Reasons Why Arrow Is Wrong
1) Arrow doesn’t allow ties. Example: For alternatives a, b and c, the
only acceptable social choice for Arrow is one of the following: abc, acb,
bac, bca, cab, cba. A tie would be of the form {abc, bca} where the
parentheses indicate that the two social rankings abc and bca are tied.
2) Arrow assumes that a tie is the same as an indifference. Let’s
assume 2 alternatives: x and y. Also that half of the individuals in society
prefer x to y – xPiy for half – and half the individuals prefer y
to x – yPix. Pi represents the preference ordering of
the ith individual. Arrow would say that society is indifferent
between the 2 alternatives – xIy. However, society is not indifferent
between the 2 alternatives. Society is evenly divided between the 2
alternatives. Half prefer x to y and, let us assume, very passionately. Half
prefer y to x and, let us assume, equally as passionately. Society would be
indifferent if every individual was indifferent between the 2 alternatives.
3) According to Arrow the only information an individual may specify
is pairwise binary comparisons. Any other information is irrelevant.
However, there is no reason why an individual shouldn’t specify as much
information as possible. For instance, if x, y and z are the alternatives,
Arrow would say that the only relevant information is in comparing x to y, x
to z and y to z. It is easy to imagine a grid with values from 1 to 100, for
example. An individual could place x, y and z on this grid in any position.
This would convey more information than binary comparisons Why shouldn’t
each individual have unlimited freedom of expression?
4) There is no
such thing as an irrelevant alternative. The number of alternatives
determines the underlying grid. If that grid changes due to the death of one
of the candidates, for instance, the individual’s preferences, if he is
required to specify them within a grid determined by the number of
candidates, may change. Preferences may become indifferences and vice versa.
See my paper, "Social
Choice, Information Theory and the Borda Count". See Figures
for "Social Choice, Information Theory and the Borda Count."
5) Since there is no such thing as an irrelevant alternative, one of
Arrow’s “rational and ethical” criteria is invalidated. Therefore, his
entire analysis is invalidated.
6) We don’t know how an individual
would vote if one candidate died, for example. If the number of candidates
changes, the individuals must be repolled or else some probabilistic
assumptions would have to be made about how they would have voted. You can’t
just assume that because an individual voted aPbIc, if a dropped out, the
individual would still be indifferent between b and c. For example, let us
assume that there are 3 candidates, a, b, c and Hitler. A voter might vote
aIbIcPHitler with the rationale that any candidate would be preferred to
Hitler. Then, if Hitler dropped out, true preferences among a, b and c might
emerge.
7) An individual shouldn’t be prevented or constrained from
using his vote in a strategic way. In any rational voting system there is
what Arrow calls the “Positive Association of Social and Individual Values.”
Therefore, an individual will express preferences based on the candidate set
in order to prevent a particular individual from being elected or to help
guarantee that another individual will be elected if he feels that strongly
about some particular candidate. The individual shouldn’t be required, as
Arrow does, to vote consistently when the candidate set changes. For
example, let’s assume that the candidate set consists of candidates a, b, c
and Jesus. It would be an entirely rational vote to specify (Jesus)PaIbIc.
Compared to Jesus, the individual is indifferent among a, b and c. Now let
us suppose that Jesus drops out of the race. Then it would be entirely
rational for the individual to vote aPbPc. In other words, the voter voted
strategically to get Jesus elected and should be allowed to do so. If Jesus
is not in the race, the individual’s true preferences among a, b and c
emerge. Arrow’s condition, “Independence of Irrelevant Alternatives” is not
rational after all.
8) “Adopting an informational perspective, then,
[Arrow’s theorem] just state[s] that procedures for three or more candidates
require more information than just the relative rankings of pairs.” Saari,
DG, (1995), Basic Geometry of Voting, Springer-Verlag, Berlin.
9)
Rankings have been considered to be cardinal or ordinal where ordinal
represents a simple ranking and cardinal allows more information. Cardinal
rankings supposedly allow “preference intensity” to be represented. It’s not
about preference intensity; it’s about freedom of expression. Let’s add a
third type of ranking: digital. An ordinal comparison for 3 candidates can
be specified by 3 bits since there are 6 possibilities. If we allow more
bits, then more information and relevant information can be gleaned from
each individual. Allowing each individual to specify his or her preferences
using the same number of bits eliminates the “interpersonal comparisons of
utility,” another of Arrow’s bugaboos. There is no preference given to one
individual over another because of supposed greater need. Therefore,
allowing more than just ordinal information is just as impersonal as
allowing only ordinal information.
10) Arrow constrains freedom of
expression.
