30Oct2012

## Notes on Arrow’s Impossibility Theorem

Economics, Game Theory, Shameless Self-Promotion48 Comments

Someone ran across my CV and asked me if I could send anything I’d written on game theory. So I dug up my class notes (for an undergrad class at Hillsdale) on Kenneth Arrow’s famous “Impossibility Theorem” regarding social choice. I haven’t looked at these in 7 years, so I hope they’re right:

=============
NOTES ON ARROW’S

IMPOSSIBILITY THEOREM

Economics 356

History of Economic Thought II

Spring 2005

Individual Preferences

Following the methodological revolution described in Hicks, by the 1940s most economists no longer believed in cardinal utility.  At the very least, most economists considered it much safer to assume that people had merely ordinal preferences, rather than to take the stronger view that people actually received units of psychic happiness (“utils” or “wantabs”) from various goods and services.

Economists developed formal techniques to rigorously develop this line of thought.  The first step is to define the set of all possible items to be valued.  Depending on the context, this set consists of different types of things.  At the most abstract level, it could be viewed as “the set of all possible universes.”  In a much more specific example, it could merely refer to “the set of all possible pizza orders the class could phone in to Hungry Howie’s.”

Once we have defined the appropriate set, we then can talk about how each individual ranks each element in the set.  Since we are not going to assume cardinal utility, we can only discuss an individual’s ordinal rankings; that is, we can only take two elements at a time, and ask the individual, “Which of these do you prefer, or are you indifferent between them?”  We can never ask—indeed, it doesn’t make sense to ask—the individual, “How much more do you like this element over this other element?”

Formally, we can summarize an individual’s answers to these questions by use of a preference relation.  Normally we indicate this by a symbol that looks like a curvy greater-than-or-equal-to sign, but here I’ll just use the symbol @.  If we take two elements, let’s call them x and y, from the set of all possible things to be valued, then the statement x@y means that the individual thinks that x is at least as good as y.  If it were not true (at the same time) that y@x, then we would conclude that x is better than y (not merely just as good), because we know x is at least as good as y, but y is not at least as good as x.  And if we knew that x@y and y@x at the same time, then we would conclude that this individual is indifferent between x and y.

NOTE:  In order to construct a coherent ranking (from best to worst) of an individual’s preferences, it is necessary that his or her @ be complete and transitive.  If @ is complete, that means it can be applied to any two elements from the set.  I.e., for any elements (call them a and b), the individual could report either a@b, b@a, or both.  If @ were incomplete, then the individual might say, “I really don’t know how I feel about those two elements; I can’t tell you which is at least as good as the other.”

If @ is transitive, then whenever x@y and y@z, it must also be the case that x@z.

“Social” Preferences

Economists often want to use their science in order to make policy recommendations, or at least to make “objective” statements about various social arrangements.  By analogy with an individual preference ranking, we can ask how “society” does (or should) value the different possible elements in the set of all valued things.

Because society is ultimately composed of individuals, most economists think that “social” preferences ought to be constructed from the preferences of the individuals in society.  But at this point, a problem emerges:  If people do not agree on how to rank, say, x with y—i.e. some people feel that x@y while other people do not—then how can we say how “society” should rank these two possible outcomes?

Economists thus began a search for plausible social welfare orderings.  These are functions that take the @ for each individual—and we could keep them distinct by putting a superscript on them, so that @1 is the preference relation of person #1, etc.—and then use this information to generate a preference relation for society, which we will label @S.

So now the question is, what types of social welfare orderings are appealing, both on logical and moral grounds?  In principle, there are billions of different rules we could invent, in order to generate a @S out of the individual @i of each member i in the society.

Arrow’s Theorem

Kenneth Arrow intended to weed out the “silly” or obviously distasteful social welfare orderings (henceforth SWO).  So he came up with a quite reasonable list of criteria that any decent SWO would need to satisfy.

One basic requirement is that it should be complete and transitive.  That is, whenever the individual @i of each person in society is complete and transitive, whatever our rule is that generates the @S, that list of social preferences should also end up being complete and transitive.

Another criterion is that the SWO should obey weak Pareto optimality.  In the present context, this means that if x@iy for every single person i, then it should also be the case that x@Sy.  In other words, if every single person in society thinks that x is at least as good as y, it would be ridiculous if our SWO then ended up saying that “society” should value y more than x.

