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Arrows Consideration of Ties and Indifference
by
John Clifton Lawrence
Web site: HYPERLINK "http://www.socialchoiceandbeyond.com" http://www.socialchoiceandbeyond.com
Blog: http://willblogforfood.com
Email: jlawrence@cox.net
( 1999 by John Clifton Lawrence
May 1, 1999
Revised: September 19, 2006
Abstract
In "Social Choice and Individual Values" Kenneth Arrow (1951) asserts that, in his model of social and individual choice, ties are considered. However, as this paper makes clear only ties among alternatives and not ties among preference orderings are taken into account although Arrow claims to consider ties between binary orderings, at least, in Axiom 1 using R, the preference or indifference relation. Since he defines indifference as the logical ANDing of the elements of a tie, Arrows treatment of ties is confined to the inclusion of indifference in a preference ordering. Moreover, Arrow clearly intends for the specification of individual and social preference orderings to be made in terms of P (the preference relation) and I (the indifference relation). Therefore, R should be defined in terms of them and not the other way around. If ties are included, Arrows conditions must be rewritten in a more general manner. The inclusion of ties provides for the existence of the Social Welfare Function (SWF) and solutions are presented for the case of three alternatives.
Introduction
In "Social Choice and Individual Values" Kenneth Arrow (1951) postulates 2 axioms and 5 conditions which a Social Welfare Function (SWF) should meet in order to be considered rational and ethical and then goes on to show that such a SWF doesnt exist. He assumes a population of n voter/consumers who specify their preferences among m alternatives using the preference or indifference operator, P or I. This then constitutes Arrows model. In order to disprove Arrows result, it would be necessary to substantially retain his model (although it can be shown to be somewhat arbitrary, incomplete and incorrect) or else a result is proven that applies to some other model and the validity of Arrows result remains. However, to the extent that Arrows model is undefined, selfcontradicting or loose, it is possible to add to, correct and tighten it in such a way as to remain within his model. Arrow seems to provide for the inclusion of ties in a general way in Axiom 1, but it is clear from the later context that ties are only included in his analysis by means of the indifference operator, I, which provides for ties between alternatives. Ties among orderings are not included.
The primary relation in Arrows model, R, means "preferred or indifferent". He states that he uses R instead of P and I "for convenience." The statement "x is preferred or indifferent to y" is symbolized by xRy. Each individual voter/consumer is numbered so that it is possible to speak of the ith voter/consumer whose choice between x and y is symbolized as xRiy. If there are m alternatives in a set V, each individual submits his or her input, however, in the following form:
Ri = x1Qi1x2 ... xj1Qi(j1) xj ... xm1Qi(m1) xm, xj ( V and Qij is chosen from the set {P,I}.
Arrow states: "The letter R, by itself, will be the name of the relation and will stand for a knowledge of all pairs such that xRy" [emphasis added]. Therefore, the use of R on the left hand side of the equals sign is correct.
Similarly, societys choice is given without the subscript i i.e. R = x1Q1x2 ... xj1 Q(j1) xj ... xm1 Q(m1) xm.
Alternatively, the preference and indifference information submitted by each individual can be translated into binary pairs of the form {xRiy, yRix}. Each pair of alternatives must be expressed in this way since just the relation xRiy is insufficient to express xPiy, yPix or xIiy. The logic is the following. If xPiy, then xRiy is true and yRix is false. If xIiy, then xRiy is true and yRix is true. If yPix, then yRix is true and xRiy is false. Arrow does not make this clear in his statement that one must have a "knowledge of all pairs such that xRy." One needs to know both xRy and yRx.
Each individual submits an ordering. A domain element is a combination of one ordering from each Individual. The totality of all possible combinations of individual orderings represents the domain. Then the SWF transforms each element of the domain into a social ordering which constitutes an element of the range.
Another issue in Arrows model is whether or not the individuals and society have exactly the same choices available to them. Is the range composed of exactly the same orderings as the domain? Arrow indicates that the social and individual orderings do not consist of exactly the same set. For individuals, "the chooser considers in turn all possible pairs of alternatives, say x and y, and for each such pair he makes one and only one of three decisions: x is preferred to y, x is indifferent to y, or y is preferred to x. The decisions made for different pairs are assumed to be consistent with each other, so, for example, if x is preferred to y and y to z, then x is preferred to z; similarly, if x is indifferent to y and y to z, then x is indifferent to z."
For society the ordering over all alternatives is specified by the SWF. According to Arrows definition: "By a SWF will be meant a process or rule which, for each set of individual orderings R1, ..., Rn for alternative social states (one ordering for each individual), states a corresponding social ordering of alternative social states, R." The choice between x and y by society is not simply a function of the individual choices between x and y but of the totality of individual choices according to the definition. However, the social ordering can be broken down into binary form due to the fact that it is transitive. For example, aPbIcPd can be broken down to the following: aPb, aPc, aPd, bIc, bPd, cPd, but aPb is not necessarily a function just of all of the aRibs and bRias. Note that with respect to the decomposition of the mth stage social ordering into binary components (where the mth stage represents the alternatives taken m at a time), these components are not necessarily the same as the 2nd stage social orderings. For example, if aP4bI4cP4d is the 4th stage social ordering, aP4b is not necessarily the same as the second stage ordering which might be bP2a where Pk represents the kth stage preference operator. This is all in accordance with Arrows definition.
If ties are taken into account, then an individual might choose {xPiy, yPix} to indicate that he or she is equally divided between the two orderings as opposed to being indifferent between them. Society might choose {xPy, yPx} to indicate that society is equally divided between the two orderings as opposed to being indifferent between them which would be indicated xIy. In this way the indifference operator represents a relation between alternatives. A tie represents an equal weighting of orderings. In general,
Rim = {(x1 Qi1mx2 ... xj1 Qi(j1)mxj ... xm1 Qi(m1)mxm)1, (x1 Qi1mx2 ... xj1 Qi(j1)mxj ... xm1 Qi(m1)mxm)2, ... , (x1 Qi1mx2 ... xj1 Qi(j1)mxj ... xm1 Qi(m1)mxm)p}, xj ( V where there are p elements of the tie. Similarly,
Rm = {(x1 Q1mx2 ... xj1 Q(j1)mxj ... xm1 Q(m1)mxm)1, (x1 Q1mx2 ... xj1 Q(j1)mxj ... xm1 Q(m1)mxm)2, ... , (x1 Q1mx2 ... xj1 Q(j1)mxj ... xm1 Q(m1)mxm)p}. For purposes of simplification, one might choose to eliminate the indifference operator and consider only the preference operator and ties or one might choose, in the interest of rationality, to restrict individual orderings from including ties. In general, one can consider both preference and indifference operators and ties for both individuals and society.
Axiom 1 and Axiom 2
Arrow chooses R as his primary relationship because it is "slightly more convenient." Accordingly, P and I are derivative relationships. He demands in Axiom 1 that any two alternatives be comparable:
Axiom 1: For all x and y, either xRy or yRx.
He states: "Note also that the word or in the statement of Axiom 1 does not exclude the possibility of both xRy and yRx. That word merely asserts that at least one of the two events must occur; both may." One assumes that Axiom 1 applies both to individuals and society. Therefore, the ith individual would specify one of the following: 1) xRiy = true; yRix = false. 2) yRix = true; xRiy = false or3) xRiy = true; yRix = true. Society would likewise specify one of the following: 1) xRy = true; yRx = false. 2) yRx = true; xRy = false or 3) xRy = true; yRx = true. One cannot have both xRy and yRx be false. Axiom 1 is a little misleading because both the truth values for xRy and yRx must be specified in order to have a 11 linkage with the underlying preference and indifference notation. So why doesn't Arrow just specify the truth table or state Axiom 1 in terms of P and I? An individual must specify the truth values for both xRiy and yRix, not just one or the other.
