The Process of Computing Election Victories

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CS110 Introduction to Computer Science Lab
Module 4
Computational Sociology Social Choice and
Voting Methods
The Process of Computing Election Victories
Quantitative skills and concepts Data
Analysis Mathematical Modeling Algorithms for
Rank determination Rank Aggregation
Prepared by Fred Annexstein University of
Cincinnati Some rights reserved.
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The Rank Aggregation Problem
B D C A F E
Consensus ranking of all
B D C A
A B D C FE
B C D A F E
Let us create our own data by ranking the
previous 3 labs.
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Submit your rankings on Bb Chat

• submit in your rank order of the three candidates
  
• Lab 1 - Napoleon
• Lab 2 - Al Gore (Mr. Global Warming)
• Lab 3 - Archimedes

4
Voting using Plurality Method

• Plurality method
• Election of 1st place votes
• Plurality candidate
• The Candidate with the most 1st place votes
• In your worksheet determine the number of 1st
  place votes for each candidate
• Is there a Majority candidate?
• A majority candidate has gt 50 of 1st place votes
  
• If not, then is the plurality candidate a good
  and fair choice?

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A Fairness Criteria

• Condorcet Criterion A candidate which wins every
  other in pairwise simple majority voting should
  be ranked first. A plurality candidate may or may
  not satisfy this.
• Does the plurality candidate in our election
  satisfy this Condorcet Criterion?
• To determine this we need to compute pairwise
  victors. 1v.2, 1v.3, 2v.3, etc.
• If a candidate wins every head-to-head comparison
  call it a Condorcet candidates. Not always
  possible! Why?

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The Method of Pairwise Comparisons

• The winner of each pairwise comparison gets 1
  point and the loser gets none in case of a tie
  each candidate gets ½ point. The winner of the
  election is the candidate with the most points
  after all the pairwise comparisons are tabulated.
• Determine the pairwise comparison scores for each
  of the three candidates.
• Is there a victorious candidate using this
  method?
• In our election between 3 candidates, there are 3
  pairwise comparison contests.
• How many comparison contests will be needed for
  an election having 6 candidates? Can you
  determine a formula c(n) for the case of n
  candidates?

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An Alternative Bordas method

• Head-to-head comparisons can get out of control.
• Borda Count Method an easy score-based method.
  Each place on a ballot is assigned points. In an
  election with N candidates we give 1 point for a
  last place, 2points for second from last place,
  and so on.
• So in our example we give 3 points for 1st, 2
  points for 2nd, and 1 point for 3rd.
• Compute Borda scores for all three candidates.

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An Alternative Kendalls Method

• Want to answer question of all potential
  orderings, which is the best?
• Use Kendall tau distance between two ranked lists
• Count the number of pairwise disagreements
  between the two lists
• Compute the Kendall Tau distances for all 3!6
  potential orderings
• This can be done by using data from part 1 on
  pairwise contests. For example, for potential
  candidate ordering (1,2,3) there are
• 3 disagreements for ordered pair (1,2)
• 3 disagreements for ordered pair (1,3)
• 2 disagreements for ordered pair (2,3)
• -gt 8 total disagreements for ordering (1,2,3)
• Which of 6 orderings gives lowest (best) score
  for our candidate election?

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A Celebrated Theorem
You might be asking yourself whether there is a
method that is superior to all others. In 1972
Kenneth Arrow won the Nobel Prize in Economics
for his social choice theory. Arrows
Impossibility Theorem It is mathematically
impossible for a democratic voting method to
satisfy a set of natural fairness criteria.
Submit your final worksheet to Blackboard Dropbox.