A third criterion is the independence of irrelevant alternatives.  This is the least intuitive of the criteria.  What it requires is that the determination of the social ranking of x and y should depend only on how each individual ranks x and y.

The final criterion is no dictatorship.  This means that there cannot be some individual j such that @S=@j no matter what every other person’s preferences are.  Note that this is a very weak requirement.  For any particular group of individual preferences @1, @2, @3, …, it’s perfectly acceptable if our SWO constructs a @S that happens to be identical to some individual @j; this alone would not christen individual j as a dictator.  What would qualify him as a dictator is if @S=@j for any possible group of individual preferences @1, @2, @3, …

What Arrow proved is that there does not exist any SWO that satisfies all four of the above conditions (if we have at least a few people and a few different elements in the set of valued things).  Specifically, Arrow proved that if we assume we are dealing with an SWO that meets the first three criteria, then that SWO necessarily must work by picking some individual j and then simply setting @S=@j.

Examples

Since Arrow’s Impossibility Theorem is a negative result, it’s best to illustrate it by showing SWOs that do not satisfy his criteria.  For simplicity, we’ll assume there are only three people, Joe, Billy, and Martha, and only three possible states of the world, x, y, and z.  We thus are looking for a set of rules to take @J, @B, and @M in order to construct a “social” ranking of the possible outcomes x, y, and z.

Suppose we have the very simple SWO that says, “No matter how Joe, Billy, and Martha rank the alternatives, @S should always be defined so that x@Sy, y@Sz, and x@Sz, and so that the reverse is not true, e.g. that it is not the case that y@Sx, etc.”

Which of Arrow’s criteria does this suggested SWO violate?  Well, it’s complete and transitive, so it’s okay on those grounds.  It doesn’t have a dictator, either (it’s always possible that any person will have preferences that differ from those indicated by @S).  Although it’s not as easy to see, I’m pretty sure that this hypothetical SWO also obeys the independence criterion.  (E.g. the social ranking of x and y will never be affected by changing the individuals’ rankings of, say, y and z.)

What this SWO does (obviously) violate is the weak Pareto condition.  For example, if Joe, Billy, and Martha all strictly prefer z to y, then our suggested SWO will still say that “society” prefers y to z.  Thus our suggested SWO does not meet Arrow’s criteria, and we must keep looking.

What about majority rule?  That is, suppose we define @S such that x@Sy only if at least two people feel this way, etc.

Majority rule violates the criterion of transitivity.  That is, there are possible preferences that Joe, Billy, and Martha could have, such that a @S constructed on the basis of majority rule would violate transitivity.  (To see this, consider the case where Joe ranks the alternatives in the order x, y, z, Billy ranks them y, z, x, and Martha ranks them z, x, y.)  Note that Arrow requires the SWO to be transitive for any possible list of individual preference relations; it’s not enough that the SWO might satisfy all four criteria for some particular list of individuals’ preferences.

Finally, let’s consider the SWO that proceeds like this:  “We will say that x@Sy only if Joe, Billy, and Martha all agree that x is at least as good as y.  If at least one of them disagrees, though, we will say that it is not the case that x@Sy.  Etc.”

Which criterion does this rule violate?  Well, it’s transitive (so long as the individual relations are); if everybody thinks x is better than y, and that y is better than z, then that means everybody thinks x is better than z, and thus so will “society.”  There is also no dictator with this proposed SWO, and it is also true (I think) that there is no violation of independence.  And of course this SWO obeys the weak Pareto condition.

But this proposed SWO is, unfortunately, incomplete.  That is, the rule we defined will not always tell us how “society” should compare, say, x and z.  For suppose that Joe thinks x is strictly better than z, but that Martha thinks that z is strictly better than x.  Then according to our rule, it can neither be true that x@Sz nor z@Sx.  And completeness requires that our preference relation be able to tell us that one (or both) of these items is at least as good the other.  Hence this proposed SWO too fails to satisfy Arrow’s criteria.

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked votingelectoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem.

The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

• If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
• If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
• There is no "dictator": no single voter possesses the power to always determine the group's preference.

Cardinal voting electoral systems are not covered by the theorem, as they convey more information than rank orders.[2][3] (See the subsection discussing the cardinal utility approach to overcoming the negative conclusion.) Arrow originally rejected cardinal utility as a meaningful tool for expressing social welfare,[4] and so focused his theorem on preference rankings, but later stated that a cardinal score system with three or four classes "is probably the best".[2]

The theorem can also be sidestepped by weakening the notion of independence.