It is obvious that to Arrow a tie and an indifference amount to the same thing. He goes on to say "The adjective weak refers to the fact that the ordering does not exclude indifference i.e. Axioms I and II do not exclude the possibility that for some distinct x and y, both xRy and yRx [are true]. A strong ordering, on the other hand, is a ranking in which no ties are possible." No! A strong ordering is one in which no indifferences are possible! Arrow's statement would only be true if ties between orderings are excluded. Consider the strong ordering P and a situation in which half the individuals specify xPiy and half specify yPix. Might not society then conclude that there is a tie between the two orderings xPy and yPx? Since Arrow considers a tie, in fact, to be identical to an indifference, he doesnt consider this possibility. But a tie is more general than an indifference. A tie between two alternatives in an individual's mind represents an indifference between the alternatives whereas a tie between the orderings of two alternatives in society represents an even division between those two orderings. To say that society is indifferent between x and y is different from saying that society is evenly divided between xPy and yPx.
Arrow defines an indifference as a tie:
Definition 2: xIy means xRy and yRx.
It is clear that this definition is identical to the case in Axiom 1 in which "both [xRy and yRx] occur." But the situation is different between individuals and society. If we're talking about individuals, it is true that an individual can be indifferent between 2 (or more) alternatives. In the case of society there are more cases to consider. Case 1: The number of individuals having {xRiy=T, yRix=T} equals the number of individuals having {xRiy=T, yRix=F}. Therefore, there is a tie between xPy and xIy or {xRy=T,yRx=T} is tied with {xRy=T,yRx=F}. Case 2: The number of individuals having {xRiy=T, yRix=T} equals the number of individuals having {xRiy=F, yRix=T}. Therefore, there is a tie between yPx and xIy=yIx or {xRy=T,yRx=T} is tied with {xRy=F,yRx=T}. Case 3: The number of individuals having {xRiy=T, yRix=F} equals the number of individuals having {xRiy=F, yRix=T}. Therefore, there is a tie between xPy and yPx or {xRy=T,yRx=F} is tied with {xRy=F,yRx=T}. Case 4: The number of individuals having {xRiy=T, yRix=T} Is greater than the number of individuals having {xRiy=T, yRix=F} and greater than the number of individuals having {xRy=F,yRx=T}. Therefore xIy or {xRy=T,yRx=T}. Case 5: The number of individuals having {xRiy=T, yRix=F} Is greater than the number of individuals having {xRiy=F, yRix=T} and greater than the number of individuals having {xRy=T,yRx=T}. Therefore xPy or {xRy=T,yRx=Y}. Case 6: The number of individuals having {xRiy=F, yRix=T} Is greater than the number of individuals having {xRiy=T, yRix=F} and greater than the number of individuals having {xRy=T,yRx=T}. Therefore yPx or {xRy=F,yRx=T}.
It is clear that for both individuals and society, we must know the truth values of both xRy and yRx so a statement like xRy or xRiy in and of itself is meaningless.
In a tie situation, the individual or society gives equal weighting to the two possibilities whereas, in an indifference situation, the individual or society cant distinguish between them. Logically, however, according to Arrows definition, an indifference and a tie are the same.
Arrows axiom of transitivity states the following:
Axiom 2: For all x, y, and z, xRy and yRz imply xRz.
The underlying axioms in terms of P and I are the following:
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
However, Arrow's statement of the axiom in terms of R is incorrect. 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 in every case.
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.
Now if ties are added to the mix, the situation gets even more complicated. Let's assume that the number of individuals voting xPiy is the same as the number voting yPix. Then the social choice would be (xPy)T(yPx) where T indicates a tie. Similarly, if N(xPiy) = N(xIiy), society would have (xPy)T(xIy) where N(xPiy) = the number of individuals voting xPiy. Also, if N(yPix) = N(xIiy), society would have (yPx)T(xIy).
One could postulate a transitivity rule for ties as follows:
If aTb and bTc, then aTc.
However, this rule is not correct and ties are not transitive. For example, if the number voting xPy equals the number voting yPx and the number voting yPx equals the number voting yPz, the number voting xPy is not necessarily equal to the number voting yPz. Therefore, transitivity can't be part of the axiomatic structure of the theory.
Interpretations of R, P and I
Arrow defines P and I in terms of R. However, the primary information submitted by the individuals is P and I information. The system then converts this into R information. In any case P and I can be defined in terms of R or R can be defined in terms of P and I. R or Ri used by itself is a representation device which indicates a complete preference and indifference relationship e.g. aPdIfPcIb. R used with two adjacent letters can indicate a binary relationship: aRb. R can only be used to indicate a binary relationship as a relationship of the form aRdRfRcRb is meaningless. In order to have complete information we must know the truth value not only of aRb but also bRa. For more than 2 alternatives we must know the truth values of all binary pairs e.g. {aRb, bRa}, {aRc, cRa}, {aRd, dRa}, ,{bRc, cRb}, {bRd, dRb}, ,{cRd, dRc} etc.
P and I Derivable from R
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.
Arrows definition of a SWF is as follows:
Definition 4: By a SWF will be meant a process or rule which, for each set of individual orderings R1, ..., Rn for alternative social states (one ordering for each individual), states a corresponding social ordering of alternative social states, R.
It is clear that Arrow intends for individuals to express their preferences as e.g. x1Qi1x2Qi2...xjQijxj+1Qi(j+1)...xm1Qi(m1)xm where the set of m alternatives is {x1, x2, ... xm} and Qij can be either Pij or Iij. Arrow also clearly intends that P and I information be derivable from the primary R information, but, as we have seen, P information is not derivable from a specification of just xRy and, as we shall see shortly, I information is only partially derivable. Heuristically, if an individual states xRiy, then we know that i prefers (or is indifferent between) x to (and) y. What we dont know is whether he actually prefers or is actually indifferent between x and y. Lets say the reality in is mind is xPiy. Then xRiy is a true statement for him. But lets say the reality in is mind is xIiy. Then xRiy is also a true statement; but yRix is an equally valid statement. If i cant make up his mind between xPiy and yPix, then xPiy tied with yPix would be a true statement. In other words, if he knows hes not indifferent, but equally divided between xPiy and yPix, then he would choose the tie as his choice. This casts doubt on Arrows definition of indifference as a tie. If i is truly indifferent, he has the option of expressing xRiy or yRix. However, if we require i to express an indifference as {xRiy=T, yRix=T} and a preference as {xRiy=T, yRix=F}, then there is no ambiguity as to what i actually means. Otherwise, i would have 3 ways to express indifference: xRiy, yRix, and {xRiyANDyRix}. In this interpretation, Arrows definition of indifference as xRiy AND yRix, would tell us only some of the indifferences and not those in which an individual expressed an indifference as xRiy or yRix. By the same token, if xPiy, then i could only express xRiy and not either yRix or {xRiy=T, yRix=T} in order to be logically unambiguous. Therefore, it can be concluded that some I information can be derived from the normal specification of an R ordering, but no P information, and this conclusion holds both heuristically and logically.
