A Fourier-Theoretic Perspective on the Condorcet Paradox and Arrow ’ s Theorem. By Gil Kalai, Institute of Mathematics, Hebrew University Presented by: Ilan Nehama

2 Basic notations n players m alternatives Each player have a preference over the alternatives R i a > i b := Player i prefers a over b Linear order I.e. Full and asymmetric:  a, b : (a>b) XOR (a<b) Transitive The vector of all preferences (R 1, R 2, …,R n ) is called a profile.

3 Basic notations The preferences are aggregated to the society preference. a > b := The society prefers a over b Full and asymmetric:  a, b : (a>b) XOR (a<b) We do not require it to be transitive The aggregation mechanism is called a social choice function

4 Basic notations Probability space For a social choice function F and a property φ Pr[φ(R N )]:=#{Profiles R N :φ(F)]}/#{Profiles}

5 Social choice function ’ s properties Social choice function is a function between profiles to relations. The social choice function is called rational on a specific profile R N if f(R N ) is an order. The social choice function is called rational if it is rational on every profile. An important property of a social choice function is Pr[F is non-rational].

6 Social choice function ’ s properties IIA – Independence of Irrelevant Alternatives. for any two alternatives a>b depends only on the players preferences between a and b. {i: a> i b} determines whether a>b

7 Social choice function ’ s properties Balanced-For any two alternatives x,y : Pr[x>y]=Pr[y>x] Neutral-The function is invariant under permutations of the alternatives

8 Social choice function ’ s properties Dictator Profile-For a profile each player i that the social aggregation over the profile agrees with his opinion is called a dictator for that profile. General-A player that is a dictator on a ‘ big portion ’ of the profiles is called a dictator. Dictatorship-A social choice function that have one dictator player is called a dictatorship.

9 Main results There exists an absolute constant K s.t.: For every  >0 and for any neutral social choice function If the probability that the function is non-rational on a random profile <  Then there exists a dictator such that for every pair of alternatives the probability that the social choice differs from the dictator ’ s choice < K 

10 Main results For the majority function the probability of getting an order as result (avoiding the Condorcet Paradox) approaches (as n approaches to infinity) to G <G<0.9192

11 Agenda Defining the mathematical base – The Discrete Cube The probability of irrational social choice for three alternatives The probability of the Condorcet paradox A Fourier-theoretic proof of Arrow ’ s theorem

12 Discrete Cube X n ={0,1} n =P([n])=[2 n ] Uniform probability f,g:X->R

13 An orthonormal basis: u s (T)=(-1) |S  T|

14 u s (T)=(-1) |S  T| form an orthonormal basis

15 For f a boolean function f:X->{0,1}. F is a characteristic function for some A  X. A  2 (2 [n] ) P[A]:=|A|/2 n Boolean functions over X

16 Agenda Defining the mathematical base – The Discrete Cube The probability of irrational social choice for three alternatives The probability of the Condorcet paradox A Fourier-theoretic proof of Arrow ’ s theorem

17 Domain definition F is a social choice function < = F(< 1, < 2, …,< n ) F is not necessarily rational Three alternatives – {a,b,c} F is IIA {i: a> i b} determines whether a>b

18 Each player preference can be described by 3 boolean variables x i =1 a> i b y i =1 b> i c z i =1 c> i a Domain definition

19 F can be described by three boolean functions of 3n variables f(x 1,..,x n,y 1,..,y n,z 1,..,z n )=1 a>b g(x 1,..,x n,y 1,..,y n,z 1,..,z n )=1 b>c h(x 1,..,x n,y 1,..,y n,z 1,..,z n )=1 c>a Domain definition

20 F is IIA {i: a> i b} determines whether a>b f,g,h are actually functions of n variables f(x)=f(x 1,..,x n ) g(y)=g(y 1,..,y n ) h(z)=h(z 1,..,z n )

21 Define F will be called balanced when p 1 =p 2 =p 3 =½

22 The domain of F is: Ψ = {all (x,y,z) that correspond to rational profiles} = {(x,y,z) |  i (x i,y i,z i )  {(0,0,0),(1,1,1)} P[Ψ] = (6/8) n

23 W=W(F)=W(f,g,h) is defined to be The probability of obtaining a non-rational outcome (from rational profile) f(x)g(y)h(z)+(1-f(x))(1-g(y))(1-h(z))=1 F(x,y,z) is non-rational W- Probability of a non- rational outcome

24 Theorem 3.1

25 Proof of Thm. 3.1 A,B are boolean functions on 3n variables Subsets of 2 3n A=Χ Ψ B=f(x)g(y)h(z)

26 Proof of Thm. 3.1

27 Proof of Thm. 3.1

28 Proof of Thm. 3.1

29 Proof of Thm. 3.1

30 Proof of Thm. 3.1

31 Agenda Defining the mathematical base – The Discrete Cube The probability of irrational social choice for three alternatives The probability of the Condorcet paradox A Fourier-theoretic prosof of Arrow ’ s theorem

32 The Condorcet Paradox There are cases that the majority voting system (which seems natural) yields irrational results. Three voters, three alternatives 1) a> 1 b> 1 c 2) b> 2 c> 2 a 3) c> 3 a> 3 b Result: a>b>c>a Marie Jean Antoine Nicolas Caritat, marquis de Condorcet

33 Computing the probability of the Condorcet Paradox 3 alternatives n=2m+1 voters f=g=h are the majority function G(n,3):=The probability of a rational outcome. G(3):=lim n →∞ G(n,3)

34 Computing the probability of the Condorcet Paradox It is known that We will prove

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40 Agenda Defining the mathematical base – The Discrete Cube The probability of irrational social choice for three alternatives The probability of the Condorcet paradox A Fourier-theoretic proof of Arrow ’ s theorem

41 Arrow ’ s Theorem At least three alternatives Let f be a social choice function which is: unanimity respecting / Pareto optimal independent of irrelevant alternatives Then f is a dictatorship. Kenneth Arrow

42 Lemma 6.1: For f a boolean function: If =0  S: |S|>1 Then exactly one of the following holds f is constant f=1 or f=0 f depends on one variable (x i ) f(x 1, x 2, …,x 1 )=x i or f(x 1, x 2, …,x 1 )=1-x i

43 =0  S: |S|>1 f is not constant => f depends on one variable Proof of Lemma 6.1

44 =0  S: |S|>1 f is not constant => f depends on one variable Proof of Lemma 6.1

45 =0  S: |S|>1 f is not constant => f depends on one variable Proof of Lemma 6.1

46 Proof of Arrow ’ s theorem (assuming neutrality) From lemma 6.1 one can prove Arrow’s theorem for neutral social choice function Instead we will use a generalization of this lemma to prove a generalization of Arrow’s theorem.

47 Proof of Arrow ’ s theorem using lemma 6.1

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50 Generalized Arrow ’ s Theorem Theorem 7.2: For every ε>0 and for every neutral social choice function on three alternatives: If the probability the social choice function if non- rational≤ε Then there is a dictator such that the probability that the social choice differs from the dictator’s choice is smaller than Kε Notice that for ε=0 we get Arrow’s theorem.

51 Proof of theorem 7.2 using theorem 7.1

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53 Corollary For f m a balanced social choice family on m alternatives For every ε>0, as m tends to infinity, If for every pair of alternatives there is no dictator with probability (1- ε) Then, the probability for a rational outcome tends to zero

54 The End

55 Proposition 5.2 If the social choice function is neutral then the probability of a rational outcome is at least 3/4

56 Proof of Proposition 5.2

57 Proof of Proposition 5.2