The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem.[5]

The practical consequences of the theorem are debatable: Arrow has said "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."[6]

## Statement of the theorem

The need to aggregate preferences occurs in many disciplines: in welfare economics, where one attempts to find an economic outcome which would be acceptable and stable; in decision theory, where a person has to make a rational choice based on several criteria; and most naturally in electoral systems, which are mechanisms for extracting a decision from a multitude of voters' 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 ranked voting electoral system, called a social welfare function (preference aggregation rule), which transforms the set of preferences (profile of preferences) into a single global societal preference order. 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 (assumed to be a reasonable requirement of a fair electoral system) at once:

Non-dictatorship
The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter.
Unrestricted domain, or universality
For any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. Thus:
• It must do so in a manner that results in a complete ranking of preferences for society.
• It must deterministically provide the same ranking each time voters' preferences are presented the same way.
Independence of irrelevant alternatives (IIA)
The social preference between x and y should depend only on the individual preferences between x and y (pairwise independence). More generally, changes in individuals' rankings of irrelevant alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset. For example, the introduction of a third candidate to a two-candidate election should not affect the outcome of the election unless the third candidate wins. (See Remarks below.)
Monotonicity, or positive association of social and individual values
If any 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.
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 surjective: It has an unrestricted target space.

A later (1963)[7] version of Arrow's theorem replaced the monotonicity and non-imposition criteria with:

Pareto efficiency, or unanimity
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 later version is more general, having weaker conditions. The axioms of monotonicity, non-imposition, and IIA together imply Pareto efficiency, whereas Pareto efficiency (itself implying non-imposition) and IIA together do not imply monotonicity.

### Independence of irrelevant alternatives (IIA)

The IIA condition has three purposes (or effects):[8]

Normative
Irrelevant alternatives should not matter.
Practical
Use of minimal information.
Strategic
Providing the right incentives for the truthful revelation of individual preferences. Though the strategic property is conceptually different from IIA, it is closely related.

Arrow's death-of-a-candidate example (1963, page 26)[7] suggests that the agenda (the set of feasible alternatives) shrinks from, say, X = {a, b, c} to S = {a, b} because of the death of candidate c. This example is misleading since it can give the reader an impression that IIA is a condition involving two agenda and one profile. The fact is that IIA involves just one agendum ({x, y} in case of pairwise independence) but two profiles. If the condition is applied to this confusing example, it requires this: Suppose an aggregation rule satisfying IIA chooses b from the agenda {a, b} when the profile is given by (cab, cba), that is, individual 1 prefers c to a to b, 2 prefers c to b to a. Then, it must still choose b from {a, b} if the profile were, say: (abc, bac); (acb, bca); (acb, cba); or (abc, cba).

In different words, Arrow defines IIA as saying that the social preferences between alternatives x and y depend only on the individual preferences between x and y (not on those involving other candidates).

## Formal statement of the theorem

Let A be a set of outcomes, N a number of voters or decision criteria. We shall denote the set of all full linear orderings of A by L(A).

A (strict) social welfare function (preference aggregation rule) is a function

which aggregates voters' preferences into a single preference order on A.[9]

An N-tuple(R1, …, RN) ∈ L(A)N of voters' preferences is called a preference profile. In its strongest and simplest form, Arrow's impossibility theorem states that whenever the set A of possible alternatives has more than 2 elements, then the following three conditions become incompatible:

Unanimity, or weak Pareto efficiency
If alternative, a, is ranked strictly higher than b for all orderings R1 , …, RN, then a is ranked strictly higher than b by F(R1, R2, …, RN). (Note that unanimity implies non-imposition.)
Non-dictatorship
There is no individual, i whose strict preferences always prevail. That is, there is no i ∈ {1, …, N} such that for all (R1, …, RN) ∈ L(A)N, a ranked strictly higher than b by Ri implies a ranked strictly higher than b by F(R1, R2, …, RN), for all a and b.
Independence of irrelevant alternatives
For two preference profiles (R1, …, RN) and (S1, …, SN) such that for all individuals i, alternatives a and b have the same order in Ri as in Si, alternatives a and b have the same order in F(R1, R2, …, RN) as in F(S1, S2, …, SN).

