## Range Voting and Arrow's General Impossibiliity Theorem

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 unnormalized 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.

### Commentary

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.

Arrow's theorem page

Another famous theorem by Gibbard & Satterthwaite