulation to single voter manipulation. (Require subtle application of FKG)

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The three steps of the proofs

Step I: The probability of “cyclic outcomes” for general social welfare functions; relies on (K. 2002)Step II: A reduction to SCF for multi-voter manipulationStep III: Relating multi-voter manipulation to single voter manipulation. (Require subtle application of FKG)

I will discuss step I and the connection to generalized social welfare functions .

I will not have time today to discuss Steps II and III and also not the notion of “generalized Condorcet winner” which is important in the proof and extensions for more alternatives.

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Another Basic Model: (Neutral) Generalized Social

Welfare Functions

We start with a voting rule between two alternatives

(Like the majority rule)

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Second Basic Model: Generalized 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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Generalized social Welfare Functions (cont.)

A generalized 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:

Our version assumes “neutrality”

We do not assume the social preferences are order relations.

Property “IIA” (independence of Irrelevant alternatives) is already assumed in the description of SWF.

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Example: Dictatorship

As for social choice functions also for social welfare functions,the social preferences agrees with the preferences of a single individual (the dictator).

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Example 2: MajorityThe 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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Condorcet’s paradox

Codorcet: Majority may lead to cyclic social preferences

Marie Jean Nicolas Caritat, marquis de Condorcet (1743-1794)

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Arrow’s theorem

Codorcet: Majority may lead to cyclic social preferences

Arrow: And so is every non-dictatorial social welfare function.

Kenneth Arrow

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The Probability for cyclic social preferencesWhat is the probability for cyclic social

preferences for random voters preferences on three alternatives ?

Gulibaud’s theorem: For the majority rule ,the probability for cyclic outcomes for the majority rule is :

1/4 - 3(/2 π )arcsin (1/3) = 0.08744

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The Probability for cyclic social preferencesTheorem (K. 2002) : There exist a

constant C >0 such that for every t>0 if f is neutral social welfare function on 3 alternatives which is t-far from dictatorship then the probability for cyclic social preferences is at least Ct .

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The Fourier ToolFourier analysis of Boolean functions is a useful 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: the above stability result

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 .

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Impossibility theorems

We talked about:

1 )Every nontrivial rule for aggregation of preferences leads to irrational choices and to manipulations .

Homework: Formulate and prove:

2 )For every nontrivial form of market economy (with or without governmental intervention), major market failures and collapses are unavoidable.

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Thank you ! תודה רבה!