## Informal proof

Based on two proofs appearing in Economic Theory. For simplicity we have presented all rankings as if ties are impossible. A complete proof taking possible ties into account is not essentially different from the one given here, except that one ought to say "not above" instead of "below" or "not below" instead of "above" in some cases. Full details are given in the original articles.

We will prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA) is a dictatorship. The key idea is to identify a pivotal voter whose ballot swings the societal outcome. We then prove that this voter is a partial dictator (in a specific technical sense, described below). Finally we conclude by showing that all of the partial dictators are the same person, hence this voter is a dictator.

### Part one: There is a "pivotal" voter for B over A

Say there are three choices for society, call them A, B, and C. Suppose first that everyone prefers option B the least: everyone prefers A to B, and everyone prefers C to B. By unanimity, society must also prefer both A and C to B. Call this situation profile 0.

On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each i let profile i be the same as profile 0, but move B to the top of the ballots for voters 1 through i. So profile 1 has B at the top of the ballot for voter 1, but not for any of the others. Profile 2 has B at the top for voters 1 and 2, but no others, and so on.

Since B eventually moves to the top of the societal preference, there must be some profile, number k, for which B moves above A in the societal rank. We call the voter whose ballot change causes this to happen the pivotal voter for B over A. Note that the pivotal voter for B over A is not, a priori, the same as the pivotal voter for A over B. In part three of the proof we will show that these do turn out to be the same.

Also note that by IIA the same argument applies if profile 0 is any profile in which A is ranked above B by every voter, and the pivotal voter for B over A will still be voter k. We will use this observation below.

### Part two: The pivotal voter for B over A is a dictator for B over C

In this part of the argument we refer to voter k, the pivotal voter for B over A, as pivotal voter for simplicity. We will show that pivotal voter dictates society's decision for B over C. That is, we show that no matter how the rest of society votes, if Pivotal Voter ranks B over C, then that is the societal outcome. Note again that the dictator for B over C is not a priori the same as that for C over B. In part three of the proof we will see that these turn out to be the same too.

In the following, we call voters 1 through k − 1, segment one, and voters k + 1 through N, segment two. To begin, suppose that the ballots are as follows:

• Every voter in segment one ranks B above C and C above A.
• Pivotal voter ranks A above B and B above C.
• Every voter in segment two ranks A above B and B above C.

Then by the argument in part one (and the last observation in that part), the societal outcome must rank A above B. This is because, except for a repositioning of C, this profile is the same as profile k − 1 from part one. Furthermore, by unanimity the societal outcome must rank B above C. Therefore, we know the outcome in this case completely.

Now suppose that pivotal voter moves B above A, but keeps C in the same position and imagine that any number (or all!) of the other voters change their ballots to move B below C, without changing the position of A. Then aside from a repositioning of C this is the same as profile k from part one and hence the societal outcome ranks B above A. Furthermore, by IIA the societal outcome must rank A above C, as in the previous case. In particular, the societal outcome ranks B above C, even though Pivotal Voter may have been the only voter to rank B above C. By IIA, this conclusion holds independently of how A is positioned on the ballots, so pivotal voter is a dictator for B over C.

### Part three: There can be at most one dictator

In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying parts one and two to the other pairs of candidates). First, the pivotal voter for B over C must appear earlier (or at the same position) in the line than the dictator for B over C: As we consider the argument of part one applied to B and C, successively moving B to the top of voters' ballots, the pivot point where society ranks B above C must come at or before we reach the dictator for B over C. Likewise, reversing the roles of B and C, the pivotal voter for C over B must be at or later in line than the dictator for B over C. In short, if kX/Y denotes the position of the pivotal voter for X over Y (for any two candidates X and Y), then we have shown

kB/C ≤ kB/AkC/B.

Now repeating the entire argument above with B and C switched, we also have

kC/BkB/C.

Therefore, we have

kB/C = kB/A = kC/B

and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election.

## Interpretations of the theorem

Although Arrow's theorem is a mathematical result, 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.[12] 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 deterministic preferential voting mechanism—that is, one where a preference order is the only information in a vote, and any possible set of votes gives a unique result—cannot comply with all of the conditions given above simultaneously.

