Rank Aggregation

1
Rank Aggregation

• Chengkai Li

2
Rank Aggregation

• Rank Aggregation
• Combine many different rank orderings on the
  same set of candidates/alternatives to obtain a
  better ordering.
• Example abcd, acdb, cabd ? acbd

3
Overview

• Desired Properties
• Rank Aggregation Methods
• Condorcet Criterion vs. Borda
• Kemeny Optimal Aggregation
• Rank Aggregation in fields other than social
  choice
• Web Search
• Machine Learning
• Information Retrieval
• Database Middleware
• Conclusion

4
Desired Properties

• Condorcet Criterion (1785)
• Condorcet alternative
• Defeats every other alternative in pairwise
  simple majority voting
• ab 21, ac 21

5
Desired Properties

• Condorcet Criterion (contd.)
• Condorcet Criterion
• Condorcet alternative wins (if one exists)
• cycle possible
• ab 21, bc 21, ca 21

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Desired Properties

• Consistency
• If the voters are divided into 2 parts and
  separate voting in the two parts get the same
  winner, then the overall voting result should be
  the same
• Independence from Irrelevant Alternatives
• X gt Y (overall), XZY ? ZXY (one voter), still
  XgtY (overall)
• Monotonicity
• X wins (overall), YZX ? YXZ (one voter), still X
  wins (overall)
• If only Xs rank is raised.

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Desired Properties

• Pareto Optimality
• X gt Y (all individual orderings) ? XgtY (overall)
• Neutrality
• Should not favor any alternative
• Manipulability
• insincere voting

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Manipulability ?

• Gibbard (1973) and Satterthwaite (1975)
• Any non-dictatorial voting systems with at least
  three possible outcomes is subject to individual
  manipulation.
• Although impossible
• Some systems tend to encourage manipulation more
  than other.
• It is easier to identify manipulation strategies
  in some systems than others

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Arrows Impossibility Theorem

• KENNETH J. ARROW Social Choice and Individual
  Values (1951). Won Nobel Prize in 1972
• No voting scheme over three or more alternatives
  can satisfies all of several reasonable and
  logical conditions.
• Universality
• no restriction on individual ordering. All
  orderings are achievable.
• Non-dictatorship
• Monotonicity
• Independence of irrelevant alternatives
• Pareto Optimality

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Rank Aggregation Methods

• Majority Rule and Majoritarian Methods
• Positional Methods
• Utilitarian Methods
• Multi-Stage Methods

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Majority Rule and Majoritarian Methods

• Use only information from pariwise comparisons
  between candidates
• Majority Rule (2 alternatives)
• The one preferred by most voters is selected.
• Majoritarian Methods (gt2 alternatives)
• select the Condorcet winner if one exists
• Various Majoritarian Methods (no Condorcet
  winner)
• The Amendment Procedure
• The Successive Procedure

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Positional Methods

• Not pairwise comparisons, but orderings.
• Plurality Voting (only top-1 alternative)
• Approval Voting (only top-k alternatives)
• Borda Count (entire ordering)

13
Borda Count
14
Borda Count
Condorcet Winner a (ab 32) Borda Winner b
(a55441, b55443)
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Borda Count
Condorcet Winner a (ab 32) Borda Winner a
(a55444, b55443) Violate Independence
from Irrelevant Alternatives
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Borda Count

• Borda (1770)
• Member of French Academy of Sciences
• Noted for work in hydraulics, optics, navigation
  instrument
• Purpose Reforming the election procedure of
  French Academy.
• Criticize plurality method
• Independency violated
• Condorcet (1785)
• Viewd Borda as enemy
• Finding best ordering by hypothesis testing
• Switch to propose Condorcet winner
• Cycles ? eliminate candidates ? monotonicity
  violated
• (Systems that eliminate before selecting violate
  monotonicity)

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Utilitarian Methods

• Require voters to assign utilities to
  alternatives
• Majoritarian pairwise xgty
• Positional position xyz ? 321
• Utilitarian scale x0.5, y0.2, z0.1,
  x0.5, y0.49, z0.1
• Direct Aggregation of Cardinal Utility Methods
• summation, multiplication
• Demand-Revealing Methods

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Demand-Revealing Methods

• Tideman and Tullock (1976)
• Each voter offers to pay money m to obtain a
  preferred alternative.
• The alternative with the largest sum of offered
  money wins.
• Voters need to pay tax proportional to their
  contribution if their choices win.
• This tax is wasted(!) or given to people not
  involved.

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Multi-Stage Methods

• Black's Method (1940)
• Condorcet Winner, if one exists, otherwise Borda
  count.
• The Runoff Election Procedure
• Nanson's Borda-Elimination Procedure
• Single Transferable Voting
• Coomb's Procedure

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Condorcets intention to order

• 3 candidates, 13 voters
• ordering abc has support 26
• agtb8, agtc7, bgtc11
• ordering bca has support 22
• altb5, cgta6, bgtc11

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Kemeny Optimal Aggregation

• John Kemeny 1959 (developed BASIC language)
• Satisfies Condorcet criterion
• Equal to Condorcets ordering

22
Kemeny Optimal Aggregation

• Kemeny distance
• Number of different pairwise orders.
• Example
• 2 ranked list r1abc, r2bac
• K(r1,r2) 1 (a gtr1 b, a ltr2 b)
• Kemeny Optimal Aggregation
• ranked list r1, r2, , rn
• aggreate ranked list r argmin ? K(r,ri)
• NP-hard

23
Application

• Web Search
• Machine Learning
• Information Retrieval
• Database Middleware

24
Web Search

• Rank Aggregation Methods for the Web. Cynthia
  Dwork, Ravi Kumar, Moni Naor, D. Sivakumar.
  WWW10.
• Spam-fighting in Meta-Search
• Local Kemenization
• efficient, satisfy Condorcet criterion.

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Conclusion

• Desired Properties
• Rank Aggregation Methods
• Condorcet Criterion vs. Borda
• Kemeny Optimal Aggregation
• Rank Aggregation in fields other than social
  choice