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?
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.