Social choice theory dates back to 1785, when French philosopher Nicolas de Condorcet (1743-1794) examined how winners were selected by voters during elections. According to Peter C.Fishburn, writing in the SIAM Journal of Applied Mathematics (1977), Condorcet's principle established that if one candidate was able to obtain a clear majority over other candidates, then the majority candidate should be elected.
The modern form of social choice theory is generally acknowledged to originate from the impossibility theorem of Nobel Prize-winning American economist Kenneth Arrow (b.1921). Arrow proved that there was no single choice procedure that could consistently come out with a fair result when choosing among three or more alternatives. In 1951, Arrow developed his impossibility theorem based on certain conditions with the aim to discover a fair and consistent election system.
The first condition is called universal admissibility, or the assumption that all voters have unique sets of rational preferences. Second, Arrow examined the unanimity condition, which states that if all individuals prefer A to B, then the whole group prefers A to B. Independence from irrelevant alternatives is the third condition, meaning that if all individuals prefer A to B, then no change in preferences that is not relevant to this relationship will affect the group preference. The fourth factor is that there are no dictators, or in other words group preferences cannot be determined by a single individual.
The key to understanding Arrow's theorem centers on understanding the meaning of so-called "intransitive preferences," a term used in both economics and politics. Transitive preferences can be arranged in a logical manner, following best-to-least order: A is preferred to B and C, and B is preferred to C. If, however, A is preferred to B and B to C, but C is preferred to A, then preferences are called cyclic or intransitive.
Arrow's theorem states that when choosing between more than two options, it is generally impossible to implement these four conditions without creating cycling group preferences. Strikingly, transitive group preferences combined with the first three conditions strongly suggest a possible dictatorship. Arrow's attempt to establish an election system that would facilitate the formation of transitive group preferences over three or more alternatives concluded that this was impossible. One way to illustrate this is by looking at a common election system, such as the plurality voting system, where voters express their preference for one favorite candidate only and that the candidate with the majority of votes is considered the winner. The problem here is that the winning candidate might have attracted less than half of all votes.
For example, in the 1992 U.S. presidential election, Bill Clinton led the score with around 43 percent of the vote, while George H. W. Bush won about 38 percent, with Ross Perot accounting for about 19 percent. Supposing that all Perot voters would prefer Bush, provided that Perot had not run at all, Bush would be the winner of the election by 57 to 43 percent. Such a result may be considered a violation of the condition for independence from irrelevant alternatives.
Every voting system is characterized by similar problems. In order to come up with a sensible election procedure, political scientists have tried to figure out how to soothe the effect of Arrow's original conditions. Since most researchers do not doubt the unanimity and no-dictator conditions, most attention is being paid to the condition for independence from irrelevant alternatives and to the question how often a particular system runs into problems.
The plurality voting system actually leads to intransitive preferences less often then generally considered. According to a study by Shepsle and Bonchek, cited by Nathan Collins in the article Arrow's Theorem Proves No Voting System is Perfect (2003), in a three-voter, three-candidate election only 12 out of 216 possible preference arrangements would cause intransitive group preferences.
There are researchers who believe that other election systems are less likely to experience difficulties similar to those in the 1992 presidential election. Examples of such systems include the "instant-runoff" voting and Cambridge's version of proportional representation. These methods eliminate low-ranked candidates while redistributing votes among those remaining in the competition. Some sports use the so-called Borda system in their ranking schemes. This system is based on voters giving ranks to candidates. However, instead of elimination, each candidate gets a certain amount of points according to rankings, with these points being used to determine the winner.