Arrow's impossibility theorem

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In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. These criteria are called unrestricted domain, non-imposition, non-dictatorship, monotonicity, and independence of irrelevant alternatives, and are defined below.

The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his Ph.D. thesis and popularized it in his 1951 book Social Choice and Individual Values.

The original paper was entitled "A Difficulty in the Concept of Social Welfare" and can be found in The Journal of Political Economy, Volume 58, Issue 4 (August, 1950), pages 328-346.

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”).

Statement of the theorem

The need to aggregate preferences occurs in many different disciplines: in welfare economics, where one attempts to find an economic outcome which would be acceptable and stable; in decision making, where a person has to make a rational choice based on several criteria; and most naturally in voting systems, which are mechanisms for extracting a decision from a multitude of voter's preferences.

The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options (outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of preferences on the set of outcomes. We are searching for a preferential voting system, called a social welfare function, which transforms the set of preferences into a single global societal preference order. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method:

• unrestricted domain or universality: the social welfare function should create a deterministic, complete societal preference order from every possible set of individual preference orders. In other words: 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 consistently give the same result for the same profile of votes—no randomness is allowed in the process.
• non-imposition or citizen sovereignty: every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is onto: It has an unrestricted target space.
• non-dictatorship: the social welfare function should not simply follow the preference order of a special individual while ignoring all others. This means that the social welfare function is sensitive to more than the wishes of a single voter.
• 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 welfare 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 (ones outside the subset) should have no impact on the societal ranking of the relevant subset. This is a restriction on the sensitivity of the social welfare function.

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 welfare function that satisfies all these conditions at once.

Another version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria 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, again, is a demand that the social welfare function will be minimally sensitive to the preference profile.

This version of the theorem is stronger—has weaker conditions—since monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency, non-imposition, and independence of irrelevant alternatives together do not imply monotonicity.

Formal statement of the theorem

Let <math> \mathrm{A} </math> be a set of outcomes, <math> \mathrm{N} </math> a number of voters or decision criteria. We shall denote the set of all full linear orderings of <math> \mathrm{A} </math> by <math> \mathrm{L(A)} </math> (this set is equivalent to the set <math> \mathrm{S_N} </math> of permutations on the elements of <math> \mathrm{A} </math>).

A social welfare function is a function <math> F \; : \; \mathrm{L(A)}^N \; \to \; \mathrm{L(A)} </math> which aggregates voters' preferences into a single preference order on <math> \mathrm{A} </math>. The n-tuple <math> (R_1,... R_N) </math> of voter's preferences is called a preference profile.

In its strongest and most simple form, Arrow's impossibility theorem states that whenever the set <math> \mathrm{A} </math> of possible alternatives has more than 2 elements, then the following three conditions become incompatible:

unanimity, or Pareto efficiency

If alternative a is ranked above b for all orderings <math> R_1 ,... R_N </math> , then a is ranked higher than b by <math> F(R_1,R_2,...R_N) </math>. (Note that unanimity implies non-imposition).

non-dictatorship

There is no individual i whose preferences always prevail. <math> \lnot\exist i \; \in \; \{1,...,N\} \; s.t. \; \forall (R_1 ,... R_N) \; \in \; \mathrm{L(A)}^N \; : \; F(R_1,R_2,...R_N) \; = \; R_i </math>.

independence of irrelevant alternatives

For two preference profiles <math> R_1 ,... R_N </math> and <math> S_1 ,... S_N </math> such that for all individuals i alternatives a and b have the same order in <math>R_i</math> and <math>S_i</math>, alternatives a and b have the same order in <math> F(R_1,R_2,...R_N)</math> and <math> F(S_1,S_2,...S_N) </math>.

Interpretations of the 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.

