How does the proof of Arrow’s theorem work?
Arrow’s theorem proves that there can be no ideal voting system. Any voting system (more precisely, any social welfare function) that generates a rank ordering of the candidates must violate at least one of the following criteria
- universal domain
- no dictatorship
- independence of irrelevant alternatives (IIA)
Universal domain is the requirement that the voting procedure must take as input any configuration of individually rational preferences. Unanimity is the very plausible condition that, if every voter prefers A over B, then the outcome of the voting procedure should rank A over B. The no dictatorship condition says that there should never be a possible configuration of preferences among the voters such that the voting procedure always outputs the very same ranking as a single voter – doing so would make that one voter dictatorial. And the independence of irrelevant alternatives condition says that the collective ranking of A and B should never depend on how one or more voters rank a third item C.
You can read elsewhere about the relative importance of these conditions and the sorts of problems that arise if you violate the last of them, in particular.
The main aim of this post is to help you to understand the gist of Arrow’s proof itself. While the details are not overwhelmingly technical, it can be hard to hold enough of them in your head at once to really get an intuition as to what is going on. This post is intended to help you get that intuition.
First: why do these conditions clash? At first inspection, they look like they are logically independent of each other, so it can be hard to see how they could generate a contradiction. Speaking very roughly, I suggest the clash comes from two key ingredients: transitivity and IIA.
Because the output of a social welfare function must be a rank ordering, it will satisfy transitivity. Transitivity tells us that if A > B and B > C, then A > C. Notice what this means: it means that the relative ranking of A and C to some extent depends on the way we rank a third alternative, B. If B were worse than C and B was better than A, then we would in fact be compelled to rank C > A, by transitivity.
And IIA is precisely the idea that the ranking of any two things in the outcome of our voting procedure should not depend on the individual preferences people have over any third thing.
Second: having got that intuition for how the conditions may clash, you still may not understand how Arrow goes about constructing the proof. The basic idea is to show that if you grant the first three ideas: universal domain, unanimity, and IIA, then you are bound to have a dictatorial situation arise. He does this by starting with a possible configuration of preferences, and then:
- progressively tweaking it in ways that must be possible, given universal domain, and then
- drawing inferences about what the output of the voting procedure must be, given unanimity, transitivity, and IIA, and then
- eventually ending up with a configuration where someone is a dictator.
In the video below I give a sketch of the strategy. Note, the video only shows how someone can come to be dictator over a single issue, but Arrow’s proof goes further, showing that someone could be dictator over every issue!
Of course, all the ingredients are necessary, not just these two. But there is some sense in which these seem to be particularly responsible for the tension that gives rise to outright impossibility. ↩
The ideas do not conflict directly, because transitivity only applies within the output of the social choice function, whereas independence of irrelevant alternatives applies across the input and the output. It says that the ranking of two things in the output should not depend on any irrelevant alternatives in the inputs. In effect, this is why you need the other conditions for the proof to go through – to make the tension between the ideas explicitly connected. ↩