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. It is the one breached in most useful electoral 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:

• 1 vote for A > B > C
• 1 vote for B > C > A
• 1 vote for C > A > B

then the pairwise majority preference of the group is that A wins over B, B wins over C, and C wins over A: these yield rock-paper-scissors preferences for any pairwise comparison. In this circumstance, any aggregation rule that satisfies the very basic majoritarian requirement that a candidate who receives a majority of votes must win the election, will fail the IIA criterion, if social preference is required to be transitive (or acyclic). To see this, suppose that such a rule satisfies IIA. Since majority preferences are respected, the society prefers A to B (two votes for A > B and one for B > A), B to C, and C to A. Thus a cycle is generated, which contradicts the assumption that social preference is transitive.

So, what Arrow's theorem really shows is that any majority-wins electoral system is a non-trivial game, and that game theory should be used to predict the outcome of most voting mechanisms.[13] 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.

### Remark: Scalar rankings from a vector of attributes and the IIA property

The IIA property might not be satisfied in human decision-making of realistic complexity because the scalar preference ranking is effectively derived from the weighting—not usually explicit—of a vector of attributes (one book dealing with the Arrow theorem invites the reader to consider the related problem of creating a scalar measure for the track and field decathlon event—e.g. how does one make scoring 600 points in the discus event "commensurable" with scoring 600 points in the 1500 m race) and this scalar ranking can depend sensitively on the weighting of different attributes, with the tacit weighting itself affected by the context and contrast created by apparently "irrelevant" choices. Edward MacNeal discusses this sensitivity problem with respect to the ranking of "most livable city" in the chapter "Surveys" of his book MathSemantics: making numbers talk sense (1994).

## Other possibilities

In an attempt to escape from the negative conclusion of Arrow's theorem, social choice theorists have investigated various possibilities ("ways out"). These investigations can be divided into the following two:

• those investigating functions whose domain, like that of Arrow's social welfare functions, consists of profiles of preferences;
• those investigating other kinds of rules.

### Approaches investigating functions of preference profiles

This section includes approaches that deal with

• aggregation rules (functions that map each preference profile into a social preference), and
• other functions, such as functions that map each preference profile into an alternative.

Since these two approaches often overlap, we discuss them at the same time. What is characteristic of these approaches is that they investigate various possibilities by eliminating or weakening or replacing one or more conditions (criteria) that Arrow imposed.

#### Infinitely many individuals

Several theorists (e.g., Kirman and Sondermann[14]) point out that when one drops the assumption that there are only finitely many individuals, one can find aggregation rules that satisfy all of Arrow's other conditions.

However, such aggregation rules are practically of limited interest, since they are based on ultrafilters, highly non-constructive mathematical objects. In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule.[14] Mihara[15][16] shows that such a rule violates algorithmic computability.[17] These results can be seen to establish the robustness of Arrow's theorem.[18]

#### Limiting the number of alternatives

When there are only two alternatives to choose from, May's theorem shows that only simple majority rule satisfies a certain set of criteria (e.g., equal treatment of individuals and of alternatives; increased support for a winning alternative should not make it into a losing one). On the other hand, when there are at least three alternatives, Arrow's theorem points out the difficulty of collective decision making. Why is there such a sharp difference between the case of less than three alternatives and that of at least three alternatives?

Nakamura's theorem (about the core of simple games) gives an answer more generally. It establishes that if the number of alternatives is less than a certain integer called the Nakamura number, then the rule in question will identify "best" alternatives without any problem; if the number of alternatives is greater or equal to the Nakamura number, then the rule will not always work, since for some profile a voting paradox (a cycle such as alternative A socially preferred to alternative B, B to C, and C to A) will arise. Since the Nakamura number of majority rule is 3 (except the case of four individuals), one can conclude from Nakamura's theorem that majority rule can deal with up to two alternatives rationally. Some super-majority rules (such as those requiring 2/3 of the votes) can have a Nakamura number greater than 3, but such rules violate other conditions given by Arrow.[19]

#### Pairwise voting

A common way "around" Arrow's 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 satisfying even Pareto efficiency, 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. Many sports use the tournament mechanism—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. This means that the person controlling the order by which the choices are paired (the agenda maker) has great control over the outcome. In any case, when viewing the entire voting process as one game, Arrow's theorem still applies.

#### Domain restrictions

Another approach is relaxing the universality condition, which means restricting the domain of aggregation rules. The best-known result along this line assumes "single peaked" preferences.