R Derivable form P and I
Despite the fact that Arrow specifies R first and derives P and I from R and also specifies his definition of a SWF in terms of R, his real primary values, as the subsequent development of his exposition shows, are P and I and his intention is that individual and social orderings are specified in terms of P and I, not R. The model under these assumptions is the following:
Axiom 1': For all x and y, one and only one of the following must be true: xPy, yPx or xIy.
Axiom 2':
1) For all x, y and z, if xPy and yPz, then xPz;
2) For all x, y and z, if xPy and yIz, then xPz;
3) For all x, y and z, if xIy and yPz, then xPz;
4) For all x, y and z, if xIy and yIz, then xIz.
Definition 1': xRy is defined to mean xPy EOR xIy.
where EOR is the exclusive OR.
Definition 4': By a SWF will be meant a process or rule which, for each set of individual orderings R1, ..., Rn for alternative social states (one ordering for each individual), states a corresponding social ordering of alternative social states, R, where Ri = x1Qi1x2 ... xj1Qi(j1) xj ... xm1Qi(m1) xm, and Qij = P or I ,
R = x1Q1x2 ... xj1 Q(j1) xj ... xm1 Q(m1) xm and Qj = P or I.
Now in this model ties can be included or excluded. It is clear that Arrows intention is that they be excluded except for ties among alternatives indicated by the indifference operator. Since he doesnt take ties among orderings into account, his theory is essentially correct although an examination of his 5 conditions shows some minor errors if this model is used. Ties among orderings in this model would be of the form, for example, {xPyIz, yIzIx, zPyPx}.
The Five Conditions
An examination of Arrows five conditions shows that they do not make sense in terms of the model in which R is primary and P and I are derived. We examine the conditions and rewrite them without reference to P and I. Later we take the opposite tack and make P and I primary and R derivative and examine the conditions again from this point of view. In this interpretation, changes are also required in the conditions.
Condition 1 requires a "free triple" of alternatives i.e. there are three alternatives among which there can be any possible combination of individual orderings. As stated it is OK.
Condition 2, the positive association of social and individual values, is as follows:
"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."
However, Arrows logical statement of the positive association of social and individual values and his verbal statement, "...if one alternative social state rises or remains still in the ordering of every individual without any other change in those orderings, we expect that it rises, or at least does not fall, in the social ordering," are not equivalent. It would suffice that one alternative rise in the social ordering without placing restrictions on the other alternatives and without requiring that that alternative does not fall in the social ordering with respect to any other particular alternative. An example will suffice to illustrate this point. Let the old orderings be denoted R and the new orderings, R'. Let the old social ordering be aQbQxQcQdQe, for example, and the new ordering be cQ'xQ'bQ'eQ'aQ'd. Then x has definitely risen in the social ordering since it has gone from third place to second place. However, when we break the old and new social choices down into their binary constituents, we have the following: aRx, bRx, xRc, xRd, xRe and cR'x, xR'b, xR'e, xR'a, xR'd. Even though x has risen in the social ordering, xRc and cR'x in violation of Arrows statement of condition 2. Therefore, condition 2, as stated, is too restrictive. Arrow requires that x be preferred or indifferent to every alternative in the new ordering that it is preferred or indifferent to in the old. Also Arrow requires that all the other alternatives besides x maintain their same places in the social orderings between old and new. In the above example, for instance, dRe and eR'd. Whether e and d rise or fall with respect to each other in the social ordering has nothing to do with whether x rises or falls in the social ordering given that x rises or remains the same for each individual ordering. Arrow is restricting the SWF unnecessarily by requiring that "if xPy, then xP'y" and also that "for x' and y' distinct from a given alternative x, x'Ri'y' if and only if x'Riy'". Therefore, Condition 2 is not general enough.
In our restatement of the condition we will not require relationships among alternatives other than x to remain constant in the new ordering nor will we require that if xRy', then xR'y' for any particular y'. We do require that if x has a certain rank in the old ordering, it will have that rank or higher in the new ordering. Rank is defined as the difference in the number of alternatives to which x is preferred or indifferent and the number of alternatives that are preferred or indifferent to x and can be positive or negative. If xRy for s values of y in the old environment, then xR'y for r e" s values of y in the new environment in order for the rank of x to increase or remain the same.
We, therefore, have a new restatement of Condition 2:
Condition 2'': Let R1, ..., Rn and R'1, ..., R'n be two sets of individual ordering relations, R and R' the corresponding social orderings. Suppose that the individual ordering relations are connected in the following ways: if si is the rank of x for R1, ..., Rn and ri is the rank for R'1, ..., R'n and ri e" si for all i, then r e" s in the social ordering.
When ties are considered, since there is in general no one to one correspondence between the elements of the tie in the old and new orderings, we can only require that the average rank of x over all members of the tie solution should increase or remain the same. An example is R = {xRyRz, xRzRy, yRxRz, yRzRx}. R' = {zR'x R'y, xR'zR'y, yR'zR'x}. The average rank of z has increased between R and R'.
Condition 3:
"Let R1, ..., Rn and R'1, ..., R'n be two sets of individual orderings and let C(S) and C'(S) be the corresponding social choice functions. If, for all individuals i and all x and y in a given environment S, xRiy if and only if xRi'y, then C(S) and C'(S) are the same."
Since this Condition doesnt involve the explicit use of P, it is acceptable under the assumption that R is primary and P and I, derivative except for the following observations. Arrow clearly intends for S to include fewer alternatives than there are in the set corresponding to R1, ..., Rn and R'1, ..., R'n. Therefore, C(S) and C'(S) are not the social choice functions corresponding to R1, ..., Rn and R'1, ..., R'n. Let U be the set corresponding to R1, ..., Rn and R'1, ..., R'n. Then Condition 3 can be restated as follows:
Condition 3': Let R1, ..., Rn and R'1, ..., R'n be two sets of individual orderings over a set U. If, for all individuals i and all x and y in a given environment S ( T , xRiy if and only if xRi'y, then C(S) and C'(S) are the same.
Arrow clearly intends for the individual data to be the same for the two sets of individual orderings over the set S. Therefore, since there are three possibilities, xRiy, yRix and {xRiy, yRix}, there must be an "if and only if" statement for yRix or else there could be some individual switching between yRix and {xRiy, yRix}. Therefore, Condition 3 must be changed to the following:
Condition 3'': Let R1, ..., Rn and R'1, ..., R'n be two sets of individual orderings over a set T. If, for all individuals i and all x and y in a given environment S ( T, xRiy if and only if xRi'y and yRix if and only if yRi'x, then C(S) and C'(S) are the same.
Its worth taking another look at this condition since it is used to justify the decomposition of the social ordering among m alternatives into binary orderings which are then assumed to be the same as the social orderings when only two alternatives are considered at a time. Arrow makes no distinction among values of R as a function of m, the number of alternatives. He states: "If, then, we know C([x,y]) for all twoelement sets, we have completely defined the relations P and I and therefore the relation R; but, by Definition 3, knowing the relation R completely determines the choice function C(S) for all sets of alternatives. Hence, one of the consequences of the assumptions of rational choice is that the choice in any environment can be determined by a knowledge of the choices in twoelement environments." (italics added) Arrow is confusing here the decomposition of R which is a function of m into binary components which can always be done by virtue of transitivity, and the binary social orderings based on binary individual orderings over all twoelement sets. They are not necessarily the same. In fact, as we shall see, Condition 3 provides for binary independence but does not imply that the binary decomposition of the social ordering over the set U containing m alternatives is equal to the set of binary social orderings based on binary individual orderings. We write Rm to emphasize the dependence of the social ordering on the number of alternatives being ordered.