11) Arrow confuses ties and indifference in the binary
case in which there are 2 alternatives and n voters. See my paper, "Neutrality
and the Possibility of Social Choice" He says majority rule when
there are only 2 alternatives is the only case where social choice actually
works. But according to his analysis, if done correctly, social choice isn’t
even possible with 2 alternatives. The key point is that when the number of
voters who prefer x to y, N(x,y), equals the number of voters who prefer y
to x, N(y,x), you have a tie between the solutions xPy and yPx which I
indicate {xPy, yPx}. This is not the same as xIy, x is indifferent to y.
13) In the binary
case, Arrow assumes that a tie in the domain of individual votes implies a
social indifference. The domain consists of all possible combinations of
votes by the individual voters. The range consists of all the possible
choices by society as a whole i.e. social choices.
14)
Arrow states (p. 12, 13 of “Social Choice and Individual Values”): “A strong
ordering…is a ranking in which no ties are possible.” WRONG! If n/2 voters
prefer y to x and n/2 voters prefer x to y (n being even), this clearly is a
tie! This section clearly shows Arrow’s confusion between the concept of a
tie and the concept of indifference. He thinks that both xRy and yRx imply a
tie. Wrong again. They imply an indifference.
15) Arrow’s R
notation. On p. 12 of “Social Choice and Individual Values,” Arrow states:
“Preference and indifference are relations between alternatives. Instead of
working with two relations, it will be slightly more convenient to use a
single relation, ‘preferred or indifferent.’ The statement ‘x is preferred
or indifferent to y’ will be symbolized by xRy.” The problem is Arrow's R notation is confusing. He means for the voter to vote xPiy, yPi or xIiy, and for the analysis to be in terms of xRiy. You have to know the truth values of both xRiy and yRix in order to have a one to one relationship with the voter's actual vote. This is all very confusing. He never makes this clear. So although it may be slightly more convenient from an analytical point of view, it's vastly more confusing than if he had just done the entire analysis in terms of P and I.
16) Definition 9 (p.
46) The case of two alternatives. “By the method of majority decision is
meant the social welfare function in which xRy holds if and only if the
number of individuals such that xRiy is at least as great as the
number of individuals such that yRix.” This is totally
ridiculous. First of all it violates one of Arrow’s five “rational and
ethical principles” which all social welfare functions must comply with: the
principle of neutrality. When the number of individuals such that
xRiy equals the number of individuals such that yRix,
why is the solution xRy? Why not yRx which is equally as valid? In fact it
is a tie between xRy and yRx, or according to Arrow’s own terminology, when
xRy and yRx, then xIy not xRy! But wait, there is more. If half the
individuals prefer x to y and half prefer y to x, we have a tie between x
and y: {xPy, yPx}. If all the individuals are indifferent between x and y,
we have a societal indifference: xIy. These are not the same thing! If more
prefer x to y than prefer y to x, we have xPy and vice versa. If some
individuals are indifferent between x and y, but more prefer x to y than
prefer y to x, we have xPy and vice versa. This pretty well covers all the
cases. Arrow is determined to ignore the significance of a tie and to turn a
societal indifference into a tie. See "Arrow's
Consideration of Ties and Indifference" Apologists for Arrow will say that what he meant to say was "for all values of x' and y', by the method of majority decision is
meant the social welfare function in which x'Ry' holds if and only if the
number of individuals such that x'Riy' is at least as great as the
number of individuals such that y'Rix', where x' and y' are variables that take on the values x and y" This does give a certain symmetry to the definition while making it even more confusing!
17) Arrow defines an
indifference as a tie.
18) Arrow's Axiom 2 p. 13 "Social Choice and Individual Values" which states "For all x, y and z, xRy and yRz imply xRz." Arrow confuses the underlying relationships involving P and I which are:
For all x, y and z,
(a) xPy and yPz imply xPz
(b) xIy and yIz imply xIz
(c) xPy and yIz imply xPz
(d) xIy and yPz imply xPz
Let's consider all the cases:
1) {xRy=T;yRx=F}AND{yRz=T;zRy=F} imply {xRz=T;zRx=F}
This is a true statement. However, the next statement is false:
2) {xRy=T;yRx=F}AND{yRz=T;zRy=F} imply {xRz=T;zRx=T}
(2) is also a possibility for the statement xRy and yRz implies xRz.
Therefore, the statement xRy and yRz implies xRz is not true if we want the underlying statements as given above to hold. Now let's consider
3) {xRy=T;yRx=T}AND{yRz=T;zRy=T} imply {xRz=T;zRx=T}
This corresponds to (b) above and is a true statement. Now consider
4) {xRy=T;yRx=T}AND{yRz=T;zRy=T} imply {xRz=T;zRx=F}
This is not a true statement compared to the underlying statements in terms of P and I. It does correspond to the statement {xRy and yRz imply xRz}. Therefore that statement is false.
