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