For example, lets consider the social ordering aR5bR5cR5dR5e which, by virtue of transitivity, can be decomposed into the following set of binary social orderings: {aR5b, aR5c, aR5d, aR5e, bR5c, bR5d, bR5e, cR5d, cR5e, dR5e}. In general, these are not the same as the social orderings among alternatives taken two at a time which, for example, might be {bR2a, aR2c, aR2d, eR2a, bR2c, dR2b, bR2e, cR2d, eR2c, dR2e}. It is not even required by Condition 3 that the set of orderings formed by the decomposition of the mary ordering into binary constituents and selecting just those orderings involving alternatives in S be the same as the set of binary social orderings involving alternatives in Sonly that the binary constituents be independent of other individual ordering information.
From Arrows definition of a SWF we have f(R1, ..., Rn) = R where f is the SWF. In general this is not the same as the recomposition of the functions fxy(R1, ..., Rn) for all values of the alternatives x and y where fxy is the binary SWF, f is the mary SWF and there are m alternatives altogether. Therefore, C(S) is not, in general, C({fxy(R1, ..., Rn)}all values of x and y in S). In words, the social choice over the set S = {x1, x2,...xm) in which the elements of S are related by an mary relationship, x1Rm x2Rm... Rmxm is not the same as the social choice over the set S = {x1, x2,...xm) in which the elements of S are related by binary relationships: x1R2 x2, x1R2 x3, ..., x2R2 x3, x2R2 x4, ..., xm1R2 xm. The reason is that it is not required by Condition 3.
In Arrows Condition 3 the social choices C(S) and C'(S) are related to the social orderings R and R' which are in turn related to the individual orderings R1, ..., Rn and R'1, ..., R'n. How the orderings over the set S are related to the orderings R and R' is never specified. Arrow does give some verbal justification for Condition 3 as follows: "Suppose an election is held, with a certain number of candidates in the field, each individual filing his list of preferences, and then one of the candidates dies. Surely the social choice should be made by taking each of the individuals preference lists, blotting out completely the dead candidates name, and considering only the orderings of the remaining names in going through the procedure of determining a winner. That is, the choice to be made among the set S of surviving candidates should be independent of the preferences of individuals not in S. To assume otherwise would be to make the result of the election dependent on the obviously accidental circumstance of whether a candidate died before or after the date of polling."
None of this requires that, if the candidate who dies is y and if the social choice is x before the candidate dies, then the choice must be x after the candidate dies, only that it be based on individual orderings that do include y in the first case and dont in the second. It is not required that the social choice be the same as alternatives are added or deletedonly that those alternatives be added or deleted from each individuals list before the SWF produces the social ordering from which the social choice is obtained. If Arrow intended that the social choice be the same in every case regardless of the number of alternatives, this should have been stated.
Arrow goes on to say: "Alternatively stated, if we consider two sets of individual orderings such that, for each individual, his ordering of those particular alternatives in a given environment is the same each time, then we require that the choice made by society from that environment be the same when individual values are given by the first set of orderings as they are when given by the second."
The problem is that Arrows words indicate that the choice in both cases be made from the "given environment," the set S. Obviously, the choice will be the same. Therefore, Arrow must be referring to the environment created by the decomposition of the Rm social ordering into binary constituents. Clearly, the social choice function could operate over that subset of the binary constituents containing alternatives belonging to the set S. However, this set of binary constituents is not necessarily equal to the set of binary constituents produced by the SWF operating on individual data containing only alternatives belonging to S. Nowhere in Condition 3 is any mention made that the binary social ordering between x and y be invariant regardless of the number of alternatives. It is not required either by Arrows verbal or formal statement of Condition 3.
Binary independence and the binary decomposition of Rm being a function of the ordering produced by the binary SWF are not the same thing. Let fm be the mth stage SWF. Then if the binary decomposition of Rm is a function of the binary social ordering, we have:
For all x and y: Rmx,y = h{f2([R1, ..., Rnxy]} where fm = SWF for m alternatives and Rx,y is the binary decomposition of R over x and y.
In general, f2([R1, ..., Rnxy] need not be equal to Rmx,y in order for binary independence to hold. Binary independence will hold if Rmx,y is a function of
[R1, ..., Rn xy] i.e. Rmx,y = f2[R1, ..., Rnxy]. Note that Arrows specification of Condition 3 does not require that Rmx,y be equal to f2([R1, ..., Rn xy] only that Rmx,y = R'mx,y.
Arrows alternative verbal statement does not elucidate the situation and should be changed to reflect the fact that the relationships among the alternatives in S referred to in Condition 3 are derived from the relationships in the social orderings R and R' over U and are not specified by the SWF operating upon individual orderings over the set S. Condition 3 does specify binary independence in the sense that for any given stage, the social ordering between x and y should only be a function of the individual data concerning x and y. However, that ordering can vary from stage to stage where the stage number represents the number of alternatives being considered. If Arrow intends that there should be stage to stage invariance, then this should be so stated. Verbally, what is required is something like the following:
If we consider a set of individual orderings over the set T and another set over the set S ( T, then the binary decomposition of the social choice is the same in each case for those alternatives common to both sets.
This could be strengthened as follows:
If we consider a set of individual orderings over the set T and another set over the set S ( T, then the set of binary decompositions of the social ordering over S is equal to the set of binary decompositions of the social ordering over T truncated to include only those alternatives in the set S.
One might also consider any function g which would transform Rm to Rm' instead of requiring strict equality between binary constituents. We call this a "reduction" from Rm to Rm' and g the reduction function. We would then have a modified Condition 3 as follows:
Condition 3''': Let R1, ..., Rn and R'1, ..., R'n represent individual orderings over S and U, respectively, (S ( U), U contains m alternatives and S contains m' alternatives) with social orderings Rm and R'm'. xRiy if and only if xRi'y and yRix if and only if yRi'x for all i and for all x and y in S. Let there be a function g that reduces Rm as follows: Rm' = g(Rm). Then Rm' and R'm' are the same.
For tie solutions, Condition 3''' is all that is necessary to insure a rational relationship among social orderings over alternative sets S and U consisting of different numbers of alternatives where S ( U. However, there is a special case corresponding to Arrows specification for "blotting out" the dead candidates. That would be to blot out the dead candidates from each element of the tie solution and then combine terms, if necessary, to obtain a solution identical (rather than a function of) to the solution for the reduced number of alternatives.
"Condition 4: The social welfare function is not to be imposed."
"Definition 5: A social welfare function will be said to be imposed if, for some pair of distinct alternatives x and y, xRy for any set of individual orderings R1, ..., Rn, where R is the social ordering corresponding to R1, ..., Rn."
Condition 4 needs no changes.
"Condition 5: The social welfare function is not to be dictatorial."
"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."
Changing P to R in Definition 6 makes it acceptable under the current assumptions. Also, "xPiy implies xPy" is incorrect. It should be "xPy if and only if xPiy."
Definition 6': A social welfare function is said to be dictatorial if there exists an individual i such that, for all x and y, xRy if and only if xRiy and yRx If and only yRix regardless of the orderings R1, ..., Rn of all individuals other than i, where R is the social preference relation corresponding to R1, ..., Rn.
