Is there a version of Arrow's theorem without unrestricted domain?

Question

To recall Arrow's theorem:

Suppose we have a finite set $X$ of voters and a finite set $Y$ of candidates.

An election is a map $\phi: X \rightarrow T$ where $T$ is the space of total orderings of $Y$. So we are voting via a total ranked list ballot. Let $E$ be the space of all elections.

A social choice function is a function $f: E' \rightarrow T$ defined on some subset $E' \subset E$.

Kenneth Arrow asks that a social choice function satisfy some reasonable-looking axioms, namely:

1. Unanimity - If every single voter prefers $a > b$, then the outcome should also give $a > b$,
2. Independence of irrelevant alternatives - Holding fixed an election $\phi$ and candidate $c$, suppose we alter the ballots by moving around only $c$ in the ranking of each ballot, without disturbing the relative order of all other candidates. Call this altered election $\phi'$. Then the outcome $f(\phi')$ should coincide with $f(\phi)$ except possibly with $c$'s place in the ranking changed. In particular, it should not reverse any other pair order, such as $a > b$ into $b > a$,
3. Non-monarchy - The social choice function is not just picking one voter, copying their ballot, and ignoring everyone else,
4. Unrestricted domain - The social choice function is defined on all of $E$.

Arrow's theorem asserts that if three or more candidates are running, then no such social choice function exists.

But I've never understood why we should adopt the axiom of unrestricted domain. It seems a wildly unrealistic thing to ask.

For example, we can easily manufacture (very low probability) election outcomes with symmetric results, such as one-third voting $a > b > c$, one-third voting $b > c > a$, and one third voting $c > a > b$. The only way I see to deal with such edge cases is to restrict our social choice function to not consider such elections.

So I find it no surprise that such a strong hypothesis as unrestricted domain would ruin any attempt to find a satisfactory social choice function. But I would like to think that the essence of Arrow's theorem is something more robust than just exploiting unrestricted domain to get a contradiction.

So my question is: is there a weaker, more reasonable substitute for unrestricted domain, and a corresponding version of Arrow's theorem in this more general setting?

Maybe I should add that I would much prefer a deterministic social choice function.

Answer 1

There are two possible directions one can take this. One is to look at weaker conditions than full domain that still allow one to obtain ArrowÄs theorem, or one can look at restrictions that allow for more positive results. A classical criterion in the first category is the following:

Chain Property: We say $E'$ has the chain property if for every two pairs of alternatives $(u,v)$ and $(x,y)$ there is a finite sequence of the form $u,v,y_1,\ldots,y_n,x,y$ and for every three consecutive candidates in the sequence, there exist profiles with all possible rankings of these three candidates.

If one replaces the assumption of full domain by the assumption that $E'$ has the chain property, Arrow's theorem still holds. For a proof (and related results), you can look at:

Campbell, Donald E., and Jerry S. Kelly. "Impossibility theorems in the Arrovian framework." Handbook of Social Choice and Welfare 1 (2002): 35-94.

On the other hand, there are so-called domain restrictions under which one obtains non-dictatorial rules and a large literature studying such restrictions.