Arrow did use the term "fair" to refer to his criteria. Indeed, the Pareto principle, as well as the demand for non-imposition, seems trivial. As for the independence of irrelevant alternatives (IIA) - suppose Dave, Chris, Bill and Agnes are running for office. And suppose Agnes has a clear advantage. Now according to Arrow's theorem, there could be a situation where if Dave steps out of the race, it will suddenly be Bill, and not Agnes, who would win the race. This would seem "unfair" by many. And yet it can happen, and Arrow's theorem states that these "unfair" situations cannot be avoided in general, without relaxing some other criterion. Something has to give. So the important question to be asked, in light of Arrow's theorem is: which condition should be relaxed?

Various theorists and hobbyists 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 dropped in most useful voting systems.

Relaxing the IIA criterion, though popular, has a distinct disadvantage: it can result in strategic voting, making the voting mechanism 'manipulable'. That is, any voting mechanism which is not IIA can yield a setup where some of the voters get a better result by mis-reporting their preferences (e.g. I prefer a to b to c, but I claim I prefer b to c to a). Clearly, any non-monotonic social welfare function is manipulable as well. If one uses a manipulable voting scheme in real life, one should expect some "dishonest" voting. What this means is that the real-life implementation of most voting mechanisms results in a complicated game of skill. The Gibbard-Satterthwaite theorem, an attempt at weakening the conditions of Arrow's paradox, replaces the IIA criterion with a criterion of non-manipulability, only to reveal the same impossibility.

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.

Other possibilities

The preceding discussion assumes that the "correct" way to deal with Arrow's paradox is to eliminate (or weaken) one of the criteria. The IIA criterion is the most natural candidate. Yet there are other "ways out".

Duncan Black has shown that if there is only one agenda by which the preferences are judged, then all of Arrow's axioms are met by the majority rule. Formally, this means that if we properly restrict the domain of the social welfare function, then all is well. Black's restriction, the "single peaked preference" principle, states that there is some predetermined linear ordering P of the alternative set. Every voter has some special place he likes best along that line, and his dislike for an alternative grows larger as the alternative goes further away from that spot.

Indeed, many different social welfare functions can meet Arrow's conditions under such restricting of the domain. It has been proved, however, that any such restriction that makes any social welfare function adhere with Arrow's criteria, will make the majority rule adhere with these criteria. So the majority rule is in some respects the fairest and most natural of all voting mechanisms.

Another common way "around" the paradox is limiting the alternative set to two alternatives. Thus, whenever more than two alternatives should be put to the test, it seems very tempting to use a mechanism that pairs them and votes by pairs. As tempting as this mechanism seems at first glance, it is generally far from meeting even the Pareto principle, not to mention IIA. The specific order by which the pairs are decided strongly influences the outcome. This is not necessarily a bad feature of the mechanism. In fact, many sports use the tournament mechanism - which is essentially a pairing mechanism - to choose a winner. This gives considerable opportunity for weaker teams to win, thus adding interest and tension throughout the tournament.

There has developed an entire literature following from Arrow's original work which finds other impossibilities as well as some possibility results. For example, if we weaken the requirement that the social choice rule must create a social preference ordering which satisfies transitivity and instead only require acyclicity (if a is preferred to b, and b is preferred to c, then it is not the case that c is preferred to a) there do exist social choice rules which satisfy Arrow's requirements.

Economist and Nobel prize winner Amartya Sen has suggested at least two other alternatives. He has offered both relaxation of transitivity and removal of the Pareto principle. He has shown the existence of voting mechanisms which comply to all of Arrow's criteria, but supply only semi-transitive results. Also, he has demonstrated another interesting impossibility result: the "impossibility of the Paretian Liberal". Sen proved that the Pareto principle is irreconcilable with even very weak liberty. That is to say, if by liberty we mean that there are some domains in life over which individuals are decisive regardless of other individuals' preferences (such as whether Joe decides to sleep on his back or his front tonight), then this is incompatible with the Pareto principle. (See Liberal paradox for details). Sen went on to argue that this demonstrates the futility of demanding Pareto optimality in relation to voting mechanisms.

See also

• Gibbard-Satterthwaite theorem
• Voting paradox
• Category:Voting_systems

External links

• Three Brief Proofs of Arrow’s Impossibility Theorem
• A Pedagogical Proof of Arrow’s Impossibility Theoremde:Arrow-Theorem

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