Arrow's R Notation
Mood:  energetic
Now Playing: Cross Blogged with Will Blog for Food
Topic: Social Choice
In the arcane world of social choice, a man by the name of Kenneth Arrow looms large. In 1951 he published a book, "Social Choice and Individual Values," in which he supposedly proved that social choice is impossible. But what is social choice? Let us say we have a society composed of N individuals numbered 1,2,3, ... . Those individuals have to order a set of M alternatives with their most preferred alternative being their first choice etc. Let's indicate the alternatives as a, b, c, ... . Then a social welfare function accepts the individual orderings as inputs and produces as output the social ordering which is an ordering of the alternatives that applies to the whole society.

If individual 1 prefers a to b, we write aP1b. If society prefers a to b, we write aPb. So far so good. But we also want to provide for the case in which an individual is indifferent between a and b. We write this aI1b and aIb, respectively. Arrow's analysis then combines these two relationships into a relationship he denotes as R which means "prefers or is indifferent to" so aR1b means individual 1 prefers a to b or is indifferent between a and b. Arrow's rationale for this is the following: "Instead of working with two relations, it will be slightly more convenient to use a single relation, 'preferred or indifferent.'" (p. 12) (emphasis added)

Arrow then goes on to postulate two axioms. Axiom 1 states that either xRy or yRx and he notes that this does not exclude the possibility that both xRy AND yRx. Axiom 2 has to do with transitivity which will not concern us here. Again Arrow states (p. 13): "Axioms 1 and 2 do not exclude the possibility that for some distinct x and y, both xRy and yRx. A strong ordering on the other hand, [one with only preferences and without indifferences] is a ranking in which no ties are possible." This is blatant nonsense. One could have half the population with xPy and half with yPx [strong orderings] and that certainly would represent a tie so a tie is possible. What Arrow is implying without coming out and saying it directly is that in his world a tie between two alternatives is to be represented as a social indifference. This is completely arbitrary and limits his entire analysis.

One must assume that in Arrow's world each individual will submit his input in terms of R. That is individual 1 would submit aR1b, aR1c etc. until all pairwise comparisons have been made. For now we will go along with Arrow's demand that only pairwise comparisons need to be submitted. It can be assumed that individuals are not permitted to submit a comparison using the indifference relation since then what would be the purpose of introducing R to make the analysis "slightly more convenient." The whole idea of "slightly more convenient" is to reduce the number of relations from 2 (P and I) to 1 (R). However, Arrow proposes (without saying so) to use the I relation in the social choice to cover the case of a tie. Therefore, the social choice could be aRb, bRa or aIb.

Now the idea of the social welfare function (or of any function for that matter) is to connect each element of the domain (consisting of all possible combinations of individual choices) to an element of the range (consisting of all possible social choices). There are a great number of possible functions. Each function will hook up elements of the domain with elements of the range differently. The important thing is that each possible element of the domain is hooked up to one and only one element of the range. Arrow implies that any element of the domain that represents a tie (such as half the population having aRb and half having bRa) should be hooked up with the range element aIb. Respectfully, I disagree with this approach for the following reason: the half of the population that has aRb could actually prefer a to b (no one is indifferent), and the half of the population that has bRa could actually prefer b to a. That represents a tie to be sure, but society is hardly indifferent between the two alternatives. Arrow has confused a tie with an indifference! By so doing he has guaranteed that his analysis will yield the result that no social choice is possible.

Secondly, I would like to point out that individual information is lost when an individual submits his input as aR1b or "I prefer a to b or I'm indifferent between a and b." The system does not know which, and this introduces ambiguity at the outset. Not only that, but say an individual is indifferent between a and b. He has two ways to express it! He can submit either aR1b or bR1a. The resulting analysis becomes meaningless as the system knows not how many of the individual aRb's represent indifferences and how many of them represent preferences. Ditto for the individual bRa's! There can be no meaningful social welfare function given these kinds of inputs.

Therefore, I suggest that Arrow's approach is not acceptable and that his conclusion that social choice is impossible is invalid. A more rigorous approach is necessary involving the possibility of ties between orderings as elements of the range. One possibility of dealing with these ties is to randomly choose among them which I think my friend, Ben, at Oxford is considering as a doctoral these.

For more on this subject, please see my blog Will Blog for Food.