__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 i^{th} voter would submit his input in the form aP_{i}bP_{i}cI_{i}dP_{i}e…
where a, b, c, d, e, f
etc. are the alternatives and not in the form aR_{i}bR_{i}cR_{i}dR_{i}e…
where R means “preferred or indifferent.” These relations can be broken down
into binary, pairwise comparisons such as aP_{i}b, aP_{i}c,
aR_{i}b, aR_{i}c
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, xP_{i}y, xI_{i}y and xPy, xIy, respectively, which are logically related to R_{i}_{ }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 R_{1}, …, R_{n}, [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 xP_{i}y
or xI_{i}y would be replaced by relations of
the form {xR_{i}y,
yR_{i}x} 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:

xR |
T |
T |
F |
F |

yR |
T |
F |
T |
F |

xP |
F |
T |
F |
F |

yP |
F |
F |
T |
F |

xI |
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 … R_{1}^{‘}, …, R_{n}^{‘ }and the social ordering corresponding
to them R^{’}, then we would certainly expect that xP^{’}y
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
R_{i}^{‘ }[will be
the same as R_{i}]; in
symbols, for all x^{’ }≠ x
and y^{’ }≠ x, x^{’}R_{i}^{‘}y^{’} if and only if x^{’}R_{i}
y^{’}. …”

The key words are *for all. *This means
that x^{’}R^{‘}_{i}y^{’} iff x^{’}R_{i}y^{’} and y^{’}R^{‘}_{i}
x^{’} iff
y^{’}R_{i}x^{’} for all pairs of alternatives.

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

for all x^{’ }≠ x , y^{’ }≠ x, x^{’}P^{’}_{i}y^{’} iff x^{’}P_{i}y^{’} and

x^{’}I_{i}^{‘}y^{’} iff x^{’}I_{i}y^{’}. ?

We must know the truth values of both x^{’}R_{i}y^{’ }and
y^{’}R_{i}x^{’}
in order to have complete P and I information. However, we need only
know which of the 3 relations: x^{’}P_{i}y^{’}, x^{’}P_{i}y^{’}^{ }or x^{’}I_{i}y^{’ }is 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 aR_{i}bR_{i}cR_{i}dR_{i}e…
is meaningless, while an expression of the form aP_{i}bP_{i}cI_{i}dP_{i}e…
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, {xR_{i}y, yR_{i}x}
for every possible binary pair of alternatives.

Arrow continues: “The condition
that x be not lower on the R_{i}^{‘ }scale than x was on the R_{i}^{ }scale means that x is preferred
on the R_{i}^{‘ }scale to any alternative to which it was preferred on the
old (R_{i}) 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^{’}, xR_{i}^{‘}y^{’} if and only if xR_{i}y^{’}.

The fact that Arrow distinguishes
here between xP_{i}y^{’ }and xI_{i}_{
}y^{’ }would 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’, xR_{i}y’ implies xR_{i}^{‘}y^{’}, and xP_{i}y’ implies
xP_{i}^{‘}y^{’}.”

Consider the truth table for xR_{i}y’ implies
xR_{i}^{‘}y^{’}:

xR |
T |
T |
T |
T |
F |
F |
F |

y’R |
T |
T |
F |
F |
T |
T |
T |

xR |
T |
T |
T |
T |
F |
T |
T |

y |
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 y’ in the new data.
This is not correct. Now
consider the truth table for xP_{i}y’ implies xP_{i}^{‘}y^{’}:

xP |
T |
F |
F |

xP |
T |
F |
T |

If xP_{i}y’ is
true, xP_{i}^{‘}y^{’ }is true. If xP_{i}y’ is
false, xP_{i}^{‘}y^{’ }may be true or false.

Now if the relation, xP_{i}y’ implies
xP_{i}^{‘}y^{’}, takes precedence over the relation, xR_{i}y’ implies
xR_{i}^{‘}y^{’}, we have the following truth table:^{ }

xR |
T |
T |
T |
T |
F |
F |
F |

y’R |
T |
T |
F |
F |
T |
T |
T |

xR |
T |
T |
T |
T |
F |
T |
T |

y |
T |
F |
F |
F |
T |
T |
F |

The only thing that changes is that
a preference for x over y’ (xR_{i}y’ true
and y’R_{i}x
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, xP_{i}y’ implies
xP_{i}^{‘}y^{’}, takes precedence over the relation, xR_{i}y’ implies
xR_{i}^{‘}y^{’}.

This condition could have been
stated more transparently and succinctly as “for all y’, xI_{i}y’ implies xR_{i}^{‘}y^{’}, and xP_{i}y’ implies xP_{i}^{‘}y^{’}.” The truth table for xI_{i}y’ implies
xR_{i}^{‘}y^{’ }is the following:

xI |
T |
T |
F |
F |
F |

xR |
T |
T |
T |
T |
F |

y |
T |
F |
T |
F |
T |

The truth table for xP_{i}y’ implies
xP_{i}^{‘}y^{’ }and
xI_{i}y’ implies
xR_{i}^{‘}y^{’} is the following:

xP |
T |
F |
F |
F |
F |
F |

xI |
F |
F |
F |
F |
T |
T |

xP |
T |
T |
F |
F |
T |
F |

xR |
T |
T |
F |
T |
T |
T |

y |
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.} aP_{i}bP_{i}cI_{i}dP_{i}e…
is transparent in terms of the logical relationships, a ballot such as aR_{i}bR_{i}cR_{i}dR_{i}e…
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

{aR_{i}b, bR_{i}a},
{ aR_{i}c, cR_{i}a},
{ aR_{i}d, dR_{i}a},
{ aR_{i}e, eR_{i}a},
{ bR_{i}c, cR_{i}b}…