 Arrow's impossibility theorem

In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives (options), no voting system can convert the ranked preferences of individuals into a communitywide (complete and transitive) ranking while also meeting a certain set of criteria. These criteria are called unrestricted domain, nondictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of election 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 Ph.D. 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".^{[1]} Arrow was a corecipient of the 1972 Nobel Memorial Prize in Economics.
In short, the theorem proves that no voting system can be designed that 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.
There are several voting systems that sidestep these requirements by using cardinal utility (which conveys more information than rank orders) and weakening the notion of independence (see the subsection discussing the cardinal utility approach to overcoming the negative conclusion). Arrow, like many economists, rejected cardinal utility as a meaningful tool for expressing social welfare, and so focused his theorem on preference rankings.
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.^{[3]}
Contents
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 theory, 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 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 preferential voting 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. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method:
 Nondictatorship
 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. (See Remarks below.)
 Positive association of social and individual values
 (or monotonicity) 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.
 Nonimposition
 (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.
Arrow's theorem says that if the decisionmaking 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.
A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and nonimposition 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.
The later version of this theorem is stronger—has weaker conditions—since monotonicity, nonimposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency, nonimposition, and independence of irrelevant alternatives together do not imply monotonicity.
Remarks on IIA
 The IIA condition can be justified for three reasons (MasColell, Whinston, and Green, 1995, page 794): (i) normative (irrelevant alternatives should not matter), (ii) practical (use of minimal information), and (iii) 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 deathofacandidate example (1963, page 26) 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) or (acb, bca) or (acb, cba) or (abc, cba).
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.^{[4]} The Ntuple of voters' preferences is called a preference profile. In its strongest and most simple 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 Pareto efficiency
 If alternative a is ranked above b for all orderings , then a is ranked higher than b by . (Note that unanimity implies nonimposition).
 nondictatorship
 There is no individual i whose preferences always prevail. That is, there is no such that .
 independence of irrelevant alternatives
 For two preference profiles and such that for all individuals i, alternatives a and b have the same order in R_{i} as in S_{i}, alternatives a and b have the same order in as in .
Informal proof
Based on the proof by John Geanakoplos of Cowles Foundation, Yale University.^{[5]}
We wish to prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA) is a dictatorship.
Part one: there is a "pivotal" voter for B
Say there are three choices for society, call them A, B, and C. Suppose first that everyone prefers option B the least. That is, everyone prefers every other option to B. By unanimity, society must prefer every option to B. Specifically, society prefers A and C to B. Call this situation Profile 1.
On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity. So it is clear that, if we take Profile 1 and, running through the members in the society in some arbitrary but specific order, move B from the bottom of each person's preference list to the top, there must be some point at which B moves off the bottom of society's preferences as well, since we know it eventually ends up at the top. When it happens, we call that voter as pivotal voter.
We now want to show that, at the point when the pivotal voter n moves B off the bottom of his preferences to the top, the society's B moves to the top of its preferences as well, not to an intermediate point.
To prove this, consider what would happen if it were not true. Then, after n has moved B to the top (i.e., when voters have B at the top and voters still have B at the bottom) society would have some option it prefers to B, say A, and one less preferable than B, say C.
Now if each person moves his preference for C above A, then society would prefer C to A by unanimity. But moving C above A should not change anything about how B and C compare, by independence of irrelevant alternatives. That is, since B is either at the very top or bottom of each person's preferences, moving C or A around does not change how either compares with B, leaving B preferred to C. Similarly, by independence of irrelevant alternatives society still prefers A to B because the changing of C and A does not affect how A and B compare. Since C is above A, and A is above B, C must be above B in the social preference ranking. We have reached an absurd conclusion.
Therefore, when the voters have moved B from the bottom of their preferences to the top, society moves B from the bottom all the way to the top, not some intermediate point.
Note that even with a different starting profile, say Profile 1' , if the order of moving preference of B is unchanged, the pivotal voter remains n. That is, the pivotal voter is determined only by the moving order, and not by the starting profile.
