Arrow's Impossibility Theorem, General Impossibilty Theorem, Arrow's paradox

8/7/2019 Arrow's Impossibility Theorem, General Impossibilty Theorem, Arrow's paradox

1/12

Arrow's impossibility theoremFrom Wikipedia, the free encyclopediaJump to: navigation , search

For Zeno's discussion of an arrow's flight in an attempted indirect proof of the impossibility andthus the illusory nature of motion, see Arrow paradox .

In social choice theory , Arrows impossibility theorem , the General Possibility Theorem , or Arrows paradox , states that, when voters have three or more discrete alternatives (options), novoting system can convert the ranked preferences of individuals into a community-wideranking 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 isoften 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 hisPh.D. thesis and popularized it in his 1951 book Social Choice and Individual Values . Theoriginal paper was titled "A Difficulty in the Concept of Social Welfare". [1] 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:

y If every voter prefers alternative X over alternative Y, then the group prefers X over Y.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).y There is no "dictator": no single voter possesses the power to always determine thegroup's preference.

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 negativeconclusion). 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 differentfrom the earlier one in voting theory, in which rules were investigated one by one. One cantherefore say that the contemporary paradigm of social choice theory started from this theorem. [3]

C ontents

[hide ]


2/12

y 1 Statement of the theorem y 2 Formal statement of the theorem y 3 Informal proof

o 3.1 Part one: there is a "pivotal" voter for B o 3.2 Part two: voter n is a dictator for AC o 3.3 Part three: there can be at most one dictator

y 4 Interpretations of the theorem y 5 Other possibilities

o 5.1 Approaches investigating functions of preference profiles 5.1.1 Infinitely many individuals 5.1.2 Limiting the number of alternatives 5.1.3 Domain restrictions 5.1. 4 Relaxing transitivity 5.1.5 Relaxing IIA 5.1.6 Relaxing the Pareto criterion

5.1.7 Social choice instead of social preference o 5.2 Approaches investigating other rules

y 6 See also y 7 Notes y 8 References y 9 External links

[edit ] 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; indecision theory , where a person has to make a rational choice based on several criteria; and mostnaturally 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 agiven set of options (outcomes). Each individual in the society (or equivalently, each decisioncriterion) 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 ), whichtransforms the set of preferences ( profile of preferences) into a single global societal preferenceorder. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method:

Non-dictatorshipThe social welfare function should account for the wishes of multiple voters. It cannotsimply mimic the preferences of a single voter.

Unrestricted domain (or universality ) For any set of individual voter preferences, the social welfare functionshould yield a unique and complete ranking of societal choices. Thus:


3/12

y It must do so in a manner that results in a complete ranking of preferences for society.

y It must deterministically provide the same ranking each time voters' preferencesare 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 impacton 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 acertain option, then the societal preference order should respond only by promoting thatsame option or not changing, never by placing it lower than before. An individual shouldnot be able to hurt an option by ranking it higher .

Non-imposition(or citizen sovereignty ) Every possible societal preference order should be achievable bysome set of individual preference orders. This means that the social welfare function issurjective : It has an unrestricted target space.

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

A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity andnon-imposition criteria with:

Pareto efficiency (or unanimity ) If every individual prefers a certain option to another, then so must theresulting societal preference order. This, again, is a demand that the social welfarefunction will be minimally sensitive to the preference profile.

The later version of this theorem is strongerhas weaker conditionssince monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereasPareto efficiency, non-imposition, and independence of irrelevant alternatives together do notimply monotonicity.

Remarks on IIA

1. The IIA condition can be justified for three reasons ( Mas-Colell , Whinston, and Green,1995, page 79 4 ): (i) normative (irrelevant alternatives should not matter), (ii) practical(use of minimal information), and (iii) strategic (providing the right incentives for thetruthful revelation of individual preferences). Though the strategic property isconceptually different from IIA, it is closely related.


4/12

2. Arrow's death-of-a-candidate example (1963, page 26) suggests that the agenda (the setof feasible alternatives) shrinks from, say, X = {a, b, c} to S = {a, b} because of the deathof a candidate c. This example is misleading since it can give the reader an impressionthat 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 aggregationrule 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).

[edit ] Formal statement of the theorem

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

A (strict) social welfare f u nction (preference aggregation r u le) is a function

which aggregates voters' preferences into a single preference order on

A. [4 ] The N- tuple of voter's preferences is called a preference profile . In itsstrongest 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 becomeincompatible:

u nanimity , 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 nosuch that

.independence of irrelevant alternatives

For two preference profiles and such that for allindividuals 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 .

[edit ] 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, andindependence of irrelevant alternatives ( IIA ) is a dictatorship.

