## Arrow’s impossibility theorem

There is an amazing result in the theory of social choice called Arrow’s impossibility theorem. Suppose we have a set  $A$  of  $n$  possible alternatives, say, election candidates, and  $N$  voters. Each voter organizes its individual preference list over  $A,$  that is, a linear ordering of  $A$  (suppose for simplicity that ties are not allowed). A voting system is a map of $N$-tuple of individual preference lists (also called profile) into a single ordering of  $A$ — a social preference list. A  dictatorship  is a voting system that has the following property: there is a fixed voter — a “dictator” — such that its individual preference coincides with the social preference for any possible profile. As Arrow’s theorem states,

Any consistent (in some sense) voting system turns out to be a dictatorship.

Here one may find the precise statement as well as several short proofs. Of course, it turns out that consistency, in the sense required by Arrow’s theorem, is not met in real-life voting systems as explained in this notice.