1. On Sensitivity and Chaos
Elchanan Mossel,
U.C. Berkeley
9/22/2018

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 : {-1,1}n ! {-1,1}.
Assume below that
f(-x1,…,-xn) = -f(x1,…,xn)
Two prime examples:
Majority vote,
Electoral college.
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3. A mathematical model of voting
At the morning of the vote:
Each voter tosses a coin.
The voters want to vote according
to the outcome of the coin.
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4. Ranking 3 candidates Each voter tosses a dice.
Vote according to the corresponding
order on A,B and C.
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5. A mathematical model of voting machines
Which voting schemes are more robust against noise?
Simplest model of noise: The voting machine flips each vote independently with probability .
Registered vote
Intended vote
prob  1 -1
prob 1- -1
prob  -1 1
prob 1- 1
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6. Formal model Assume x is chosen uniformly in {-1,1}n. Let  = 1-2.
Let y = N(x) is obtained from x by flipping each of x coordinates with probability .
Equivalently: yi = xi with probability ; otherwise an
independent coin-toss.
Question: What is the probability that the population voted for who they meant to vote for?
What is S(f) = E[f(x) f(N(x))]?
Which is the most sensitive / stable f?
Is there an f which is both stable and sensible?
Maj? Electoral college?
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7. Predicting the outcome of the elections
How many of the votes do we need to see to predict the outcome of the elections?
Let yi = 0 (unknown) with probability  and
yi = xi = § 1 with probability 1-.
How large can
V(f,) = Py[ E[f(x) |y] > 1 - ]
be for small ?
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8. Arrow’s Paradox for ranking
Assume x is chosen uniformly in S3n
Let xABi = 1 if voter i prefers A to B.
Suppose we declare A preferable to B if
f(xAB) = 1 and similarly for the preference of A to C and B to C.
What is the probability of a paradox:
PDX(f) = P[f(xAB) = f(xBC) = f(xCA)]?
Arrow’s: If f is non-trivial – probability is non-zero.
Kalai 02: PDX(f) = ¼ - ¾ S1/3(f) B A C
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9. Examples: majority & elec. college
It is easy to calculate that:
For f = Majority on n voters:
limn ! 1 Er(f) = ½ - arcsin(1 – 2 )/
When  is small Er(f) » 2 1/2/.
Machine error ~ 0.01% ) voting error ~ 1%.
Result is essentially due to Sheppard (1899!): “On the application of the theory of error to cases of normal distribution and normal correlation”
An n1/2 £ n1/2 electoral college gives Er(f) = (1/4).
Machine error ~ 0.01% ) voting error ~ 10%!.
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10. Some easy answers
Noise Theorem (folklore): Dictatorship, f(x) = xi is the most stable balanced voting scheme.
In other words, for all schemes:
%(error in voting) ¸ %(error in machines)
Paradox Theorem (Arrow’s):
f(x) = xi minimizes the probability of a non-rational outcome.
Exit poll calculations:
For dictator we know the outcome with
probability 1-.
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11. Low influences and democracy
But in fact, we do not care about Dictators or Juntas
Want functions that depend on all (many) coordinates.
The influence of i’th variable of f : {-1,1}n ! {-1,1} is
Ii(f) = P[f(x1,…,xi,…,xn)  f(x1,…,-xi,…,xn)]
More generally for f : {-1,1}n ! R let Ii(f) = i 2 S fS2.
Examples: Ii(f(x) = xj) = i,j, Ii(Maj) ~ n-1/2
Notion introduced by BenOr-Linial, Later KKL …
From now on look at function with low influences –
these are not determined by small # of coordinates. X X
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12. Low influences and PCPs
Khot 02 suggested a paradigm for proving that problems are hard to approximate.
(Very) Roughly speaking the hardness of approximation factor is given by c/s where
c = lim ! 0 supn,f E[f(x) f(y) : 8 i, Ii(f) · , E[f] = a}
s = supn,f E[f(x) f(y) : E[f] = a}
x and y are correlated inputs. The correlation between them is related to the problem for which one wants to
prove hardness.
It is not known if Khots paradigm is equivalent to classical NP hardness.
But the paradigm have given sharp hardness factors for many problems.
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13. Some Conjectures – now theorems
Let I(f) = max Ii(f).
Conj (Kalai-01) Thm: (M-O’Donnell-Oleskiewicz-05):
For f with low influences – “it ain’t over until it’s over:”
As  ! 1, ( ! 0 and  ! 0):
sup[V(f,) : I(f) · , E[f] = 0] = 0.