It can be seen as following. If we concentrate on a pair of B and one of other choices, during each step on the process, preferences in the pair are unchanged whether we start from Profile 1 and Profile 1' for every person. Therefore by IIA, preference in the pair should be unchanged. Since it applies to every other choices, for Profile 1' , the position of B remains at bottom before n and remains at top after and including n, just as Profile 1.
Part two: voter n is a dictator for A–C
We show that voter n dictates society's decision between A and C. In other words, we show that n is a (local) dictator over the set {A, C} in the following sense: if n prefers A to C, then the society prefers A to C and if n prefers C to A, then the society prefers C to A.
Let p1 be any profile in which voter n prefers A to C. We show that society prefers A to C. To show that, construct two profiles from p1 by changing the position of B as follows: In Profile 2, all voters up to (not including) n have B at the top of their preferences and the rest (including n) have B at the bottom. In Profile 3, all voters up to (and including) n have B at the top and the rest have B at the bottom.
Now consider the profile p4 obtained from p1 as follows: everyone up to n ranks B at the top, n ranks A above B above C, and everyone else ranks B at the bottom. As far as the A–B decision is concerned, p4 is just as in Profile 2, which we proved puts A above B (in Profile 2, B is actually at the bottom of the social ordering). C's new position is irrelevant to the B–A ordering for society because of IIA. Likewise, p4 has a relationship between B and C that is just as in Profile 3, which we proved has B above C (B is actually at the top). We can conclude from these two observations that society puts A above B above C at p4. Since the relative rankings of A and C are the same across p1 and p4, we conclude that society puts A above C at p1.
Similarly, we can show that if q1 is any profile in which voter n prefers C to A, then society prefers C to A. It follows that person n is a (local) dictator over {A, C}.
Remark. Since B is irrelevant (IIA) to the decision between A and C, the fact that we assumed particular profiles that put B in particular places does not matter. This was just a way of finding out, by example, who the dictator over A and C was. But all we need to know is that he exists.
Part three: there can be at most one dictator
Finally, we show that the (local) dictator over {A, C} is a (global) dictator: he also dictates over {A, B} and over {B, C}. We will use the fact (which can be proved easily) that if is a strict linear order, then it contains no cycles such as . We have proved in Part two that there are (local) dictators i over {A, B}, j over {B, C}, and k over {A, C}.
 If i, j, k are all distinct, consider any profile in which i prefers A to B, j prefers B to C and k prefers C to A. Then the society prefers A to B to C to A, a contradiction.
 If one of i, j, k is different and the other two are equal, assume i=j without loss of generality. Consider any profile in which i=j prefers A to B to C and k prefers C to A. Then the society prefers A to B to C to A, a contradiction.
It follows that i=j=k, establishing that the local dictator over {A, C} is a global one.
Interpretations of the theorem
Arrow's theorem is a mathematical result, but it is often expressed in a nonmathematical 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 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.
Arrow did use the term "fair" to refer to his criteria. Indeed, Pareto efficiency, as well as the demand for nonimposition, seems acceptable to most people.
Various theorists^{[who?]} 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 voting 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 rockpaperscissors 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 majoritywins voting system is a nontrivial game, and that game theory should be used to predict the outcome of most voting mechanisms.^{[7]} 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 decisionmaking 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, 1972^{[8]}) 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 nonconstructive mathematical objects. In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule. Mihara (1997,^{[9]} 1999^{[10]}) shows that such a rule violates algorithmic computability.^{[11]} These results can be seen to establish the robustness of Arrow's theorem.^{[13]}
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 supermajority 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.^{[14]}
Remark. 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 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. 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 bestknown 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 "singlepeaked" 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 singlepeaked 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 (^{[6]}) aggregation rules, which includes majority rule, have an acyclic (defined below) social preference, hence "best" alternatives.^{[16]} 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^{[17]}). Under singlepeaked preferences, the majority rule is in some respects the most natural voting mechanism.
