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Sensitivity of voting schemes and coin tossing protocols

Sensitivity of voting schemes and coin tossing protocols Elchanan Mossel, U.C. Berkeley mossel@stat.berkeley.edu , http://www.stat.berkeley.edu/~mossel/ Based on joint works with Ryan O’Donnell X 2 Gil Kalai and Olle Haggstrom Ryan O’Donell, Oded Regev, Jeff Steif and Benny Sudakov

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Sensitivity of voting schemes and coin tossing protocols

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  1. Sensitivity of voting schemes and coin tossing protocols Elchanan Mossel, U.C. Berkeley mossel@stat.berkeley.edu, http://www.stat.berkeley.edu/~mossel/ • Based on joint works with • Ryan O’Donnell X 2 • Gil Kalai and Olle Haggstrom • Ryan O’Donell, Oded Regev, Jeff Steif and Benny Sudakov

  2. Definition of voting schemes • A population of size n is to choose between two options / candidates. • A voting scheme is a function that associates to each configuration of votes which option to choose. • Formally, a voting scheme is a function f : {0,1}n! {0,1}. • Two prime examples: • Majority vote, • Electoral college.

  3. Properties of voting schemes • Some properties of voting schemes: • We will always assume that candidates are treated equally: The function f is anti-symmetric: f(~x) = ~f(x) where ~(z1,…,zn) = (1 – z1,…,1-zn). • We always assume that stronger support in a candidate shouldn’t heart her: The function f is monotone:x ¸ y ) f(x) ¸ f(y), where x ¸ y if xi¸ yi for all i. • Note that both majority and the electoral college are anti-symmetric and monotone.

  4. Democracy and voting schemes • Two interpretations of democracy: • “Weak democracy” – each voter has the same power: There exists a transitive group ½ Sn such that for all 2 and all x it holds that f((x(i))) = f((xi)) (***) • “Strong democracy” – each set of voters has the same power – (***) holds for all 2 Sn. • Easy: Monotonicity + antisymmetry + strong democracy) f =majority. • But: Electoral college is weak democracy (mathematically)

  5. LLN and voting schemes • LLN: If  is an i.i.d. measure on {0,1}n, [Xi] = p > ½ and f = maj then P[f(X1,…Xn) = 1] ! 1. • Q1: What if f  maj? (e.g. electoral college). • Q2: What if  is not a product measure? • Example: – if (1,…,1) = p and (0,…,0) = 1-p, then [f] = p for all f! • Conclusion: We need the outcome not to depend on few coordinates.

  6. The effect and weighted majorities • Define the effect of voter i as ei = [f | xi = 1] - [f | xi = 0]. • Suppose that [xi] = p > ½ and ei <  for all i. • Q: For which functions small effects )[f] ~ 1? • Def: We call f weighted majority if f(x1,…xn) = sign(i=1n wi (xi- ½)), wi2 R, and f is anti-symmetric.

  7. The effect and weighted majorities • Thm[Haagstrom-Kalai-M]: • If f is a weighted majority • [Xi] ¸ p> ½ and • ei· for all i, then [f] ¸ max{1 -  p (1-p)/(p – ½), p} • If fis not a weighted majority, then there exits a measure with • [Xi] ¸ p > ½ and • [f] = 0 and • ei = 0 for all i. • Cor: f is “good” and weak democracy) f = maj.

  8. Weighted majorities are good • Two proofs: • Probabilistic: Let Yi = p - Xi and g = 1-f then • E[g*Yi] = p(1-p) ei. • p(1-p) i wi¸p(1-p)( wi ei) = [( wi Yi)g] ¸ (p – ½)( wi)(1 - [f]). • Get [f] ¸ 1 -  p (1-p) /(p – ½). • Linear Program: Minimize [f] under the constraints: • [Xi] ¸ p and ei·. • Reductions to f = maj,  = symmetric measure. • Gives a tight bound and minimizers.

  9. Non majorities are no good • Proof for week democracies: • Need to show: f  maj ) fno good. • f  maj )9 x s.t. f(x0) = 0 and maj(x0) = 1. • Take  to be a uniform measure on the orbit of x0. • [Xi] > ½ for all i but • [f] = 0 and • ei = 0 for all i. + = 0, - = 1

  10. Non majorities are no good • General proof via Linear programing duality • Let G = ([n],E) be a hyper graph where • S 2 E iff f(xE) = 0, where xE(i) = 1 iff i 2 S. • Let *(H), be the fractional covering number: min {S 2 H{S} : 8 S, [S] ¸ 0 and8 i, i 2 S[S] ¸ 1 } • By duality¤(H) is also • max{i wi : 8 i, wi¸ 0, 8 S in H, i 2 S wi· 1}. • ¤(H) ¸ 2 ) f = weighted majority. • If ¤(H) = s < 2, then take  to be the minimizer in the first definition and /s is the desired measure. • Open problems: Which f are good when  is a general FKG measure (or variants of FKG property)?