[edit ] Part one: there is a "pivotal" voter for B


5/12

Say there are three choices for society, call them A , B , and C . Suppose first that everyone prefersoption 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 throughthe 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 anintermediate 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 atthe bottom) society would have some option it prefers to B, say A, and one less preferable thanB , say C .

Now if each person moves his preference for C above A, then society would prefer C to A byunanimity. But moving C above A should not change anything about how B and C compare, byindependence 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 tothe 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 preferenceof B is unchanged, the pivotal voter remains n. That is, the pivotal voter is determined only bythe 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, duringeach 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 beforen and remains at top after and including n, just as Profile 1 .

[edit ] Part two: voter n is a dictator for A C


6/12

We show that voter n dictates society's decision between A and C . In other words, we show thatn is a (local) dictator over the set { A , C } in the following sense: if n prefers A to C , then thesociety 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 through (and including) n have B at the top andthe rest have B at the bottom.

Now consider the profile p 4 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 AB decision isconcerned, p 4 is just as in Profile 2 , which we proved puts A above B (in Profile 2 , B is actuallyat the bottom of the social ordering). C ' s new position is irrelevant to the BA ordering for society because of IIA . Likewise, p 4 has a relationship between B and C that is just as in Profile3, 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 p 4 , 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 findingout, by example, who the dictator over A and C was. But all we need to know is that he exists.

[edit ] 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 strictlinear order, then it contains no cycles such as . We have proved in Parttwo that there are (local) dictators i over { A, B }, j over { B, C }, and k over { A, C }.

y If i, j, k are all distinct, consider any profile in which i prefers A to B, j prefers B to C andk prefers C to A. Then the society prefers A to B to C to A, a contradiction.

y 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.

[edit ] Interpretations of the theorem

Arrow's theorem is a mathematical result, but it is often expressed in a non-mathematical waywith a statement such as "N o voting method is fair " , "E very ranked voting method is flawed " , or "T he only voting method that isn't flawed is a dictatorship " . These statements are simplifications


7/12

of Arrow's result which are not universally considered to be true. What Arrow's theorem doesstate is that a voting mechanism, which is defined for all possible preference orders, cannotcomply 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 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. Itis the one breached in most useful voting systems . Advocates of this position point out thatfailure of the standard IIA criterion is trivially implied by the possibility of cyclic preferences. If voters cast ballots as follows:

y 1 vote for A > B > Cy 1 vote for B > C > Ay 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 Cwins over A: these yield rock-paper-scissors preferences for any pairwise comparison. In thiscircumstance, any aggregation rule that satisfies the very basic majoritarian requirement that acandidate 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 rulesatisfies 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 theassumption that social preference is transitive.

So, what Arrow's theorem really shows is that voting is a non-trivial game, and that game theory

should be used to predict the outcome of most voting mechanisms.[7]

This could be seen as adiscouraging result, because a game need not have efficient equilibria, e.g. , a ballot could resultin an alternative nobody really wanted in the first place, yet everybody voted for.

Remark: Scalar rankings from a vector of attrib u tes and the IIA property . The IIA propertymight not be satisfied in human decision-making of realistic complexity because the scalar

preference ranking is effectively derived from the weightingnot usually explicitof 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 evente.g. how does onemake scoring 600 points in the discus event "commensurable" with scoring 600 points in the1500 m race) and this scalar ranking can depend sensitively on the weighting of differentattributes, with the tacit weighting itself affected by the context and contrast created byapparently "irrelevant" choices. Edward MacNeal discusses this sensitivity problem with respectto the ranking of "most livable city" in the chapter "Surveys" of his book MathSemantics:making numbers talk sense (199 4 ).

[edit ] Other possibilities


8/12

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

y those investigating functions whose domain, like that of Arrow's social welfare functions,

consists of profiles of preferences;y those investigating other kinds of rules.

[edit ] Approaches investigating f u nctions of preference profiles

This section includes approaches that deal with

y aggregation rules (functions that map each preference profile into a social preference),and

y 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 ischaracteristic of these approaches is that they investigate various possibilities by eliminating or weakening or replacing one or more conditions (criteria) that Arrow imposed.

[edit ] Infinitely many individ u als

Several theorists (e.g., Kirman and Sondermann, 1972 [8 ]) point out that when one drops theassumption that there are only finitely many individuals, one can find aggregation rules thatsatisfy 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 Sondermannargue that there is an "invisible dictator" behind such a rule. Mihara (1997 [9], 1999 [10] ) shows thatsuch a rule violates algorithmic computability. [11] These results can be seen to establish therobustness of Arrow's theorem. [13]

[edit ] Limiting the n u mber of alternatives

When there are only two alternatives to choose from, May's theorem shows that only simplemajority 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). Onthe 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?