5) {xRy=T;yRx=F}AND{yRz=T;zRy=T} imply {xRz=T;zRx=F}
This statement is true since it corresponds to (c) above.
6) {xRy=T;yRx=F}AND{yRz=T;zRy=T} imply {xRz=T;zRx=T}
This statement is false.
7) {xRy=T;yRx=T}AND{yRz=T;zRy=F} imply {xRz=T;zRx=F}
This statement is true.
8) {xRy=T;yRx=T}AND{yRz=T;zRy=F} imply {xRz=T;zRx=T}
This statement is false.
Therefore, Arrow's Axiom 2 is incorrect since it doesn't correspond to the underlying situation with P and I.
Arrow's basic mistake is in not deriving R from P and I. Instead he derives P and I from R. He states: It might be felt that the two axioms in question do not completely characterize the concept of a preference pattern. [They do not.] For example, we ordinarily feel that not only the relation R but also the relations of (strict) preference and of indifference are transitive. We shall show that, by defining preference and indifference suitably in terms of R, it will follow that all the usually desired properties of preference patterns obtain."
Here Arrow is putting the cart before the horse. If the voters submit their votes in terms of P and I, then P and I are the primary relationships and R should be derived from them not the other way around.
See "Arrow's
Consideration of Ties and Indifference"
19) Page 14 "Social Choice and Individual Values"
In Definition 1, Arrow defines the P relationship in terms of the primary relationship R as follows:
Definition 1: xPy is defined to mean not yRx.
I is defined as follows:
Definition 2: xIy means xRy and yRx.
Also Arrow states the following:
Lemma 1(e): For all x and y, either xRy or yPx.
However, lemma 1(e) and axiom 1, which is restated here, are in conflict:
Axiom 1: For all x and y, either xRy or yRx.
In fact lemma 1(e), axiom 1 and definition 1 cannot all be true. One cannot have lemma 1(e) and axiom 1 both true unless yRx = yPx. The problem has to do with definition 1. According to axiom 1, one of the following must be true: xRy, yRx, xRyANDyRx. Therefore, if yRx is not true (NOT yRx is true), then xRy must be true — not xPy as Arrow states in lemma 1(e). Definition 1 overrides the implications of axiom 1. However, perhaps definition 1 can be salvaged by writing it as follows:
Definition 1': xPy is defined to mean xRy and not yRx.
or, alternatively,
Definition 1'': xPy is defined to mean xRy and not xRyANDyRx.
20) Arrow states on pp. 25-26: "The condition that x be not lower on the Ri' scale than x was on the Ri scale means that x is preferred on the Ri' scale to any alternative to which it was preferred on the old (Ri) scale and also that x is preferred or indifferent to any alternative to which it was formerly indifferent. The two conditions of the last sentence, taken together, are equivalent to the following two conditions: (1) x is preferred on the new scale to any alternative to which it was formerly preferred; (2) x is preferred or indifferent on the new scale to any alternative to which it was formerly preferred or indifferent."
This paragraph contradicts itself. Clearly, the statements "x is preferred or indifferent to any alternative to which it was formerly indifferent." and "x is preferred or indifferent on the new scale to any alternative to which it was formerly preferred or indifferent." are contradictory.
21) Arrow's Definition 6 is incorrect:
"Definition 6: A social welfare function is said to be dictatorial if there exists an individual i such that, for all x and y, xPiy implies xPy regardless of the orderings R1, ..., Rn of all individuals other than i, where P is the social preference relation corresponding to R1, ..., Rn."
In particular, "xPiy implies xPy" should read "xPiy if and only if xPy". If he's a dictator, xPy not only when xPiy, but also when yPix.
22) Arrow's First Condition is wrong:
"Condition 1: Among all the alternatives there is a set S of three alternatives such that, for any set of individual orderings T1, ..., Tn of the alternatives in S, there is an admissible set of individual orderings R1, ..., Rn of all the alternatives such that, for each individual i, xRiy if and only if xTiy for x and y in S."
Clearly, if xPiy, in the T orderings, we must have xPiy in the R orderings. But this is not necessarily the case as it is stated in Condition 1 since the condition xRiy if and only if xTiy will hold if xRiy is xPiy and xTiy is xIiy since xRiy is defined as xPiy EOR xIiy and xTiy is defined as xPiy EOR xIiy. Therefore, the condition as stated is incorrect.
23) Arrow's Condition 2 is stated wrong:
"Condition 2: Let R1, ..., Rn and R'1, ..., R'n be two sets of individual ordering relations, R and R' the corresponding social orderings, and P and P' the corresponding social preference relations. Suppose that for each i the two individual ordering relations are connected in the following ways: for x' and y' distinct from a given alternative x, x'Ri'y' if and only if x'Riy'; for all y', xRiy' implies xRi'y'; for all y', xPi y' implies xPi'y'. Then, if xPy, xP'y."