Conj (Kalai-02) Thm: (M-O’Donnell-Oleskiewicz-05):
“The probability of an Arrow Paradox”
As  ! 0
sup[PDX(f) : I(f) · , E[f] = 0]
is minimized by the Majority function
Recall: PDX(f) = ¼ - ¾ S1/3(f)
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14. Some Conjectures – now theorems
Conj (Khot-Kindler-M-O’Donnell-04)
Thm(M-O’Donnell-Oleskiewicz-05): “Majority is Stablest”
As  ! 0 the quantity
sup[S(f) : I(f) · , E[f] = 0] = (2 arcsin )/ p
is maximized by the majority function for all  > 0.
Motivated by “MAX-CUT” (more later).
Thm (Dinur-M-Regev): “Large independent set”
Look at {0,1,2}n with edges between x and y if xi  yi for all i. Then an independent set of size ¸ 10-6 has an influencial variable.
Motivated by hardness of coloring.
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15. What is MAX-CUT? G = (V,E) C = (Sc,S), partition of V
w(C) = |(SxSc)  E|
w : E ―> R+
weighted  unweighted
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16. What is MAX-CUT? OPT = OPT(G) = maxc {|C|}
MAX-CUT problem: find C with w(C)= OPT
-approximation: find C with w(C) ≥ ·OPT
In KhotKindlerMO’Donnel, given Unique Games and Maj is Stablest we obtain hardness of MAX-CUT that matches GW factor ~
1st time: Hardness result matches semi-definite alg.
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17. The Fourier connection
Look at {-1,1}n.
Define (Tf)(x) = E[f(Nx) | x] and
S(f) = E[f(x) f(N x)] = E[f T f].
T has the eigenvectors uS(x) = i 2 S xi, corresponding to the eigenvalues |S|.
Proof: Txi = E[N xi | xi] = (1-) xi -  xi =  xi
{Us} is an orthogonal basis.
f(x) = S fs uS(x) := Fourier expansion of f.
fs = E[f*uS] := Fourier coefficient at S.
E[f*g] = S fs gs ) S fS2 = 1
Conclusion: S(f) = S fS2 |S| .
Thm (Kalai): PDX(f) = ¼ - ¾ S1/3(f) U1
U1,2,3
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18. The Fourier connection
For “it ain’t over until it’s over”.
Look at Zn where Z = {--1/2,0,-1/2} with probabilities (/2,1-,/2).
Writing f(x) = S fs uS(x) we show that
E[f(x) | y] has the same distribution as
T1/2 S fs vS(z) where
vs(z) = i 2 S zi
T1/2 vs(z) = |S|/2 vs(z).
Bottom line: Have to understand maximizers of norms and tail probabilities of general T operators.
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19. Fourier proof – dictator is stablest
Write f(x) = S fS uS(x).
E[f*1] = 0 ) f = S  ; fSUS ) Tf = S  ; fS |S| US
S(f) = E[f*Tf] = S  ; fS2 |S| · 
Dictatorship, f(x) = xi is the only optimal function. =
S(f) = S  ; fS2 |S| ) in general: stability is determined by how much of the Fourier mass lies on “low degree” coefficients.
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20. Reminder: Majority is Stablest
From now on we will try to prove it …
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21. From discrete to Gaussian stability
Consider the Gaussian measure on Rk.
Suppose f : Rk ! [-1,1] is “smooth” and EG[f] = 0.
Let fn : {-1,1}kn ! [-1,1] be defined by
fn(x) = f((x1+…+xn)/n1/2,…,(x(k-1)n+1+…+xkn)/n1/2)
By smoothness: limn ! 1 max1 · i · kn Ii(fn) = 0.
By the Central Limit Theorem: limn ! 1 EU[fn(x)] = 0 &
limn ! 1 EU[fn*Tfn] = EG[f(X) f(Y)], where
X Gaussian vector and Y = U X.
U X =  X + (1-2)1/2 Z, where Z independent Gaussian.
(X,Y) = (X1,…,XK,Y1,…,Yk) is a normal vector with
E[Xi Xj] = E[Yi Yj] = i,j and E[Xi Yj] =  i,j
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22. From discrete to Gaussian stability 2
“Majority is Stablest” )
for all smooth f: Rk ! [-1,1] with EG[f] = 0:
EG[f U f] = EG[f(X) f(Y)] · 1 – (2/) arcos 
By “density” the same should hold for all f 2 L2.