N akamura's theorem (about the core of simple games) gives an answer more generally. Itestablishes that if the number of alternatives is less than a certain integer called the Nakam u ran u mber , then the rule in question will identify "best" alternatives without any problem; if thenumber of alternatives is greater or equal to the Nakamura number, then the rule will not alwayswork, since for some profile a voting paradox (a cycle such as alternative A socially preferred to


9/12

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 majorityrule can deal with up to two alternatives rationally. Some super-majority rules (such as thoserequiring 2/3 of the votes) can have a Nakamura number greater than 3, but such rules violateother conditions given by Arrow. [14 ]

Remark . A common way "around" Arrow's paradox is limiting the alternative set to twoalternatives. Thus, whenever more than two alternatives should be put to the test, it seems verytempting to use a mechanism that pairs them and votes by pairs. As tempting as this mechanismseems at first glance, it is generally far from meeting even the Pareto principle, not to mentionIIA. The specific order by which the pairs are decided strongly influences the outcome. This isnot necessarily a bad feature of the mechanism. Many sports use the tournament mechanism essentially a pairing mechanismto choose a winner. This gives considerable opportunity for weaker teams to win, thus adding interest and tension throughout the tournament. This meansthat the person controlling the order by which the choices are paired (the agenda maker) hasgreat control over the outcome. In any case, when viewing the entire voting process as one game,

Arrow's theorem still applies.

[edit ] Domain restrictions

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

Duncan Black has shown that if there is only one dimension on which every individual has a"single-peaked" preference, then all of Arrow's conditions are met by majority rule . Suppose thatthere 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 fromthat spot (i.e., the graph of his utility function has a single peak if alternatives are placedaccording to the linear ordering on the horizontal axis). For example, if voters were voting onwhere 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 theywould be increasingly dissatisfied. If the domain is restricted to profiles in which everyindividual 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 themedian of all the peaks of the individuals (Black's median voter theorem [17] ). 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 [18 ]): for any x and y, one can find a sequence of alternatives such that x is beaten by x1 by amajority, x1 by x2, , xk by y.


10/12

[edit ] Relaxing transitivity

By relaxing the transitivity of social preferences, we can find aggregation rules that satisfyArrow'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 thatthe strict preference ("better than") is transitive: if and , then . Then,there do exist non-dictatorial aggregation rules satisfying Arrow's conditions, but such rules areoligarchic (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 aveto (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 existalternatives that form a cycle ( , , , ,

). Then, provided that there are at least as many alternatives as individuals, an aggregat ion rulesatisfying Arrow's other conditions is collegial (Brown, 1975 [19]). This means that there areindividuals who belong to the intersection ("collegium") of all decisive coalitions. If there issomeone 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 caseusing the N akamura number . See #Limiting the number of alternatives .

[edit ] Relaxing IIA

There are numerous examples of aggregat ion rules satisfying Arrow's conditions except IIA. TheBorda rule is one of them. These rules, however, are susceptible to strategic manipulation byindividuals (Blair and Muller, 19 8 3[20] ).

See also #Interpretations of the theorem .

[edit ] 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 thesociety 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 arguethat this demonstrates the futility of demanding Pareto optimality in relation to votingmechanisms.


11/12

[edit ] Social choice instead of social preference

In social decision making, to rank all alternatives is not usually a goal. It often suffices to findsome 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 statesthat 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, weshould 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 firstapproach 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 relatedto #Relaxing transitivity . The second approach drops the assumption of acyclic preferences.Kumabe and Mihara (2010 [22] ) adopt this approach. They make a more direct assumption thatindividual preferences have maximal elements, and examine conditions for the social preferenceto have a maximal element. See Nakamura number for details of these two approaches.

[edit ] Approaches investigating other r u les

Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfycompleteness and transitivity) on the set of alternatives. This means that if the preferences arerepresented 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, whichin turn is the same as having 99, 9 8 , 1, .997. They all represent the ordering in which a is

preferred to b to c to d. We can argue that the assumption of ordinal preferences, which precludes interpersonal comparisons of utility, is crucial for the impossibility that Arrow'stheorem establishes.

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 intoconsideration, 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 (whichevaluates alternatives in terms of the sum of individual utilities), originating from JeremyBentham . Hammond (1976) gives a justification of the maximin principle (which evaluatesalternatives 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. [23] One can view some of such methods as using information that only cardinal utility can convey. In that case, it is notsurprising if some of them satisfy all of Arrow's conditions that are reformulated. [24 ] Warren


12/12

Smith claims that range voting is such a method. [25][26] Whether such a claim is correct dependson how each condition is reformulated. [28 ]

Finally, though not an approach investigating some kind of rules, there is a criticism by JamesM. 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.

Documents

Arrow's Impossibility Theorem, General Impossibilty Theorem, Arrow's paradox