One can define the notion of "singlepeaked" 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 (1976^{[18]}): for any x and y, one can find a sequence of alternatives such that x is beaten by x_{1} by a majority, x_{1} by x_{2}, , x_{k} 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 quasitransitive (instead of transitive); this means that the strict preference ("better than") is transitive: if and , then . Then, there do exist nondictatorial aggregation rules satisfying Arrow's conditions, but such rules are oligarchic (Gibbard, 1969). 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 does 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 (Brown, 1975^{[19]}). 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.^{[6]}
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 (Blair and Muller, 1983^{[20]}).
See also Interpretations of the theorem below.
Relaxing the Pareto criterion
Wilson (1972) shows that if an aggregation rule is nonimposed and nonnull, 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.
Remark. Amartya Sen offered both relaxation of transitivity and removal of the Pareto principle.^{[21]} 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 wellknown, which states that if a social choice function whose range contains at least three alternatives is strategyproof, 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 (2011^{[22]}) 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 voting systems 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) gives a justification of utilitarianism (which evaluates alternatives in terms of the sum of individual utilities), originating from Jeremy Bentham. Hammond (1976) gives a justification of the maximin principle (which evaluates alternatives in terms of the utility of the worstoff individual), originating from John Rawls.
Not all voting methods use, as input, only an ordering of all candidates.^{[23]} Methods which don't, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") voting systems, 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.^{[24]} Warren Smith claims that range voting is such a method.^{[25]}^{[26]} Whether such a claim is correct depends on how each condition is reformulated.^{[28]} Other rated voting systems which pass certain generalizations of Arrow's criteria include Approval voting and Majority Judgment. Note that although Arrow's theorem does not apply to such methods, the Gibbard–Satterthwaite theorem still does: no system is fully strategyfree, so the informal dictum that "no voting system is perfect" still has a mathematical basis.
Finally, though not an approach investigating some kind of rules, there is a criticism by James M. Buchanan and others. It argues that it is silly to think that there might be social preferences that are analogous to individual preferences. Arrow (1963, Chapter 8) answers this sort of criticisms seen in the early period, which come at least partly from misunderstanding.
See also
Notes
 ^ Arrow, K.J., "A Difficulty in the Concept of Social Welfare", Journal of Political Economy 58(4) (August, 1950), pp. 328–346.
 ^ ^{a} ^{b} Suzumura, Kōtarō; Arrow, Kenneth Joseph; Sen, Amartya Kumar (2002). Handbook of social choice and welfare, vol 1. Amsterdam, Netherlands: Elsevier. ISBN 9780444829146.
 ^ Suzumura, 2002,^{[2]} Introduction, page 10.
 ^ 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.
 ^ Three Brief Proofs of Arrow’s Impossibility Theorem
 ^ ^{a} ^{b} ^{c} ^{d} AustenSmith, David; Banks, Jeffrey S. (1999). Positive political theory I: Collective preference. Ann Arbor: University of Michigan Press. ISBN 9780472087211.
 ^ 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 AustenSmith and Banks (1999^{[6]}), Section 7.2.
 ^ Kirman, A.; Sondermann, D. (1972). "Arrow's theorem, many agents, and invisible dictators". Journal of Economic Theory 5: 267. doi:10.1016/00220531(72)901068.
 ^ Mihara, H. R. (1997). "Arrow's Theorem and Turing computability". Economic Theory 10 (2): 257–276. doi:10.1007/s001990050157. http://econpapers.repec.org/paper/wpawuwppe/9408001.htm.
 ^ 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/S03044068(98)000615. http://econpapers.repec.org/paper/wpawuwppe/9705001.htm.
 ^ Mihara's definition of a computable aggregation rule is based on computability of a simple game (see Rice's theorem).
 ^ Taylor, Alan D. (2005). Social choice and the mathematics of manipulation. New York: Cambridge University Press. ISBN 0521008832.
 ^ See Taylor (2005,^{[12]} Chapter 6) for a concise discussion of social choice for infinite societies.
 ^ AustenSmith and Banks (1999,^{[6]} Chapter 3) gives a detailed discussion of the approach trying to limit the number of alternatives.
 ^ Campbell, D.E., Kelly, J.S., "A simple characterization of majority rule", Economic Theory 15 (2000), pp. 689–700.
