The Process of Computing Election Victories

1. 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.

2. The Rank Aggregation Problem B D C A F E “Consensus” ranking of all A B D C FE B D C A B C D A F E Let us create our own data by ranking the previous 3 labs.

3. 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 > 50% of 1st place votes • If not, then is the plurality candidate a good and fair choice?

5. 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?

6. 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?

7. An Alternative: Borda’s 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.

8. An Alternative: Kendall’s 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 disagreementsbetween 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) -> 8 total disagreements for ordering (1,2,3) • Which of 6 orderings gives lowest (best) score for our candidate election?

9. 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. Arrow’s 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.