Impossibility theorem

An impossibility theorem is a theorem that takes a set of seemingly simple criteria that a voting method should adhere to, and then mathematically prove that the criteria are mutually exclusive. There are a lot of impossibility theorems in the electoral space. It would seem that the topic is a lot of fun for people who are good with math.

Arrow's theorem

Main article: Arrow's impossibility theorem

One very important modern impossibility theorem is "Arrow's impossibility theorem" devised by Kenneth Arrow at Stanford. Arrow's impossibility theorem is a key result in social choice showing that no order or rank-based social welfare function can produce a rational measure of society's well-being when there are more than two options.

Arrow's original theorem neglected to account for ratings-based voting systems ("cardinal systems"), so there are many people who think that cardinal systems are inherently superior to ordinal systems, and try to jump through the loophole in Arrow's work from the 1950s in order to imply that cardinal methods are superior because Arrow's theorem doesn't apply to cardinal methods.

Gibbard's theorem

Main article: Gibbard's theorem

A year after Dr. Kenneth Arrow won the Nobel prize in economics, Alan Gibbard proved that cardinal methods (and pretty much all group decision making systems) are subject to an impossibility theorem. In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It posits that for any deterministic process of collective decision that at least one of three properties must hold, and then proves that the three properties are mutually exclusive.

Chichilnisky's theorem

See also: Graciela Chichilnisky

Graciela Chichilnisky created a notable impossibility theorem in the same vein as Arrow's and Gibbard's theorems.