Arrow's impossibility theorem

Arrow's theorem is a mathematical result, but it is often expressed in a non-mathematical way with a statement such as "No voting method is fair", "Every ranked voting method is flawed", or "The only voting method that isn't flawed is a dictatorship". These statements are simplifications of Arrow's result which are not universally considered to be true. What Arrow's theorem does state is that a voting mechanism cannot comply with all of the conditions given above simultaneously for all possible preference orders.Arrow did use the term "fair" to refer to his criteria. Indeed, Pareto efficiency, as well as the demand for non-imposition, seems trivial. Various theorists have suggested weakening the IIA criterion as a way out of the paradox. Proponents of ranked voting methods contend that the IIA is an unreasonably strong criterion, which actually does not hold in most real-life situations. Indeed, the IIA criterion is the one breached in most useful voting systems.Advocates of this position point out that failure of the standard IIA criterion is trivially implied by the possibility of cyclic preferences. If voters cast ballots as follows:7 votes for A > B > C6 votes for B > C > A5 votes for C > A > Bthen the net preference of the group is that A wins over B, B wins over C, and C wins over A. In this circumstance, any system that picks a unique winner, and satisfies the very basic majoritarian rule that a candidate who receives a majority of all first-choice votes must win the election, will fail the IIA criterion. Without loss of generality, consider that if a system currently picks A, and B drops out of the race (as e.g. in a two-round system), the remaining votes will be:7 votes for A > C11 votes for C > AThus, C will win, even though the change (B dropping out) concerned an "irrelevant" alternative candidate who did not win in the original circumstance.So, what Arrow's theorem really shows is that voting is a non-trivial game, and that game theory should be used to predict the outcome of most voting mechanisms. This could be seen as a discouraging result, because a game need not have efficient equilibria, e.g., a ballot could result in an alternative nobody really wanted in the first place, yet everybody voted for.Note, however, that not all voting systems require (or even allow), as input, a strict ordering of all candidates. These systems may then trivially fail the universality criterion. Some systems may satisfy a version of Arrow's theorem with some reformulation of universality and independence of irrelevant alternatives; Warren Smith claims that Range voting is such a system.