Voting

Voting System

  • Given Individual preferences
  • Construct Social preference
  • Example: Majority Outcome

Individual Preferences

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Individual Preferences

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Individual Preferences

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Societal Preference

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, Right?

Individual Preferences

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Condorcet's Paradox

  • Not really the first guy to notice
  • Pairwise breaks transitivity
  • Transitivity is pretty reasonable

Can we do better?

What do we want?

  • Transitive
  • Unanimous
  • No dictators
  • Independence of Irrelevant Alternatives
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}

Pretty reasonable

 

 

 

  • Independence of Irrelevant Alternatives

???

1992 election

  • Plurality - Candidate with the most votes wins
  • Bush vs. Perot vs. Clinton

43%

19%

38%

IIA (Independence ... )

  • A vs. B should not depend on A vs. C or B vs. C
  • Bush would have won with Perot's votes
  • Violates IIA; Clinton > Bush regardless of Perot
  • Perot = Spoiler

Systems we looked at

  • Majority: Candidate that is preferred over every other candidate wins.
  • Plurality: Candidate with the most votes wins.

40%

X

Y

Z

35%

Y

Z

X

25%

Z

Y

X

40%

X

Y

Z

35%

Y

Z

X

25%

Z

Y

X

Plurality

X wins, because X has the highest number of votes putting him as the first choice

40%

X

Y

Z

35%

Y

Z

X

25%

Z

Y

X

Majority

Y wins

(40+35 = 75)% put Y over Z

(35+25 = 60)% put Y over X

Which is right?

Neither

What do we want?

  • Transitive
  • Unanimous
  • No dictators
  • Independence of Irrelevant Alternatives

Majority Rule

Plurality

Is it even possible?

  • Given the axioms, no system satisfies all of them
  • Proof shows that dictator exists

"LOL, nope!"

The catch

  • Only applies to ranked systems
  • Too restrictive
  • Arrow thought that cardinal utility was too fuzzy
  • Range voting

So?

  • In practice, Majority Rule works reasonably well
  • Situations that violate transitivity are fairly rare, IRL
  • 2000 election:
    • Nader - left wing
    • Bush - right wing
    • Al Gore - Somewhere in the middle
    • Bush > Nader > Gore
    • Nader > Bush > Gore
  • ​ P. Dasgupta and E. Maskin, “On the Robustness of Majority Rule”
\}
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Pretty Unlikely

Voting Methods

  • Wikipedia suggest a LOT:
    • Highest averages
    • D'Hondt
    • Sainte Lague
    • ...
  • Mapping methods to situations = Open Problem
  • Vote?

Collaborative Filtering

  • Recommendation systems (Netflix, Amazon)
  • You like things that others like you like
  • Start with a set of assumptions about the CF algorithm:
    • Following Arrow's reasoning, your preferences depend only on one other user (dictatorship)
    • Nearest neighbor
  • Weaken demands, you get weighted average, used in practice.

CF Axioms

  • Universal property: Always output a rating
  • Unanimity
  • IIA: Rating of "Frozen" shouldn't depend on rating of "Toy Story" or "Cars"
  • Scale Invariance: If the original rating of an item     by user      is         , then replacing that rating with        , where                                  ,shouldn't change anything.
r_i(j)
r​i​​(j)
i
i
j
j
r_i'(j)
r​i​′​​(j)
r_i'(j) = \alpha_i \times r_i(j) + \beta_i
r​i​′​​(j)=α​i​​×r​i​​(j)+β​i​​
  • As it stands, only nearest neighbor works.
  • Change SI to TI, gives weighted average

Clustering

  • A little like CF
  • Relaxing them gives k-means, k-medians, sum-of-pairs single-linkage
  • Simple axioms result in impossibility result

More clusters

  • Lots of clustering algorithms - 100+
  • Kleinberg's Axioms
    • Scale Invariance: Replace                with 
    • Richness: Any partition can be created by some metric
    • Monotonicity: Shrink distances between points in a cluster, increase distances between clusters
d(x,y)
d(x,y)
\alpha \times d(x,y)
α×d(x,y)

Kannan and Hopcroft

  • Replace Richness with Richness II 
  • Richness II: For any set of     distinct points      has another set of points      such that clustering on     gives you       clusters whose centers are the points of 
k
k
K
K
N
N
N
N
k
k
K
K

Ensemble Learning

  • Supervised Learning
  • Combine many weak learners
  • Form a strong learner
  • Watson: 100+ question generators 
  • Similar results:
    • Under strong set of requirements; dictatorship is the only result
    • Under weaker set; weighted average

Ensemble Learning Axioms

  • Universal property: Always output a classification
  • Unanimity
  • IIA: banana vs. apple shouldn't depend on pear. 
  • Scale Invariance: If the original rating of an item     by user      is         , then replacing that rating with        , where                                  ,shouldn't change anything.
r_i(j)
r​i​​(j)
i
i
j
j
r_i'(j)
r​i​′​​(j)
r_i'(j) = \alpha_i \times r_i(j) + \beta_i
r​i​′​​(j)=α​i​​×r​i​​(j)+β​i​​
  • No ensemble learning algorithm satisfies all conditions
  • Change SI to TI, gives weighted average

The END

No, really

STAAAHP

Impossibility

By Manas George

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Impossibility

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