P and I Primary
Although Arrow clearly specifies the possibility of ties in his model as stated, it is not his intention to include ties among orderings but only ties among alternatives via the indifference operator I. In addition, despite the fact that he specifies R first and derives P and I from R and also specifies his definition of a SWF in terms of R, his real primary values, as the subsequent development of his theory shows, are P and I and his intention is that individual and social data be specified in terms of P and I, not R. The model under these assumptions is really the following:
Axiom 1: For all x and y, one and only one of the following must be true: xPy, yPx or xIy.
Axiom 2:
1) For all x, y and z, if xPy and yPz, then xPz;
2) For all x, y and z, if xPy and yIz, then xPz;
3) For all x, y and z, if xIy and yPz, then xPz;
4) For all x, y and z, if xIy and yIz, then xIz.
Definition 1: xRy is defined to mean xPy EOR xIy.
where EOR is the exclusive or.
Now in this model ties can be included or excluded. It is clear that Arrows intention is that they be excluded except for ties among alternatives. Since he doesnt take ties among orderings into account, his theory is essentially correct although an examination of his 5 Conditions shows some minor errors.
The Five Conditions in the P and I Model
"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."
Analysis: For Ti we can have the following:
Ti = x1Qi1x2 ... xj1Qi(j1) xj ... xm1Qi(m1) xm, and Qij is chosen from the set {P,I}.
Let the set S be { x1,x2, x3}
Then
Ri = x1Qi1x2Qi2x3
where Qi1 and Qi2 are chosen from the set {P,I}
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 ( xPiy EOR xIiy and xTiy ( xPiy EOR xIiy. Therefore, the condition as stated is incorrect. The correct statement would be the following:
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 Q1, ..., Qn (where Q1 and Q are the names of the relations in terms of P and I for individual and social orderings, respectively) of all the alternatives such that, for each individual i, xPiy if and only if xP'iy for x and y in S and xIiy if and only if xI'iy for x and y in S, where Pi and Ii refer to orderings in S and P'i and I'i refer to orderings in T.
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.
Analysis: Again substitute Qi for Ri in the individual orderings where Q is a an ordering relation in terms of P and I. Arrow doesnt 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 xIy, then xR'y.
x'Ri'y' if and only if x'Riy' can be satisfied in any of the following ways: x'Pi'y' and x'Piy', x'Pi'y' and x'Iiy', x'Ii'y' and x'Piy', x'Ii'y' and x'Iiy', y'Pi'x' and y'Pix', y'Pi'x' and y'Iix', y'Ii'x' and y'Pix'. Clearly, Arrow intends for the exact relationship to hold in each of the two cases ie x'Pi'y' if and only if x'Piy' and x'Ii'y' if and only if x'Iiy'.
Arrow states: "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."
The second condition causes problems as follows: for all y', xRiy' implies xRi'y' can be satisfied by xPiy' and xPi'y', xPiy' and xIi'y', xIiy' and xPi'y' or xIiy' and xIi'y'. Clearly, Arrow doesnt intend for this to be the case. Also, clearly, his statements (1) "x is preferred or indifferent to any alternative to which it was formerly indifferent" and (2) "x is preferred or indifferent on the new scale to any alternative to which it was formerly preferred or indifferent" are incompatible. Clearly, Arrow intends for (1) in the last sentence to be implemented. This can be implemented by the following statement: for all y', xIi y' implies xRi'y'.
The truth table is as follows:
xPi'y ' xIi'y' y'Pi'x
xPiy' 1 0 0
xIiy' 1 1 0
y'Pix 1 1 1
Condition 2 can be restated as follows:
Condition 2': Let Q1, ..., Qn and Q'1, ..., Q'n be two sets of individual ordering relations, Q and Q' 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'Pi'y' if and only if x'Piy' and x'Ii'y' if and only if x'Iiy'; for all y', xPi y' implies xPi'y'; for all y', xIiy' implies xRi'y';. Then, if xPy, xP'y and if xIy, xR'y.
When ties are considered, since we may not be dealing with the same set of ties as x rises, Condition 2' should hold on average over the set of ties.
However, Arrows analysis of the Positive Association of Social and Individual values is too narrow. We take at face value his statement: "...if one alternative social state rises or remains still in the ordering of every individual without any other change in those orderings, we expect that it rises, or at least does not fall, in the social ordering." This general statement of positive association, however, does not translate exactly into Condition 2. An example will suffice to point out Arrows lack of generality. Suppose in all the individual orderings alternative x rises or remains the same. The old orderings are denoted Q and the new orderings, Q'. Let the old social ordering be aQbQxQcQdQe, for example, and the new ordering be cQ'xQ'bQ'eQ'aQ'd. Then x has definitely risen in the social ordering since it has gone from third place to second place. However, when we break the old and new social choices down into their binary constituents, we have the following: aQx, bQx, xQc, xQd, xQe and cQ'x, xQ'b, xQ'e, xQ'a, xQ'd. Even though x has risen in the social ordering, xQc and cQ'x in violation of Arrows statement of Condition 2 which is too restrictive.
Arrow requires that x be preferred to every alternative in the new ordering that it is preferred to in the old ordering and preferred or indifferent in the new ordering to every alternative to which it is indifferent in the old ordering while his verbal statement only requires that x rise in the social ordering. Note that Arrow doesnt require that all the other alternatives besides x maintain their same places in the social orderings between old and new, however. In the above example, for instance, dQe and eQ'd which is in accordance with Arrows statement of Condition 2. Whether e and d rise or fall with respect to each other has nothing to do with whether x rises or falls in the social ordering given that x rises or remains the same for each individual. Arrow is restricting the SWF unnecessarily by requiring that if xPy, then xP'y, and therefore, his proof does not apply to the more general case which is indicated by his verbal statement of the condition. There does not have to be an exact correspondence between the binary relationships of the old and the new orderings although the old and new orderings can each be decomposed into binary relationships by virtue of transitivity.
In our restatement of the condition we will not require relationships among alternatives other than x to remain constant in the new ordering nor will we require that if xPy', then xP'y' for any specific y'. We do require that if x has a certain rank in the old ordering, it will have that rank or higher in the new ordering. Rank is defined as the difference in the number of alternatives to which x is preferred and the number of alternatives that are preferred to x and can be positive or negative. Considering only preferences for the moment, if xPy for s values of y in the old environment, then xP'y for r e" s values of y in the new environment in order for the rank to increase or remain the same. Considering preferences and indifferences, let S be the set of alternatives such that xPs, s(S and U be the set of alternatives such that uPx, u(U. Then x will stay the same or increase in rank if O(S)  O(U) stays the same or increases where O(U) stands for the order of the set U. This takes into account that x may be indifferent to a set of alternatives which may increase or decrease in the new environment as compared with the old. The difference between the number of alternatives to which x is preferred and the number of alternatives which are preferred to it must remain the same or increase in the new environment in order for the rank of x to remain the same or increase. If there are alternatives in the old environment to which x is indifferent, and if the number of these alternatives increases or decreases in the new environment, then this increase or decrease must be such that the rank of x in the new environment does not decrease. At least as many of the increase in the indifference set must come from the set of alternatives that are preferred to x as from the set of alternatives to which x is preferred. Likewise, at least as many of the decrease in the indifference set must go to the set of alternatives to which x is preferred as go to the set of alternatives which are preferred to x.
Taking these restrictions away we have a new statement of Condition 2:
Condition 2'': Let Q1, ..., Qn and Q'1, ..., Q'n be two sets of individual ordering relations, Q and Q' the corresponding social orderings (where Q refers to an ordering in terms of P and I), P and P' the corresponding social preference relations and I and I' the corresponding social indifference relations. Suppose that for each i the two individual ordering relations are connected in the following ways: for all y', xPi y' implies xPi'y'; for all y', xIiy' implies xRi'y'. Then the rank of x in the social ordering R' will increase over its rank in R or remain the same where rank is defined as the difference in the number of alternatives to which x is preferred and the number of alternatives which are preferred to x.
