Sensitivity of voting schemes and coin tossing protocols

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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

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
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.
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.
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)

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.
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.
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.
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.
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

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)?
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?
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.
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 …
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
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.
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

x 01010001011011011111 (n bits)

y1 01011000011011011111

y2 01010001011110011011

y311010001011010011111

° ° °

yk 01010011011001010111

first bit

0

0

1

0

Alice

Bob

Cindy

o o o

Kate

x 01010001011011011111 (n bits)

y1 01011000011011011111

y2 01010001011110011011

y311010001011010011111

° ° °

yk 01010011011001010111

majority

1

1

1

1

Alice

Bob

Cindy

o o o

Kate

1

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.

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).

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

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.
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.
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.
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’()
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.
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