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_at_stat.berkel
ey.edu, http//www.stat.berkeley.edu/mossel/

• Based on joint works with
• Ryan ODonnell X 2
• Gil Kalai and Olle Haggstrom
• Ryan ODonell, Oded Regev, Jeff Steif and Benny
  Sudakov

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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
  0,1n ! 0,1.
• Two prime examples
• Majority vote,
• Electoral college.

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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 shouldnt 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.

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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)

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LLN and voting schemes

• LLN If ? is an i.i.d. measure on 0,1n, ?Xi
  p gt ½ and f maj then Pf(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.

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The effect and weighted majorities

• Define the effect of voter i as
• ei ?f xi 1 - ?f xi
  0.
• Suppose that ?xi p gt ½ and ei lt ? for all i.
• Q For which functions small effects ) ?f 1?
• Def We call f weighted majority if f(x1,xn)
  sign(?i1n wi (xi- ½)), wi 2 R, and f is
  anti-symmetric.

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The effect and weighted majorities

• ThmHaagstrom-Kalai-M
• If f is a weighted majority
• ?Xi pgt ½ and
• ei ? for all i,
• then ?f max1 - ? p (1-p)/(p ½),? p
• If f is not a weighted majority, then there exits
  a measure ? with
• ?Xi p gt ½ and
• ?f 0 and
• ei 0 for all i.
• Cor f is good and weak democracy ) f maj.

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Weighted majorities are good

• Two proofs
• Probabilistic Let Yi p - Xi and g 1-f then
• EgYi 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.

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Non majorities are no good

• Proof for week democracies
• Need to show f ? maj ) f no 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 gt ½ for all i but
• ?f 0 and
• ei 0 for all i.

0, - 1
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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 and 8 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 lt 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)?

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To the uniform measure

• Partial answer For the uniform measure all
  monotone fs are good (Friedgut, Kalai).
• But which monotone function is better?
• Assume x is chosen uniformly in 0,1n.
• 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(?) Pf(x) f(y)?
• Which is the most sensitive / stable f?
• Is there an f which is both stable and sensible?
• Maj? Elctoral college?

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Stability without democracy

• We now work with -1,1n instead of 0,1n.
• N(x, y) PN?(x) y, ? 1 2 ?
• and Z(f,?) ltf,Nfgt.
• N has the eigenvectors uS(x) ?i 2 S xi,
  corresponding to the eigenvalues ?S.
• Pf(x) f(N?(x)) ½ Ef(x)f(N?(x))/2
  ½ ltf,Nfgt/2.
• Write f(x) ?S fS uS(x).
• Since ltf,1gt 0, ltf,Nfgt ?S ? fS2 ?S ?
  and therefore Sf(?) 1 - ?.
• Dictatorship, f(x) xi is the only optimal
  function.

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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

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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 Russos 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

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Getting sensitive

• Kalai Are there any functions that are so
  sensitive?
• Kalai Is it enough to flip n1/2 of the votes in
  order to flip outcome with probability ?(1)?
• ThmM-ODonnell
• rec-maj-k satisfies ltf,Nfgt ??(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).
• Talgrands random function gives that it is
  enough to flip c n1/2.

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Tossing coins from cosmic source
x 01010001011011011111 (n bits)
y1 01010001011011011111 y2 01010001011011011
111 y3 01010001011011011111
yk 01010001011011011111
first bit
0 0 0 0
Alice
Bob
Cindy
o o o
Kate
0
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Broadcast with e errors
x 01010001011011011111 (n bits)
y1 01011000011011011111 y2 01010001011110011
011 y3 11010001011010011111
yk 01010011011001010111
first bit
0 0 1 0
Alice
Bob
Cindy
o o o
Kate
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Broadcast with e errors
x 01010001011011011111 (n bits)
y1 01011000011011011111 y2 01010001011110011
011 y3 11010001011010011111
yk 01010011011001010111
majority
1 1 1 1
Alice
Bob
Cindy
o o o
Kate
1
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The parameters

• n bit uniform random source string x
• k parties who cannot communicate, but wish to
  agree on a uniformly random bit
• e each party gets an independently corrupted
  version yi, each bit flipped independently with
  probability e
• f (or f1 fk) balanced protocol functions

Our goal
For each n, k, e, find the best protocol
function f (or functions f1fk) which maximize
the probability that all parties agree on the
same bit.
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Our goal
For each n, k, e, find the best protocol
function f (or functions f1fk) which maximize
the probability that all parties agree on the
same bit.
Coins and voting schemes

• For k2 we want to maximize Pf1(y1) f2(y2),
  where y1 and y2 are related by applying ? noise
  twice.
• Optimal protocol f1 f2 dictatorship.
• Same is true for k3 (M-ODonnell).

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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).
• k2 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
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Notation

• We write
• S(f1, , fk e) Prf1(y1)
  fk(yk),
• Sk(f e) 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)kk where T?
  is the Bonami-Beckner operator.

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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
• ThmM-ODonnell 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.

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More results M-ODonnell

• When k 2 or 3, the first-bit function is best.
• For fixed n, when k?8 majority is best
  (exercise!).
• For fixed n and k when e?0 and e?½, the first-bit
  is best.
• Proof for e?0 uses isoperimetric inq for edge
  boundary.
• Proof for e? ½ uses Fourier.
• For unbounded n, things get harder in general we
  dont know the best function, but we can give
  bounds for Sk(f e).
• Main open problem for finite n (odd) Is optimal
  protocol always a majority of a subset?
• Conjecture M No
• Conjecture O Yes.

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Unbounded n

• Fixing ? and n 1, how does h(k,?) Pf1
  fk decay as a function of k?
• First guess h(k,?) decays exponentially with k.
• But!
• PropM-ODonnell h(k,?) k-c(?) where c(?) gt
  0.
• ConjM-ODonnell h(k,?) ! 0 as k ! 1.
• ThmM-ODonnell-Regev-Steif-Sudakov h(k,?)
  k-c(?)

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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? fq fp if p gt 1 (Bonami-Beckner)
• T? fq fp if p lt 1 and f gt 0 (Borell).
• Our application uses 2nd in particular implies
  that for all A and B Px 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.

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Coins on other trees

• We can define the coin problem on trees.
• So far we have only discusses the star.

Y1
Y4
x
x
xy1
Y2
Y4
Y5
Y3
Y4
Y2
Y1
Y2
Y3
Y3
Y5

• Some highlights from MORSS
• On line dictator is always optimal (new result in
  MCs).
• For some trees, different fis needed.

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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.

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Open problems

1. Does f monotone anti-symmetric, ? FKG and
   ?Xi p gt ½, ei lt ? ) ?f
   1 - ??
2. For ? the i.i.d. measure the (almost) most stable
   f with ei o(1) is maj (for k2? All k?).
3. The most stable f for Gaussian coin problem is
   f(x) sign(x) and result is robust.
4. For the coin problem, the optimal f is always a
   majority of a subset.