 ^ 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.^{[15]}
 ^ Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 0898381894.
 ^ McKelvey, R. (1976). "Intransitivities in multidimensional voting models and some implications for agenda control". Journal of Economic Theory 12 (3): 472–482. doi:10.1016/00220531(76)900405.
 ^ Brown, D. J. (1975). Aggregation of Preferences. Quarterly Journal of Economics 89: 456469.
 ^ Blair, D. (1983). "Essential aggregation procedures on restricted domains of preferences*1". Journal of Economic Theory 30: 34–00. doi:10.1016/00220531(83)900923.
 ^ Sen, Amartya. 1979. Personal Utilities and Public Judgements: Or What's Wrong With Welfare Economics. The Economic Journal, 89, 537588.
 ^ Kumabe, M.; Mihara, H. R. (2011). "Preference aggregation theory without acyclicity: the core without majority dissatisfaction". Games and Economic Behavior 72: 187–201. doi:10.1016/j.geb.2010.06.008. http://econpapers.repec.org/paper/pramprapa/11728.htm.
 ^ It is sometimes asserted that such methods may trivially fail the universality criterion. However, it is more appropriate to consider that such methods fails Arrow's definition of an aggregation rule (or that of a function whose domain consists of preference profiles), if preference orderings cannot uniquely translate into a ballot.
 ^ However, a modified version of Arrow's theorem may still apply to such methods (e.g., Brams and Fishburn, 2002,^{[2]} Chapter 4, Theorem 4.2).
 ^ Warren D. Smith, et al.. "How can range voting accomplish the impossible?". http://rangevoting.org/ArrowThm.html.
 ^ New Scientist 12 April 2008 pages 3033
 ^ Sen, Amartya Kumar (1979). Collective choice and social welfare. Amsterdam: NorthHolland. ISBN 9780444851277.
 ^ No voting method that nontrivially uses cardinal utility satisfies Arrow's IIA (in which preference profiles are replaced by lists of ballots or lists of utilities). For this reason, a weakened notion of IIA is proposed (e.g., Sen, 1979,^{[27]} page 129). The notion requires that the social ranking of two alternatives depend only on the levels of utility attained by individuals at the two alternatives. (More formally, a social welfare functional F is a function that maps each list of utility functions into a social preference. F satisfies IIA (for social welfare functionals) if for all lists u, u' and for all alternatives x, y, if u_{i}(x) = u'_{i}(x) and u_{i}(y) = u'_{i}(y) for all i, then xF(u)y iff xF(u')y.) Many cardinal voting methods (including Range voting) satisfy the weakened version of IIA.
References
 Campbell, D.E. and Kelly, J.S. (2002) Impossibility theorems in the Arrovian framework, in Handbook of social choice and welfare (ed. by Kenneth J. Arrow, Amartya K. Sen and Kotaro Suzumura), volume 1, pages 35–94, Elsevier. Surveys many of approaches discussed in #Approaches investigating functions of preference profiles.
 The Mathematics of Behavior by Earl Hunt, Cambridge University Press, 2007. The chapter "Defining Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof. URL to CUP information on this book
 Why flip a coin? : the art and science of good decisions by Harold W. Lewis, John Wiley, 1997. Gives explicit examples of preference rankings and apparently anomalous results under different voting systems. States but does not prove Arrow's theorem. ISBN 0471296457
 Sen, A. K. (1979) “Personal utilities and public judgements: or what's wrong with welfare economics?” The Economic Journal, 89, 537558, arguing that Arrow's theorem was wrong because it did not incorporate nonutility information and the utility information it did allow was impoverished http://www.jstor.org/stable/2231867
External links
 Three Brief Proofs of Arrow’s Impossibility Theorem
 A Pedagogical Proof of Arrow’s Impossibility Theorem
 Another Graphical Proof of Arrow’s Impossibility Theorem
Categories: Voting systems
 Voting theory
 Economics theorems
 Economics paradoxes
 Theorems in discrete mathematics
 Paradoxes
 Social choice theory
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