"for x' and y' distinct from a given alternative x, x'Ri'y' if and only if x'Riy'" would be true if x'Ri'y'= x'Pi'y'and x'Riy'= x'Iiy'. Clearly, this is not what Arrow intends. Also, Arrow doesn’t cover the case, xIy. If x rises or remains the same in the judgment of each individual, then in the new state x would either rise to a preference or remain as an indifference. If xIiy, then xRi'y.
See "Arrow's Consideration of Ties and Indifference"
24) Arrow (Social Choice and Individual Values, p. 13-14) claims to treat ties. He asserts: “...Axioms I and II do not exclude the possibility that for some distinct [alternatives] x and y, both xRy and yRx. A strong ordering, on the other hand, is a ranking in which no ties are possible.” This is not correct. Clearly, ties are possible for the strong ordering P. If half the voters prefer x to y and half prefer y to x, there is a tie between xPy and yPx.
25) Regarding the General Possibility Theorem for two alternatives, on p. 46 of "Social Choice and Individual Values," Arrow states: “Clearly, always either N(x,y) ≥ N(y,x) or N(y,x) ≥ N(x,y), so that, for all x and y, xRy or yRx.” This is an incorrect statement. One could say correctly that ‘either N(x,y) ≥ N(y,x) or N(y,x) > N(x,y)’; or ‘either N(x,y) > N(y,x) or N(y,x) ≥ N(x,y)’; or ‘either N(x,y) > N(y,x) or N(y,x) > N(x,y) or N(y,x) = N(x,y).’ The latter restatement then would suggest the conclusion that either xRy or yRx or {xRy, yRx} which would be consistent with Axiom 1. However, Arrow’s definition of majority rule would have to be changed to allow for the tie case. With these changes one could then go on to prove that a social ordering is indeed possible for the case of two alternatives, but not allowing the acceptance of the tie case leads to the conclusion that a social ordering is impossible for the binary case as well. It is also counterintuitive to reality! Again the apologists would say that the above statements should be in terms of x' and y' and prefaced with "for all values of x' and y'." x' and y' take on the values x and y. Confused yet?
26) Arrow's discussion on The Ordering of Social States starting on p. 17 talks about ordering with respect to tastes and ordering with respect to values. Here Arrow is discussing the economic implications of social choice. Ordering with respect to tastes takes place when an individual submits his preferences consisting of his own personal work/consumption baskets. Ordering with respect to values occurs when an individual submits his preferences consisting of not only his own personal work/consumption preferences but those he prefers for everybody else! Arrow goes on to imply that his analysis will be in terms of values not tastes. First of all, any practical system would be overwhelmed if everyone submitted preferences not only for himself but for everyone else too! This is ridiculous on the face of it! Second, Arrow's model is invalidated if individuals submit preferences in terms of tastes since they would not be then ordering social states but only part of a social state insofar as their participation in it is concerned while the social ordering would be in terms of entire social states. Arrow's model insists that individual orderings and social orderings must consist of the same alternatives. That's probably why he insists on ordering according to values. His whole analysis doesn't pertain to practical economic systems. Therefore, his impossibility result does not pertain to any social choice economic systems where ordering is according to individuals' submissions of their personal economic preferences and not everyone else's.
27) Arrow's model doesn't correspond to practical political or voting systems where the intent is to elect a winner and not an ordering of winners. For instance, the election of a President does not require an ordering of all the candidates from top to bottom, only the selection of the winner. Therefore, Arrow's model, his analysis and impossibility result don't apply. Neither would they apply to the election of a Congress because here again the intent would be the election of a winning Congress not the ordering of all possible Congresses. Also, individual voters couldn't be expected to order all potential Congresses, only to vote for the individuals that they would prefer to see in the final Congress. Again Arrow's model doesn't apply!
28) Arrow's model doesn't apply to any economic system in which the goal is to choose the best economic situation rather than an ordering of economic situations. It does not even apply to judging Olympic figure skating where you do seek an ordered ranking of the contestants from first to last. However, Arrow's Condition 3, Independence of Irrelevant Alternatives, does not apply since only the contestants that actually show up and compete are judged. The judging is independent of any candidate who drops out for any reason before the competition takes place.
29) See Arrow's Example Doesn't Apply to Economic Case on January 19, 2009 entry of Social Choice Blog. On page 27 of "Social Choice and Individual Values," Arrow gives an example which violates his Condition 3, Independence of Irrelevant Alternatives. However, a corresponding economic example shows this condition to be irrelevant in the economic case.
30) More economics examples.