When ties are considered, the average rank of x over the set of ties should increase or remain the same.
Got To Here
Some Other Useful Relationships
There is a 11 relationship between {P, I} and R as shown in the following truth table:
xRiy T T F FyRix T F T FxPiy F T F FyPix F F T FxIiy T F F F
When Arrow makes the following statement, it could have been stated more simply in terms of P and I rather than R.
We have still to express formally the condition that x be not lower on each individuals scale while all other comparisons remain unchanged. The last part of the condition can be expressed by saying that, among pairs of alternatives neither of which is x, the relation Ri [will be the same as Ri]; in symbols, for all x `" x and y `" x, x Ri y if and only if x Ri y . &
The key words are for all. This means that x R iy iff x Riy and y R i x iff y Rix for all pairs of alternatives.
Wouldn t it have been simpler to have said the following:
for all x `" x , y `" x, xPiy iff xPiy and
xIiy iff xIiy. ?
We must know the truth values of both xRiy and yRix in order to have complete P and I information. However, we need only know which of the 3 relations: xPiy, xPiy or xIiy is true in order to have complete information. Why didnt Arrow make this explicit? Since we must know the truth values of both xRy and yRx or {xRy, yRx}, an expression of the form aRibRicRidRie is meaningless, while an expression of the form aPibPicIidPie conveys the full logical meaning of the voters intentions. For the voter to vote using R notation he or she would have to specify the dual truth values, {xRiy, yRix} for every possible binary pair of alternatives.
Arrow continues: 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.
This statement would seem to indicate that Arrow is dealing with the underlying P and I data and that there is a one to one relationship between P and I on the one hand and R on the other. Otherwise, he would have stated that x is preferred or indifferent on the new scale to any alternative that it was preferred or indifferent to on the old scale. That condition would be the following:
for all y, xRiy implies xRiy
The fact that Arrow distinguishes here between xPiy and xIi y would seem to indicate that he assumes specific P and I knowledge of the data and not just R knowledge of the data. And his statement is correct that, if x is preferred to y in the old data, it must remain preferred to y in the new data; and, if x is indifferent to y in the old data, it may remain indifferent or it may be preferred to y in the new data. If y is preferred to x in the old data, it may remain preferred to x in the new data or the relationship may change such that x is preferred to y in the new data.
Arrow continues: In symbols, for all y, xRiy implies xRiy, and xPiy implies xPiy.
Consider the truth table for xRiy implies xRiy:
xRiyTTTTFFFyRixTTFFTTTxRiyTTTTFTTyRixTFTFTTF
According to the fourth column of this table, when x is preferred to y in the old data, x is indifferent to y in the new data. This is not correct. Now consider the truth table for xPiy implies xPiy:
xPiyTFFxPiyTFT
If xPiy is true, xPiy is true. If xPiy is false, xPiy may be true or false.
Now if the relation, xPiy implies xPiy, takes precedence over the relation, xRiy implies xRiy, we have the following truth table:
xRiyTTTTFFFyRixTTFFTTTxRiyTTTTFTTyRixTFFFTTF
The only thing that changes is that a preference for x over y (xRiy true and yRix false) cannot go to an indifference in the new data. It must go to a preference for x. Other than that nothing changes in the truth table.
Arrows statement of the conditions leads to a contradiction between the two relations unless the relation, xPiy implies xPiy, takes precedence over the relation, xRiy implies xRiy.
This condition could have been stated more transparently and succinctly as for all y, xIiy implies xRiy, and xPiy implies xPiy. The truth table for xIiy implies xRiy is the following:
xIiyTTFFFxRiyTTTTFyRixTFTFT
The truth table for xPiy implies xPiy and xIiy implies xRiy is the following:
xPiyTFFFFFxIiy FFFFTTxPiy TTFFTFxRiyTTFTTTyRixFFTTFT
Condition 3:
"Let R1, ..., Rn and R'1, ..., R'n be two sets of individual orderings and let C(S) and C'(S) be the corresponding social choice functions. If, for all individuals i and all x and y in a given environment S, xRiy if and only if xRi'y, then C(S) and C'(S) are the same."
Aside from the fact that "xRiy if and only if xRi'y" should be stated "xPiy if and only if xPi'y and xIiy if and only if xIi'y" the major problem with this condition is that there must be a set of ordering relations over the set S, but Arrow doesnt say what these ordering relations are or how they are to be derived from R and R'. Obviously, these orderings cannot be produced by just considering the individual orderings over the set S and then applying the SWF to them because this would be tautological. Obviously, the SWF must be applied to the full set of orderings for R and R', and then the orderings over the set S derived from the orderings over R and R'. In other words the orderings over the set S are a function of the orderings over the full set, lets call it T. Its clear that this function must be the same for both R and R' so that g(RT) = RS and f(R'T) = R'S and that RS and R'S should be the same although Arrow only requires that C(S) and C'(S) be the same. Note, however, that RS and R'S need not be the same as the ordering produced by the SWF applied to the individual orderings for the set T. That is not required by Condition 3. C(S) is the "top slot" of the ordering RS and C'(S) is the "top slot" of the ordering R'S. Clearly, Arrow intends for us to take each of the orderings R and R' and "blot out" the alternatives not in S with the resultant ordering being the ordering over the set S. But Arrow never specifies this function mathematically. A specification of the function, g, is the following:
Let RT be an ordering over the set T [O(T) = t] and RST be a reduction of RT which is an ordering over the set S [O(S) = s], S(T. RT = a1Q1a2Q2...aiQi...at1Qt1at and RST = b1Q1b2Q2...bjQj...bs1Qs1bs. g is given by the following algorithm:
RT0 = 1; Rt = 1; j=0.
For i=1 to i=t,
If ai(S, then {RTj+1 = RTj aiQi; j=j+1, Rj = Ri}
If ai(S, if Qj =I AND Qi = P, RTj = RTj(P/I), Rj=P
As an example, let RT = aPbIcIdIePf and R'T = aIbPdIcIePf and S = {a,c,e,f}.
RT0 =1; R6=1; j=0
i=1, a1(S, RT1= 1. aP, j=1, R1=P
i=2, a2(S, R1=P, R2=I
i=3, a3(S, RT2=aPcI, j=2, R2=I
i=4, a4(S, R2=I, R3=I
i=5, a5(S, RT3=aPcIeP, j=3, R3=P
i=6, a6(S, RT4=aPcIePf .1
RST=aPcIePf
R'T0=1; R6=1; j=0.
i=1, a1(S, RT1= 1.aI, j=1, R1=I
i=2, a2(S, R1=I, R2=P, RT1=aP, R1=P
i=3, a3(S, R1=P, R3=I
i=4, a4(S, RT2 = aPcI, j=2, R2 = I
i=5, a5(S, RT3 = aPcIeP, j=3, R3=P
i=6, a6(S, RT4 = aPcIePf .1
R'ST=aPcIePf = RST
The function g will work so long as RT and R'T have the set S in the same order. But there is no requirement that they be in the same order in RT as in R'T, only that they be transformed into the same order by whatever function converts RT and R'T into the reduced form of RT and R'T, RST and R'ST. Therefore, the reduction function, as we shall call it, is related to the SWF and is not necessarily the same for every SWF.
