[EM] Lewis Carroll and limited vote

[Forest Simmons]

On Tue, 26 Mar 2002, Michael Rouse wrote:

> While I was surfing the net for pros and cons of different voting systems, I
> came upon this blurb about Lewis Carroll:
>
> "Writing in 1884, he praised the unpopular 'limited vote' which then
> operated in the big English cities, in which each voter had fewer votes than
> there were seats to fill. Most people thought that this was less democratic
> than giving each voter as many votes as there were seats. Dodgson proved
> that it was more democratic. To do so, he used concepts we would today label
> 'game theoretic', although such concepts were not formalised until decades
> after his death."
>
> This raised a few questions in my mind, such as:
> 1.) Is there a fairly simple description of this proof?

I don't know about his proof, but here's an heuristic argument.  If the
voters are allowed only one vote, then the method is strategically
equivalent to cumulative voting, a well known, if some what crude method
of Proportional Representation.

So the closer to one vote, the closer to PR.

The game theoretic part is about the strategic equivalence.  Consider
voters (who are limited to one mark per ballot) coordinating their votes
or appropriately randomizing among the ones that they would have voted for
under the cumulative method. [Alternately, start with cumulative voting
and apply the corner principle of linear optimization, to get the lone
mark result.]

Bart recently pointed out the irony of Approval versus Plurality voting.
Plurality, which is bad for single winner and is OK as a PR method for
multiwinner election, is mainly used for single winner elections.

Approval, which is good for single winner and not a PR method (unless the
PAV count rule is used), is most commonly approximated in multiwinner
elections (in "at large" elections where the instructions are "vote for up
to seven candidates," when there are seven candidates, for example).

A mathematician of Dodgson's wit could easily see through this irony.

This is related to the difference between IRV and Coombs. Both of these
methods eliminate one candidate at each stage.  Coombs goes by the
candidate you would most like to eliminate (the last ranked candidate).
IRV effectively gives a vote of elimination to all except your first
ranked candidate.  If you are going to keep n-1 for the second round, then
Coombs in effect gives you n-1 votes, while IRV gives you one vote for
whom to keep, even though you are going to keep n-1.

In other words, IRV is a kind of crude PR runoff.

But a moment's reflection shows that PR runoff is not likely to give the
best single winner results. Suppose for example that candidates A,B,C, and
D each represent (as favorite) approximately 25 percent of the electorate
in a PR election. That doesn't mean that one of them would be the best
candidate in a single winner election.  A non-favorite approved by sixty
percent of the voters would be a better choice for a single winner office,
than a favorite of 26 percent who was disliked by the other 74 percent.

Forest