Election! Voting rules, noise sensitivity and stability and the anomaly of majority Gil Kalai Einstein Institute of Mathematics Hebrew University of Jerusalem Distinguished Lecture Series Department of Mathematics Indiana University Bloomington Fall 2016

Two theorems by Condorcet: 1) cyclic preferences Codorcet’s Paradox: Majority may lead to cyclic social preferences Marie Jean Nicolas Caritat, marquis de Condorcet (1743-1794)

Two theorems by Condorcet 2) Aggregation of information Codorcet’s Jury Theorem: Let p>1/2 be a real number. Consider an election between Alice and Bob (according to the majority voting rule) 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.

Majority is noise stable Sheppard Theorem: (1899) Suppose that every voter votes at random for each candidate with probability ½ (independently) Suppose that there is a probability t for a mistake in counting each vote. (Mistake= the count gives random unbiased result.) Then the probability that the outcome of the election are reversed is: (1+o(1)) arccos(1-t)/π When t is small this behaves like t1/2

General voting rules Codorcet: Majority may lead to cyclic social preferences Arrow: And so is every non-dictatorial voting rule! Kenneth Arrow

Boolean functions (voting rules) Boolean functions are functions f(x1,x2,…,xn) of n Boolean variables (xi=1 or xi=-1) so that the value of f is in {-1,1}. Boolean functions are of importance in combinatorics, probability theory, computer science, voting, and other areas.

Noise Sensitivity and stability: The Primal description We consider a BOOLEAN FUNCTION f :{-1,1}n  {-1,1} f(x1 ,x2,...,xn) Given x1 ,x2,...,xn we define y1 ,y2,...,yn as follows: xi = yi with probability 1-t xi = -yi with probability t Let C(f;t) be the correlation between f(x1 , x2,...,xn) and f(y1,y2,...,yn)

Noise Sensitivity and stability: The Primal description (cont.) A class of balanced Boolean function is (uniformly) noise stable if for every t > 0 The correlation between f(x) and f(Nt(x)) is bounded away from 0. A sequence of Boolean function (fn ) is (completely) noise- sensitive if for every t>0, C(fn,t) tends to zero with n.

Examples 1) Dictatorship f(x1 ,x2,...,xn) =x1 Ik(f) = 0 for k>1 I1(f)=1 2) Majority f(x1 ,x2,...,xn) =1 iff x1 + x2+...+xn > 0 Ik(f) behaves like n-1/2 for every k.

Examples (cont.) 3) Recursive majority I(f)= K n 1-log2/log3 4) Tribes I(f)= K log n

HEX (critical percolation)

Examples (cont.) 5) HEX (The crossing event for percolations) For percolation, every hexagon corresponds to a variable. xi =-1 if the hexagon is white and xi =1 if it is grey. f=1 if there is a left to right grey crossing. Ik(f) behaves like n-3/8 for every k but few.

Hex and Nate Silver

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

The Shapley-Shubik power index 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! 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). This is not the original definition which was axiomatic rather than probabilistic.

Information of aggregation and power Theorem (Friedgut and Kalai 1996): A weakly symmetric sequence of monotone voting rules satisfies the the conclusion of Condorcet’s jury theorem. 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.

Noise sensitivity and noise stability Theorem (Benjamini, Kalai, Schramm 99): A class of monotone balanced Boolean functions is noise sensitive unless Σ(Ik(f))2 is bounded away from 0 Theorem (Benjamini, Kalai, Schramm 99): A class of monotone balanced Boolean functions is noise sensitive unless it has a uniformly positive correlation with a weighted majority functions.

Nate Silver Information aggregation and Noise Sensitivity

Majority is stablest! Theorem: Mossel, O’Donnell and Oleszkiewicz : (2005) Let (fn ) be a sequence of Boolean functions with diminishing maximum influence. I.e., lim max Ik(f) -> 0 Then the probability that the outcome of the election are reversed when for every vote there is a probability t it is flipped is at least (1-o(1)) arccos(1-t)/π

Noise sensitivity of Fourier of HEX Benjamini, Kalai, and Schramm: Most Fourier Coefficients are above log n Schramm and Steif: Most Fourier coefficients are above nb (b>0) Schramm and Smirnov: Scaling limit for spectral distribution for Percolation exists Garban, Pete and Schramm: Spectral distributions concentrated on sets of size n3/4(1+o(1)). The scaling limit for the spectral distribution of percolation is described by Cantor sets of dimension ¾.

Quantum Election? Can we find a mechanism for n voters to reach a state representing quantum outcome a | > + b| > With 1) Information aggregation (error correction) 2) Noise stability (representation via low degree polynomials) The fact that no such quantum mechanisms are available appears to be crucial for the understanding of why classical information and computation is possible but quantum computation is not!