When ties are considered, Condition 3 need not be changed at all. The reduction function must convert the solutions RT and R'T where one or both of RT and R'T consist of ties to RST and R'ST where RST and R'ST consist of the same identical orderings whether ties or singular solutions.
Condition 4: The social welfare function is not to be imposed.
"A social welfare function will be said to be imposed if, for some pair of distinct alternatives x and y, xRy for any set of individual orderings R1, ..., Rn, where R is the social ordering corresponding to R1, ..., Rn."
This means that, no matter what the individual orderings, society can never choose yPx. But it can choose xPy or xIy. This seems to be an ambiguous imposition. If it were truly imposed then the choice would be xPy regardless of individual orderings or xIy regardless of individual orderings, but the way the condition is stated, individual orderings can decide between xPy and xIy. A better definition of imposition would be the following:
Definition 5': A social welfare function will be said to be imposed if, for some pair of distinct alternatives x and y, xPy for any set of individual orderings Q1, ..., Qn, or xIy for any set of individual orderings Q1, ..., Qn where Q refers to the set {P,I}.
Condition 5: The social welfare function is not to be dictatorial.
"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."
The only change to be made here in Definition 5 would be to include the case of the dictator being indifferent between x and y as follows:
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 and xIiy implies xIy regardless of the orderings Q1, ..., Qn of all individuals other than i, where P is the social preference relation and I is the social indifference relation corresponding to Q1, ..., Qn.
The Implications of Inclusion of Ties
Let's assume in the first model that only the P relation is possible. In other words we don't consider indifferences. Arrow says that in this model ties are not possible. Not true. If half the voters prefer x to y and half prefer y to x, then there is a tie.
We now proceed to demonstrate solutions which are social orderings for a specific SWF for the case m = 3. Much of this follows Lawrence, 1998. Let us assume alternatives x, y and z and n (odd) voter/consumers. We will assume that "knowing the social choices made in pairwise comparisons determines the entire social ordering," although, as weve seen, this is not guaranteed by Condition 3. Accordingly, we consider the social choices of the alternatives two by two. Our SWF is as follows. If N(x,y) > N(y,x), then xPy. If N(y,x) > N(x,y), then yPx. N(x,y) is the number of voters who prefer x to y. At the ternary level we have 8 cases:
Case 1: xPy, xPz, yPz
Case 2: xPy, xPz, zPy
Case 3: xPy, zPx, yPz
Case 4: xPy, zPx, zPy
Case 5: yPx, xPz, yPz
Case 6: yPx, xPz, zPy
Case 7: yPx, zPx, yPz
Case 8: yPx, zPx, zPy
According to the Condorcet (1785) method for determining the outcome of an election, we consider each of the alternatives in pairs, determine the winner for each pair and then determine the final social ordering by combining these results. We use the Condorcet method in our SWF for the above cases in which it actually produces a result. Therefore, we have the following:
Case Social Ordering
1 xPyPz
2 xPzPy
4 zPxPy
5 yPxPz
7 yPzPx
8 zPyPx
This leaves only cases 3 and 6. Consider the solution {xPyPz, yPzPx, zPxPy} for Case 3. We call a reduced ordering or reduced solution an ordering with one or more alternatives removed. If we consider {xPyPz, yPzPx, zPxPy} and remove z, we get {xPy, yPx, xPy}. Combining terms we have {2xPy, yPx}. If we choose the most numerous of xPy and yPx as the solution, we get xPy by 2 to 1 which we know to be true.
Likewise, if we reduce {xPyPz, yPzPx, zPxPy} by y, we get {xPz, zPx, zPx} or {xPz, 2zPx}. 2zPx > xPz and we take zPx as the reduced solution which agrees with the known binary solution. Similarly, if we remove x from the social solution, we have {yPz, yPz, zPy} which yields yPz. Accordingly, our SWF algorithm is as follows:
1) Choose the Condorcet solution if it exists.
2) If the Condorcet solution doesnt exist, construct a solution
such that, when the solution is reduced by any single alternative,
the most numerous of the remaining binary relationships is the
same as the known binary solution.
Notice that our algorithm will always produce consistent results if the ternary solution is generated from the binary solution in such a way that there is a 2 to 1 ratio between the correct binary solution and the incorrect binary solution and then we take the larger of the two as our reduced solution. We construct our solutions in this manner in order to be compliant with Arrows Condition 3. Satisfying the other Conditions is then trivial as can be shown. Whether or not such a solution always exists will be answered affirmatively elsewhere (Lawrence, 1998). Here all we need to show is the existence of a solution for Case 6. Consider the solution {yPxPz, xPzPy, zPyPx}. Reduction by z yields yPx; by y, xPz; by x, zPy which agrees with the known binary case and is consistent with the above definition.
Therefore, we have demonstrated a consistent algorithm for the SWF which yields the same social orderings when reduced from the ternary case to the binary case as those produced at the binary level directly from the domain. There is complete consistency of social orderings and not just of alternatives produced by the choice function. The choice function only produces the top position in an ordering. We demand consistency over all orderings which can be produced by reducing a social ordering and this strengthens Arrow's Condition 3.
The Ternary Case n even
When n is even we have a total of 27 cases. We have already considered the first 8 cases above. For convenience we define {xPy, yPx} as xTy. In addition there is one more tie possibility, a three way tie: N(x,y) = N(y,x) = N(y,z) = N(z,y) = N(x,z) = N(z,x). We write this as {xPy, yPx, yPz, zPy, xPz, zPx} and define this as xTyTz. Solutions for the remaining cases are shown below.
Case Binary Solutions Ternary Solution
9: xPy, xPz, yTz xPyTz
10: xPy, zPx, yTz {zPxPy, xPyTz, yTzPx}
11: yPx, xPz, yTz {yPxPz, xPyTz, yTzPx}
12: yPx, zPx, yTz yTzPx
13: xPy, xTz, yPz {xPyPz, yPxTz, xTzPy}
14: xPy, xTz, zPy xTzPy
15: yPx, xTz, yPz yPxTz
16: yPx, xTz, zPy {zPyPx, yPxTz, xTzPy}
17: xTy, xPz, yPz xTyPz
18: xTy, xPz, zPy {xPzPy, xTyPz, zPxTy}
19: xTy, zPx, yPz {yPzPx, xTyPz, zPxTy}
20: xTy, zPx, zPy zPxTy
21: xPy, xTz, yTz {xPyTz, xTzPy, xTyTz}
22: yPx, xTz, yTz {yPxTz, yTzPx, xTyTz}
23: xTy, xPz, yTz {xPyTz, xTyPz, xTyTz}
24: xTy, zPx, yTz {zPxTy, yTzPx, xTyTz}
25: xTy, xTz, yPz {yPxTz, xTyPz, xTyTz}
26: xTy, xTz, zPy {zPxTy, xTzPy, xTyTz}
27: xTy, xTz, yTz xTyTz
P and I Model
In the P and I model, from Arrows point of view, there is no need to mention ties since they are covered by the I operator although, again, the I operator only provides for ties between alternatives and not for ties among orderings. If this model is assumed, then Arrows Impossibility Theorem may be true. In general xIy is not the same as {xPy, yPx}. xIy means that an individual or society considers the alternatives x and y to be indistinguishable. {xPy, yPx} means that an individual or society is divided between xPy and yPx. The consideration of ties in this model would include the following possibilities: {xPy, yPx}, {xPy, xIy}, {yPx, xIy}, {xPy, yPx, xIy}.
