Arrow’s R Notation

Is Arrow’s R notation “only a representation device” or is it a logical relation or both? Will real voters who vote in real elections using the preference relation, P, and the indifference relation, I or will they use R? Using P and I, the ith voter would submit his input in the form aPibPicIidPie… where a, b, c, d, e, f  etc. are the alternatives and not in the form aRibRicRidRie… where R means “preferred or indifferent.” These relations can be broken down into binary, pairwise comparisons such as aPib, aPic, aRib, aRic etc.   By convention aRb, aPb and aIb without the subscripts refer to the social choices.

Arrow says this on p. 12 of “Social Choice and Individual Values”: “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 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]

So R, by itself, is the representation device, but in the statement, xRy, R is a logical relation with a truth value – either true or false. We need to avoid confusion by keeping these two different meanings of R clear. Also, when R is used as a logical relation, it needs to be used in a mathematically correct manner. In this paper we’ll check out these assumptions.

The social welfare function maps every element of the domain consisting of all possible combinations of individual votes into the social choice which is an element of the range which consists of all possible permutations of an individual vote. If tie solutions are allowed, the range would also include elements that would represent all possible ties. There’s no a priori reason why the range has to contain the same exact elements as the range. I think Arrow intends that both the domain and the range contain the elements, xPiy, xIiy and xPy, xIy, respectively, which are logically related to Ri and R. However, there is no reason why the range could not contain an element, T, for example, which would indicate a tie, xTy. There would be a lack of symmetry between domain and range if this were so, but so what?

On p. 14 Arrow defines xPy and xIy in terms of R.

“Definition 1: xPy is defined to mean not yRx.”

“Definition 2: xIy means xRy and yRx.”

On p. 15 Arrow describes the choice function, C(S):

“Definition 3: C(S) is the set of all alternatives x in S such that, for every y in S, xRy.”

Arrow makes the case that there is a 1-1 relationship between P and I, on the one hand, and R on the other. This is true if we know the truth values of both xRy and yRx. If xRy is true and yRx is false, then xPy is true. If xRy is true and yRx is true, then xIy is true. If yRx is true and xRy is false, then yPx is true.

Consider Axiom 1 (p.13): “For all x and y, either xRy or yRx

Now consider Lemma 1(e) (p.14): “For all x and y, either xRy or yPx.”

Is it any wonder we’re confused? In the first case, Arrow must mean the inclusive OR and in the second he must mean the exclusive OR. The following are the truth tables:

Inclusive OR

 xRy T T F F yRx T F T F xRy(IOR)yRy T T T F

Exclusive OR

 xRy T T F F yPx T F T F xRy(EOR)yPx F T T F

First we examine some of Arrow’s Conditions to see if they are consistent with the assumption that the input data is in terms of P and I. For instance, let’s consider the Positive Association of Social and Individual Values. Arrow says on p. 25 of “Social Choice and Individual Values”:

“Hence, if one alternative social state [a, b, c, d,…] 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 condition can be reformulated as follows: Suppose, in the initial position, that individual values are given by a set of individual orderings R1, …, Rn, [We assume there are n voters expressing votes in terms of P and I. These are then translated to terms of R.] and suppose that the corresponding social ordering R is such that xPy, where x and y are two given alternatives and P is the preference relation corresponding to R, i. e., defined in terms of R in accordance with Definition 1.”

In an actual election relations of the form xPiy or xIiy would be replaced by relations of the form {xRiy, yRix} since it would be important to know both truth values in order to maintain the 1-1 relationship between {P, I} and R. Here’s the truth table:

 xRiy T T F F yRix T F T F xPiy F T F F yPix F F T F xIiy T F F F

Arrow continues: “Suppose values subsequently change in such a way that for each individual the only change in relative rankings, if any, is that x is higher in the scale than before. If we call the new individual orderings … R1, …, Rnand the social ordering corresponding to them R, then we would certainly expect that xPy where P is the preference relation corresponding to R. This is a natural requirement since no individual ranks x lower than he formerly did; if society formerly ranked x above y, we should certainly expect that it still does.

