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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

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

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

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

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).

- 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

- Some highlights from MORSS:
- On linedictator is always optimal (new result in MCs).
- For some trees, different fi’s needed.

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.

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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