  A crucial aspect of democracy is the notion of free and fair elections. While
much attention has been placed on factors such as voter turnout and the quality
of the candidates, the voting system used in an election is just as
consequential in determining the winner. In Social Choice Theory, voting
systems are evaluated based on whether or not they violate fairness criteria.
Arrow's Impossibility Theorem states that no voting method can jointly satisfy
five fairness criteria: unanimity, transitivity, independence of irrelevant
alternatives, unrestricted domain, and non-dictatorship. However, voting
systems are not equally likely to violate criteria. This paper takes a
statistical approach to determine which of 12 voting systems is the fairest
based on their probabilistic likelihood of violating nine social choice
criteria. The criteria include Arrow's five conditions, the Condorcet winner
and loser criteria, the majority criteria, and citizen sovereignty. Elections
with up to 10 million voters and between three and six candidates are simulated
using Impartial Culture and Impartial Anonymous Culture. We first compute the
frequencies with which voting systems violate individual criteria and then
compute Joint Satisfaction frequencies, the likelihood that a voting system
jointly satisfies all nine criteria. We find that Pairwise Majority is the most
likely to jointly satisfy the nine conditions, but increasingly violates
transitivity as the number of candidates increases. Of the systems that satisfy
transitivity, the Baldwin method has the highest Joint Satisfaction frequency
in elections with three candidates. As the number of candidates increase, the
Joint Satisfaction frequencies decrease rapidly for all systems.
