Arrow impossibility theorem

In 1951, K. Arrow [a1] discovered a troubling result about decisions involving three or more alternatives. After posing innocuous conditions that seemingly are satisfied by all reasonable procedures, he proved that they require a dictator. This counter-intuitive assertion is one of the better known results in the social and decision sciences; it is part of Arrow's 1972 Nobel Prize.

Arrow's first condition, unrestricted domain, allows each voter to rank the candidates in any complete, transitive manner (without ties). The Pareto condition requires that when the voters unanimously agree on the strict ranking of two candidates, then that is society's relative ranking of this pair. Independence of irrelevant alternatives, (IIA), requires society's relative ranking of a pair to depend only on how the voters rank them; it is not influenced by their views of other "irrelevant" alternatives. This condition eliminates paradoxes, where an $ A \succ C $ relative outcome can change by voters varying their $ B $ ranking but keeping their $ \{ A, C \} $ rankings fixed. Finally, the procedure has transitive outcomes.

Arrow proved that with more than two alternatives, a dictator is the only procedure satisfying these conditions, i.e., society's ranking is determined by the dictator. If $ P ^ {n} $ is the set of all $ n! $ transitive rankings of the $ n $ candidates, then decision procedures are mappings $ F : {\prod P ^ {n} } \rightarrow {P ^ {n} } $, where a dictator is an $ F $ that is the identity mapping on one variable; e.g., there is a component $ j $ so that for any profile $ \mathbf p = ( \mathbf p _ {1} \dots \mathbf p _ {a} ) \in \prod P ^ {n} $, one has $ F ( \mathbf p ) \equiv \mathbf p _ {j} $.

Explanation.

With the many extensions (see [a3]) and mathematical proofs of Arrow's theorem, ranging from ultrafilters to geometry [a4] to algebraic topology [a2], it is surprising that it admits an elementary explanation with a benign re-interpretation [a4], [a5]. To explain this, notice that the theorem is meaningless unless voters have transitive preferences. (For instance, if all voters had the same cyclic preferences, then Pareto forces a non-transitive outcome.) However, IIA vitiates this critical transitivity assumption. To see why, for each pair $ \{ A _ {i} , A _ {j} \} $, let $ F _ {i, j } ( \mathbf p ) $ be the $ \{ A _ {i} , A _ {j} \} $- relative ranking of $ F ( \mathbf p ) $; e.g., if $ F ( \mathbf p ) = A _ {2} \succ A _ {3} \succ A _ {1} $, then $ F _ {1, 3 } ( \mathbf p ) = A _ {3} \succ A _ {1} $. IIA requires $ F $ to be decomposed into the $ ( \begin{array}{c} n \\ 2 \end{array} ) $ mappings $ \{ F _ {i,j } \} _ {i < j } $, where the $ F _ {i,j } $ domain depends only on the voters' rankings of $ \{ A _ {i} , A _ {j} \} $. Letting $ B ^ {n} $ denote the set of all

$$ 2 ^ {\left ( \begin{array}{c} n \\ 2 \end{array} \right ) } $$

ways to list the strict rankings of the $ ( \begin{array}{c} n \\ 2 \end{array} ) $ pairs, IIA requires $ F $ to be expressible as $ F : {\prod B ^ {n} } \rightarrow {B ^ {n} } $. This domain allows IIA-admissible procedures to be used by unsophisticated voters without transitive rankings. Similarly, the implicit independence condition conferred on the $ \{ F _ {i,j } \} $ functions by IIA renders them incapable of sequencing the $ ( \begin{array}{c} n \\ 2 \end{array} ) $ pairwise outcomes into transitivity.

Thus, the transitivity assumption is a domain restriction; the goal is to identify those IIA-procedures where the $ \prod P ^ {n} $ restriction is sufficiently severe to force the outcomes to be in $ P ^ {n} $. Trivially, this includes a dictator. However, when the rankings of two or more voters are involved, some $ F $ level sets contain transitive and non-transitive profiles, where the non-transitive profile determines a non-transitive $ F $ outcome. Arrow's assertion, then, holds, because IIA prohibits procedures from recognizing transitivity. This permits the alternative interpretation that rational (transitive) outcomes cannot be expected from procedures intended for highly irrational voters.

Resolution.

To resolve the problem, one replaces IIA with a condition that recognizes rational preferences. One is intensity of irrelevant alternatives (IIIA), where society's relative ranking of two candidates is determined by each voter's relative ranking and the number of alternatives separating them. Replacing Arrow's negative conclusion is that [a4], [a5] the Borda count is a procedure with transitive outcomes satisfying unrestricted domain, IIIA and Pareto. (The Borda count assigns $ n - j $ points to a voter's $ j $ th ranked candidate, $ j = 1 \dots n $; the candidates are ranked according to the sum of assigned points.) Thus, if rational outcomes are desired, use procedures designed for rational voters.

References

[a1]	K. Arrow, "Social choice and individual values" , Wiley (1963) (Edition: Second)
[a2]	G. Chichilnisky, "The topological equivalence of the Pareto condition and the existence of a dictator" J. Econ. Theory , 9 (1982) pp. 223–234
[a3]	J. Kelly, "Social choice bibliography" Soc. Choice Welfare , 8 (1991) pp. 97–169
[a4]	D.G. Saari, "Basic geometry of voting" , Springer (1995)
[a5]	D.G. Saari, "Arrow's and Sen's theorems revisited and resolved" Social Choice Welfare , to appear (1997)