Duncan Black has shown that if there is only one dimension on which every individual has a "single-peaked" preference, then all of Arrow's conditions are met by majority rule. Suppose that there is some predetermined linear ordering of the alternative set. An individual's preference is single-peaked with respect to this ordering if he has some special place that he likes best along that line, and his dislike for an alternative grows larger as the alternative goes further away from that spot (i.e., the graph of his utility function has a single peak if alternatives are placed according to the linear ordering on the horizontal axis). For example, if voters were voting on where to set the volume for music, it would be reasonable to assume that each voter had their own ideal volume preference and that as the volume got progressively too loud or too quiet they would be increasingly dissatisfied. If the domain is restricted to profiles in which every individual has a single peaked preference with respect to the linear ordering, then simple[20] aggregation rules, which includes majority rule, have an acyclic (defined below) social preference, hence "best" alternatives.[21] In particular, when there are odd number of individuals, then the social preference becomes transitive, and the socially "best" alternative is equal to the median of all the peaks of the individuals (Black's median voter theorem[22]). Under single-peaked preferences, the majority rule is in some respects the most natural voting mechanism.

One can define the notion of "single-peaked" preferences on higher-dimensional sets of alternatives. However, one can identify the "median" of the peaks only in exceptional cases. Instead, we typically have the destructive situation suggested by McKelvey's Chaos Theorem:[23] for any x and y, one can find a sequence of alternatives such that x is beaten by x1 by a majority, x1 by x2, up to xk by y.

#### Relaxing transitivity

By relaxing the transitivity of social preferences, we can find aggregation rules that satisfy Arrow's other conditions. If we impose neutrality (equal treatment of alternatives) on such rules, however, there exists an individual who has a "veto". So the possibility provided by this approach is also very limited.

First, suppose that a social preference is quasi-transitive (instead of transitive); this means that the strict preference ("better than") is transitive: if and , then . Then, there do exist non-dictatorial aggregation rules satisfying Arrow's conditions, but such rules are oligarchic.[24] This means that there exists a coalition L such that L is decisive (if every member in L prefers x to y, then the society prefers x to y), and each member in L has a veto (if she prefers x to y, then the society cannot prefer y to x).

Second, suppose that a social preference is acyclic (instead of transitive): there do not exist alternatives that form a cycle (). Then, provided that there are at least as many alternatives as individuals, an aggregation rule satisfying Arrow's other conditions is collegial.[25] This means that there are individuals who belong to the intersection ("collegium") of all decisive coalitions. If there is someone who has a veto, then he belongs to the collegium. If the rule is assumed to be neutral, then it does have someone who has a veto.

Finally, Brown's theorem left open the case of acyclic social preferences where the number of alternatives is less than the number of individuals. One can give a definite answer for that case using the Nakamura number. See limiting the number of alternatives.

#### Relaxing IIA

There are numerous examples of aggregation rules satisfying Arrow's conditions except IIA. The Borda rule is one of them. These rules, however, are susceptible to strategic manipulation by individuals.[26]

#### Relaxing the Pareto criterion

Wilson (1972)[27] shows that if an aggregation rule is non-imposed and non-null, then there is either a dictator or an inverse dictator, provided that Arrow's conditions other than Pareto are also satisfied. Here, an inverse dictator is an individual i such that whenever i prefers x to y, then the society prefers y to x.

##### 

Amartya Sen offered both relaxation of transitivity and removal of the Pareto principle.[28] He demonstrated another interesting impossibility result, known as the "impossibility of the Paretian Liberal" (see liberal paradox for details). Sen went on to argue that this demonstrates the futility of demanding Pareto optimality in relation to voting mechanisms.

#### Social choice instead of social preference

In social decision making, to rank all alternatives is not usually a goal. It often suffices to find some alternative. The approach focusing on choosing an alternative investigates either social choice functions (functions that map each preference profile into an alternative) or social choice rules (functions that map each preference profile into a subset of alternatives).

As for social choice functions, the Gibbard–Satterthwaite theorem is well-known, which states that if a social choice function whose range contains at least three alternatives is strategy-proof, then it is dictatorial.

