Americans are so used to “one person, one vote” that they often imagine this is the only sensible way to vote. It’s not. (In fact, we’ll see that it’s about the

leastsensible way to vote!) – W. Poundstone, “Gaming the Vote: Why Elections Aren’t Fair (and What We Can Do About It)”

Every four years, the United States presidential electoral process gets renewed scrutiny and criticism, usually focused on the arcane goofiness of the electoral college, particularly the non-uniform voting power of people in different states, as well as the possible (and in several cases, actual) difference between the outcomes of the popular and electoral votes, Gore vs. Bush in 2000 being the most recent example.

I think a nationwide popular vote for president makes sense, at least more sense than the current system. However, it is likely that we are stuck with what we have. The presidential electoral process is specified in the Constitution; an amendment would require ratification (by legislature or convention) by at least three-fourths, or 38, of the states. The electoral college provides voters in the smaller states significantly more voting power than in a nationwide popular vote. There are quite a few more than 13 such “small” states that would oppose such an amendment, Wyoming being the most extreme example, with voters there controlling over three times the per-capita electoral votes compared to those in California.

But with either the electoral vote *or* a popular vote, problems– and solutions– still remain. The quote above refers to the problems with the simple *plurality* voting system in use for almost every single-seat election in the country. That is, a plurality vote assigns one point per ballot to the corresponding voter’s top-ranked candidate; the candidate with the most points wins the election.

Simple and obvious, right? Indeed, as Poundstone suggests, people often do not even consider that other voting procedures are possible and perhaps better, particularly when there are three or more candidates in the election. For example, the *anti-plurality* vote (discussed in an earlier post) awards one point per ballot to each voter’s *bottom*-ranked candidate, with the *least* number of points winning the election. With just two candidates, these two procedures are equivalent; with three or more candidates, however, these and other procedures can yield different outcomes. (Note that I am not suggesting the antiplurality vote for the presidential election; this is just an example of a different procedure that can yield different election outcomes.)

So which procedure is best? A common mathematical approach to finding an optimal “something” is to specify a few “obvious” desirable properties of that something, then figure out what thing or things have those properties. Economist Kenneth Arrow took that approach, resulting in his now-famous “Impossibility Theorem.”

The theorem is rather simple to state: consider an election involving three (or more) candidates, call them *A*, *B*, and *C*. Suppose we want to find a voting procedure whose inputs are each voter’s strict transitive ranking of the candidates (e.g., *A* > *B* > *C*), and whose output is a ranking representing the election outcome. There are several properties that such a procedure should (presumably) have:

- (U)
*Unanimity*: For any pair of candidates*x*and*y*, if all voters prefer*x*over*y*, then*x*>*y*in the outcome. - (IIA)
*Independence of irrelevant alternatives*: For any pair of candidates*x*and*y*, the ranking of*x*and*y*in the outcome should depend only on the voters’ rankings of*x*and*y*(i.e., and not on the ranking of any other candidate). - (D)
*Non-dictatorship*: The election outcome should not be a function of a single distinguished voter’s preferences.

Arrow’s Theorem states that *there is no voting procedure* with these three simple properties. Furthermore, any voting procedure with properties (U) and (IIA) is a dictatorship!

It turns out that the situation is not quite as grim as it seems. My point in this post is to refer you to an interesting paper by Donald Saari, “Connecting and Resolving Sen’s and Arrow’s Theorems.” I think the paper does a good job of explaining why Arrow’s Theorem is not surprising, and how a minor modification of the property (IIA) yields more optimistic results.

Also mentioned in the paper is a somewhat technical generalization of the theorem that I had not seen before (and that motivated this post and its title). The unanimity condition is stronger than it needs to be; Saari describes a weaker property that he calls “involvement” whose substitution for (U) yields essentially the same result. The involvement condition (I) requires that there be at least two pairs of candidates *x* and *y* such that there is a set of ballots (inputs) yielding the outcome *x* > *y*, and another set of ballots yielding *y* > *x*.

The problem: I can see how unanimity is a special case of involvement (exercise for the reader: show that this is in fact the case)… but I have not quite wrapped my head around the paper to see how this weaker property still excludes all possible voting procedures.