Voting-Rules

Hypothesis

1. For each of Copeland, Borda and Plurality, the winner is the candidate securing the highest score.
2. In Copeland, modifying the preferecne of a single voter, say v, might not necessarily lead to an increase in score of v's preferred candidate. For instance, consider an example where the head to head scores for every pair of candidates differ by at least 2 without considering the vote ofv. In such a scenario, vote of v won't have any effect on the final scores of candidates.
3. In Borda and Plurality, on the other hand, the vote of v can increase the score of v's preferred candidates (other than v's most preferred candidate). Thus, the extent of manipulability appears higher in Borda and Plurality.
4. In Borda, there is a possibility to increase the maximum score of v's more preferred candidates (more preferred than original winner) and at the same time decrease the maximum score of v's lesser preferred candidates (original winner and lesser preferred candidates). However, in Plurality, the vote of v can only increase the maximum score of v's more preferred candidates. So, it appears that Borda in more manipulable than Plurality. So, the expected trend in manipulability is Borda > Plurality > Copeland

Experimental Setup

Experiments

1. Analyzing the convergence of fraction of manipulable preferences with changing sample size. We fixed the number of candidates to 5, number of voters to 100 and varied the sample size as [100, 500, 1000, 2000, 5000, 10000].
2. Analyzing the fraction of manipulable preferences with changing candidate count. We fixed the number of voters to 100 and number of samples to 5000 and varied the candidate count as [2, 3, 4, 5, 6, 8, 10]
3. Analyzing the fraction of manipulable preferences with changing voter count. We fixed the number of candidates to 5 and number of samples to 5000 and varied the voter count as [1, 2, 5, 10, 20, 50, 100, 200]

Justifications

1. A setting of 5 candidates and 100 voters looks natural for an election, and at the same time is small enough to be efficiently computable for manipulability.
2. Based on the convergence plot, 5000 samples appears to be sweet spot for both high confidence of manipulable fractions as well as efficiency of computation.

Algorithm for checking manipulability

1. Greedy strategy for f-Manipulation, discussed in lecture 14, is being used for checking manipulability.
2. If the original winner is w, then for each voter v, denote by G(v, w) the set of candidates which are more preferred than w by v. For each voter v, and each candidate c in G(v, w), we check if v can manipulate its vote to make c win, using the greedy startegy.

Experimental Results

Inference

1. We observe that the manipulability is 0 for candidates = 2. This is because if some voters' best candidate doesn't win, then he/she can has only one way to change his/her preference and that makes his/her best candidate even worse off. We also observe that on increasing the number of candidates, manipulability increases almost linearly.
2. At candidates = 6, the fraction becomes nearly equal for all the voting rules. This happens probably due to small sample size.
3. We observe that for candidates=2, Copeland and Borda have same manipulability.
4. The order of manipulability observed from the plots - Copeland < Borda < Plurality. However, as the number of candidates increase (keeping the number of voters fixed), we observe that the manipulability for Copeland overshoots the other two. We conjecture that this happens when the number of candidates comes close to the number of voters.
5. As the number of voters increase, the manipulability decreases. This is expected because more voters means less weight of a single vote. So, wrong preferences given by one voter is less likely to change the outcome.
6. We observe that on increasing the number of voters (keeping the number of candidates fixed), the manipulability first increases rapidly and then decays exponentially later on.