As for social choice rules, we should assume there is a social preference behind them. That is, we should regard a rule as choosing the maximal elements ("best" alternatives) of some social preference. The set of maximal elements of a social preference is called the core. Conditions for existence of an alternative in the core have been investigated in two approaches. The first approach assumes that preferences are at least acyclic (which is necessary and sufficient for the preferences to have a maximal element on any finite subset). For this reason, it is closely related to relaxing transitivity. The second approach drops the assumption of acyclic preferences. Kumabe and Mihara[29] adopt this approach. They make a more direct assumption that individual preferences have maximal elements, and examine conditions for the social preference to have a maximal element. See Nakamura number for details of these two approaches.

### Rated electoral system and other approaches

Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and transitivity) on the set of alternatives. This means that if the preferences are represented by a utility function, its value is an ordinal utility in the sense that it is meaningful so far as the greater value indicates the better alternative. For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as having 1000, 100.01, 100, 0, which in turn is the same as having 99, 98, 1, .997. They all represent the ordering in which a is preferred to b to c to d. The assumption of ordinal preferences, which precludes interpersonal comparisons of utility, is an integral part of Arrow's theorem.

For various reasons, an approach based on cardinal utility, where the utility has a meaning beyond just giving a ranking of alternatives, is not common in contemporary economics. However, once one adopts that approach, one can take intensities of preferences into consideration, or one can compare (i) gains and losses of utility or (ii) levels of utility, across different individuals. In particular, Harsanyi (1955)[30] gives a justification of utilitarianism (which evaluates alternatives in terms of the sum of individual utilities), originating from Jeremy Bentham. Hammond (1976)[31] gives a justification of the maximin principle (which evaluates alternatives in terms of the utility of the worst-off individual), originating from John Rawls.

Not all voting methods use, as input, only an ordering of all candidates.[32] Methods which don't, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") electoral system, can be viewed as using information that only cardinal utility can convey. In that case, it is not surprising if some of them satisfy all of Arrow's conditions that are reformulated.[33]Range voting is such a method.[6][34] Whether such a claim is correct depends on how each condition is reformulated.[35] Other rated electoral system which pass certain generalizations of Arrow's criteria include approval voting and majority judgment. Note that Arrow's theorem does not apply to single-winner methods such as these, but Gibbard's theorem still does: no non-defective electoral system is fully strategy-free, so the informal dictum that "no electoral system is perfect" still has a mathematical basis.[36]

Finally, though not an approach investigating some kind of rules, there is a criticism by James M. Buchanan, Charles Plott, and others. It argues that it is silly to think that there might be social preferences that are analogous to individual preferences.[37] Arrow (1963, Chapter 8)[38] answers this sort of criticism seen in the early period, which come at least partly from misunderstanding.

A multi-pronged refutation of Arrow's theorem was published by philosophy Professor Howard DeLong in 1991.[39][40]

He challenges the theorem on the basis that Arrow wrongly assumes Preference is transitive property and that Collective Preference is the same as summing up individual preferences. He also claims that Arrow's model fails to model democracy as it exists in the real world as the model ignores the possibility of consensual temporary dictatorships (ie: Greek tyrants in times of war) and the effect of allowing lotteries to decide tie breakers and to avoid the problem of the tyranny of the majority, the example used being a group of campers at a summer camp, 8 of whom prefer cake, 7 who prefer ice cream, but funds are limited to one choice or the other on a weekly basis. Under the collective preference of majority rule each week the group would select cake. By drawing lots the choices would more accurately reflect the preferences of the collective than the majority.[41] DeLong also criticizes Arrow's model for being hedonistic and not incorporating a sense of morality and social justice.

### Analogs to Arrow's theorem

Some applications of Arrow's impossibility theorem have been made to domains other than social choice. For example, Stegenga argues that amalgamating evidence from multiple sources faces an impossibility result analogous to Arrow's theorem.[42]