  11. To the uniform measure • Partial answer: For the uniform measure all monotonef’s are good (Friedgut, Kalai). • But which monotone function is better? • Assume x is chosen uniformly in {0,1}n. • Let y = N(x) is obtained from x by flipping each of x coordinates with probability . • Question: What is the probability that the population voted for who they meant to vote for? • What is Sf() = P[f(x) = f(y)]? • Which is the most sensitive / stable f? • Is there an f which is both stable and sensible? • Maj? Elctoral college?

  12. Stability without democracy • We now work with {-1,1}n instead of {0,1}^n. • N(x, y) = P[N(x) = y], = 1 – 2  and Z(f,) = <f,Nf>. • N has the eigenvectorsuS(x) = i 2 S xi, corresponding to the eigenvalues|S|. • P[f(x) = f(N(x))] = ½ + E[f(x)f(N(x))]/2 = = ½ + <f,Nf>/2. • Write f(x) = S fS uS(x). • Since <f,1> = 0, <f,Nf> = S ; fS2|S|· and therefore Sf() = 1 - . • Dictatorship, f(x) = xi is the only optimal function.

  13. Stability with democracy • How stable can we get with democracy? • limn !1 Sf() = ½ - arcsin(1 – 2 )/for f = majn. • When  is small Sf() » 2 1/2/. • An n1/2£ n1/2 electoral college gives Sf() = (1/4). • Conjecture (???): If f = fn satisfies • max {ei : 1 · i · n} = o(1), then • limn !1 Sf() ¸ ½ - arcsin(1 – 2 )/. • )most stable “weak democracy” = maj. • Much stronger than recent results of Bourgain. • Would give interesting results elsewhere …

  14. Getting sensitive • How sensitive can a monotone function be? • Interesting in learning, neural networks, hardness amplification … • majn maximizes the isoperimetric edge bounds among all monotone functions and I(majn)2 ~ 2n/. • By Russo’s formula: I(f) = Z’(f,1). • But Z’(f,1) = S |S| fS2. • Consider the following relaxation of the problem: minimize S aS|S| under the constraints: • S aS = 1, aS¸ 0, S |S| aS ~· (2n/)1/2 :=  • We get that Z(f,) ~¸a

  15. Getting sensitive • Kalai: Are there any functions that are so sensitive? • Kalai: Is it enough to flipn1/2 of the votes in order to flip outcome with probability (1)? • Thm[M-O’Donnell]: • rec-maj-k satisfies <f,Nf> ~·(n,k) where (n,k) ~ n(k) and (k) ! ½ as k !1 (enough to flip n1-(k)) • rec-maj with increasing arities gives that it is enough to flip logt(n)*n1/2 where t = ½ log2(/2). • Talgrand’s random function gives that it is enough to flip c n1/2.

  16. Tossing coins from cosmic source x 01010001011011011111 (n bits) y1 01010001011011011111 y2 01010001011011011111 y3 01010001011011011111 ° ° ° yk 01010001011011011111 first bit 0 0 0 0 Alice Bob Cindy o o o Kate 0

  17. Broadcast with ε errors x 01010001011011011111 (n bits) y1 01011000011011011111 y2 01010001011110011011 y311010001011010011111 ° ° ° yk 01010011011001010111 first bit 0 0 1 0 Alice Bob Cindy o o o Kate

  18. Broadcast with ε errors x 01010001011011011111 (n bits) y1 01011000011011011111 y2 01010001011110011011 y311010001011010011111 ° ° ° yk 01010011011001010111 majority 1 1 1 1 Alice Bob Cindy o o o Kate 1

  19. The parameters n bit uniform random “source” string x k parties who cannot communicate, but wish to agree on a uniformly random bit ε each party gets an independently corrupted version yi, each bit flipped independently with probability ε f (or f1… fk): balanced “protocol” functions Our goal For each n, k, ε, find the best protocol function f (or functions f1…fk) which maximize the probability that all parties agree on the same bit.