“We have still to express formally the condition that x be not lower on each individual’s 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, xRiy if and only if  xRi y. …”

The key words are for all. This means that xRiy iff xRiy and yRi x iff yRix for all pairs of alternatives.

Wouldn’t it have been simpler to have said the following:

for all xx , y≠ x, xPiy iff   xPiy and

xIiy iff  xIiy. ?

We must know the truth values of  both xRiyand yRix in order to have complete P and I information. However, we need only know which of the 3 relations: xPiy,  xPiy or xIiyis true in order to have complete information. Why didn’t  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 voter’s 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 Riscale than x was on the Ri  scale means that x is preferred on the Riscale 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 if and only if  xRiy.

The fact that Arrow distinguishes here between xPiyand xIi ywould seem to indicate that he assumes specific P and I knowledge of the data consistent with our original assumption 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, xRiyimplies xRiy, and xPiyimplies xPiy.”

Consider the truth table for xRiyimplies xRiy:

 xRiy’ T T T T F F F y’Rix T T F F T T T xRi‘y’ T T T T F T T y’Ri‘x T F T F T T F

According to the fourth column of this table, when x is preferred to y, x is indifferent to yin the new data. This is not correct. Now  consider the truth table for xPiyimplies xPiy:

 xPiy’ T F F xPi‘y’ T F T

If  xPiy is true, xPiyis true. If  xPiy is false, xPiymay be true or false.

Now if the relation, xPiyimplies xPiy, takes precedence over the relation, xRiyimplies xRiy, we have the following truth table:

 xRiy’ T T T T F F F y’Rix T T F F T T T xRi‘y’ T T T T F T T y’Ri‘x T F F F T T F

The only thing that changes is that a preference for x over y(xRiytrue 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.

Arrow’s statement of the conditions leads to a contradiction between the two relations unless the relation, xPiyimplies xPiy, takes precedence over the relation, xRiyimplies xRiy.

This condition could have been stated more transparently and succinctly as “for all y, xIiyimplies xRiy, and xPiyimplies xPiy.” The truth table for  xIiy implies xRiyis the following:

 xIiy’ T T F F F xRi‘y’ T T T T F y’Ri‘x T F T F T

The truth table for xPiyimplies xPiy  and xIiyimplies xRiy is the following:

 xPiy’ T F F F F F xIiy’ F F F F T T xPi‘y’ T T F F T F xRi‘y’ T T F T T T y’Ri‘x F F T T F T

Conclusion

Arrow has stated that he was using the R notation as a representation device when the letter stood alone and as a logical operator when used between alternatives such as the following: xRy. Also we have assumed that Arrow’s intent was to maintain a 1-1 relationship between P and I, on the one hand, and R on the other so that individual voters would submit their ballots in terms of P and I. These ballots could then be translated in terms of R as long as one knew both xRy and yRx. The dichotomy between the two notations is that one only need know xPy, yPx or xIy since they are all mutually exclusive. If you know that xPy is true, for example, you need not know the truth values of xIy or yPx. However, you do need to know the truth values for both xRy and yRx in order to maintain the 1-1 relationship between R and {P,I}.

The key statement in this regard is on p. 12 of “Social Choice and Individual Values”: “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 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] When he says “all pairs” he means xRy and yRx.

We have shown that it is more transparent and less confusing to use the P and I notation instead of the R notation . Arrow’s use of the R notation because it is, according to him, “slightly more convenient,” turns out to be more cumbersome and more confusing. The same proofs could be done using P and I instead of  R. Whereas a voter’s ballot stated in terms of P and I e.g. aPibPicIidPie… is transparent in terms of the logical relationships, a ballot such as aRibRicRidRie… makes no sense unless you know the truth values of aRb and bRa and similarly for all binary pairings. So the voter’s input would have to be translated into binary pairs such as the following

{aRib, bRia}, { aRic, cRia}, { aRid, dRia}, { aRie, eRia}, { bRic, cRib}…