Voting Paradoxes

Arrow's Impossibility Theorem

A theorem was invented by Kenneth Arrow, describing the impossibility of a perfect voting system in a situation when there are at least three candidates, that sill meets a certain set of rules. Arrow's Impossibility Theorem states that there can be no voting system that satisfies: • The outcome of a vote should not always be identical to one voter's preference list. -No dictators • If every voter prefers candidate A to B, then A should be ranked above B -Pareto Efficiency • If every voter's preference between A and B remains unchanged, then the group's preference between A and B will also remain unchanged, even if other candidates are introduced into the ballet.

Voting Paradox

A voting paradox occurs when the result of a vote is contradictory, or opposite of the expected outcome. There are many different types of voting paradoxes, such as the Condorcet Paradox, credited to Marquis de Condorcet, in 1785. Other voting paradoxes include popular voting paradoxes, paradoxes in plurality voting, and many others. Many of these paradoxes have been illustrated in history as well, proving just how unreliable our voting systems may be.

Popular Vote Paradox

A popular vote paradox is probably more common than a Condorcet Paradox, and results in the least popular candidate in an election actually winning. For example, let's again say we have three candidates running for the same position, only this time, A and B have similar beliefs, and 60% of the voters have the same beliefs as A and B, leaving C with only 40% of the votes. Theoretically though, A's votes and B's votes are actually ½ of 60% of the votes. Now, A and B, the more popular candidates among the voters each have 30%, leaving C with 40% of the votes. Therefore, C actually wins the election by relative majority.

Plurality Voting

Another voting paradox can appear in plurality voting. This specific paradox occurs under certain circumstances, in which one of three or more candidates is removed from the ballet, completely changing the vote. For example, once more lets say there are three candidates, A, B, and C, and nine voters. Four voters vote A > B > C. Two voters vote B > C > A. Three voters vote C > A > B. So, the results are A > C > B. Now, because B was the candidate with the least amount of votes, he is removed from the election. Seemingly, A should win just as before, but because we removed B from the equation, the listings are now four votes for A (just as before), but now five votes for C, because candidate B's voters went with their second choice, C. Now, because one unimportant candidate was removed, the most popular candidate lost the election. In plurality voting, there is another small paradox that occurs based on absolute majority vs. relative majority. Once more, lets say there are three candidates, A, B, and C, and seven voters. Three voters vote A > B > C. Two voters vote B > C > A. Two voters vote C > B > A. It is clear that A got the most votes of any of the candidates (3), although four of the voters, the absolute majority, believed that A was the worst of the candidates. So, although more of the voters believed that candidate A was the worst candidate in the election, A still won because he/she had the relative majority, or the most votes.

Marquis de Condorcet

Marquis de Condorcet was a famous mathematician/philosopher born in Ribemont, France, on September 17, 1743, and died in 1794. Condorcet's two most popular contributions to political science are the Condorcet's Voting Paradox, and the Condorcet Method. Condorcet's Paradox occurs when "majority rules" actually fails. For example, lets say there are three voters, and three different candidates all running for the same position. Voter One's preference list is A > B > C. Voter Two's preference list is B > C > A. Voter Three's preference list is C > A > B. Because every voter voted a different candidate, A will randomly be elected. Now though, voters 2 and 3 can argue that they each voted C over A, so now C is elected. Now, voters 1 and 2 argue that they each voted B over C, so now B is elected instead of C. This cycle repeats itself because now voters 1 and 3 argue that they each voted A over B, so A is elected once again. One example of a Condorcet Paradox that occurred in 1956 resulted due to a vote in Congress regarding school funding. The outcome of the vote resulted the same way as the example above, ultimately resulting in a Condorcet Paradox. Another voting paradox can occur in a popular vote, in which the least voted candidate of three or more candidates actually wins the election.

Jean-Charles de Borda

Other than Condorcet, another Frenchman, Jean-Charles de Borda, created a system of voting he believed fair, which revolved around the idea of ranking the candidates, using Borda points. This method of voting was called the Borda Count. Both Condorcet and Borda argued about whose system of voting were more efficient and fair, but each still has its weaknesses. Borda's system suggests each candidate should receive points based on their ranking. So, using the same example from above lets again say we have the same three candidates, and the same voters, with the same preferences. Using the Borda Method, the table would be arranged as such: Place 3 2 2 1st A(9) B(6) C(6) 2nd B(6) C(4) B(4) 3rd C(3) A(2) A(2) Any 1st place vote counts as three points, then two points, and then one point. So, after solving for the number of points each candidate received, you add all of candidate A's points together (13), all of B's points together (16), and all of C's points together (13). So, from the results of the Borda Count Method, candidate B would win the election because he/she has the most points. There are different variations of the Borda Count Method, and the winner might change as a result to which variation is used.

Conclusions

So, after seeing the different kinds of voting paradoxes and different aspects/theorems concerning voting paradoxes, it can be noticed the significance of these voting paradoxes by viewing real examples, which have happened. Voting paradoxes are significant or numerous reasons. Voting paradoxes apply math concerning game theory, and are a very bid deal in political science. The most important aspect of voting paradoxes is that they prove democracy fails. This can be observed by each of the voting paradoxes. In Condorcet's Paradox, majority fails, because any of the candidates can be declared the winner. In the popular vote paradox, the least wanted candidate is elected. In absolute majority vs. relative majority, the actual majority of the voters aren't strong enough to counteract the relative majority. So, are our voting systems as reliable as we think they are?