# Why teach Arrow's impossibility theorem?

As part of a Macro class on social welfare, I am about to teach a very brief introduction to Arrow's impossibility theorem.

The classic demonstration of this involves three voters choosing between three alternatives, whose preferences are as follows:

  A B C
1 x y z
2 y z x
3 z x y


We are then shown the supposedly interesting result that, given these obviously incompatible preferences, no possible voting system can pick a favoured option in a non-arbitrary way.

More generally, any voting system risks ending up in this kind of three-way tie situation.

But this seems impossibly theoretical a concern. Clearly if voters' preferences split n alternatives precisely n ways, as in this example, there's no outcome that satisfies a majority. But in an electorate, the probability of this is vanishingly small. Also, if each alternative is joint first in preference, we surely don't care which one is selected.

Why do we teach this mysterious theorem at all? Is it just because it's got a cool name? Has it ever been usefully applied to a real-world problem?

• The theorem is junk food. While "true" it doesn't really apply because if you introduce the requirement of coherent voters it falls, and the non-coherent version of the problem is not interesting. Apr 18 '16 at 19:13

I see the important lesson of the impossibility theorem as establishing that it is not generally speaking possible to have nicely behaved preferences of groups, even if individuals have nicely behaved preferences. Therefore a social welfare function may not exist . Attempts to improve aggregate welfare by maximizing the outcome of a preference aggregation mechanism may result in unfair or irrational outcomes.

The impossibility theorem was supposedly a major reason for reforms in the papal election process. It also launched important research into if the issues Arrow raised were important for people's actual preferences and institutions. Interestingly, they are for the most common forms of preference aggregation (e.g., see Bush v. Gore. v. Nader or the notorious differences in seats and votes in UK elections). It also shows (along with Condorcet) that agenda control (in what order do we vote on the choices) is a powerful force for determining final outcomes. Finally, it launched research into which aggregation mechanisms were the least bad, seeming to settle on something like Single Transferable Vote or Instant Runoff mechanisms in most situations.

But for a more critical overview (with which the authors disagree), see pages 14-15 of Social Choice and Legitimacy: The Possibilities of Impossibility.

• When teaching it, I expect it's also worth having an answer ready to the objection, "we can escape the preconditions of Arrow's Theorem, and also the conclusion, by simply adding up the individual's absolute utility values to define group utility. So why are we restricting ourselves to using only individual preferences and not individual absolute utility?" Basically Arrow's Theorem is only as important as preferences are, and the response to the objection has to explain why absolute utilities are undesirable and/or impractical. Apr 19 '16 at 8:25
• Arrow's Impossibility Theorem only applies to ranked-choice preferences. If you use a cardinal preference system like Range Voting, all of Arrow's criteria can be solved - ie a well behaved group preference function can exist as long as its not a ranked-order system. Read more: governology.wordpress.com/2017/09/05/kenneth-arrow-is-a-dick
– B T
Sep 5 '17 at 3:19

What you describe has not much to do with Arrow's impossibility theorem. This is called the Condorcet paradox. The preference profile you gave is used to demonstrate that even if all individual preferences are transitive group judgement may not be. Using majority voting y beats z, z beats x and x beats y.

Arrow's impossibility theorem is a more nuanced theorem. If you have not studied it do read up on it, it is fascinating! (You probably have to get a book on Social Choice to get familiar with the concepts the theorem uses.)

Judgement aggregation is covered in basic micro courses to show that it is not clear which of the efficient allocations a social planer should use, because society may not agree.

There is a temptation for economists to be utilitarians - to go around trying to maximise some measure (total, average, minimax as in Rawls ...) of utility. I think this happens even to economists who've never studied any moral or political philosophy because we often end up using welfare tests.

It seems to me that Arrow is very useful (more useful than Condorcet) in pointing out that quite natural extensions of the axioms of choice that we use as foundations of economics in thinking about individual choices do not yield attractive rules for making social choices. In other words, I think it is really useful in saying to economics students: "there is a world of social choice out there that is not as simple as the (rational) individual choice that economists think they have got straight".

• All models are wrong; some models are useful.(George Box) Apr 18 '16 at 13:51
• Excellent answer. Just pipped at the post by this one for its inclusion of specific real-world example. Apr 18 '16 at 15:47