Arrow’s Impossibility Theorem
Let’s right away jump in and see the theorem that is the focus of this chapter. Theorem (Arrow1) There does not exist any social choice rule satisfying all of: 1. the standard domain constraint; 2. the strong Pareto condition; 3. independence of irrelev
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Let's right away jump in and see the theorem that is the focus of this chapter. Theorem (Arrow1) There does not exist any social choice rule satisfying all of: 1. the standard domain constraint; 2. the strong Pareto condition; 3. independence of irrelevant alternatives; 4. has transitive explanations; 5. absence of a dictator. It is not just that it is difficult to find a social choice rule that satisfies all these. We
already know it is difficult; we have looked at more than a dozen rules many of which seemed at first to embody important aspects of our ideas of democracy and collective choice but none have satisfied all five properties. No, it isn't just that it is hard; it is impossible. This is really an astonishing result. The conditions do not, on first consideration, seem at odds with one another. And we know from the first section of the previous chapter that we have "countless" billions on billions of conceivable social choice rules to work with. What's worse, this is a very short list of criteria for a social choice rule. We might want much more than this. We might want each Cu(v) to be a singleton; we wish to provide incentives for individuals to tell the truth about preferences. We might, for that matter, wish to introduce very special assignments of power to individuals or groups of individuals - to endow some people with decisiveness between some alternatives. For example, we might want to endow some individuals the right to decline being chosen to elective office even if we don't endow them with the right to be the one chosen. But if we can't find a social choice rule satisfying all of Arrow's criteria, then we certainly can't find a social choice rule satisfying those plus some more! What is especially disturbing about Arrow's theorem is that each of the properties seems on the surface to be desirable, as we have discussed in Chapter 6. But if you still are bothered by some, like having transitive explanations or indepen1 The theorem first appeared in "A Difficulty in the Concept of Social Welfare, .. Journal ofPolitical Economy, Vol. 58, No.4 (August, 1950) and then in the first edition of Arrow's Social Choice and Individual Values (Wiley, 1951). Those versions, however, were shown to be incorrect by Julian Blau; a
corrected version appeared in the second edition of Arrow's book (Wiley, 1968).
J. S. Kelly, Social Choice Theory © Springer-Verlag Berlin Heidelberg 1988
81 dence of irrelevant alternatives, be on warning that there are other impossibility results that don't use them. In Chapter 10 we will present an impossibility result concerning the criterion of no one having an incentive to submit false preferences; in Chapter 8 we will present an impossibility result about endowing some people with narrowly constrained power. Neither of these later impossibility results will use either transitive explanations or independence of irrelevant alternatives. In order to separate out some of the messy details from a proof of Arrow's theorem, we will first establish preliminary results called contagio
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