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Topic: Social Choice
The following quotation is from "Social Choice and Individual Values," by Kenneth Arrow. We want to examine the "reasonableness" of Arrow's example of the Borda count which he maintains violates his Condition3: Independence of Irrelevant Alternatives.
THE INDEPENDENCE OF IRRELEVANT ALTERNATIVES 27
CONDITION 3: Let R1', ... , Rn' and R1', ... , Rn' be two sets of individual orderings and let G(S) and G'(S) be the corresponding social choice functions. If, for all individuals i and all x and y in a given environment S,
x Ri Y if and only if x Ri'y, then G(S) and G'(S) are the same (independence of irrelevant alternatives).
The reasonableness of this condition can be seen by consideration of the possible results in a method of choice which does not satisfy Condition 3, the rank-order method of voting frequently used in clubs.2 With a finite number of candidates, let each individual rank all the candidates, i.e., designate his first-choice candidate, second-choice candidate, etc. Let preassigned weights be given to the first, second, etc., choices, the higher weight to the higher choice, and then let the candidate with the highest weighted sum of votes be elected. In particular, suppose that there are three voters and four candidates, x, y, z, and w. Let the weights for the first, second, third, and fourth choices be 4, 3, 2, and 1, respectively. Suppose that individuals 1 and 2 rank the candidates in the order x, y, z, and w, while individual 3 ranks them in the order z, w, x, and y. Under the given electoral system, x is chosen. Then, certainly, if y is deleted from the ranks of the candidates, the system applied to the remaining candidates should yield the same result, especially since, in this case, y is inferior to x according to the tastes of every individual; but, if y is in fact deleted, the indicated electoral system would yield a tie between x and z.
A similar problem arises in ranking teams in a contest which is essentially individual, e.g., a foot race in which there are several runners from each college, and where it is desired to rank the institutions on the basis of the rankings of the individual runners. This problem has been studied by Professor E. V. Huntington,3 who showed by means of an example that the usual method of team scoring in those circumstances, a method analogous to the rank-order method of voting, was inconsistent with a condition analogous to Condition 3, which Huntington termed the postulate of relevancy.
The condition of the independence of irrelevant alternatives implies that in a generalized sense all methods of social choice are of the type of
2 This example was suggested by a discussion with G. E. Forsythe, National Bureau of Standards.
'E. V. Huntington, "A Paradox in the Scoring of Competing Teams," Science, Vol. 88, September 23, 1938, pp. 287-288. I am indebted for this reference to J. Marschak.
Now we examine this example in some detail.
Let's define social utility as the sum of the individual utilities where the individual utility for an alternative is equal to the point value of that alternative according to the individual's rating. For example, in the first case individuals 1 and 2 assign a point value of 4 to alternative x, 3 to alternative y, 2 to alternative z and 1 to alternative w. Individual 3 assigns a point value of 4 to alternative z, 3 to alternative w, 3 to alternative x and 1 to alternative y. The social utility then is 10 (4+4+2) for alternative x, 7 (3+3+1) for y, 8 (2+2+4) for z and 5 (1+1+3) for w. So the winner is x.
Now we consider the case in which y is removed from the election. With the Borda count, the social utility for x is 7 (3+3+1); for z: 7 (2+2+3); for w: 4 (1+1+2). As Arrow observes, there is a tie between alternatives x and z. Although Arrow thinks that the result should still be x, if you interpret the situation that z is now considered first by one individual and second by 2 while x is considered first by 2 and third by 1, a tie between x and z is not unreasonable!
However, let us consider the above example using Range Voting instead of the Borda count. With Range Voting and point assignments between 1 and 4, the point assignments for each individual remain the same. Then the maximum social utility is 12 in both cases (including y and excluding y). The result in the first case is the same: x wins with a social utility of 10. With y removed, the social utility remains the same since the individual point values remain the same. So x still wins with a social utility of 10! With Range Voting the election is truly independent of irrelevant alternatives at least in this example. This is explored more fully in my paper, "Social Choice, Information Theory and the Borda Count."