1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 0.9092<G<0.9192

11. 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. 12 Discrete Cube X n ={0,1} n =P([n])=[2 n ] Uniform probability f,g:X->R

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

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

15. 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. 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. 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. 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. 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. 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. 21 Define F will be called balanced when p 1 =p 2 =p 3 =½

22. 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. 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. 24 Theorem 3.1

25. 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. 26 Proof of Thm. 3.1

27. 27 Proof of Thm. 3.1

28. 28 Proof of Thm. 3.1

29. 29 Proof of Thm. 3.1

30. 30 Proof of Thm. 3.1

31. 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. 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. 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. 34 Computing the probability of the Condorcet Paradox It is known that We will prove

35. 35

36. 36

37. 37

38. 38

39. 39

40. 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. 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. 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. 43 =0  S: |S|>1 f is not constant => f depends on one variable Proof of Lemma 6.1

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

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

46. 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. 47 Proof of Arrow ’ s theorem using lemma 6.1

48. 48

49. 49

50. 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. 51 Proof of theorem 7.2 using theorem 7.1

52. 52

53. 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. 54 The End

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

56. 56 Proof of Proposition 5.2

57. 57 Proof of Proposition 5.2