Arrow's impossibility theorem

In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates the non-existence a set of rules for social decision making that would meet all of a certain set of criteria.

The theorem is named after economist Kenneth Arrow, who proved the theorem in his Ph.D. thesis and popularized it in his 1951 book Social Choice and Individual Values. Arrow was a co-recipient of the 1972 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (popularly known as the “Nobel Prize in Economics”).

The theorem’s content, somewhat simplified, is as follows. Suppose that a society needs to agree on a preference order among several different options. Each individual in the society has a particular personal preference order. The problem is to find a general mechanism, called a social choice function, which transforms the set of preference orders, one for each individual, into a global societal preference order. The theorem considers the following properties, assumed by Arrow to be reasonable requirements of a fair voting method:

• unrestricted domain or universality: the social choice function should create a deterministic, complete societal preference order from every possible set of individual preference orders. (The vote must have a result that ranks all possible choices relative to one another, the voting mechanism must be able to process all possible sets of voter preferences, and it should always give the same result for the same votes, without random selection.)
• non-imposition or citizen sovereignty: every possible societal preference order should be achievable by some set of individual preference orders. (Every result must be achievable somehow.)
• non-dictatorship: the social choice function should not simply follow the preference order of a single individual while ignoring all others.
• positive association of social and individual values or monotonicity: if an individual modifies his or her preference order by promoting a certain option, then the societal preference order should respond only by promoting that same option or not changing, never by placing it lower than before. (An individual should not be able to hurt an option by ranking it higher.)
• independence of irrelevant alternatives: if we restrict attention to a subset of options, and apply the social choice function only to those, then the result should be compatible with the outcome for the whole set of options. (Changes in individuals’ rankings of “irrelevant” alternatives [i.e., ones outside the subset] should have no impact on the societal ranking of the “relevant” subset.)

Arrow’s theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social choice function that satisfies all these conditions at once.

Another version of Arrow’s theorem can be obtained by replacing the monotonicity criterion with that of:

• unanimity or Pareto efficiency: if every individual prefers a certain option to another, then so must the resulting societal preference order.

This statement is stronger, because assuming both monotonicity and independence of irrelevant alternatives implies Pareto efficiency.

Interpretations of Arrow's 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, because they require strong assumptions about what makes a voting method "fair".

Arrow used the term "fair" to refer to his criteria, but there is no inherent reason that these criteria should be considered a requirement for fairness. For one thing, Arrow's requirements implicitly require that the voting method uses a ranked ballot, but it would be narrow-minded to state that any voting method that does not use a ranked ballot is "unfair". If a method requires a random decision (such as to break a tie), it is also enough to be deemed "unfair" by Arrow's criteria.

Proponents of ranked voting methods contend that independence of irrelevant alternatives is an unreasonably strong criterion, and therefore Arrow's theorem is an uninteresting statement. Various theorists and hobbyists have suggested replacements for this criterion, but none have become largely accepted. The Gibbard-Satterthwaite theorem also depends on the independence of irrelevant alternatives criterion, so opponents of IIA would conclude that it is uninteresting as well.

Arrow's theorem can still be seen as a useful result even if one considers its criteria to be too stringent: it provided the incentive to look for criteria that are looser but still reasonable.

See also

• Gibbard-Satterthwaite theorem
• Voting paradox

External links

• Three Brief Proofs of Arrow’s Impossibility Theorem (http://ideas.repec.org/p/cwl/cwldpp/1123r3.html)
• A Pedagogical Proof of Arrow’s Impossibility Theorem (http://citeseer.nj.nec.com/dardanoni99pedagogical.html)
• Discussion of Arrow’s Theorem and Condorcet’s method (http://www.electionmethods.org/Arrow.htm)de:Arrow-Theorem

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