Incoming flashbacks to my CTY course on Probability and Game Theory.
This is a classic study of what I call election theory: the study of electoral systems’ legitimacy. The above represent three kinds of elections: plurality, approval voting, and score voting.
Plurality is simple: each voter chooses a single candidate, and the one with the most votes wins. The first mock election is a clear plurality system, with 3 voters voting A, 4 voters voting B, 1 voter voting C, and none for D. In such a system, B would win, and become what is known as the Condorcet Candidate. The plurality system satisfies Pareto Efficiency, where because every voter prefers B, B wins. In addition, irrelevant candidates for each voter do not affect the final result. However, there is plenty of room for dictators, or people who have swing vote power, to form.
Approval voting is a similar idea, only with the ability to vote for more than one candidate per voter. The second mock election shows approval voting, and since A received 4 votes, B received 4 votes, C received 4 votes, and D received 7 votes, D would win. Dictators can’t form in such a system because no one person can swing vote here. Furthermore, candidates that are not considered by voters are irrelevant to the final result. However, Pareto Efficiency is not satisfied because not every person prefers candidate D as their number one choice judging by the other elections, and yet D still won.
Lastly, score voting (or cardinal voting) is when people score candidates on a certain scale. In the case of the third mock election, with A obtaining 45 points, B obtaining 41 points, C obtaining 50 points, and D obtaining 46 points, C would win. Score voting prevents swing votes again, while also providing a more complete picture behind voting. However, Pareto Efficiency fails because 4 voters prefer C and B alike in such a case and yet C won by a large margin. In addition, changing around opinions of “irrelevant candidates” (not your top two choices) can significantly affect such an election.
Many people claim that there exists a perfect election method, especially with recent outrage against elections such as the 2016 Presidential Election and the UK Referendum. Mathematically, such an election would satisfy all three criteria I alluded to above: No dictators, Pareto Efficiency, and irrelevant candidates not playing a role in voting. That simply is not the case; in fact, it was actually proven by Nobel Laureate Kenneth Arrow in what is known as Arrow’s Impossibility Theorem.