By John C.Lawrence

One position regarding
Arrow's
General Impossibility Theorem
is that it doesn't apply to
utility-based voting systems and, therefore, range voting "escapes" Arrow's
conclusion.
I believe a stronger position should be taken: that
Arrow *intended* for his General Impossibility Theorem to apply to range
voting but that range voting *refutes* it.
Certainly, Arrow's Theorem is
valid given certain assumptions, but I don't believe that it's
"general." Instead it should be called "Arrow's *Special* Impossibility
Theorem" since it only applies to ranking procedures.

Arrow certainly considers utility based systems and claims that they are invalid given his acceptance criteria. Consider the following quote from pp. 32-33 of Arrow's book "Social Choice and Individual Values":

"It may be of interest, however, to consider a particular rule for
assigning utility indicators to individual orderings. Assume that the
individual orderings for probability distributions over alternatives
obey the axioms of von Neumann and Morgenstern; then there is a method
of assigning utilities to the alternatives, unique up to a linear
transformation, which has the property that the probability
distributions over alternatives are ordered by the expected value of
utility. Assume that for each individual there is always one alternative
which is preferred or indifferent to all other conceivable alternatives
and one to which all other alternatives are preferred or indifferent.
Then, for each individual, the utility indicator can be defined uniquely
among the previously defined class, which is unique up to a linear
transformation, by assigning the utility 1 to the *best conceivable*
alternative and 0 to the *worst conceivable* alternative. [Our italics.]
This assignment
of values is designed to make individual alternatives interpersonally
comparable.

"It is not hard to see that the suggested assignment of utilities is extremely unsatisfactory. Suppose there are altogether three alternatives and three individuals. Let two of the individuals have the utility 1 for alternative x, .9 for y, and 0 for z; and let the third individual have the utility 1 for y, .5 for x and 0 for z. According to the above criterion, y is preferred to x. Clearly, z is a very undesirable alternative since each individual regards it as worst. If z were blotted out of existence, it should not make any difference to the final outcome; yet, under the proposed rule for assigning utilities to alternatives, doing so would cause the first two individuals to have utility 1 for x and 0 for y, while the third individual has utility 0 for x and 1 for y, so that the ordering by sum of utilities would cause x to be preferred to y."

"A simple modification of the above argument shows that the proposed rule does not lead to a sum-of-utilities social welfare function consistent with Condition 3. Instead of blotting z out of existence, let the individual orderings change in such a way that the first two individuals find z indifferent to x and the third now finds z indifferent to y, while the relative positions of x and y are unchanged in all individual orderings. Then the assignment of utilities to x and y becomes the same as it became in the case of blotting out z entirely, so that again the choice between x and y is altered, contrary to Condition 3."

"The above result appears to depend on the particular method of choosing the units of utility. But this is not true, although the paradox is not so obvious in other cases. The point is, in general, that the choice of two particular alternatives to produce given utilities (say 0 and 1) is an arbitrary act, and this arbitrariness is ultimately reflected in the failure of the implied social welfare function to satisfy one of the conditions laid down."

**My Response:**
Clearly, Arrow is setting up the rules so as to
produce normalized range
votes which violates his own assumption that 1 should correspond to the
*best possible* alternative and 0 to the *worst possible* alternative. Had
he actually used *un*normalized range voting, the ratings for x and y
would not have changed and this would have lead to a contradiction of
his theory. However, his theory would still apply to ranking procedures.

My point is that Arrow's *general* Impossibility Theorem is not general at
all but in actuality is a *special* Impossibility Theorem. This deflates
Arrow's balloon and should encourage those seeking new forms in
political science and economics. I think the Center for Range Voting
should maintain that range voting invalidates or refutes Arrow's *general*
Impossibility Theorem.

**Arrow-rescue-attempt (by Warren D. Smith):**
Lawrence makes a good case. But one could counter-argue that Arrow was not that dumb!
In that view, the key phrase in the Arrow quote is "unique up to a linear transformation."
In other words, Arrow does not believe that a "best conceivable" and "worst conceivable"
alternative actually *exist* that we can all clearly agree on; instead Bob's and
Alice's utility scales
might be related by some unknown and unknowable linear transformation.
Then unnormalized range votes are inherently impossible to produce and meaningless.

**But Lawrence counter-counter-argues:**
It seems to me that range voting invalidates Arrow's Theorem even if each
voter has his *own* unique "best conceivable" and "worst conceivable"
alternative. In that case, for example, his ratings would not change if a
candidate dropped out;
IIA (Arrow's "independence of irrelevant alternatives" condition)
is satisfied; and there need not be any agreement
among the voters as to what are the best and worst conceivable alternatives.

**Conclusion:** In order for Arrow's views to be justified, it seems he needs
to very powerfully deny
the existence of "utility." He is not merely denying the ability to compare
different people's utilities, he may actually be denying
that any individual can even *have* a real-valued utility function – or at least,
Arrow denies the existence of best and worst conceivable scenarios for people (or
at least denies that they have real-valued utilities).

**About Bayesian Regret:**
But even if we *highly accept* anti-utilitarian Arrovian views – i.e. even if the magic
linear transformation relating Alice's to Bob's utility scale is unmeasurable and inapproximable
by any possible
physical/biological process –
then please note that the Bayesian Regret
methodology for comparing voting systems based on overall human happiness,
still is 100% valid. That is because that methodology can be used
to measure (to arbitrary accuracy)
the Bayesian regret of any algorithmically-defined voting method (Instant Runoff,
Normalized Range Voting, Approval Voting, Plurality voting, whatever)
under any given utility-generator and voter-strategic-behavior model,
*without ever finding any utility for any human.*

To those who wish to attack Bayesian Regret methodology because they believe
human utility is unmeasurable, I suggest considering the following sentence:
"If Mary likes apples and grapes then Mary likes some fruit."
Even if the notion of Mary "liking" something is regarded as inherently unmeasurable
and scientifically unknowable, *that does not matter* –
the sentence still is 100% valid.