  20. Our goal For each n, k, ε, find the best protocol function f (or functions f1…fk) which maximize the probability that all parties agree on the same bit. Coins and voting schemes • For k=2 we want to maximize P[f1(y1) = f2(y2)], where y1 and y2 are related by applying  noise twice. • Optimal protocol: f1 = f2 = dictatorship. • Same is true for k=3 (M-O’Donnell).

  21. Some variants • Conjecture(???): For all k, if n = 1 and maxi ei = o(1) “optimal protocol is given by” majn. • Gaussian version:x = N(0,1)yi = x + Ni(0,). • Conjecture(???): For all k, optimal protocol given by f(x) = sign(x). • k=2 recently proved by Wenbo Li (but not robust) using isoperimetric shifting of Gaussian measure. • Variants motivated by recent results of Bourgain and needed in computational complexity. x

  22. Notation We write: S(f1, …, fk; ε) = Pr[f1(y1) = ··· = fk(yk)], Sk(f; ε) in the case f = f1 = ··· = fk. Further motivation • Noise in “Ever-lasting security” crypto protocols (Ding and Rabin). • Variant of a decoding problem. • Study of noise sensitivity: |T(f)|kkwhere T is the Bonami-Beckner operator.

  23. protocols • Recall that we want the parties’ bits, when agreed upon, to be uniformly random. • To get this, we restricted to balanced functions. • However this is neither necessary nor sufficient! • In particular, for n = 5 and k = 3, there is a balanced function f such that, if all players use f, they are more likely to agree on 1 than on 0!. • To get agreed-upon bits to be uniform, it suffices for functions be antisymmetric: • Thm[M-O’Donnell]: In optimal f1 = … = fk = f and f is monotone (Pf uses convexity and symmetrization). • We are thus in the same setting as in the voting case.

  24. More results [M-O’Donnell] • When k = 2 or 3, the first-bit function is best. • For fixed n, when k→∞ majority is best (exercise!). • For fixed n and k when ε→0 and ε→½, the first-bit is best. • Proof for ε→0 uses isoperimetric inq for edge boundary. • Proof for ε→ ½ uses Fourier. • For unbounded n, things get harder… in general we don’t know the best function, but we can give bounds for Sk(f; ε). • Main open problem for finite n (odd): Is optimal protocol always a majority of a subset? • Conjecture M: No • Conjecture O: Yes.

  25. Unbounded n • Fixing and n = 1, how does h(k,) := P[f1 = … = fk] decay as a function of k? • First guess:h(k,) decays exponentially with k. • But! • Prop[M-O’Donnell]:h(k,) ¸ k-c() where c() > 0. • Conj[M-O’Donnell]:h(k,) ! 0 as k !1. • Thm[M-O’Donnell-Regev-Steif-Sudakov]:h(k,) · k-c’()

  26. Harmonic analysis of Boolean functions • To prove “hard” results need to do harmonic analysis of Boolean functions. • Consists of many combinatorial and probabilistic tricks + “Hyper-contractivity”. • If p-1=2(q-1) then • |T f|q· |f|p if p > 1 (Bonami-Beckner) • |T f|q¸ |f|p if p < 1 and f > 0 (Borell). • Our application uses 2nd – in particular implies that for all A and B: P[x 2 A, N(x) 2 B] ¸ P(A)1/p P(B)q. • Similar inequalities hold for Ornstein-Uhlenbeck processes and “whenever” there is a log-sob inequality.

  27. Coins on other trees • We can define the coin problem on trees. • So far we have only discusses the star. Y1 Y4 x x x=y1 Y2 Y4 Y5 Y3 Y4 Y2 Y1 Y2 Y3 Y3 Y5 • Some highlights from MORSS: • On linedictator is always optimal (new result in MCs). • For some trees, different fi’s needed.

  28. Wrap-up • We have seen a variety of “stability” problems for voting and coins tossing. • Sometimes it is “easy” to show that dictator is optimal. • Sometimes majority is (almost) optimal, but typically hard to prove (why?). • Recursive majority is really (the most) unstable.

  29. Open problems • Does f monotone anti-symmetric, FKG and [Xi] = p > ½, ei < ) [f] ¸ 1 - ? • For  the i.i.d. measure the (almost) most stable f with ei = o(1) is maj (for k=2? All k?). • The most stable f for Gaussian coin problem is f(x) = sign(x) and result is robust. • For the coin problem, the optimal f is always a majority of a subset.

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