Arrow's impossibility theorem
Encyclopedia
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 community-wide (complete and transitive) ranking while also meeting a certain set of criteria. These criteria are called unrestricted domain
, non-dictatorship
, 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". Arrow was a co-recipient 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:
There are several voting systems that side-step 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.
s 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:
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:
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.
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.
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.
A later (1963) version of Arrow's theorem can be obtained by replacing 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.
The later version of this 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.
Remarks on IIA
be a set of outcomes, 
a number of voters or decision criteria. We shall denote the set of all full linear orderings
of
by 
.
A (strict) social welfare function (preference aggregation rule) is a function
which aggregates voters' preferences into a single preference order on
.
The
-tuple
of voters' preferences is called a preference profile. In its strongest and most simple form, Arrow's impossibility theorem states that whenever the set 
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 non-imposition).
non-dictatorship: 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 
as in 
, alternatives a and b have the same order in 
as in 
.
of Cowles Foundation
, Yale University
.
We wish to prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA) is a dictatorship.
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.
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.
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}.
It follows that i=j=k, establishing that the local dictator over {A, C} is a global one.
Arrow did use the term "fair" to refer to his criteria. Indeed, Pareto efficiency
, as well as the demand for non-imposition, seems acceptable to most people.
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 voting system
s.
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:
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 voting system 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.
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).
These investigations can be divided into the following two:
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.
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 ultrafilter
s, highly nonconstructive mathematical objects.
In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule.
Mihara (1997, 1999)
shows that such a rule violates algorithmic computability.
These results can be seen to establish the robustness of Arrow'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.
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.
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 aggregation rules, which includes majority rule, have an acyclic (defined below) social preference,
hence "best" alternatives.
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).
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
(1976):
for any x and y, one can find a sequence of alternatives such that
x is beaten by
by a majority, 
by 
, 
,
by y.
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 (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).
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.
The Borda rule
is one of them.
These rules, however, are susceptible to strategic manipulation by individuals
(Blair and Muller, 1983).
See also Interpretations of the theorem below.
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.
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.
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 (2011) 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.
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 worst-off individual), originating from John Rawls
.
Not all voting methods use, as input, only an ordering of all candidates.
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.
Warren Smith claims that range voting
is such a method.
Whether such a claim is correct depends on how each condition is reformulated. 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 strategy-free, 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.
Social choice theory
Social choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
, 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 community-wide (complete and transitive) ranking while also meeting a certain set of criteria. These criteria are called unrestricted domain
Unrestricted domain
In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters are allowed...
, non-dictatorship
Non-dictatorship
In voting theory, non-dictatorship is a property of social choice functions, where the results cannot simply mirror that of any ONE single person's preferences without consideration of the other voters. Fairness requires that the social welfare function take into account the desires of more than...
, Pareto efficiency
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
, and independence of irrelevant alternatives
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
. 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
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
, who demonstrated the theorem in his Ph.D. thesis and popularized it in his 1951 book Social Choice and Individual Values
Social Choice and Individual Values
Kenneth Arrow's monograph Social Choice and Individual Values and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor...
. The original paper was titled "A Difficulty in the Concept of Social Welfare". Arrow was a co-recipient 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 side-step these requirements by using cardinal utility
Cardinal utility
In economics, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations...
(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.
Statement of the theorem
The need to aggregate preferencePreference
-Definitions in different disciplines:The term “preferences” is used in a variety of related, but not identical, ways in the scientific literature. This makes it necessary to make explicit the sense in which the term is used in different social sciences....
s occurs in many different disciplines: in welfare economics
Welfare economics
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
, where one attempts to find an economic outcome which would be acceptable and stable; in decision theory
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
, 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
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...
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:
Non-dictatorship
Non-dictatorship
In voting theory, non-dictatorship is a property of social choice functions, where the results cannot simply mirror that of any ONE single person's preferences without consideration of the other voters. Fairness requires that the social welfare function take into account the desires of more than...
: The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter.
Unrestricted domain
Unrestricted domain
In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters are allowed...
: (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 deterministicallyDeterministic system (mathematics)In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...
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
Monotonicity criterion
The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below.Douglas R...
) 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
Surjective function
In mathematics, a function f from a set X to a set Y is surjective , or a surjection, if every element y in Y has a corresponding element x in X so that f = y...
: It has an unrestricted target space.
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.
A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria with:
Pareto efficiency
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
: (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, 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.
Remarks on IIA
- The IIA condition can be justified for three reasons (Mas-ColellAndreu Mas-ColellAndreu Mas-Colell is a Spanish economist, an expert in microeconomics and one of the world's leading mathematical economists. He is the founder of the Barcelona Graduate School of Economics and a professor in the department of economics at Pompeu Fabra University in Barcelona, Catalonia, Spain...
, 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 death-of-a-candidate 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 be a set of outcomes, a number of voters or decision criteria. We shall denote the set of all full linear orderingsTotal order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
of
by .A (strict) social welfare function (preference aggregation rule) is a function
which aggregates voters' preferences into a single preference order on
.The
-tupleTuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
of voters' preferences is called a preference profile. In its strongest and most simple form, Arrow's impossibility theorem states that whenever the set 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 non-imposition).non-dictatorship: 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 as in , alternatives a and b have the same order in as in .Informal proof
Based on the proof by John GeanakoplosJohn Geanakoplos
John Geanakoplos is the James Tobin Professor of Economics at Yale University. Before the Late-2000s financial crisis, he was known primarily for his contributions to General equilibrium theory, particularly Incomplete markets general equilibrium theory...
of Cowles Foundation
Cowles Foundation
The Cowles Commission for Research in Economics is an economic research institute, founded in Colorado Springs in 1932 by Alfred Cowles, a businessman and economist. In 1939, the Cowles Commission moved to the University of Chicago under the directorship of Theodore O. Yntema. Jacob Marschak took...
, Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
.
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 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 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
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
, as well as the demand for non-imposition, seems acceptable to most people.
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 voting system
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
s.
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
Rock-paper-scissors
Rock-paper-scissors is a hand game played by two people. The game is also known as roshambo, or another ordering of the three items ....
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 voting system is a non-trivial game, and that game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
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.
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
Decathlon
The decathlon is a combined event in athletics consisting of ten track and field events. The word decathlon is of Greek origin . Events are held over two consecutive days and the winners are determined by the combined performance in all. Performance is judged on a points system in each event, not...
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) 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 ultrafilter
Ultrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
s, highly nonconstructive mathematical objects.
In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule.
Mihara (1997, 1999)
shows that such a rule violates algorithmic computability.
These results can be seen to establish the robustness of Arrow's theorem.
Limiting the number of alternatives
When there are only two alternatives to choose from, May's theoremMay's theorem
In social choice theory, May's theorem states that simple majority voting is the only anonymous, neutral, and monotone choice function between two alternatives. Further, this procedure is resolute when there are an odd number of voters and ties are not allowed. Kenneth May first published this...
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
Nakamura number
In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....
,
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.
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
Pareto principle
The Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes.Business-management consultant Joseph M...
, 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 domainDomain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
of aggregation rules.
The best-known result along this line assumes "single peaked" preferences.
Duncan Black
Duncan Black
Duncan Black was a Scottish economist who laid the foundations of social choice theory. In particular he was responsible for unearthing the work of many early political scientists, including Charles Dodgson, and was responsible for the Black electoral system, a Condorcet method whereby, in the...
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
Majority rule
Majority rule is a decision rule that selects alternatives which have a majority, that is, more than half the votes. It is the binary decision rule used most often in influential decision-making bodies, including the legislatures of democratic nations...
.
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 aggregation rules, which includes majority rule, have an acyclic (defined below) social preference,
hence "best" alternatives.
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).
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
(1976):
for any x and y, one can find a sequence of alternatives such that
x is beaten by
by a majority, by , , 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 (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).
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
Borda count
The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...
is one of them.
These rules, however, are susceptible to strategic manipulation by individuals
(Blair and Muller, 1983).
See also Interpretations of the theorem below.
Relaxing the Pareto criterion
Wilson (1972) 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.
Remark. Amartya Sen
Amartya Sen
Amartya Sen, CH is an Indian economist who was awarded the 1998 Nobel Prize in Economic Sciences for his contributions to welfare economics and social choice theory, and for his interest in the problems of society's poorest members...
offered both relaxation of transitivity and removal of the Pareto principle.
He demonstrated another interesting impossibility result, known as the "impossibility of the Paretian Liberal". (See liberal paradox
Liberal paradox
The liberal paradox is a logical paradox advanced by Amartya Sen, building on the work of Kenneth Arrow and his impossibility theorem, which showed that within a system of menu-independent social choice, it is impossible to have both a commitment to "Minimal Liberty", which was defined as the...
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 (2011) 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
Nakamura number
In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....
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
Intensity of preference
"Intensity of preference" is a term popularized by the work of the economist Kenneth Arrow, who was a co-recipient of the 1972 Nobel Memorial Prize in Economics...
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
Utilitarianism
Utilitarianism is an ethical theory holding that the proper course of action is the one that maximizes the overall "happiness", by whatever means necessary. It is thus a form of consequentialism, meaning that the moral worth of an action is determined only by its resulting outcome, and that one can...
(which evaluates alternatives in terms of the sum of individual utilities), originating from Jeremy Bentham
Jeremy Bentham
Jeremy Bentham was an English jurist, philosopher, and legal and social reformer. He became a leading theorist in Anglo-American philosophy of law, and a political radical whose ideas influenced the development of welfarism...
.
Hammond (1976) gives a justification of the maximin principle (which evaluates alternatives in terms of the utility of the worst-off individual), originating from John Rawls
John Rawls
John Bordley Rawls was an American philosopher and a leading figure in moral and political philosophy. He held the James Bryant Conant University Professorship at Harvard University....
.
Not all voting methods use, as input, only an ordering of all candidates.
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.
Warren Smith claims that range voting
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...
is such a method.
Whether such a claim is correct depends on how each condition is reformulated. Other rated voting systems which pass certain generalizations of Arrow's criteria include Approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...
and Majority Judgment
Majority Judgment
Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner. If more than one candidate has the...
. Note that although Arrow's theorem does not apply to such methods, the Gibbard–Satterthwaite theorem still does: no system is fully strategy-free, 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
James M. Buchanan
James McGill Buchanan, Jr. is an American economist known for his work on public choice theory, for which he received the 1986 Nobel Memorial Prize in Economic Sciences. Buchanan's work initiated research on how politicians' self-interest and non-economic forces affect government economic policy...
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.