1. 1 Social choice: Information, power, indeterminacy and chaos Gil Kalai, Hebrew University of Jerusalem HU Economics summer school 2007 Center for rationality

2. 2 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.

3. 3 PART 0: Our basic model: social welfare functions

4. 4 The Basic Model: (Neutral) Social Welfare Functions We start with a voting rule between two alternatives (Like the majority rule)

5. 5 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.

6. 6 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.

7. 7 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.

8. 8 PART I: Irrationality of social preferences

9. 9 Example I: Dictatorship The social preferences agrees with the preferences of a single individual (the dictator).

10. 10 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.)

11. 11 Phenomenon I: Cyclic Outcomes (irrationality) Codorcet: Majority may lead to cyclic social preferences Marie Jean Nicolas Caritat, marquis de Condorcet (1743-1794)

12. 12 Phenomenon I: Cyclic Outcomes (irrationality) Codorcet: Majority may lead to cyclic social preferences Arrow: And so is every non- dictatorial social welfare function. Kenneth Arrow

13. 13 Example 3: Junta The outcome is determined by the preferences of a small number of individuals

14. 14 PART II: Aggregation of information Weak individual signals aggregate to the correct outcome.

15. 15 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.

16. 16 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.

17. 17 Asymptotic paradigm Note: Condorcet’s Jury Theorem is an asymptotic result.

18. 18 Stochastic paradigm In Condorcet’s theorem, voters preference relations are random variables.

19. 19 Q: What kind of condition on an election rule would guarantee asymptotically complete aggregation of information? A: It has to do with power!

20. 20 How to measure power?

21. 21 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?

22. 22 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).

23. 23 The Shapley-Shubik power index Lloyd Shapley and Martin Shubik 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?

24. 24 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.

25. 25 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.)

26. 26 PART III: Indeterminacy Everything can happen

27. 27 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...)

28. 28 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

29. 29 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.)

30. 30 Phenomenon III’: Weak (bi- partite) 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.

31. 31 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.

32. 32 Random Uniform Voter profile assumption Next we consider the situation where voters preferences are random, uniformly distributed, and statistically independent.

33. 33 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

34. 34 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. )

35. 35 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).

36. 36 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.

37. 37 Example 5: Majority and a Judge There are n voters The rule is majority But if the gap between two candidates is smaller than n 2/3 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.

38. 38 Part IV :The Noise Stability/Noise sensitivity Dichotomy

39. 39 COMPLETE CHAOS STOCHASTIC STABILTY PARTIALLY CHAOTIC

40. 40 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.

41. 41 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

42. 42 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.

43. 43 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.

44. 44 Interpretation: Some “stochastic rationality” for the outcomes of the majority voting method.

45. 45 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,...

46. 46 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 election systems that were prevalent in the 70's.

47. 47 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)

48. 48 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.

49. 49 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.

50. 50 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.

51. 51 Equivalence Theorem Theorem (Kalai, 2004): Noise sensitivity is equivalent to social chaos for any (fixed) number of alternatives.

52. 52 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!

53. 53 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!

54. 54 A moment of reflection: why uniform distribution on voters preferences

55. 55 Noise Stability Noise Sensitivity dictatorshop Juntas Diminishing individual Shapley Shubik Power Diminishing individual Banzhaf Power Complete social chaos Simple Majority Two levels majority Three levels majority Uniformly chaotic SWFs Diminishing correlation with weighted majority functions Uniformly noise stable SWFs Asymptotically complete aggregation of information Figure 3: Asymptotic picture of social wefare functions

56. 56 DICTATORSHIPS AND JUNTAS MAJORITY MAJORITY OF MAJORITY MAJORITY OF MAJORITY OF MAJORITY Powerful individual exist Figure 2: Stochastically stable SWF’s Shapley-Shubik Diminishing Individual Power (Banzhaf) Powerful individual exist banzhaf

57. 57 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).

58. 58 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.)

59. 59 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;

60. 60 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.

61. 61 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, superiority of the majority rule, find more!

62. 62 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

63. 63 Thank you! תודה רבה !

64. 64 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?)

65. 65 Finer understanding of the noise-sensitive regime Tribes and bounded depth Boolean circuits; Power-low sensitivity.