We now proceed to develop a set of solutions for the P and I model for m=3. In the P and I model, we have two relationships to deal with. Let N(x,y) be the number of individual voter/consumers who prefer x to y, and M(x,y) be the number who are indifferent between x and y. There are then 13 possibilities as follows:
Case 1 : N(x,y) > N(y,x) > M(x,y)
Case 2: N(y,x) > N(x,y) > M(x,y)
Case 3: N(x,y) > M(x,y) > N(y,x)
Case 4: N(y,x) > M(x,y) > N(x,y)
Case 5: M(x,y) > N(x,y) > N(y,x)
Case 6: M(x,y) > N(y,x) > N(x,y)
Case 7: N(x,y) > N(y,x) = M(x,y)
Case 8: N(y,x) > N(x,y) = M(x,y)
Case 9: M(x,y) > N(x,y) = N(y,x)
Case 10: N(x,y) = N(y,x) > M(x,y)
Case 11: M(x,y) = N(x,y) > N(y,x)
Case 12: M(x,y) = N(y,x) > N(x,y)
Case 13: M(x,y) = N(x,y) = N(y,x)
One possible binary decision rule might the following. If N(x,y) > N(y,x) and M(x,y), then xPy. If N(y,x) > N(x,y) and M(x,y), then yPx. If M(x,y) > N(x,y) and N(y,x), then xIy. If N(x,y) = N(y,x) > M(x,y), then {xPy, yPx}. If N(x,y) = M(x,y) > N(y,x), then {xPy, xIy}. If N(y,x) = M(x,y) > N(x,y), then {yPx, xIy}. If N(x,y) = N(y,x) = M(x,y), then {xPy, yPx, xIy}. There would be 7 possible social orderings at the binary level. At the ternary level would be 73 possible combinations each of which would require a social ordering.
However, the SWF need not make use of every possible range element in providing a mapping from domain to range. We only need to make sure that there is at least one set of connections which satisfy Arrow's criteria and axioms. Accordingly, we only consider the following binary social orderings: xPy, yPx, xTy = {xPy, yPx}, xIy, and the following binary decision rule.
Social Ordering
Case 1 : N(x,y) > N(y,x) > M(x,y) xPy
Case 2: N(y,x) > N(x,y) > M(x,y) yPx
Case 3: N(x,y) > M(x,y) > N(y,x) xPy
Case 4: N(y,x) > M(x,y) > N(x,y) yPx
Case 5: M(x,y) > N(x,y) > N(y,x) xIy
Case 6: M(x,y) > N(y,x) > N(x,y) xIy
Case 7: N(x,y) > N(y,x) = M(x,y) xPy
Case 8: N(y,x) > N(x,y) = M(x,y) yPx
Case 9: M(x,y) > N(x,y) = N(y,x) xIy
Case 10: N(x,y) = N(y,x) > M(x,y) xTy
Case 11: M(x,y) = N(x,y) > N(y,x) xPy
Case 12: M(x,y) = N(y,x) > N(x,y) yPx
Case 13: M(x,y) = N(x,y) = N(y,x) xTy
At the ternary level we have 64 = 43 cases to consider as follows. We present the solutions in Appendix 1.
THEOREM For m = 3 and any n there exists a SWF relative to the relations P and I for which the social orderings consist of either unique rankings or of ties of at most three orderings.
PROOF By inspection.
Conclusions
We have shown that Arrows assertion that he included the consideration of ties in his model boils down to the inclusion of ties among alternatives and not ties among orderings. A closer examination reveals that, while a tie between two binary orderings, {xRy, yRx}, is acceptable in his model, the indifference operator, I, is defined as xRy AND yRx. Therefore, ties need not be considered further since they have been defined as indifferences. Heuristically, it doesnt make sense to define a social choice as an indifference, xIy, if half the individuals specify xPiy and half specify yPix. In such a case society is not indifferent at all but evenly divided.
We examined a SWF for 3 alternatives when ties among orderings are allowed for the two cases: (1) P is the only ordering relation; and (2) P and I are both considered as ordering relations. We conclude that such a function exists for this special case which complies with the reformulated axioms, definitions and conditions. Elsewhere, it is shown that such a function exists for any number of alternatives and voters (Lawrence, 1998).
Appendix 1
Case Binary Solutions Ternary Solution
1 xPy, xPz, yPz xPyPz
2 xPy, xPz, zPy xPzPy
3 xPy, zPx, yPz {xPyPz, yPzPx, zPxPy}
4 xPy, zPx, zPy zPxPy
5 yPx, xPz, yPz yPxPz
6 yPx, xPz, zPy {yPxPz, xPzPy, zPyPx}
7 yPx, zPx, yPz yPzPx
8 yPx, zPx, zPy zPxPy
9 xPy, xPz, yIz xPyIz
10 xPy, zPx, yIz {zPxPy, xPyIz, yIzPx}
11 yPx, xPz, yIz {yPxPz, xPyIz, yIzPx}
12 yPx, zPx, yIz yIzPx
13 xPy, xIz, yPz {xPyPz, yPxIz, xIzPy}
14 xPy, xIz, zPy xIzPy
15 yPx, xIz, yPz yPxIz
16 yPx, xIz, zPy {zPyPx, yPxIz, xIzPy}
17 xIy, xPz, yPz xIyPz
18 xIy, xPz, zPy {xPzPy, xIyPz, zPxIy}
19 xIy, zPx, yPz {yPzPx, xIyPz, zPxIy}
20 xIy, zPx, zPy zPxIy
21 xPy, xIz, yIz {xPyIz, xIzPy, xIyIz}
22 yPx, xIz, yIz {yPxIz, yIzPx, xIyIz}
23 xIy, xPz, yIz {xPyIz, xIyPz, xIyIz}
24 xIy, zPx, yIz {zPxIy, yIzPx, xIyIz}
25 xIy, xIz, yPz {yPxIz, xIyPz, xIyIz}
26 xIy, xIz, zPy {zPxIy, xIzPy, xIyIz}
27 xIy, xIz, yIz xIyIz
28 xPy, xPz, yTz xPyTz
29 xPy, zPx, yTz {zPxPy, xPyTz, yTzPx}
30 yPx, xPz, yTz {yPxPz, xPyTz, yTzPx}
31 yPx, zPx, yTz yTzPx
32 xPy, xTz, yPz {xPyPz, yPxTz, xTzPy}
33 xPy, xTz, zPy xTzPy
34 yPx, xTz, yPz yPxTz
35 yPx, xTz, zPy {zPyPx, yPxTz, xTzPy}
36 xTy, xPz, yPz xTyPz
37 xTy, xPz, zPy {xPzPy, xTyPz, zPxTy}
38 xTy, zPx, yPz {yPzPx, xTyPz, zPxTy}
39 xTy, zPx, zPy zPxTy
40 xPy, xTz, yTz {xPyTz, xTzPy, xTyTz}
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References
1. K. J. Arrow, "Social Choice and Individual Values," John Wiley & Sons Inc., New York, 1951.
2. J. C. Lawrence, The Possibility of Social Choice for 3 Alternatives, 1998 unpublished.
3. J. C. Lawrence, The Possibility of Social Choice, 1998 unpublished.
4. Y. Murakami, "Logic and Social Choice," Routledge & Kegan Paul Ltd., London, 1968.
5. A. K. Sen, "Collective Choice and Social Welfare," HoldenDay, San Francisco, 1970.
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