A Fourier-Theoretic Perspective on the Condorcet Paradox and Arrow ’ s Theorem.

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. Basic notations • n players • m alternatives • Each player have a preference over the alternatives Ri • 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 (R1, R2,…,Rn) 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(RN)]:=#{Profiles RN: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 RN if f(RN) 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>ib} 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 • 0.9092<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 • Xn={0,1}n=P([n])=[2n] • Uniform probability • f,g:X->R

13. An orthonormal basis: • us(T)=(-1)|ST|

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

15. Boolean functions over X • 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|/2n

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>ib} determines whether a>b

18. Domain definition • Each player preference can be described by 3 boolean variables • xi=1 <=> a>ib • yi=1 <=> b>ic • zi=1 <=> c>ia

19. Domain definition • F can be described by three boolean functions of 3n variables • f(x1,..,xn,y1,..,yn,z1,..,zn)=1 <=> a>b • g(x1,..,xn,y1,..,yn,z1,..,zn)=1 <=> b>c • h(x1,..,xn,y1,..,yn,z1,..,zn)=1 <=> c>a

20. F is IIA {i: a>ib} determines whether a>b • f,g,h are actually functions of n variables • f(x)=f(x1,..,xn) • g(y)=g(y1,..,yn) • h(z)=h(z1,..,zn)

21. Define • F will be called balanced when • p1=p2=p3=½

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

23. W- Probability of a non-rational outcome • 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

24. Theorem 3.1

25. Proof of Thm. 3.1 • A,B are • boolean functions on 3n variables • Subsets of 23n • A=ΧΨ • 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>1b>1c • 2) b>2c>2a • 3) c>3a>3b • Result: a>b>c>a • Marie Jean Antoine Nicolas Caritat, marquisde 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):=limn→∞G(n,3)

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

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 • Kenneth Arrow • 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.

42. Lemma 6.1: For f a boolean function: If <f,uS>=0 S: |S|>1 Then exactly one of the following holds • f is constant • f=1 or f=0 • f depends on one variable (xi) • f(x1, x2,…,x1)=xi or f(x1, x2,…,x1)=1-xi

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

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

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

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

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.