Arrow's R Notation Continued

We promised to do an examination of Arrow's R notation to resolve the differences between my friend, Ben at Oxford, and myself. Ben contended that R was only a "representation device." (See Comments.) After delving into this subject I would both agree and disagree. Arrow says 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 is both a representation device (when it stands alone) and a logical relation when it stands between two letters representing alternatives. A relation of the form aPbPcIdPf... (where a,b,c... stand for alternatives; P stands for preference and I stands for indifference) makes perfect sense since the logical relationships are clear. However, a relation of the form aRbRcRdRf... makes no sense since one must know the truth values of aRb and bRa, aRc and cRa, aRd and dRa etc. etc.

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 to 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}.

We think 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. The in-depth analysis of this conumdrum continues here.

Comments

[Ben]

Hi John, sorry for time taken to reply.

I don't know why you make such a big deal out of this R thing, I can't really see it's importance.

I agree there is a sense in which it's easier to know xPy and infer that not xIy or yPx.

BUT, I think there's also a sense in which it's easier to use just one relation (R) rather than two (P and I) - as I think we see when we try to formulate something like Pareto or positive responsiveness:

If for all i xRiy then xRy

Rather than: if for all i xPiy or xIiy then xPy or xIy.

And (more roughly): If xRy and x rises in one person's ranking then xRy.

If xPy or xIy and x rises in one person's ranking then xPy.

So I think there are cases where it's a bit more cumbersom, and others where it's simpler - it depends what you want your representational device to represent.

p.s. Finally got round to buying my own copy of Arrow. Am intending to post about each of his conditions over the next few weeks.

[John Lawrence]

Hi Ben, Well I guess the reason I made such a big deal out of the R thing is that I didn't fully understand it. But now, having delved into it more deeply, I think I've straightened out my thinking on the subject in part thanks to you for pointing me in the right direction.

I think this is important because I think it's obvious that in the binary case of two alternatives, Arrow throws out individual indifferences and defines a tie among individual voters as a social indifference. This might make sense in a practical way because, in terms of utility at least, as you've pointed out, society is truly indifferent in this case, but, if you're trying to prove in general that social choice is impossible, I don't think it's fair to define away the case of a tie.

Now that I understand the notation better, I'll scrutinize Arrow's math to make sure this is what he's done.

I'm looking forward to your posts about Arrow's conditions.

Thanks again.

[Ben]

I do think the tie or indifference question is a much more interesting one. If xPiy and yPjx then Arrow is indifferent between x and y, or in other words between satisfying i and j. Of course, i and j aren't indifferent between which of them is satisfied - this I think is why we need some process for fairly adjudicating competing claims.

My thoughts on Universal Domain already published (http://bensaunders.blogspot.com/2006/05/universal-domain.html ), Pareto coming soon...