mark wang john sturm sanjeev kulkarni paul cuff. basic background – what is the problem? ...
TRANSCRIPT
![Page 1: Mark Wang John Sturm Sanjeev Kulkarni Paul Cuff. Basic Background – What is the problem? Condorcet = IIA Survey Data Pairwise Boundaries = No](https://reader033.vdocument.in/reader033/viewer/2022052702/56649c8b5503460f9494524d/html5/thumbnails/1.jpg)
Requirements for Fair and Robust Voting Systems
Mark WangJohn Sturm
Sanjeev KulkarniPaul Cuff
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Basic Background – What is the problem? Condorcet = IIA Survey Data Pairwise Boundaries = No strategic voting
Outline
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How to make group decisions when you must.◦ Corporations◦ Organizations◦ Academic Departments◦ Politics
Democracy
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Majority Decision No real controversy
Three candidates or more --- not so obvious
Only Two Choices
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Voters express preferences as a list
Preferential Voting
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Vote Profile (entire preferences of population)
◦ Overly simplified for illustration
Voting Example
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Vote Profile
Each voter submits only one name on ballot The candidate named the most wins Ann wins with 37% of the vote.*
Plurality
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Vote Profile
Rounds of plurality voting, eliminating one loser each round.
Betty loses first, then Carl wins with 63%.
Instant Run-off
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Vote Profile
Score given for place on ballot Ann = .74, Betty = 1.3, Carl = .96 Betty wins
Borda
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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 System
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Natural Assumptions:1. (Anonymous) Each voter is treated equally2. (Neutral) Each candidate is treated equally3. (Majority) Two candidates = majority4. (Scale invariant – to be defined)
Assumptions
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1. Robust to Candidate adjustments◦ Independence of Irrelevant Alternatives (IIA)
2. Robust to Voter adjustments◦ Immunity to Strategic Voting
Golden Properties
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The voting outcome would not be different if any of the non-winning candidates had not run in the race.
Our Definition of IIA
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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 = IIA
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Arrow’s Impossibility Theorem
Condorcet winner may not exist
Problem – IIA is impossible!
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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 Representation
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Simplex Representation
(A > B > C)
(A > C > B)
(B > A > C)
(B > C > A)
Vote Profile
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Simplex
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Some 3-candidate voting systems can be visualized through projections of the simplex
Example: Three dimensional space of pair-wise comparisons◦ Condorcet Regions◦ Many Condorcet Methods◦ Borda
Projections of the Simplex
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Condorcet Regions
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Condorcet Regions
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Condorcet Regions
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Borda
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Black Method
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Baldwin Method
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Kemeny-Young Method
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What really happens?◦ In real life, does a Condorcet winner exist or not?
Item 3, Data
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Tideman Data (1987, supported by NSF grant)◦ UK Labour Party Leader◦ Ice Skating
American Psychology Association Elections Debian Leader Election (2001-2007) Various City Elections Our collection: Survey responses from 2012
US Presidential Election
Sources of Data
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Visualization of Tideman Data
Sets of 3 candidates
Top View Side View
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Visualization of Tideman Data
Sets of 3 candidates with enough voters
Top View Side View
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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 Voting
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Immunity to Strategic Voting is a property of the shape of the boundaries.
Some boundaries may not give incentive for strategic voting.
Boundary Property
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BordaAll boundaries incentivize strategic voting
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Kemeny-Young Method
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B
Consider a planar boundary
Requirement for Robust Boundary
A
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Robust boundaries are based only on pairwise comparisons
Robust Boundaries
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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 Theorem
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Condorcet RegionsIn some sense, this is uniquely non-strategic.
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Voters are allowed to modify their vote based on feedback◦ The population is sampled◦ One random voter is allowed to change his vote◦ Update the feedback… pick another random voter
Dynamic Setting
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US Presidential Election Survey
Results2012 Poll
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All Results
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Google Poll
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Mercer County Results
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Mercer County
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Mechanical Turk Poll
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Mechanical Turk
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NYT Blog Poll
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Google – Condorcet
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Google – Plurality
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Google – Plurality
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Google – Instant Run-off
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Google – Borda
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Google – Condorcet(Democrat voters)
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Google – Condorcet(Republican voters)
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Google – Borda(Republican voters)
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Google – Range Voting(Republican voters)
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Google – Condorcet(Independent voters)
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Google – Borda(Independent voters)
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Condorcet winner is uniquely:◦ IIA
Whenever possible◦ Robust to Strategic Voting
Within the Condorcet regions
Upon strategy, some methods become Condorcet
Disagreement among voting systems is non-negligible.
Summary