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Social Choice and Beyond
Thursday, April 3, 2008
Range Voting and Arrow's General Impossibiliity Theorem
Mood:  incredulous
Now Playing: Wagner: Der Ring des Nibelungen
Topic: Range Voting

The position taken by the Center for Range Voting regarding Arrow's General Impossibility Theorem is, I believe, that it doesn't apply to utility based systems and, therefore, range voting "escapes" Arrow's conclusion. I believe a much stronger position should be taken: that Arrow intends that his General Impossibilty Theorem DOES apply to range voting and 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 "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. 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."

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.

Posted by jclawrence at 2:35 PM PDT
Updated: Thursday, April 3, 2008 2:41 PM PDT

Thursday, April 3, 2008 - 8:11 PM PDT

Name: "Greg"