Mark Wang John Sturm Sanjeev Kulkarni Paul Cuff Basic Background What is the problem Condorcet IIA Survey Data Pairwise Boundaries No strategic voting Outline How to make group decisions when you must ID: 675791
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Slide1
Requirements for Fair and Robust Voting Systems
Mark Wang
John Sturm
Sanjeev
Kulkarni
Paul CuffSlide2
Basic Background – What is the problem?
Condorcet = IIA
Survey DataPairwise Boundaries = No strategic voting
OutlineSlide3
How to make group decisions when you must.
Corporations
OrganizationsAcademic Departments
Politics
DemocracySlide4
Majority DecisionNo real controversy
Three candidates or more --- not so obvious
Only Two ChoicesSlide5
Voters express preferences as a list
Preferential VotingSlide6
Vote Profile
(entire preferences of population)
Overly simplified for illustration
Voting ExampleSlide7
Vote Profile
Each voter submits only one name on ballot
The candidate named the most wins
Ann wins
with 37% of the vote.*
PluralitySlide8
Vote Profile
Rounds of plurality voting, eliminating one loser each round.
Betty loses first, then
Carl wins
with 63%.
Instant Run-offSlide9
Vote Profile
Score given for place on ballot
Ann = .74, Betty = 1.3, Carl = .96
Betty wins
BordaSlide10
Voting systems can give different outcomes
Do they really?
Is there a best system?How do we arrive at it?
Item 2, A Perfect Voting SystemSlide11
Natural Assumptions:
(Anonymous) Each voter is treated equally
(Neutral) Each candidate is treated equally
(Majority) Two candidates = majority
(Scale invariant – to be defined)
AssumptionsSlide12
Robust to Candidate adjustments
Independence of Irrelevant Alternatives (IIA)
Robust to Voter adjustments
Immunity to Strategic Voting
Golden PropertiesSlide13
The voting outcome would not be different if any of the non-winning candidates had not run in the race.
Our Definition of IIASlide14
One assumption:Given someone’s honest preference order, we can assume they would vote for the higher ranked candidate in a two-candidate election.
IIA => Winner must win in any subset
=> winner must win pairwise against everyone
=> Condorcet winner must exist and win election
Condorcet = IIASlide15
Arrow’s Impossibility Theorem
Condorcet winner may not exist
Problem – IIA is impossible!Slide16
Anonymous:Outcome only depends on number of voters who cast each possible ballot.
i.e. ( #(A,B,C), #(A,C,B), … )
Equivalent: ( #(A,B,C
)/n,
#(A,C,B
)/n,
…
) and n.
Scale Invariant:
Only depends on
( #(A,B,C)/n, #(A,C,B)/n, … )
i.e. Empirical distribution of votes
Graphical RepresentationSlide17
Simplex Representation
(A > B > C)
(A > C > B)
(B > A > C)
(B > C > A)
Vote ProfileSlide18
SimplexSlide19
Some 3-candidate voting systems can be visualized through projections of the simplex
Example: Three dimensional space of pair-wise comparisons
Condorcet RegionsMany Condorcet Methods
Borda
Projections of the SimplexSlide20
Condorcet RegionsSlide21
Condorcet RegionsSlide22
Condorcet RegionsSlide23
BordaSlide24
Black MethodSlide25
Baldwin MethodSlide26
Kemeny
-Young MethodSlide27
What really happens?
In real life, does a Condorcet winner exist or not?
Item 3, DataSlide28
Tideman Data (1987, supported by NSF grant)
UK
Labour Party LeaderIce Skating
American Psychology Association Elections
Debian
Leader Election (2001-2007)
Various City Elections
Our collection: Survey responses from 2012 US Presidential Election
Sources of DataSlide29
Visualization of Tideman Data
Sets of 3 candidates
Top View
Side ViewSlide30
Visualization of Tideman Data
Sets of 3 candidates with enough voters
Top View
Side ViewSlide31
Visualization of Tideman Data
Sets of top 3 candidates
Top View
Side ViewSlide32
Do voters have an incentive to vote strategically?
A strategic vote is a vote other than the true preferences. Does this ever benefit the voter?
Gibbard-Satterthwaite
Theorem
Yes
Strategic VotingSlide33
Immunity to Strategic Voting is a property of the shape of the boundaries.
Some boundaries
may not give incentive for strategic voting.
Boundary PropertySlide34
Borda
All boundaries incentivize strategic votingSlide35
Kemeny
-Young MethodSlide36
B
Consider a planar boundary
Requirement for Robust Boundary
ASlide37
Robust boundaries are based only on pairwise comparisons
Robust BoundariesSlide38
Assume anonymity(more than G-S assumes)
Assume at least 3 candidates can win
Not possible to partition using only non-strategic boundaries.
Geometric Proof of G-S TheoremSlide39
Condorcet Regions
In some sense, this is uniquely non-strategic.Slide40
Voters are allowed to modify their vote based on feedbackThe population is sampled
One random voter is allowed to change his vote
Update the feedback… pick another random voter
Dynamic SettingSlide41
US Presidential Election Survey Results
2012 PollSlide42
All ResultsSlide43
Google PollSlide44
Mercer County ResultsSlide45
Mercer CountySlide46
Mechanical Turk PollSlide47
Mechanical TurkSlide48
NYT Blog PollSlide49
Google – CondorcetSlide50
Google – PluralitySlide51
Google – PluralitySlide52
Google – Instant Run-offSlide53
Google –
BordaSlide54
Google – Condorcet
(Democrat voters)Slide55
Google – Condorcet
(Republican voters)Slide56
Google –
Borda
(Republican voters)Slide57
Google – Range Voting (Republican voters)Slide58
Google – Condorcet
(Independent voters)Slide59
Google –
Borda
(Independent voters)Slide60
Condorcet winner is uniquely:
IIA
Whenever possibleRobust to Strategic Voting
Within the Condorcet regions
Upon strategy, some methods become Condorcet
Disagreement among voting systems is non-negligible.
Summary