Note that we obtain equality when m(x) = sgn(x1) –
m(x) is the “limit” of the majority functions.
So “Majority is Stablest” implies Gaussian results that may be easier to verify.
Indeed was proved by Borell85 using Erhard symmetrization.
Easiest to see via 2pt symmetrization on the sphere.
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23. From Gaussian to discrete stability
Is there a way to deduce the discrete results from the Gaussian result?
Let’s look at the CLT theorem again:
CLT: If |a|2 = 1 and supi |ai| ·  then
 ai xi ~ N where ~ means
supx |P[i ai xi · x] – P[N · x]| ·  and  ! 0 as  ! 0
Different formulation:
Let f : {-1,1}n ! R be a linear function: f(x) =  ai xi and
|f|2 = 1.
Ii(f) ·  for all i.
Then f ~  ai Ni where
Ni are i.i.d. Gaussians.
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24. From Gaussian to discrete stability
A new limit theorem [M+O’Donnell+Oleszkiewicz]:
Let f = 0 < |S| · k aS i 2 S xi be a degree k polynomial such that
|f|2 = 1
Ii(f) ·  for all i.
Then f ~ 0 < |S| · k aS i 2 S Ni
Similar result for other discrete spaces.
Generalizes:
CLT
Gaussian chaos results for U and V statistics.
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25. A proof sketch : maj is stablest
Idea: Truncate and follow your nose.
Suppose f : {-1,1}n ! [-1,1] has small influences but E[f T f] = is large.
Then the same is true for g = T f (() < 1).
Let h = |S| · k gS uS then |h-g|2 is small.
Let h’ = |S| · k gS i 2 S Ni
Then: <h,T h> = <h’, U h’> is large and by the new limit theorem:
h’ is close in L2 to a [-1,1] R.V.
Take g’(x) = h’(x) if |h’(x)| · 1 and g’(x) = sgn(h’(x)).
E[g’ U g’] is too large – contradiction! +
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26. A proof sketch : new limit theorem
Recall: p a degree k multi-linear polynomial with:
|p|2 = 1 and Ii(p) ·  for all i.
Want to show p(x1,…,xn) ~ p(N1,…,Nn).
Step 1 (classical): Suffices to show that for every smooth F --|F’’’| · C, it holds that
E[F(p(x1,…,xn)] is close to E[F(p(N1,…,Nn))].
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27. Sketch of proof of Lemma
p(…,xi-1,Ni,…) = Ri + Ni Si and p(…,xi,Ni+1,…) = Ri + xi Si
By Calculus and independence:
|E[F(Ri + Ni Si)] – E[F(Ri + xi Si)] |·
sup |F’’’| (E[|xi|3] + E[|Ni|3]) E[|Si|3] /6 · C E[|Si|3]
If we could say E[|Si|3] · C’ E[Si2]3/2
then we’re done since E[Si2]3/2 = Ii3/2.
This is a “hyper-contractive” inequality.
So all that is left is to prove:
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28. Sketch of proof of Lemma
This is a standard argument.
Somewhat similar results (same proof idea)
Rotar (75) Slightly different setting. No Berry-Essen bounds. Lindenberg conditions instead of hypercontractivity.
Chaterjee (04) Elegant but conditions are too strong – uses worst case influences instead of average case.
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29. Conclusion
We’ve seen how Gaussian techniques can help solve discrete stability problems.
Future work:
Better understanding of the dependency on all parameters for general prob. spaces.
Applications to
Social choice.
PCP’s
Learning.
Sometimes need better Gaussian understanding.
Example:
Suppose we want to partition Gaussian space to 3 parts of equal measure – what is the most stable way?
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30. 9/22/2018

31. Properties of voting schemes
Some properties of voting schemes:
Some properties we may require from voting schemes:
The function f is anti-symmetric: f(~x) = ~f(x) where ~(z1,…,zn) = (1 – z1,…,1-zn).
The function f is balanced: EUnif[f] = 0.
stronger support in a candidate shouldn’t hurt 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.
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32. Stability of voting schemes
Which voting schemes are more robust against noise?
Simplest model of noise: The voting machine flips each vote independently with probability  (not realistic).
Simplest model of voter distribution: i.i.d. distribution where each voter votes 0/1 with probability ½.
Very far from reality …
Buy maybe good model for “critical voting”.
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