## Notes

1. ^ ab"Interview with Dr. Kenneth Arrow". The Center for Election Science. October 6, 2012.
2. ^Sen, Amartya (1999). "The Possibility of Social Choice". American Economic Review. 89 (3): 349–378. doi:10.1257/aer.89.3.349. JSTOR 117024.
3. ^"Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina, The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, 20, Routledge, pp. 21–45
4. ^Suzumura, Kōtarō. "Introduction". In Arrow, Sen & Suzumura (2002), p. 10.
5. ^ abMcKenna, Phil (12 April 2008). "Vote of no confidence". New Scientist. 198 (2651): 30–33. doi:10.1016/S0262-4079(08)60914-8.
6. ^ abArrow, Kenneth Joseph Arrow (1963). Social Choice and Individual Values(PDF). Yale University Press. ISBN 0300013647.
7. ^Mas-Colell, Andreu; Whinston, Michael Dennis; Green, Jerry R. (1995). Microeconomic Theory. Oxford University Press. p. 794. ISBN 978-0-19-507340-9.
8. ^Note that by definition, a social welfare function as defined here satisfies the Unrestricted domain condition. Restricting the range to the social preferences that are never indifferent between distinct outcomes is probably a very restrictive assumption, but the goal here is to give a simple statement of the theorem. Even if the restriction is relaxed, the impossibility result will persist.
9. ^Cockrell, Jeff (2016-03-08). "What economists think about voting". Capital Ideas. Chicago Booth. Archived from the original on 2016-03-26. Retrieved 2016-09-05.
10. ^This does not mean various normative criteria will be satisfied if we use equilibrium concepts in game theory. Indeed, the mapping from profiles to equilibrium outcomes defines a social choice rule, whose performance can be investigated by social choice theory. See Austen-Smith & Banks (1999) Section 7.2.
11. ^ abKirman, A.; Sondermann, D. (1972). "Arrow's theorem, many agents, and invisible dictators". Journal of Economic Theory. 5: 267–277. doi:10.1016/0022-0531(72)90106-8.
12. ^Mihara, H. R. (1997). "Arrow's Theorem and Turing computability"(PDF). Economic Theory. 10 (2): 257–276. doi:10.1007/s001990050157. JSTOR 25055038. Archived from the original(PDF) on 2011-08-12. Reprinted in Velupillai, K. V.; Zambelli, S.; Kinsella, S., eds. (2011). Computable Economics. International Library of Critical Writings in Economics. Edward Elgar. ISBN 978-1-84376-239-3.
13. ^Mihara, H. R. (1999). "Arrow's theorem, countably many agents, and more visible invisible dictators". Journal of Mathematical Economics. 32: 267–277. doi:10.1016/S0304-4068(98)00061-5.
14. ^Mihara's definition of a computable aggregation rule is based on computability of a simple game (see Rice's theorem).
15. ^See Taylor (2005, Chapter 6) for a concise discussion of social choice for infinite societies.
16. ^Austen-Smith & Banks (1999, Chapter 3) gives a detailed discussion of the approach trying to limit the number of alternatives.
17. ^Austen-Smith, David; Banks, Jeffrey S. (1999). Positive political theory I: Collective preference. Ann Arbor: University of Michigan Press. ISBN 978-0-472-08721-1. Retrieved 2016-02-16.
18. ^Indeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proved, however, that under any such restriction, if there exists any social welfare function that adheres to Arrow's criteria, then the majority rule will adhere to Arrow's criteria. See Campbell, D. E.; Kelly, J. S. (2000). "A simple characterization of majority rule". Economic Theory. 15 (3): 689–700. doi:10.1007/s001990050318. JSTOR 25055296.
19. ^Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 0-89838-189-4.
20. ^McKelvey, Richard D. (1976). "Intransitivities in multidimensional voting models and some implications for agenda control". Journal of Economic Theory. 12 (3): 472–482. doi:10.1016/0022-0531(76)90040-5.
21. ^Gibbard, Allan F. (2014) [1969]. "Intransitive social indifference and the Arrow dilemma". Review of Economic Design. 18 (1): 3–10. doi:10.1007/s10058-014-0158-1.
22. ^Brown, D. J. (1975). "Aggregation of Preferences". Quarterly Journal of Economics. 89 (3): 456–469. doi:10.2307/1885263. JSTOR 1885263.
23. ^Blair, Douglas; Muller, Eitan (1983). "Essential aggregation procedures on restricted domains of preferences". Journal of Economic Theory. 30 (1): 34–53. doi:10.1016/0022-0531(83)90092-3.
24. ^
Part one: Successively move B from the bottom to the top of voters' ballots. The voter whose change results in B being ranked over A is the pivotal voter forBoverA.
Part two: Switching A and B on the ballot of voter k causes the same switch to the societal outcome, by part one of the argument. Making any or all of the indicated switches to the other ballots has no effect on the outcome.
Part three: Since voter k is the dictator for B over C, the pivotal voter for B over C must appear among the first k voters. That is, outside of segment two. Likewise, the pivotal voter for C over B must appear among voters k through N. That is, outside of Segment One.